Mémoire

déposé en vue de l’obtention de

l’habilitation à diriger des recherches

par

Stefan Neuwirth

Aspects quantitatifs de l’inconditionnalité

et soutenu le 9 décembre 2008 devant le jury composé de – Christian Le Merdy,

– Daniel Li,

– Gilles Pisier et

– Éric Ricard, les rapporteurs étant

– Aline Bonami,

– Vern Paulsen et

– Vladimir Peller.

Table des matières

Liste des articles et notes 6

Résumé 7 1 Trois cas de figure pour l’inconditionnalité . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Définition des suites basiques inconditionnelles . . . . . . . . . . . . . . . . . . . . . . 7 3 Matrices et multiplicateurs de Schur pour les classes de Schatten . . . . . . . . . . . . 8 4 Suites de matrices élémentaires et graphes bipartis . . . . . . . . . . . . . . . . . . . . 8 5 Matrices lacunaires et inconditionnalité . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6 Matrices lacunaires et 1-inconditionnalité . . . . . . . . . . . . . . . . . . . . . . . . . 10 7 Graphes bipartis sans cycle de longueur donnée . . . . . . . . . . . . . . . . . . . . . . 12 8 Matrices lacunaires et ensembles lacunaires . . . . . . . . . . . . . . . . . . . . . . . . 13 9 Sous-espaces S_{I}^{p} 1-complémentés . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 10 Matrices de rang 1 partiellement spécifiées . . . . . . . . . . . . . . . . . . . . . . . . . 13 11 Propriété d’approximation métriquement inconditionnelle dans S_{I}^{p} . . . . . . . . . . . 13 12 Inégalités matricielles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 13 Transfert entre multiplicateurs de Schur et de Fourier . . . . . . . . . . . . . . . . . . 14 14 Ensembles lacunaires somme de deux ensembles infinis . . . . . . . . . . . . . . . . . . 15 15 Problèmes extrémaux pour les polynômes trigonométriques . . . . . . . . . . . . . . . 15 16 Problèmes extrémaux pour les trinômes trigonométriques . . . . . . . . . . . . . . . . 16 17 Points extrémaux et exposés de la boule unité de l’espace C_{Λ} . . . . . . . . . . . . . . 18 18 La variation du module maximum en fonction de l’argument . . . . . . . . . . . . . . . 19 19 Problèmes extrémaux pour les quadrinômes trigonométriques . . . . . . . . . . . . . . 20

A Lacunary matrices 21 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 σ(p) sets as matrix Λ(p) sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4 The intersection of a σ(p) set with a finite product set . . . . . . . . . . . . . . . . . . 26 5 Circuits in graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6 A random construction of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

B Cycles and 1-unconditional matrices 31 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 Relative Schur multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Idempotent Schur multipliers of norm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 Unconditional basic sequences in S^{p} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5 Varopoulos’ characterisation of unconditional matrices in S^{∞} . . . . . . . . . . . . . . 40 6 Closed walk relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7 Schur multipliers on a cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 8 1-unconditional matrices in S^{p}, p not an even integer . . . . . . . . . . . . . . . . . . . 45 9 1-unconditional matrices in S^{p}, p an even integer . . . . . . . . . . . . . . . . . . . . . 47 10 Metric unconditional approximation property for S_{I}^{p} . . . . . . . . . . . . . . . . . . . 48 11 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 C Matrix inequalities with applications to iterated kernels 55

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2 Matrix inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Asymptotic matrix inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4 Asymptotic kernel inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

D The size of bipartite graphs with girth eight 64 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2 Uncoloured graphs and bipartite graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Expanding a graph to a bipartite graph . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Contracting a bipartite graph to an uncoloured graph . . . . . . . . . . . . . . . . . . 65

3 Bipartite graphs of girth six . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4 Bipartite graphs of girth eight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 A generalisation of an inequality of Atkinson et al. . . . . . . . . . . . . . . . . . . . . 67 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

E Ordering simultaneously the columns and lines of 0,1 matrices 71

F Transfer of Fourier into Schur multipliers and sumsets in a group 74 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2 Transfer between Fourier and Schur multipliers . . . . . . . . . . . . . . . . . . . . . . 78 3 Local embeddings of L^{p} into S^{p} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4 Transfer of lacunary sets into lacunary matrix patterns . . . . . . . . . . . . . . . . . . 84 5 Toeplitz Schur multipliers on S^{p} for p < 1 . . . . . . . . . . . . . . . . . . . . . . . . . 86 6 The Riesz projection and the Hilbert transform . . . . . . . . . . . . . . . . . . . . . . 87 7 Unconditional approximating sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8 Relative Schur multipliers of rank one . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

G The Sidon constant of sets with three elements 93 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3 A solution to Extremal problem (†) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4 A solution to Extremal problem (‡) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5 Some consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6 Three questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

H The maximum modulus of a trigonometric trinomial 99 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2 Isometric relative Fourier multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3 The arguments of the Fourier coefficients of a trinomial . . . . . . . . . . . . . . . . . 104 4 The frequencies of a trigonometric trinomial . . . . . . . . . . . . . . . . . . . . . . . . 104 5 Location of the maximum point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6 Uniqueness of the maximum point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7 The maximum modulus points of a trigonometric trinomial . . . . . . . . . . . . . . . 109 8 Exposed and extreme points of the unit ball of C_{Λ} . . . . . . . . . . . . . . . . . . . . 110 9 Dependence of the maximum modulus on the arguments . . . . . . . . . . . . . . . . . 112 10 The norm of unimodular relative Fourier multipliers . . . . . . . . . . . . . . . . . . . 114 11 The Sidon constant of integer sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

I On the Sidon constant of {0, 1, 2, 3} 117 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 2 Necessary conditions for solutions to Extremal problem (†) . . . . . . . . . . . . . . . 118 3 Necessary conditions for solutions to Extremal problem (‡) . . . . . . . . . . . . . . . 119 4 The case {0, 1, 2, 3}: a distinguished family of polynomials . . . . . . . . . . . . . . . . 120 5 The real unconditional constant of {0, 1, 2, 3} . . . . . . . . . . . . . . . . . . . . . . . 121 6 Trigonometric polynomials of degree 3 with real coefficients . . . . . . . . . . . . . . . 122

Bibliographie 124

Liste des articles et notes

[A] A. Harcharras, S. Neuwirth et K. Oleszkiewicz, Lacunary matrices, Indiana Univ. Math. J. 50 (2001), 1675–1689.

[B] S. Neuwirth, Cycles and 1-unconditional matrices, Proc. London Math. Soc. (3) 93 (2006), 761–790.

[C] W. Banks, A. Harcharras, S. Neuwirth et É. Ricard, Matrix inequalities with applications to the theory of iterated kernels, Linear Algebra Appl. 362 (2003), 275–286.

[D] S. Neuwirth, The size of bipartite graphs with girth eight. http://arxiv.org/abs/math/

0102210, 2001.

[E] N. Kosmatov et S. Neuwirth, Ordering simultaneously the columns and lines of 0,1 matrices, manuscrit (2002).

[F] S. Neuwirth et É. Ricard, Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group, manuscrit (2008).

[G] S. Neuwirth, The Sidon constant of sets with three elements. http://arxiv.org/abs/math/

0102145, 2001.

[H] , The maximum modulus of a trigonometric trinomial, J. Anal. Math. 104 (2008), 371–396.

[I] , On the Sidon constant of {0, 1, 2, 3}, manuscrit (2003).

[J] , Metric unconditionality and Fourier analysis, Studia Math. 131 (1998), 19–62.

[K] , Two random constructions inside lacunary sets, Ann. Inst. Fourier (Grenoble) 49 (1999), 1853–1867.

Résumé

1 Trois cas de figure pour l’inconditionnalité

Toutes nos recherches sont liées à la notion d’inconditionnalité, motivée par la question suivante :

Question 1.1. Lorsqu’un élément x d’un espace normé admet une représentation comme combinaison linéaire ∑ c_{q}e_{q} d’éléments e_{q}, de quelle manière la norme de x dépend-elle du signe des coefficients c_{q}?

Les réponses que nous obtiendrons seront en termes du support de x, c’est-à-dire de l’ensemble I d’indices q pour lesquels c_{q} ≠ 0.

Selon la situation, un changement du signe des coefficients c_{q}

– fait varier la norme de x de manière bornée et on dira que (e_{q})_{q∈I} est une suite basique inconditionnelle ;

– multiplie au plus la norme de x par un facteur D explicite et D sera la constante d’inconditionnalité de la suite (e_{q})_{q∈I} ;

– ne change pas la norme de x et on parlera de suite basique (e_{q})_{q∈I} 1-inconditionnelle ; Lorsque nous chercherons à déterminer une constante d’inconditionnalié exacte, nous devrons

préciser de quelle manière nous nous permettons de changer le signe des coefficients : – de manière réelle en multipliant certains coefficients par −1, ou

– de manière complexe en les faisant tourner d’un angle t_{q}. Voici trois cas de figure dans lesquels cette question se pose.

(a) Si x est une fonction sur un groupe abélien compact et les e_{q} sont les caractères de ce groupe, cette représentation est la série de Fourier de x et un changement du signe des coefficients de Fourier est une convolution ou multiplication de Fourier unimodulaire.

(b) Si x est un opérateur et les e_{q} sont les matrices élémentaires, cette représentation est la matrice de x et un changement du signe des coefficients matriciels est une multiplication de Schur unimodulaire.

(c) Si x est un élément de l’algèbre d’un groupe discret G et les e_{q} sont les fonctions indicatrices des éléments de G, alors un changement du signe des coefficients est une multiplication de Herz-Schur unimodulaire.

2 Définition des suites basiques inconditionnelles

Voici une définition formelle qui reprend la discussion ci-dessus.

Définition 2.1. Soit X un espace vectoriel quasi-normé muni d’une suite distinguée (e_{q}) dans X. Soit (e_{q})_{q∈I} une sous-suite. Soit 𝕊 = 𝕋 = {z ∈ ℂ : |z| = 1} (vs. 𝕊 = {−1, 1}.)

(a) I est inconditionnelle dans X s’il y a une constante D telle que

∥∑_{q∈I} ϵ_{q}a_{q}e_{q}∥ ⩽ D∥∑_{q∈I} a_{q}e_{q}∥  (1)

pour tout choix de signes ϵ_{q} ∈ 𝕊 et toute suite de coefficients complexes a_{q} de support fini. Sa constante d’inconditionnalité complexe (vs. réelle) est le minimum de telles constantes D.

(b) I est 1-inconditionnelle complexe (vs. réelle) dans X si sa constante d’inconditionnalité complexe (vs. réelle) vaut 1. Cela veut dire que l’inégalité (1) devient l’égalité

∥∑_{q∈I} ϵ_{q}a_{q}e_{q}∥ = ∥∑_{q∈I} a_{q}e_{q}∥.

3 Matrices et multiplicateurs de Schur pour les classes de Schatten-von-Neumann

Cette thématique de recherche correspond au cas de figure (b) ci-dessus : (e_{q}) est la suite des matrices élémentaires.

Notons C l’ensemble des indices colonne et R l’ensemble des indices ligne de matrices, en général deux copies de ℕ, et soit I une partie de R × C. La propriété d’inconditionnalité de I peut aussi se formuler ainsi : une suite I est inconditionnelle de constante D si et seulement si, pour toute matrice φ à coefficients complexes (vs. réels) et pour tout x dont les coefficients de matrice sont nuls hors de I (dont l’espace sera noté X_{I}) on a

∥φ ∗ x∥ ⩽ D sup|φ_{rc}| ∥x∥,

où φ ∗ x est le produit de Schur (ou de Hadamard) défini par

(φ ∗ x)_{rc} = φ_{rc}x_{rc}.

L’opérateur de multiplication par φ est un multiplicateur de Schur relatif. On peut aussi décrire les multiplicateurs de Schur relatifs comme les opérateurs diagonaux sur la suite (e_{q})_{q∈I} .

Notre étude se concentrera sur les classes de Schatten-von-Neumann X = S^{p} dont la quasi-norme

est donnée par ∥x∥ = (tr(x^{∗}x)^{p/2})^{1/p} : il s’agit de la contrepartie non commutative des espaces ℓ^{p}. Lorsque p ⩾ 1, l’espace de Banach S^{p} admet une structure d’espace d’opérateurs canonique qui rend la définition suivante naturelle.

Définition 3.1. I est complètement inconditionnelle dans S^{p} s’il y a une constante D telle que (1) vaut pour tout choix de signes ϵ_{q} ∈ 𝕊 et toute suite de coefficients opérateurs a_{q} ∈ S^{p} à support fini. Sa constante d’inconditionnalité complète complexe (vs. réelle) est le minimum de telles constantes D.

De la même manière, on parle de la norme complète de multiplicateurs de Schur relatifs. On ne sait pas si on définit vraiment ainsi une classe nouvelle ; ce serait répondre à la conjecture de Gilles Pisier qu’il existe des multiplicateurs de Schur bornés sur S^{p} (p ≠ 1, 2, ∞) qui ne sont pas complètement bornés.

Une suite inconditionnelle de matrices élémentaires dans S^{∞} est en fait un ensemble V-Sidon,

classe que Varopoulos a introduite dans l’étude des algèbres tensorielles c_{0}(C) ⊗^{∧} c_{0}(R) sur des espaces discrets (voir le théorème B.5.1 page 40 qui rassemble les résultats connus : I doit être réunion finie d’ensembles qui soit contiennent au plus un élément par ligne, soit contiennent au plus un élément par colonne.) Notre étude généralise ainsi les résultats de Varopoulos à toutes les classes de Schattenvon-Neumann.

4 Suites de matrices élémentaires et graphes bipartis

Les suites inconditionnelles de matrices élémentaires forment la contrepartie matricielle des ensembles Λ(p) de Walter Rudin étudiés en analyse de Fourier (le cas de figure (a) de la section 1.) Alors que l’étude des ensembles Λ(p) voit surgir naturellement leurs propriétés arithmétiques (de théorie additive des nombres,) l’inconditionnalité de I se traduit avantageusement en termes de théorie des graphes.

Nous allons donc considérer I comme un graphe biparti dont les deux classes (« couleurs ») de sommets sont C et R, dont les éléments seront appelés respectivement « sommets colonne » et « sommets ligne. » Ses arêtes (non dirigées) relient seulement des sommets ligne r ∈ R avec des sommets colonne c ∈ C, et cela exactement lorsque (r, c) ∈ I. La matrice (χ_{I} (r, c))_{(r,c)∈R×C} fonction caractéristique de I est la matrice d’incidence de ce graphe biparti.

Voici deux exemples importants.

Exemple 4.1. Soit s un entier. Considérons l’ensemble

I = { (r, c) ∈ ℤ/sℤ × ℤ/sℤ : r − c ∈ {0, 1} }.

Le graphe biparti associé est le cycle (ligne 0, colonne 0, ligne 1, colonne 1, …, ligne s−1, colonne s−1) de longueur 2s. La matrice d’incidence de ce graphe est

( \begin{matrix}& 0 & 1 & ··· & s−2 & s−1 \\ 0 & 1 & 0 & ... & 0 & 1 \\ 1 & 1 & 1 & ... & 0 & 0 \\ ⋮ & ... & ... & ... & ... & ... \\ s−2 & 0 & 0 & ... & 1 & 0 \\ s−1 & 0 & 0 & ... & 1 & 1 \end{matrix} ) .

Exemple 4.2. Considérons l’ensemble

I = { (r, c) ∈ ℤ/7ℤ × ℤ/7ℤ : r + c ∈ {0, 1, 3} }.

Le graphe biparti associé est le graphe de Heawood.

La matrice d’incidence de ce graphe est

( \begin{matrix}& 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 2 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ 3 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 4 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ 5 & 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 6 & 0 & 1 & 1 & 0 & 1 & 0 & 0 \end{matrix} ) .

5 Matrices lacunaires et inconditionnalité

Dans l’article Lacunary matrices, nous montrons que ces sous-suites doivent satisfaire la condition de densité suivante, qui est l’analogue de la condition de maille de Walter Rudin [88, Theorem 3.5] pour les ensembles Λ(p).

Théorème 5.1 (page 26). Si I est inconditionnelle de constante D dans S^{p}, alors la taille #I′ de tout sous-graphe I′ induit par m sommets colonne et n sommets ligne, c’est-à-dire le cardinal de toute partie I′ = I ∩ R′× C′ avec #C′ = m et #R′ = n, satisfait

#I′ ⩽ D^{2}(m^{1/p}n^{1/2} + m^{1/2} n^{1/p})^{2}  (2)
⩽ 4D^{2} min(m, n)^{2/p} max(m, n).

Les exposants de cette inégalité sont optimaux dans les trois cas suivants :

(a) si m ou n est fixé (trivial ;)

(b) si p = 4 (voir le graphe aléatoire ci-dessous ;)

(c) si p est un entier pair et m = n (voir [37, Theorem 4.8].)

Si m ≠ n, nous construisons des graphes aléatoires qui testent l’inégalité (2) sans en montrer toujours l’optimalité.

Théorème 5.2 (page 28). Pour tout ε > 0 et tout entier pair p ⩾ 4, il existe un graphe I de taille

#I ∼ \{ \begin{matrix}max(m, n)^{1−ε} min(m, n)^{1/2} & si p = 4 \\ max(m, n)^{1/2−ε} min(m, n)^{1/2+2/p} & si p ⩾ 6. \end{matrix}

et de constante d’inconditionnalité indépendante de m et n lorsque mn → ∞.

Si p est un entier pair, nous donnons aussi une condition suffisante en termes de sentiers sur un graphe biparti : un sentier de longueur s dans I est une suite (v_{0}, v_{1}, …, v_{s}) de sommets alternativement dans R et C telle que les arêtes reliant v_{0} à v_{1}, v_{1} à v_{2}, etc. correspondent à des élements deux à deux distincts de I (alors qu’un chemin est requis d’avoir même tous ses sommets distincts et que les sommets d’une promenade sont admis à se répéter.) Le théorème suivant est aussi l’analogue d’un théorème de Walter Rudin.

Théorème 5.3 (page 24). Soit p un entier pair. Si le nombre de sentiers dans I de longueur p/2 entre deux sommets donnés admet une borne uniforme, alors I est inconditionnelle dans S^{p}.

Le calcul suivant montre le lien étroit entre la norme S^{p} avec p = 2s un entier pair et les promenades fermées de longueur p dans ce graphe.

tr|∑_{q∈I} a_{q}e_{q}|^{p} = tr( ∑_{(r,c),(r′,c′)∈I} (a_{rc} e_{rc})^{∗}(a_{r′c′} e_{r′c′}))^{s}

= tr ∑_{\begin{matrix}(r_{1} ,c_{1} ),(r′_{1},c′_{1} ),..., \\ (r_{s} ,c_{s} ),(r′_{s},c′_{s})∈I \end{matrix}} ∏_{i=1}^{s} (\overline{a_{r_{i} c_{i}}} e_{c_{i} r_{i}} )(a_{r′_{i}c′_{i}} e_{r′_{i}c′_{i}} )

= ∑_{\begin{matrix}(r_{1} ,c_{1} ),(r_{1} ,c_{2} ),..., \\ (r_{s} ,c_{s} ),(r_{s} ,c_{s+1} )∈I \end{matrix}}  ∏_{i=1}^{s} \overline{a_{r_{i} c_{i}}} a_{r_{i} c_{i+1}} (où c_{s+1} = c_{1}.)

Cette dernière somme est indexée par les promenades fermées (c_{1}, r_{1}, c_{2}, …, c_{s}, r_{s}) de longueur p dans le graphe associé à I!

La conjonction des théorèmes 5.1 et 5.3 donne une nouvelle preuve du théorème de Paul Erdős selon lequel un graphe sur v sommets sans circuit de longueur paire p est de taille bornée par v^{1+2/p} , à une constante près (un circuit est un sentier fermé.) La généralisation de ce théorème des circuits aux cycles (chemins fermés) par Bondy et Simonovits [12] échappe à notre méthode. L’existence de graphes qui montreraient l’optimalité de cette estimation est une question ouverte posée par Erdős en 1963.

6 Matrices lacunaires et 1-inconditionnalité

L’article Cycles and 1-unconditional matrices aboutit à une caractérisation des suites basiques 1-in-conditionnelles dans S^{p}.

Un des ingrédients est l’étude des multiplicateurs de Schur unimodulaires sur un cycle. Nous obtenons en particulier la proposition suivante.

Proposition 6.1 (page 44). Si p n’est pas un entier pair, alors ϵ est un multiplicateur de Schur unimodulaire isométrique sur un cycle I pour S^{p} si et seulement si ϵ peut être interpolée par une matrice de rang 1 : ϵ_{rc} = ζ_{c}η_{r} pour (r, c) ∈ I, où ζ ∈ 𝕋^{C} et η ∈ 𝕋^{R}.

Esquisse de démonstration. La condition est bien suffisante : on a alors

ϵ ∗ x = ( \begin{matrix}. & . & . \\ & & & η_{r} \\ & & & & . & . & . \end{matrix}) (x_{rc}) ( \begin{matrix}. & . & . \\ & & & ζ_{c} \\ & & & & . & . & . \end{matrix}) .

Étudions la nécessité. On peut supposer que le cycle I soit donné comme dans l’exemple 4.1. Soit ϵ ∈ 𝕋^{I} une matrice de nombres unimodulaires partiellement spécifiée. Il est possible de multiplier les lignes et les colonnes de ϵ par des nombres complexes de module 1 de sorte que ϵ devienne la matrice circulante

ϵ̃=  ( \begin{matrix}& 0 & 1 & ··· & s−2 & s−1 \\ 0 & 1 & 0 & ... & 0 & ϑ \\ 1 & ϑ & 1 & ... & 0 & 0 \\ ⋮ & ... & ... & ... & ... & ... \\ s−2 & 0 & 0 & ... & 1 & 0 \\ s−1 & 0 & 0 & ... & ϑ & 1 \end{matrix} )  (3)

avec ϑ racine sième de \overline{ϵ_{00}}ϵ_{10} … \overline{ϵ_{s−1 ,s−1}} ϵ_{0,s−1} . Un argument de transfert montre que la norme du multiplicateur de Schur par ϵ̃ sur I borne le multiplicateur de Fourier relatif μ: a + bz ↦ a + ϑbz dans le groupe G des racines sièmes de l’unité, où on norme a + bz par la norme L^{p} : ∥a + bz∥ = (s^{−1} ∑ _{z^{s} =1} |a + bz|^{p}) ^{1/p} : voir la proposition 13.2 (a). Si μ est une isométrie, le théorème d’équimesurabilité de Plotkin-Rudin montre que z et ϑz ont même distribution et donc ϑ^{s} = 1.

On peut calculer la norme exacte du multiplicateur de Schur relatif ϵ̃ sur S_{I}^{1} et sur S_{I}^{∞} : elle égale la norme de μ sur L_{Λ}^{1} (G) et sur L_{Λ}^{∞} (G) avec Λ = {1, z} et cette norme est

\frac{max_{z^{s} =−1} |ϑ + z|}{|1 + e^{iπ/s} |}

(proposition B.7.1(d) page 44.)

Cette proposition est une des étapes dans la démonstration du théorème suivant.

Théorème 6.2 (page 45). Soit p un nombre réel strictement positif qui ne soit pas un entier pair. Les propriétés suivantes sont équivalentes.

– I est complètement 1-inconditionnelle complexe dans S^{p}.

– I est 1-inconditionnelle réelle dans S^{p}.

– I est une réunion disjointe d’arbres, c’est-à-dire que I ne contient aucun cycle.

– Toute suite de signes complexes ϵ ∈ 𝕋^{I} peut être interpolée par une matrice de rang 1.

– I est un ensemble de Varopoulos de V-interpolation de constante 1 : toute suite φ ∈ ℓ_{I}^{∞} peut

être interpolée par un tenseur u ∈ ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} avec ∥u∥ = ∥φ∥.

– I est un ensemble d’interpolation isométrique pour les multiplicateurs de Schur : toute suite φ ∈ ℓ_{I}^{∞} est la restriction d’un multiplicateur de Schur sur S^{∞} de norme ∥φ∥.

Dans le cas où p est un entier pair, la combinatoire devient plus compliquée : cela se reflète dans la proposition suivante.

Proposition 6.3 (page 44). Si p est un entier pair, alors ϵ est un multiplicateur de Schur unimodulaire isométrique sur un cycle I de longueur 2s pour S^{p} si et seulement si p/2 ∈ {1, 2, …, s − 1} ou si ϵ peut être interpolée par une matrice de rang 1.

Cette proposition est une des étapes dans la démonstration de la caractérisation suivante.

Théorème 6.4 (page 47). Soit p un entier pair. Les propriétés suivantes sont équivalentes. – I est complètement 1-inconditionnelle complexe dans S^{p}.

– I est 1-inconditionnelle réelle dans S^{p}.

– I ne contient aucun cycle de longueur paire inférieure ou égale à p.

Illustrons ce théorème sur l’exemple 4.2. Le graphe de Heawood ne contient aucun cycle de longueur 4 : donc la norme S^{4} de sa matrice d’incidence ne varie pas si on change le signe de ses coefficients.

La propriété de ne pas contenir de cycle de longueur paire donnée a été très étudiée en théorie des graphes. Quelle conséquence a-t-elle pour la taille du graphe ? La section suivante en propose un résumé, à comparer aux résultats du théorème 5.1.

7 Graphes bipartis sans cycle de longueur donnée

Proposition 7.1 (page 53). Soient 2 ⩽ n ⩽ m, I ⊆ R × C avec # C = n et # R = m, et e = # I.

(a) Si I est 1-inconditionnelle dans S^{4} — I ne contient pas de cycle de longueur 4 — alors

n ⩾ 1 + ( \frac{e}{m} − 1) + ( \frac{e}{m} − 1) ( \frac{e}{n} − 1) ,

c’est-à-dire e^{2} − me − mn(n − 1) ⩽ 0. On a égalité si et seulement si I est le graphe d’incidence d’un système de Steiner S(2, e/m; n) sur n points et m blocs (voir [9, Def. I.3.1] pour la définition des systèmes de Steiner.)

(b) Si I est 1-inconditionnelle dans S^{6} — I ne contient pas de cycle de longueur 4 ni 6 — alors

n ⩾ 1 + ( \frac{e}{m} − 1) + ( \frac{e}{m} − 1) ( \frac{e}{n} − 1) + ( \frac{e}{m} − 1)^{2}( \frac{e}{n} − 1) ,

c’est-à-dire e^{3} − (m + n)e^{2} + 2mne − m^{2}n^{2} ⩽ 0. On a égalité si et seulement si I est le graphe d’incidence du quadrangle (le cycle de longueur 8) ou d’un quadrangle généralisé avec n points et m lignes (voir [56, Def. 1.3.1] pour la définition des polygones généralisés ; l’exemple 4.2 décrit le plus petit quadrangle généralisé.)

(c) Si I est 1-inconditionnelle dans S^{p} avec p = 2k un entier pair — I ne contient pas de cycle de longueur inférieure ou égale à p — alors

n ⩾ ∑_{i=0}^{k} ( \frac{e}{m} − 1)^{⌈ \frac{i}{2} ⌉}( \frac{e}{n} − 1)^{⌊ \frac{i}{2} ⌋}.  (4)

On a égalité si et seulement si I est le graphe d’incidence du (k + 1)gone (le cycle de longueur 2k + 2) ou d’un (k + 1)gone généralisé avec n points et m lignes.

Les résultats (a) et (b) ci-dessus ont été obtenus dans la note The size of bipartite graphs with girth eight (voir pages 65 et 66), alors que le cas général résulte de travaux de Noga Alon, Shlomo Hoory et Nathan Linial (voir [44]).

L’inégalité (4) montre que si I est 1-inconditionnelle dans S^{2k}, alors # I ⩽ n^{1+1/k} + (s − 1)n/s. Si k ∉ {2, 3, 5, 7}, il n’existe pas de (k + 1)gones généralisés de taille arbitrairement grande et l’existence de graphes arbitrairement grands sans cycle de longueur 2k de taille minorée par n^{1+1/k} à une constante près est une question importante en théorie des graphes extrémaux.

La recherche pratique de graphes extrémaux nous a amenés à écrire un algorithme implémenté en langage C qui énumère tous les graphes bipartis d’un nombre de sommets donnés et teste l’existence de cycles. La proposition suivante, démontrée indépendamment par Adolf Mader et Otto Mutzbauer [55], réduit le nombre de matrices d’incidence de graphes bipartis à tester.

Proposition 7.2 (page 71). Toute matrice à coefficients 0 ou 1 peut être simultanément ordonnée selon les ordres lexicographiques des lignes et des colonnes (c’est-à-dire l’ordre des lignes et des colonnes lues comme des nombres binaires) par une permutation des lignes et des colonnes.

En effet, une permutation des lignes et des colonnes de la matrice d’incidence d’un graphe biparti consiste juste à réindexer les sommets de ce graphe.

8 Matrices lacunaires et ensembles lacunaires d’un groupe abélien discret

Voici la traduction naturelle entre inconditionnalités de Fourier et matricielle (les cas (a) et (b) de la section 1.)

Proposition 8.1 (page 52). Soit I ⊆ R × C. Soit p ∈ [1, ∞] : les propriétés suivantes sont équivalentes.

– I est complètement inconditionnelle dans S^{p}.

– La suite de produits de Walsh de longueur deux {ϵ_{c}ϵ′_{r} : (r, c) ∈ I} est complètement inconditionnelle dans L^{p}({−1, 1}^{C} × {−1, 1}^{R}).

– La suite de produits de deux fonctions de Steinhaus {z_{c}z′_{r} : (r, c) ∈ I} est complètement incon-

ditionnelle dans L^{p}(𝕋^{C} × 𝕋^{R}). Soit p ∈ (0, ∞] : les propriétés suivantes sont équivalentes.

– I est 1-inconditionnelle dans S^{p}.

– La suite {ϵ_{c}ϵ′_{r} : (r, c) ∈ I} est 1-inconditionnelle dans L^{p}({−1, 1}^{C} × {−1, 1}^{R}).

– La suite {z_{c}z′_{r} : (r, c) ∈ I} est 1-inconditionnelle dans L^{p}(𝕋^{C} × 𝕋^{R}).

La proposition B.11.1 (page 52) décrit dans quelle mesure cette proposition reste vraie pour d’autres groupes discrets.

9 Sous-espaces S_{I}^{p} 1-complémentés

En route pour ces résultats, nous obtenons aussi la caractérisation suivante.

Proposition 9.1 (page 37). Le sous-espace S_{I}^{p} de S^{p} formé des opérateurs à support dans I est 1-complémenté si et seulement si I est la réunion disjointe de graphes bipartis complets R_{j} × C_{j} : sa matrice d’incidence est, à une permutation des colonnes et des lignes près, bloc-diagonale :

( \begin{matrix}& C_{1} & C_{2} & C_{3} & ··· \\ R_{1} & (1) & (0) & (0) & ⋯ \\ R_{2} & (0) & (1) & (0) & ... \\ R_{3} & (0) & (0) & (1) & ... \\ ⋮ & ⋮ & ... & ... & ... \end{matrix}) .

10 Matrices de rang 1 partiellement spécifiées

Si φ est une matrice de rang 1, φ = x ⊗ y, alors l’opérateur de multiplication de Schur par φ est de norme sup|x_{r}| sup|y_{c}|. Or les exemples de calcul exact de normes de tels opérateurs sont très rares et nous avons voulu savoir comment cette norme change lorsque φ agit sur un sous-espace S_{I}^{p}.

Théorème 10.1 (page 91). Soit I ⊂ R×C et considérons (x_{r})_{r∈R} et (y_{c})_{c∈C} . Alors le multiplicateur de Schur relatif S_{I}^{p} donné par (x_{r}y_{c})_{(r,c)∈I} est de norme sup_{(r,c)∈I} |x_{r}y_{c}|.

11 Propriété d’approximation métriquement inconditionnelle dans S_{I}^{p}

Même si I n’est pas 1-inconditionnelle dans l’espace S^{p}, l’espace S_{I}^{p} pourrait néanmoins admettre une autre base 1-inconditionnelle. Pour approcher de telles questions, Peter G. Casazza et Nigel J. Kalton ont introduit la propriété (c) ci-dessous.

Définition 11.1. Soit X un espace de Banach séparable et 𝕊 = 𝕋 (vs. 𝕊 = {−1, 1}.)

(a) Une suite (T_{k}) d’opérateurs sur X est une suite approximante si chaque T_{k} est de rang fini et T_{k}x → x pour chaque x ∈ X.

(b) ([73].) Posons ΔT_{k} = T_{k} − T_{k−1} . L’espace X a la propriété d’approximation inconditionnelle s’il existe une suite approximante (T_{k}) telle que pour une certaine constante D

∥∑_{k=1}^{n} ϵ_{k}ΔT_{k} ∥ ⩽ D pour tout n et tous ϵ_{k} ∈ 𝕊.

La constante d’inconditionnalité complexe (vs. réelle) de (T_{k}) est la plus petite des constantes D.

(c) ([22, § 3], [32, § 8].) L’espace X a la propriété d’approximation métriquement inconditionnelle complexe (vs. réelle) si, pour tout δ > 0, X admet une suite approximante de constante d’inconditionnalité complexe (vs. réelle) 1 + δ.

Voici notre description des sous-espaces S_{I}^{p} métriquement inconditionnels :

Théorème 11.2. On a deux cas.

– Si p ∈ [1, ∞] \ {2, 4, 6, …} et S_{I}^{p} a la propriété d’approximation métriquement inconditionnelle réelle, alors la distance d’un sommet colonne à un sommet colonne est asymptotiquement infinie dans I : leur distance devient arbitrairement grande en effaçant un nombre fini d’arêtes de I.

– Si p ∈ {2, 4, 6, …}, l’espace S_{I}^{p} a la propriété d’approximation métriquement inconditionnelle complexe, ou réelle, si et seulement si deux sommets à distance 2j + 1 ⩽ p/2 sont à distance asymptotiquement supérieure ou égale à p − 2j + 1.

12 Inégalités matricielles

L’article Matrix inequalities with applications to the theory of iterated kernels montre l’inégalité matricielle suivante.

Théorème 12.1 (page 55). Soit A une matrice de taille n × m à coefficients positifs et notons somme(A) la somme de tous ses coefficients. On a

somme( \overbrace{AA^{∗}A …A^{(∗)}}^{k termes}) ⩾ \frac{somme(A)^{k}}{n^{⌊^{\frac{k−1}{2}} ⌋}m^{⌊^{\frac{k}{2}} ⌋}}

où A^{(∗)} est A^{∗} ou A selon la parité de k.

Cette inégalité est la version discrète d’un théorème sur les itérés d’un noyau (voir la remarque C.1.4 page 56.) Si on applique cette inégalité à la matrice d’incidence d’un graphe biparti I, on obtient une minoration optimale du nombre de promenades de longueur k en termes de la taille de I. Dans le cas k = 3, une généralisation de cette inégalité (le théorème D.4.4 page 67) donne une minoration optimale du nombre de chemins (voir le corollaire D.4.6 page 68.)

13 Transfert entre multiplicateurs de Schur et de Fourier

Le théorème suivant est bien connu.

Proposition 13.1. Soit Γ un groupe discret et R, C ⊆ Γ. À Λ ⊂ Γ associons I = { (r, c) ∈ R × C : rc ∈ Λ }. À ϱ ∈ ℂ^{Λ} associons φ ∈ ℂ^{I} défini par φ(r, c) = ϱ(rc) pour tout (r, c) ∈ I.

– Soit p > 0. La norme complète du multiplicateur de Schur relatif φ sur S_{I}^{p} est bornée par la norme complète du multiplicateur de Fourier relatif ϱ sur L_{Λ}^{p} (τ).

– La norme du multiplicateur de Schur relatif φ sur S_{I}^{∞} est bornée par la norme du multiplicateur de Fourier relatif ϱ sur L_{Λ}^{∞} (τ).

Une forme de réciproque peut être déduite du théorème limite de Szegő matriciel (voir le théorème F.2.1 page 79.)

Proposition 13.2. Soit Γ un groupe discret moyennable et soit I ⊆ Γ × Γ un ensemble toeplitzien au sens que I = {(r, c) ∈ Γ × Γ : rc^{−1} ∈ Λ} pour une partie Λ de Γ. Soit φ ∈ ℂ^{I} une matrice toeplitzienne au le sens qu’il existe ϱ ∈ ℂ^{Λ} tel que φ(r, c) = ϱ(rc^{−1} ) pour tout (r, c) ∈ I.

(a) Soit p > 0. La norme du multiplicateur de Fourier relatif ϱ sur L_{Λ}^{p} (τ) est le supremum de la norme du multiplicateur de Schur relatif φ sur des sous-espaces de matrices de Toeplitz tronquées dans S_{I}^{p}.

(b) De plus, la norme complète du multiplicateur de Fourier relatif ϱ sur L_{Λ}^{p} (τ) et la norme complète du multiplicateur de Schur relatif φ sur S_{I}^{p} sont égales.

(c) La norme du multiplicateur de Fourier relatif ϱ sur L_{Λ}^{∞} (τ) et la norme du multiplicateur de Schur relatif φ sur S_{I}^{∞} sont égales.

14 Ensembles lacunaires somme de deux ensembles infinis

Il est bien connu que les ensembles de Sidon ne peuvent contenir la somme de deux ensembles infinis ; Daniel Li a obtenu la même conclusion pour les ensembles Λ tels que C_{Λ} admet une suite approximante métriquement inconditionnelle. Nous rassemblons ces deux résultats dans le théorème suivant.

Théorème 14.1. Soit Γ un groupe abélien discret de caractères sur un groupe abélien compact G. Soit Λ ⊂ Γ. Si Γ contient la somme R + C de deux ensembles infinis R et C, alors l’espace C_{Λ} (G) n’admet pas de suite approximante inconditionnelle.

Esquisse de preuve. On utilise l’hypothèse pour montrer qu’il existe des parties infinies R′ et C′ sur lesquelles une somme à blocs sautés ∑ (T_{k_{j+1}} − T_{k_{j}} ) agit comme la projection sur la « partie triangulaire supérieure » de R′+ C′. Or ce multiplicateur de Fourier relatif se transfère en le multiplicateur de Schur qu’est la projection de Riesz sur les matrices triangulaires supérieures, qui est notoirement non bornée.

Nous obtenons ainsi une preuve élémentaire que l’algèbre du disque C_{ℕ}(𝕋) n’a pas la propriété d’approximation inconditionnelle, ni l’espace engendré par les fonctions de Walsh de longueur deux (les produits {r_{i}r_{j} } de deux fonctions de Rademacher) dans C({−1, 1}^{∞} ), ni l’espace engendré par les produits {s_{i}s_{j} } de deux fonctions de Steinhaus dans C(𝕋^{∞} ).

Nous montrons aussi qu’un ensemble « complètement Λ(p) » ne peut contenir la somme de deux ensembles infinis (théorème F.4.8 de la page 85.)

La preuve ci-dessus montre aussi que la constante d’inconditionnalité réelle pour les espaces L_{Λ}^{p} (G) est minorée par la norme complète de la projection de Riesz sur S^{p}. Cela nous a motivés pour calculer cette norme et, à défaut, la norme complète de la transformation de Hilbert matricielle.

Théorème 14.2. La norme complète de la projection de Riesz et de la transformation de Hilbert matricielle sur S^{p} coïncident avec leur norme.

– Si p est un entier pair, la norme de la transformation de Hilbert matricielle est cot(π/2p) (voir page 87).

– La norme de la projection de Riesz sur S^{4} est \sqrt{2} (voir page 88).

15 Problèmes extrémaux pour les polynômes trigonométriques

Cette thématique de recherche correspond au cas de figure (a) de la question 1.1, avec ℝ/ℤ comme groupe abélien et le module maximum comme norme.

Soit Λ = {λ_{1}, λ_{2}, …, λ_{n}} un ensemble de n entiers. Pour n nombres réels positifs r_{1}, r_{2}, …, r_{n} et n nombres réels t_{1}, t_{2}, …, t_{n}, considérons le polynôme trigonométrique

f(x) = r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + ⋯+ r_{n} e^{i(t_{n} +λ_{n} x)} .  (5)

La question de l’inconditionnalité dans l’espace C des fonctions continues est alors celle de la dépendance du module maximum du polynôme trigonométrique f par rapport aux arguments (phases) t_{1}, t_{2}, …, t_{n}.

Voici quatre problèmes qui éclairent divers aspects de l’inconditionnalité.

Problème extrémal 15.1 (problème de Mandel′shtam complexe — voir [26, page 2 et le supplément]). Trouver le minimum du module maximum du polynôme trigonometrique f pour des modules de coefficients de Fourier r_{1}, r_{2}, …, r_{n} donnés :

min_{t_{1} ,t_{2} ,...,t_{n}} max_{x}|r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + ⋯+ r_{n} e^{i(t_{n} +λ_{n} x)} |.

Problème extrémal 15.2. Trouver le minimum du module maximum du polynôme trigonometrique f pour un spectre Λ, des arguments t_{1}, t_{2}, …, t_{n} et la somme des modules r_{1} + r_{2} + ⋯+ r_{n} des coefficients de Fourier donnés :

min_{r_{1} ,r_{2} ,...,r_{n}} max_{x} \frac{|r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + ⋯+ r_{n} e^{i(t_{n} +λ_{n} x)} |}{r_{1} + r_{2} + ⋯+ r_{n}} .

Problème 15.3. Trouver le maximum de la variation du module maximum du polynôme trigonometrique f pour un spectre Λ et une variation des arguments Δt_{1}, Δt_{2}, …, Δt_{n} donnée. En d’autres termes, trouver la norme du multiplicateur de Fourier relatif unimodulaire par les signes e^{iΔt_{1}} , e^{iΔt_{2}} , …, e^{iΔt_{n}} :

min_{\begin{matrix}r_{1} ,r_{2} ,...,r_{n} \\ t_{1} ,t_{2} ,...,t_{n} \end{matrix}} \frac{max_{x}|r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + ⋯+ r_{n} e^{i(t_{n} +λ_{n} x)} |}{max_{x}|r_{1}e^{i(t_{1} +Δt_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +Δt_{2} +λ_{2} x)} + ⋯+ r_{n} e^{i(t_{n} +Δt_{n} +λ_{n} x)} |} .

Problème extrémal 15.4 (constante de Sidon). Trouver le minimum du module maximum du polynôme trigonometrique f pour un spectre Λ et la somme des modules r_{1} + r_{2} + ⋯+ r_{n} des coefficients de Fourier donnés :

min_{\begin{matrix}r_{1} ,r_{2} ,...,r_{n} \\ t_{1} ,t_{2} ,...,t_{n} \end{matrix}} max_{x} \frac{|r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + ⋯+ r_{n} e^{i(t_{n} +λ_{n} x)} )|}{r_{1} + r_{2} + ⋯+ r_{n}} .

L’inverse de ce minimum est la constante d’inconditionnalité de Λ dans l’espace des fonctions continues : c’est la constante de Sidon de Λ.

Littlewood [52] et Salem [91] se sont intéressés à ces problèmes. Ils sont aussi apparus dans la théorie du circuit électrique, comme le raconte N. G. Chebotarëv : « L. I. Mandel′shtam m’a communiqué un problème sur le choix des phases de courants électriques de fréquences différentes de sorte que la capacité du courant résultant de faire sauter les fusibles soit minimal » [24, p. 396]. Ce problème est une de ses motivations pour proposer une formule pour la valeur des dérivées directionnelles d’une fonction maximum en fonction des dérivées des fonctions dont on prend le maximum, qui a été un peu oubliée malgré son caractère naturel.

Formule de N. G. Chebotarëv ([26, Theorem VI.3.2, (3.6)]). Soit I un ouvert de ℝ^{n} et soit K

un espace compact. Soit F (t, x) une fonction réelle sur I × K qui soit continue, ainsi que \frac{∂F}{∂t} (t, x). Soit

F^{∗}(t) = max_{x∈K}F (t, x).

Alors F^{∗}(t) admet le développement limité suivant en tout t ∈ I :

F^{∗}(t + h) = F^{∗}(t) + max_{F (t,x)=F^{∗} (t)} ⟨h, \frac{∂F}{∂t} (t, x)⟩ + o(h).  (6)

Chebotarëv utilise en particulier cette formule pour résoudre le problème d’approximation polynomiale de Chebyshev et le problème du minimum d’une forme quadratique sur les entiers de Korkin et Zolotarev.

16 Problèmes extrémaux pour les trinômes trigonométriques

Ces problèmes sont déjà intéressants dans le cas n = 3. Dans l’article The Sidon constant of sets with three elements, nous avons résolu les problèmes extrémaux 15.1 et 15.4 pour ce cas. Nous allons supposer, en toute généralité, que Λ est un ensemble de trois entiers λ_{1} < λ_{2} < λ_{3} tels que λ_{2} − λ_{1} et λ_{3} − λ_{2} sont premiers entre eux.

Il s’avère que les arguments t_{1}, t_{2}, t_{3} d’un trinôme trigonométrique

f(x) = r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + r_{3}e^{i(t_{3} +λ_{3} x)}  (7)

donnent lieu à un paramètre unique, que nous appellerons l’argument du trinôme f : la détermination principale τ ∈ ]−π, π] de

(λ_{3} − λ_{2})t_{1} + (λ_{1} − λ_{3})t_{2} + (λ_{2} − λ_{1})t_{3} mod 2πℤ.  (8)

Nous avons ainsi établi que les arguments minimaux du problème extrémal 15.1 correspondent à des multiples de π :

Théorème 16.1 (page 96). Soit Λ un ensemble de trois entiers. Soient r_{1}, r_{2} et r_{3} trois réels strictement positifs. Les arguments t_{1}, t_{2}, t_{3} résolvent le problème extrémal 15.1 si et seulement si l’argument τ du trinôme égale π. En particulier, t_{1}, t_{2} et t_{3} peuvent être choisis parmi 0 et π.

Ce théorème permet de déterminer les coefficients de Fourier minimaux pour le problème extrémal 15.4 :

Proposition 16.2 (page 97). Le polynôme suivant résout le problème extrémal 15.4 :

f(x) = ϵ_{1}(λ_{3} − λ_{2})e^{iλ_{1} x} + ϵ_{2}(λ_{3} − λ_{1})e^{iλ_{2} x} + ϵ_{3}(λ_{2} − λ_{1})e^{iλ_{3} x}

où ϵ_{1}, ϵ_{2}, ϵ_{3} ∈ {−1, 1} sont trois signes réels tels que

– ϵ_{1}ϵ_{2} = −1 si λ_{2} − λ_{1} est pair ;

– ϵ_{1}ϵ_{3} = −1 si λ_{3} − λ_{1} est pair ;

– ϵ_{2}ϵ_{3} = −1 si λ_{3} − λ_{2} est pair.

La constante de Sidon de Λ égale donc cos(π/2(λ_{3} − λ_{1})) ^{−1} et les constantes d’inconditionnalité complexes et réelles de Λ dans l’espace des fonctions continues coïncident donc pour les ensembles Λ à trois éléments.

Nous avions entamé cette direction de recherche pour vérifier que les constantes d’inconditionnalité complexes et réelles d’un ensemble Λ étaient bien différentes ; les ensembles à trois éléments ne fourniront pas de contre-exemple et la question demeure ouverte.

En fait, les problèmes extrémaux 15.2 et 15.4 admettent une solution élémentaire : on ramène le trinôme trigonométrique à la forme « normale »

r_{1}e^{−ikx} + r_{2}e^{iτ/(k+l)} + r_{3}e^{ilx}  (9)

avec k et l positifs et premiers entre eux et τ ∈ ]−π, π]. Alors

\frac{max_{x} |r_{1}e^{−ikx} + r_{2}e^{iτ/(k+l)} + r_{3}e^{ilx} |}{r_{1} + r_{2} + r_{3}} ⩾ \frac{|r_{1} + r_{2}e^{iτ/(k+l)} + r_{3} |}{r_{1} + r_{2} + r_{3}}

= \sqrt{1 − \frac{4(r_{1} + r_{3})r_{2}}{(r_{1} + r_{2} + r_{3}) ^{2}} sin^{2} \frac{τ}{2(k + l)}}

⩾ \sqrt{1 − sin^{2} \frac{τ}{2(k + l)}} = cos \frac{τ}{2(k + l)} ,

et si r_{1} : r_{2} : r_{3} = l : k + l : k, le trinôme trigonométrique (9) atteint son module maximum en 0 et satisfait r_{1} + r_{3} = r_{2}.

17 Points extrémaux et points exposés de la boule unité de l’espace C_{Λ}

Si on cherche à résoudre les problèmes extrémaux 15.1, 15.2 et 15.4 par une application de la formule de Chebotarëv (6), il est utile d’obtenir des informations sur les points x tels que « F (t, x) = F^{∗}(t), » c’est-à-dire les points maximum de F (t, ·) = |f|. Par exemple, on peut déduire de cette formule qu’il y en a plus d’un car sinon (t, x) serait un point critique de F ; or un petit calcul (lemme G.3.1 page 95) montre que ce n’est pas possible.

Voici un argument d’analyse fonctionnelle qui démontre la même chose. Comme les problèmes ci-dessus sont linéaires, on peut limiter la recherche de polynômes trigonométriques extrémaux aux points exposés de la boule unité K de l’espace C_{Λ} (rappelons qu’un point P de K est exposé par un hyperplan H si H ne coupe K qu’en P.) Pourquoi un point exposé P de K atteint-il son module maximum en au moins deux points ? parce que la forme linéaire qui définit l’hyperplan H s’étend en une mesure μ qui atteint sa norme sur P et on sait que P doit être de module maximum sur le support de μ; la mesure μ n’est pas une masse de Dirac puisqu’elle atteint sa norme uniquement en P, de sorte que le support de μ a au moins deux points.

Dans l’article The maximum modulus of a trigonometric trinomial, nous obtenons une description très complète des points de module maximum d’un trinôme trigonométrique (voir le théorème H.7.1 page 109) dont voici le point saillant.

Théorème 17.1. Soit Λ un ensemble de trois entiers λ_{1} < λ_{2} < λ_{3} tels que λ_{2} − λ_{1} et λ_{3} − λ_{2} soient premiers entre eux. Soient r_{1}, r_{2}, r_{3} trois réels strictement positifs. Le trinôme trigonométrique

f(x) = r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + r_{3}e^{i(t_{3} +λ_{3} x)}

atteint son module maximum en un point unique modulo 2π, de multiplicité 2, sauf si son argument τ égale π : Si f atteint son module maximum en deux points modulo 2π, c’est parce que son graphe admet un axe de symétrie.

Esquisse de démonstration. On ramène le trinôme trigonométrique f à la forme normale (9) avec de plus τ ∈ [0, π] et on montre alors que f doit atteindre son module maximum sur le petit intervalle [−τ/k(k + l), τ/l(k + l)] en trouvant, pour tout y hors de cet intervalle, un point x qui y soit pour lequel |f(x)| ⩾ |f(y)|. De plus, on peut rendre cette inégalité stricte sauf si τ = π. Il reste alors à étudier les variations de |f| sur [−τ/k(k + l), τ/l(k + l)].

La formule de Chebotarëv donne alors une nouvelle solution pour le problème extrémal 15.1.

Proposition 17.2. Le module maximum de r_{1}e^{−ikx} +r_{2}e^{iτ/(k+l)} +r_{3}e^{ilx} est une fonction strictement décroissante de τ sur [0, π].

Démonstration. Restons dans le contexte de l’esquisse de démonstration ci-dessus et soit τ ∈ ]0, π[. Soit x^{∗} l’unique point de module maximum pour f : on a vu que x^{∗} ∈ [−τ/k(k + l), τ/l(k + l)]. Mais alors

|f(x)|^{2} = r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + 2r_{1}r_{3} cos((k + l)x) + 2r_{2}(r_{1} cos(τ/(k + l) + kx) + r_{3} cos(τ/(k + l) − lx))

et

\frac{k + l}{2r_{2}} \frac{∂|f|^{2}}{∂τ} (x^{∗}) = −r_{1} sin(τ/(k + l) + kx^{∗}) − r_{3} sin(τ/(k + l) − lx^{∗}) < 0

car τ/(k + l) + kx^{∗} ∈ [0, τ/l] et τ/(k + l) − lx^{∗} ∈ [0, τ/k] ne s’annulent pas simultanémanent.

Illustrons notre propos : le module maximum de f(x) = 4e^{−i2x} +e^{it}+e^{ix} est la distance maximum de points de l’hypotrochoïde d’équation z = 4e^{−i2x} + e^{ix} à un point donné −e^{it} du plan complexe. Nous avons donc montré que si deux points de H sont simultanément à distance maximum de −e^{it}, alors −e^{it} est sur un axe de symétrie de H, c’est-à-dire t ≡ π/3 mod 2π/3.

\begin{matrix}−1 \\ −e^{iπ/3} \end{matrix}

Notre étude aboutit au théorème suivant, dont on peut espérer une généralisation à des ensembles Λ plus grands.

Théorème 17.3 (page 100). Soit Λ un ensemble à trois éléments. Soit K la boule unité de l’espace C_{Λ} et soit P ∈ K.

– Le point P est un point exposé de K si et seulement si P est un monôme trigonométrique e^{iα} e^{iλx} avec α ∈ ℝ et λ ∈ Λ ou un trinôme trigonométrique qui atteint son module maximum, 1, en deux points modulo 2π/d. Toute forme linéaire sur C_{Λ} atteint sa norme en un point exposé de K.

– Le point P est un point extrémal de K si et seulement si P est un monôme trigonométrique e^{iα} e^{iλx} avec α ∈ ℝ et λ ∈ Λ ou un trinôme trigonométrique tel que 1 − |P |^{2} a quatre zéros modulo 2π, comptés avec leur multiplicité.

18 La variation du module maximum en fonction de l’argument

Nous utilisons aussi la formule de Chebotarëv pour montrer que le module maximum d’un trinôme trigonométrique est une fonction décroissante de la valeur absolue |τ| de son argument (voir (8)) et pour borner cette décroissance. Nous obtenons les inégalités suivantes.

Théorème 18.1 (pages 101, 101 et 102). Soit f un trinôme trigonométrique comme en (7) et varions ses arguments de Δt_{1}, Δt_{2}, Δt_{3}. Notons f̃ le trinôme qui en résulte, τ̃ l’argument du trinôme f̃, et Δτ la variation de l’argument :

Δτ ≡ (λ_{3} − λ_{2})Δt_{1} + (λ_{1} − λ_{3})Δt_{2} + (λ_{2} − λ_{1})Δt_{3} mod 2πℤ.

Si |τ̃| > |τ|, alors

max_{x}|f̃(x)| < max_{x}|f(x)|
⩽ \frac{|r_{1} + r_{2}e^{iτ/|λ_{3} −λ_{1} |} + r_{3}|}{|r_{1} + r_{2}e^{iτ|λ_{3} −λ_{1} |} + r_{3}|} max_{x}|f̃(x)|  (10)
⩽ \frac{cos(τ/2(λ_{3} − λ_{1}))}{cos(τ̃/2(λ_{3} − λ_{1}))} max_{x}|f̃(x)|  (11)
⩽ \frac{cos((π − |Δτ|)/2(λ_{3} − λ_{1}))}{cos(π/2(λ_{3} − λ_{1}))} max_{x}|f̃(x)|;  (12)

l’inégalité (10) est une égalité si et seulement si r_{1} : r_{3} = λ_{3} − λ_{2} : λ_{2} − λ_{1}, l’inégalité (11) si et seulement si r_{1} : r_{2} : r_{3} = λ_{3}−λ_{2} : λ_{3}−λ_{1} : λ_{2}−λ_{1} et l’inégalité (12) si et seulement si de plus τ = π. La norme du multiplicateur de Fourier relatif unimodulaire par les signes e^{iΔt_{1}} , e^{iΔt_{2}} , …, e^{iΔt_{n}} est donc le facteur dans l’inégalité (12).

En particulier, le module maximum de f comme fonction de τ admet les deux minorants |r_{1} + r_{2}e^{iτ/|λ_{3} −λ_{1} |} + r_{3}| et (r_{1} + r_{2} + r_{3}) cos(τ/2(λ_{3} − λ_{1})) sur l’intervalle [−π, π]. Illustrons-le dans le cas particulier f(x) = 4e^{−i2x} + e^{iτ/3} + e^{ix} :

0 m 5  τ π m = max_{x} |4e^{−i2x} + e^{iτ/3} + e^{ix} |  m = |4 + e^{iτ/3} + 1| m = (4 + 1 + 1) cos(τ/6)

19 Problèmes extrémaux pour les quadrinômes trigonométriques

Dans le cas Λ = {0, 1, 2, 3}, le problème extrémal 15.4 est un problème ouvert posé par Harold S. Shapiro en 1951. Par des moyens heuristiques, nous avons conjecturé que les polynômes extrémaux sont de la forme

\frac{i2\sqrt{2}cosu − 1 − 3 sinu}{15} + \frac{3 + sinu}{10} e^{ix} + \frac{3 − sinu}{10} e^{i2x} + \frac{i2\sqrt{2}cosu − 1 + 3 sinu}{15} e^{i3x}

où u parcourt [0, 2π[. Ces polynômes sont étudiés dans la note On the Sidon constant of {0, 1, 2, 3}, section I.4. On en déduirait que la constante de Sidon de Λ vaut 5/3, qui est sa constante d’inconditionnalité réelle (voir la proposition I.5.4 page 122.) Il s’agit de le démontrer et d’étudier plus généralement les polynômes trigonométriques à quatre termes dont le module atteint son maximum en trois points.

Lacunary matrices

with Asma Harcharras and Krzysztof Oleszkiewicz.

Abstract

We study unconditional subsequences of the canonical basis (e_{rc} ) of elementary matrices in the Schatten class S^{p}. They form the matrix counterpart to Rudin’s Λ(p) sets of integers in Fourier analysis. In the case of p an even integer, we find a sufficient condition in terms of trails on a bipartite graph. We also establish an optimal density condition and present a random construction of bipartite graphs. As a byproduct, we get a new proof for a theorem of Erdős on circuits in graphs.

1 Introduction

We study the following question on the Schatten class S^{p}.

(†) How many matrix coefficients of an operator x ∈ S^{p} must vanish so that the norm of x has a bounded variation if we change the sign of the remaining nonzero matrix coefficients?

Let C be the set of columns and R be the set of rows for coordinates in the matrix, in general two copies of ℕ. Let I ⊆ R × C be the set of matrix coordinates of the remaining nonzero matrix coefficients of x. Property (†) means that the subsequence (e_{rc})_{(r,c)∈I} of the canonical basis of elementary matrices is an unconditional basic sequence in S^{p}: I forms a σ(p) set in the terminology of [37, §4].

It is natural to wonder about the operator valued case, where the matrix coefficients are themselves operators in S^{p}. As the proof of our main result carries over to that case, we shall state it in the more general terms of complete σ(p) sets.

We show that for our purpose, a set of matrix entries I ⊆ R × C is best understood as a bipartite graph. Its two vertex classes are C and R, whose elements will respectively be termed “column vertices” and “row vertices”. Its edges join only row vertices r ∈ R with column vertices c ∈ C, this occurring exactly if (r, c) ∈ I.

We obtain a generic condition for σ(p) sets in the case of even p (Theorem 3.2) that generalises [37, Prop. 6.5]. These sets reveal in fact as a matrix counterpart to Rudin’s Λ(p) sets and we are able to transfer Rudin’s proof of [88, Theorem 4.5(b)] to a non-commutative context: his number r_{s}(E, n) is replaced by the numbers of Def. 2.4(b) and we count trails between given vertices instead of representations of an integer.

We also establish an upper bound for the intersection of a σ(p) set with a finite product set R′×C′ (Theorem 4.2): this is a matrix counterpart to Rudin’s [88, Theorem 3.5]. In terms of bipartite graphs, this intersection is the subgraph induced by the vertex subclasses C′ ⊆ C and R′ ⊆ R.

The bound of Theorem 4.2 provides together with Theorem 3.2 a generalisation of a theorem by Erdős [29, p. 33] on graphs without circuits of a given even length. In the last part of this article, we present a random construction of maximal σ(p) sets for even integers p.

Terminology. C is the set of columns and R is the set of rows, in general both indexed by ℕ. The set V of all vertices is their disjoint union R ⨿ C. An edge on V is a pair {v, w} ⊆ V. A graph on V is given by its set of edges E. A bipartite graph on V with vertex classes C and R has only edges {r, c} such that c ∈ C and r ∈ R and may therefore be described alternatively by the set I = { (r, c) ∈ R × C : {r, c} ∈ E }. A trail of length s in a graph is a sequence (v_{0}, …, v_{s}) of s + 1 vertices such that {v_{0}, v_{1}}, …, {v_{s−1} , v_{s}} are pairwise distinct edges of the graph. A trail is a path if its vertices are pairwise distinct. A circuit of length p in a graph is a sequence (v_{1}, …, v_{p}) of p vertices such that {v_{1}, v_{2}}, …, {v_{p−1} , v_{p}}, {v_{p}, v_{1}} are pairwise distinct edges of the graph. A circuit is a cycle if its vertices are pairwise distinct.

Notation. 𝕋 = { z ∈ ℂ : |z| = 1 }. Let q = (r, c) ∈ R × C. The transpose of q is q^{∗} = (c, r). The entry (elementary matrix) e_{q} = e_{rc} is the operator on ℓ_{2} that maps the cth basis vector on the rth basis vector and all other basis vectors on 0. The matrix coefficient at coordinate q of an operator x on ℓ_{2} is x_{q} = tr e_{q}^{∗}x and its matrix representation is (x_{q})_{q∈R×C} = ∑ _{q∈R×C} x_{q}e_{q}. The Schatten class S^{p}, 1 ⩽ p < ∞, is the space of those compact operators x on ℓ_{2} such that ∥x∥_{p}^{p} = tr|x|^{p} = tr(x^{∗}x)^{p/2} < ∞. For I ⊆ R × C, the entry space S_{I}^{p} is the space of those x ∈ S^{p} whose matrix representation is supported by I: x_{q} = 0 if q ∉ I. S_{I}^{p} is also the closed subspace of S^{p} spanned by (e_{q})_{q∈I} . The S^{p}-valued Schatten class S^{p}(S^{p}) is the space of those operators x from ℓ_{2} to S^{p} such that ∥x∥_{p}^{p} = tr(tr|x|^{p}) < ∞, where the inner trace is the S^{p}-valued analogue of the usual trace. The S^{p}-valued entry space S_{I}^{p}(S^{p}) is the closed subspace spanned by the x_{q}e_{q} with x_{q} ∈ S^{p} and q ∈ I: x_{q} = tr e_{q}^{∗}x is the operator coefficient of x at matrix coordinate q. Thus, for even integers p and x = (x_{q})_{q∈I} = ∑ _{q∈I} x_{q}e_{q} with x_{q} ∈ S^{p} and I finite,

∥x∥_{p}^{p} = ∑_{q_{1} ,...q_{p} ∈I} trx_{q_{1}}^{∗}x_{q_{2}} …x_{q_{p−1}}^{∗} x_{q_{p}} tr e_{q_{1}}^{∗} e_{q_{2}} … e_{q_{p−1}}^{∗} e_{q_{p}}.

A Schur multiplier T on S_{I}^{p} associated to (μ_{q})_{q∈I} ∈ ℂ^{I} is a bounded operator on S_{I}^{p} such that T e_{q} = μ_{q}e_{q} for q ∈ I. T is furthermore completely bounded (c.b. for short) if T is bounded as the operator on S_{I}^{p}(S^{p}) defined by T (x_{q}e_{q}) = μ_{q}x_{q}e_{q} for x_{q} ∈ S^{p} and q ∈ I.

We shall stick to this harmonic analysis type notation; let us nevertheless show how these objects are termed with tensor products: S^{p}(S^{p}) is also S^{p}(ℓ_{2} ⊗_{2} ℓ_{2}) endowed with ∥x∥_{p}^{p} = tr⊗ tr|x|^{p}; one should write x_{q}⊗ e_{q} instead of x_{q}e_{q}; here x_{q} = Id_{S^{p}} ⊗tr((Id_{ℓ_{2}} ⊗ e_{q}^{∗})x); T is c.b. if Id_{S^{p}} ⊗T is bounded on S^{p}(ℓ_{2} ⊗_{2} ℓ_{2}).

Acknowledgement. The first-named and last-named authors undertook this research at the Équipe d’Analyse de l’Université Paris 6. It is their pleasure to acknowledge its kind hospitality.

2 Definitions

We use the notion of unconditionality in order to define the matrix analogue of Rudin’s “commutative” Λ(p) sets.

Definition 2.1. Let X be a Banach space. The sequence (y_{n}) ⊆ X is an unconditional basic sequence in X if there is a constant D such that

∥∑ ϑ_{n}c_{n}y_{n} ∥_{X} ⩽ D∥∑ c_{n}y_{n} ∥_{X}

for every real (vs. complex) choice of signs ϑ_{n} ∈ {−1, 1} (vs. ϑ_{n} ∈ 𝕋) and every finitely supported sequence of scalar coefficients (c_{n}). The optimal D is the real (vs. complex) unconditionality constant of (y_{n}) in X.

Real and complex unconditionality are isomorphically equivalent: the complex unconditionality constant is at most π/2 times the real one. The notions of unconditionality and multipliers are intimately connected: we have

Proposition 2.2. Let (y_{n}) ⊆ X be an unconditional basic sequence in X and let Y be the closed subspace of X spanned by (y_{n}). The real (vs. complex) unconditionality constant of (y_{n}) in X is exactly the least upper bound for the norms ∥T ∥_{ℒ (Y)} , where T is the multiplication operator defined by T y_{n} = μ_{n}y_{n}, and the μ_{n} range over all real (vs. complex) numbers with |μ_{n}| ⩽ 1.

Let us encompass the notions proposed in Question (†).

Definition 2.3. Let I ⊆ R × C and p > 2.

(a) [37, Def. 4.1] I is a σ(p) set if (e_{q})_{q∈I} is an unconditional basic sequence in S^{p}. This amounts to the uniform boundedness of the family of all relative Schur multipliers by signs

T_{ϑ} : S_{I}^{p} → S_{I}^{p}, x = (x_{q})_{q∈I} ↦ T_{ϑ}x = (ϑ_{q}x_{q})_{q∈I} with ϑ_{q} ∈ {−1, 1}.  (A.1)

By [37, Lemma 0.5], this means that there is a constant D such that for every finitely supported operator x = (x_{q})_{q∈I} = ∑ _{q∈I} x_{q}e_{q} with x_{q} ∈ ℂ

D^{−1} ∥x∥_{p} ⩽ |||x|||_{p} ⩽ ∥x∥_{p},  (A.2)

where the second inequality is a convexity inequality that is always satisfied (see [96, Theorem 8.9]) and

|||x|||_{p}^{p} = ∑_{c} ( ∑_{r} |x_{rc}|^{2})^{p/2} ∨ ∑_{r} ( ∑_{c} |x_{rc}|^{2})^{p/2}.  (A.3)

(b) [37, Def. 4.4] I is a complete σ(p) set if the family of all relative Schur multipliers by signs (A.1) is uniformly c.b. By [37, Lemma 0.5], I is completely σ(p) if and only if there is a constant D such that for every finitely supported operator valued operator x = (x_{q})_{q∈I} = ∑ _{q∈I} x_{q}e_{q} with x_{q} ∈ S^{p}

D^{−1} ∥x∥_{p} ⩽ |||x|||_{p} ⩽ ∥x∥_{p},  (A.4)

where the second inequality is a convexity inequality that is always satisfied and

|||x|||_{p}^{p} = ∑_{c} ∥( ∑_{r} x_{rc}^{∗}x_{rc})^{1/2} ∥_{p}^{p} ∨ ∑_{r} ∥( ∑_{c} x_{rc}x_{rc}^{∗})^{1/2} ∥_{p}^{p}.

The notion of a complete σ(p) set is stronger than that of a σ(p) set: Inequality (A.2) amounts to Inequality (A.4) tested on operators of the type x = ∑ _{q∈I} x_{q}e_{q} with each x_{q} acting on the same one-dimensional subspace of ℓ_{2}. It is an important open problem to decide whether the notions differ. An affirmative answer would solve Pisier’s conjecture about completely bounded Schur multipliers [79, p. 113].

Notorious examples of 1-unconditional basic sequences in all Schatten classes S^{p} are single columns, single rows, single diagonals and single anti-diagonals — and more generally “column sets” (vs. “row sets”) I such that for each (r, c) ∈ I, no other element of I is in the column c (vs. row r). These sets are called sections in [100, Def. 4.3]

We shall try to express these notions in terms of trails on bipartite graphs. We proceed as announced in the Introduction: then each example above is a union of disjoint star graphs in which one vertex of one class is connected to some vertices of the other class: trails in a star graph have at most length 2.

Definition 2.4. Let I ⊆ R × C and s ⩾ 1 an integer. We consider I as a bipartite graph: its vertex set is V = R ⨿ C and its edge set is E = { {r, c} ⊆ V : (r, c) ∈ I }.

(a) The sets of trails of length s on the graph I from the column (vs. row) vertex v_{0} to the vertex v_{s} are respectively

𝒞^{s}(I; v_{0}, v_{s}) = { (v_{0}, …, v_{s}) ∈ V^{s+1} : v_{0} ∈ C & all {v_{i}, v_{i+1} } ∈ E are distinct},

ℛ^{s}(I; v_{0}, v_{s}) = { (v_{0}, …, v_{s}) ∈ V^{s+1} : v_{0} ∈ R & all {v_{i}, v_{i+1} } ∈ E are distinct}.

(b) We define the Rudin numbers of trails starting respectively with a column vertex and a row vertex by c_{s}(I; v_{0}, v_{s}) = #𝒞^{s}(I; v_{0}, v_{s}) and r_{s}(I; v_{0}, v_{s}) = #ℛ^{s}(I; v_{0}, v_{s}).

Remark 2.5. In other words, for an integer l ⩾ 1,

c_{2l−1} (I; v_{0}, v_{2l−1} ) = # \left[{(r_{1}, c_{1}), (r_{1}, c_{2}), (r_{2}, c_{2}), (r_{2}, c_{3}), …, (r_{l}, c_{l}) \atop pairwise distinct in I : c_{1} = v_{0}, r_{l} = v_{2l−1}}\right]
c_{2l}(I; v_{0}, v_{2l}) = # \left[{(r_{1}, c_{1}), (r_{1}, c_{2}), …, (r_{l}, c_{l}), (r_{l}, c_{l+1} ) \atop pairwise distinct in I : c_{1} = v_{0}, c_{l+1} = v_{2l}}\right]

and similarly for r_{s}(I; v_{0}, v_{s}). If s is odd, then c_{s}(I; v_{0}, v_{s}) = r_{s}(I; v_{s}, v_{0}) for all (v_{0}, v_{s}) ∈ C × R. But if s is even, one Rudin number may be bounded while the other is infinite: see [37, Rem. 6.4(ii)].

3 σ(p) sets as matrix Λ(p) sets

We claim the following result.

Theorem 3.1. Let I ⊆ R × C and p = 2s be an even integer. If I is a union of sets I_{1}, …, I_{l} such that one of the Rudin numbers c_{s}(I_{j}; v_{0}, v_{s}) or r_{s}(I_{j}; v_{0}, v_{s}) is a bounded function of (v_{0}, v_{s}), for each j, then I is a complete σ(p) set.

This follows from Theorem 3.2 below: the union of two complete σ(p) sets is a complete σ(p) set by [37, Rem. after Def. 4.4]; furthermore the transposed set I^{∗} = { q^{∗} : q ∈ I } ⊆ C × R is a complete σ(p) set provided I is. Note that the case of σ(∞) sets (see [37, Rem. 4.6(iii)]) provides evidence that Theorem 3.1 might be a characterisation of complete σ(p) sets for even p.

Theorem 3.2. Let I ⊆ R × C and p = 2s be an even integer. If the Rudin number c_{s}(I; v_{0}, v_{s}) is a bounded function of (v_{0}, v_{s}), then I is a complete σ(p) set.

This is proved for p = 4 in [37, Prop. 6.5]. We wish to emphasise that the proof below follows the scheme of the proof of [37, Theorem 1.13]. In particular, we make crucial use of Pisier’s idea to express repetitions by dependent Rademacher variables ([37, Prop. 1.14]).

Proof. Let x = ∑ _{q∈I} x_{q}e_{q} with x_{q} ∈ S^{p}. We have the following expression for ∥x∥_{p}.

∥x∥_{p}^{p} = tr⊗ tr(x^{∗}x)^{s} = ∥y∥_{2}^{2} with y = \overbrace{x^{∗}xx^{∗} ⋯ x^{(∗)}}^{s terms},

i.e., y is the product of s terms which are alternatively x^{∗} and x, and we set x^{(∗)} = x for even s, x^{(∗)} = x^{∗} for odd s. Set C^{(∗)} = C for even s and C^{(∗)} = R for odd s. Let (v_{0}, v_{s}) ∈ C × C^{(∗)} and y_{v_{0} v_{s}} = tr e_{v_{0} v_{s}}^{∗} y be the matrix coefficient of y at coordinate (v_{0}, v_{s}). Then we obtain by the rule of matrix multiplication

y = ∑_{q_{1} ,...,q_{s} ∈I} (x_{q_{1}}^{∗} e_{q_{1}}^{∗})(x_{q_{2}} e_{q_{2}}) …(xq_{s}^{(∗)} eq_{s}^{(∗)} )

y_{v_{0} v_{s}} = ∑_{(v_{1} ,v_{0} ), (v_{1} ,v_{2} ),... ∈I} x_{v_{1} v_{0}}^{∗} x_{v_{1} v_{2}} x_{v_{3} v_{2}}^{∗} …x(v_{s−1} ,v_{s} )^{(∗)}^{(∗)} .  (A.5)

Let ℰ be the set of equivalence relations on {1, …, s}. Then

y = ∑_{∼∈ℰ} ∑_{i∼j⇔q_{i} =q_{j}} (x_{q_{1}}^{∗} e_{q_{1}}^{∗})(x_{q_{2}} e_{q_{2}}) …(xq_{s}^{(∗)} eq_{s}^{(∗)} ).  (A.6)

We shall bound the sum above in two steps.

(a) Let ∼ be equality and consider the corresponding term in the sum (A.6). The number of terms in the sum (A.5) such that {v_{i−1} , v_{i}} ≠ {v_{j−1} , v_{j} } if i ≠ j is c_{s}(I; v_{0}, v_{s}). If c is an upper bound for c_{s}(I; v_{0}, v_{s}), we have by the expression of the Hilbert–Schmidt norm and the Arithmetic-Quadratic Mean Inequality

∥ ∑_{q_{1} ,...,q_{s} pairwise distinct} (x_{q_{1}}^{∗} e_{q_{1}}^{∗})(x_{q_{2}} e_{q_{2}}) …(xq_{s}^{(∗)} eq_{s}^{(∗)} )∥_{2}^{2}

= ∑_{(v_{0} ,v_{s} )∈C×C^{(∗)}}  ∥ ∑_{v∈𝒞^{s} (I;v_{0} ,v_{s} )} x_{v_{1} v_{0}}^{∗} x_{v_{1} v_{2}} x_{v_{3} v_{2}}^{∗} …x(v_{s−1} ,v_{s} )^{(∗)}^{(∗)} ∥_{2}^{2}

⩽ c ∑_{(v_{0} ,v_{s} )∈C×C^{(∗)}}  ∑_{v∈𝒞^{s} (I;v_{0} ,v_{s} )} ∥x_{v_{1} v_{0}}^{∗} x_{v_{1} v_{2}} x_{v_{3} v_{2}}^{∗} …x(v_{s−1} ,v_{s} )^{(∗)}^{(∗)} ∥_{2}^{2}

= c ∑_{q_{1} ,...,q_{s} pairwise distinct} ∥(x_{q_{1}}^{∗} e_{q_{1}}^{∗})(x_{q_{2}} e_{q_{2}}) …(xq_{s}^{(∗)} eq_{s}^{(∗)} )∥_{2}^{2}

⩽ c ∑_{q_{1} ,...,q_{s}} ∥(x_{q_{1}}^{∗} e_{q_{1}}^{∗})(x_{q_{2}} e_{q_{2}}) …(xq_{s}^{(∗)} eq_{s}^{(∗)} )∥_{2}^{2}
= c∥ ∑_{q_{1} ,...,q_{s}} |(x_{q_{1}}^{∗} e_{q_{1}}^{∗})(x_{q_{2}} e_{q_{2}}) …(xq_{s}^{(∗)} eq_{s}^{(∗)} )|^{2} ∥_{1}

3. σ(p) SETS AS MATRIX Λ(p) SETS  25

Now this last expression may be bounded accordingly to [37, Cor. 0.9] by

c( ∥∑ (x_{q}^{∗}e_{q}^{∗})(x_{q}e_{q})∥_{s} ∨ ∥∑ (x_{q}e_{q})(x_{q}^{∗}e_{q}^{∗})∥_{s})^{s} = c|||x|||_{p}^{p} :  (A.7)

see [37, Lemma 0.5] for the last equality.

(b) Let ∼ be distinct from equality. The corresponding term in the sum (A.6) cannot be bounded directly. Consider instead

Ψ(∼) = ∥ ∑_{i∼j⇒q_{i} =q_{j}} (x_{q_{1}}^{∗} e_{q_{1}}^{∗})(x_{q_{2}} e_{q_{2}}) …(xq_{s}^{(∗)} eq_{s}^{(∗)} )∥_{2} = ∥ ∑_{i∼j⇒q_{i} =q_{j}} ∏_{i=1}^{s} f_{i}(q_{i})∥_{2}

with f_{i}(q) = x_{q}e_{q} for even i and f_{i}(q) = x_{q}^{∗}e_{q}^{∗} for odd i. We may now apply Pisier’s Lemma [37, Prop. 1.14]: let 0 ⩽ r ⩽ s − 2 be the number of one element equivalence classes modulo ∼; then

Ψ(∼) ⩽ ∥x∥_{p}^{r}(B|||x|||_{p})^{s−r} ,  (A.8)

where B is the constant arising in Lust-Piquard’s non-commutative Khinchin inequality. In order to finish the proof, one does an induction on the number of atoms of the partition induced by ∼, along the lines of step 2 of the proof of [37, Theorem 1.13].

The Moebius inversion formula for partitions enabled Pisier [78] to obtain the following explicit bounds in the computation above:

∥y∥_{2} ⩽ c^{1/2} |||x|||_{p}^{s} + ∑_{0⩽r⩽s−2} \binom{s}{r} (s − r)!∥x∥_{p}^{r}((3π/4)|||x|||_{p})^{s−r}

∥x∥_{p} ⩽ ((4c)^{1/p} ∨ 9πp/8)|||x|||_{p}.  (A.9)

Let us also record the following consequence of his study of p-orthogonal sums. The family (x_{q}e_{q})_{q∈I} is p-orthogonal in the sense of [78] if and only if the graph associated to I does not contain any circuit of length p, so that we have by [78, Theorem 3.1]:

Theorem 3.3. Let p ⩾ 4 be an even integer. If I does not contain any circuit of length p, then I is a complete σ(p) set with constant at most 3πp/2.

Remark 3.4. Pisier proposed to us the following argument to deduce a weaker version of Theorem 3.2 from [37, Theorem 1.13]. Let Γ = 𝕋^{V} and z_{v} denote the vth coordinate function on Γ. Associate to I the set Λ = { z_{r}z_{c} : (r, c) ∈ I }. Let still p = 2s be an even integer. Then I is a complete σ(p) set if Λ is a complete Λ(p) set as defined in [37, Def. 1.5], which in turn holds if Λ has property Z(s) as given in [37, Def. 1.11]. It turns out that this condition implies the uniform boundedness of

c_{t}(I; v_{0}, v_{t}) ∨ r_{t}(I; v_{0}, v_{t}) for t ⩽ s, v_{0}, v_{t} ∈ V.

For p ⩾ 8, this implication is strict: in fact, the countable union of disjoint cycles of length 4 (“quadrilaterals”)

I = ⋃ _{i⩾0}{(2i, 2i), (2i, 2i + 1), (2i + 1, 2i + 1), (2i + 1, 2i)}

satisfies c_{t}(I; v_{0}, v_{t}) ∨ r_{t}(I; v_{0}, v_{t}) ⩽ 2 whereas Λ does not satisfy Z(s) for any s ⩾ 4.

Remark 3.5. Theorem 3.1 is especially useful to construct c.b. Schur multipliers: by [37, Rem. 4.6(ii)], if I is a complete σ(p) set, there is a constant D (the constant D in (A.4)) such that for every sequence (μ_{q}) ∈ ℂ^{R×C} supported by I and every operator T_{μ} : (x_{q}) ↦ (μ_{q}x_{q}) we have

∥T_{μ}∥_{ℒ (S^{p} (S^{p}))} ⩽ D sup_{q∈I}|μ_{q}|.

4 The intersection of a σ(p) set with a finite product set

Let I ⊆ R × C considered as a bipartite graph as in the Introduction and let I′ ⊆ I be the subgraph induced by the vertex set C′⨿ R′, with C′ ⊆ C a set of m column vertices and R′ ⊆ R a set of n row vertices. In other words, I′ = I ∩ R′× C′. Let d(v) be the degree of the vertex v ∈ C′⨿ R′ in I′: in other words,

∀c ∈ C′ d(c) = #[I′∩ R′× {c}],
∀r ∈ R′ d(r) = #[I′∩ {r} × C′].

Let us recall that the dual norm of (A.3) is

|||x|||_{p′} = inf_{\begin{matrix}α,β∈S^{p′} \\ α+β=x \end{matrix}} ( ∑_{c} ( ∑_{r} |α_{rc}|^{2})^{p′/2}) ^{1/p′} + ( ∑_{r} ( ∑_{c} |β_{rc}|^{2})^{p′/2}) ^{1/p′} ,

where p ⩾ 2 and 1/p + 1/p′ = 1 (see [37, Rem. after Lemma 0.5]).

Lemma 4.1. Let 1 ⩽ p′ ⩽ 2 and x = ∑ _{q∈I′} x_{q}. Then

|||x|||_{p′}^{p′} ⩾ ∑ ∑_{(r,c)∈I′} ( max(d(c), d(r))^{1/2−1/p′} |x_{rc}|)^{p′}.

Proof. By the p′-Quadratic Mean Inequality and by Minkowski’s Inequality,

( ∑_{c∈C′} ( ∑_{(r,c)∈I′} |α_{rc}|^{2})^{p′/2}) ^{1/p′} + ( ∑_{r∈R′} ( ∑_{(r,c)∈I′} |β_{rc}|^{2})^{p′/2}) ^{1/p′}

⩾ ( ∑_{c∈C′} d(c)^{p′/2−1} ∑_{(r,c)∈I′} |α_{rc}|^{p′})^{1/p′} + ( ∑_{r∈R′} d(r)^{p′/2−1} ∑_{(r,c)∈I′} |β_{rc}|^{p′})^{1/p′}

⩾ ( ∑ ∑_{(r,c)∈I′} (d(c)^{1/2−1/p′} |α_{rc}| + d(r)^{1/2−1/p′} |β_{rc}|)^{p′})^{1/p′}

The lemma follows by taking the infimum over all α, β with α_{q} + β_{q} = x_{q} for q ∈ I′ as one can suppose that α_{q} = β_{q} = 0 if q ∉ I; note further that 1/2 − 1/p′ ⩽ 0.

Theorem 4.2. If I is a σ(p) set with constant D as in (A.2), then the size #I′ of any subgraph I′ induced by m column vertices and n row vertices, in other words the cardinal of any subset I′ = I ∩ R′× C′ with #C′ = m and #R′ = n, satisfies

#I′ ⩽ D^{2}(m^{1/p}n^{1/2} + m^{1/2} n^{1/p})^{2}  (A.10)
⩽ 4D^{2} min(m, n)^{2/p} max(m, n).

The exponents in this inequality are optimal even for a complete σ(p) set I in the following cases: (a) if m or n is fixed;

(b) if p is an even integer and m = n.

Bound (A.10) holds a fortiori if I is a complete σ(p) set. Density conditions thus do not so far permit to distinguish σ(p) sets and complete σ(p) sets. One may conjecture that Inequality (A.10) is also optimal for p not an even integer and m = n: this would be a matrix counterpart to Bourgain’s theorem [14] on maximal Λ(p) sets.

Proof. If (A.2) holds, then ∥x|_{I′}∥_{p} ⩽ D|||x|||_{p} for all x ∈ S^{p} by Remark 3.5 applied to (μ_{q}) the indicator function of I′, and by duality |||x|_{I′}|||_{p′} ⩽ D∥x∥_{p′} for all x ∈ S^{p′} (compare with [37, Rem. 4.6(iv)]).

Let

y = ∑ ∑_{(r,c)∈I′} d(c)^{1/p′−1/2} e_{rc},

z = ∑ ∑_{(r,c)∈I′} d(r)^{1/p′−1/2} e_{rc},

Then the n rows of y are all equal, as well as the m columns of z: y and z have rank 1 and a single singular value. By the norm inequality followed by the (2/p′− 1)-Arithmetic Mean Inequality,

∥y + z∥_{p′} ⩽ ∥y∥_{p′} + ∥z∥_{p′}
= n^{1/2} ( ∑_{c∈C′} d(c)^{2/p′−1} )^{1/2} + m^{1/2} ( ∑_{r∈R′} d(r)^{2/p′−1} )^{1/2}
⩽ n^{1/2} m^{1−1/p′} (#I′)^{1/p′−1/2} + m^{1/2} n^{1−1/p′} (#I′)^{1/p′−1/2} .

We used that ∑ _{c∈C′} d(c) = ∑ _{r∈R′} d(r) = #I′. By Lemma 4.1 applied to x = y + z,

(#I′)^{1/p′} ⩽ D(n^{1/2} m^{1−1/p′} + m^{1/2} n^{1−1/p′} )(#I′)^{1/p′−1/2} ,

and we get therefore the first part of the theorem.

Let us show optimality in the given cases.

(a) Suppose that n is fixed and C′ = C: I′ = R′× C is a complete σ(p) set for any p as a union of n rows and #I′ = n · m.

(b) is proved in [37, Theorem 4.8].

Remark 4.3. If n ≁ m, the method used in [37, Theorem 4.8] does not provide optimal σ(p) sets but the following lower bound. Let p = 2s with s ⩾ 2 an integer. Consider a prime q and let k = s^{s−1} q^{s}. By [88, 4.7] and [37, Theorem 2.5], there is a subset F ⊆ {0, …, k − 1} with q elements whose complete Λ(2s) constant is independent of q. Let m ⩾ k and 0 ⩽ n ⩽ m and consider the Hankel set

I = { (r, c) ∈ {0, …, n − 1} × {0, …, m − 1} : r + c ∈ F + m − k }.

Then the complete σ(p) constant of I is independent of q by [37, Prop. 4.7] and

#I ⩾ \{ \begin{matrix}nq & if n ⩽ m − k + 1 \\ (m − k + 1)q & if n ⩾ m − k + 1. \end{matrix}

If we choose m = (s + 1)k − 1, this yields

#I ⩾ \frac{s^{1/s}}{(s + 1)^{1+1/s}} min(n, m) max(m, n)^{1/s} .

Random construction 6.1 provides bigger sets than this deterministic construction; however, it also does not provide sets that would show the optimality of Inequality (A.10) unless s = 2.

5 Circuits in graphs

Non-commutative methods yield a new proof to a theorem of Erdős [29, p. 33]. Note that its generalisation by Bondy and Simonovits [12] is stronger than Theorem 5.1 below as it deals with cycles instead of circuits. By Theorem 3.3 and (A.10)

Theorem 5.1. Let p ⩾ 4 be an even integer. If G is a nonempty graph on v vertices with e edges without circuit of length p, then

e ⩽ 18π^{2}p^{2} v^{1+2/p} .

If G is furthermore a bipartite graph whose two vertex classes have respectively m and n elements, then

e ⩽ 9π^{2}p^{2} min(m, n)^{2/p} max(m, n).  (A.11)

Proof. For the first assertion, recall that a graph G with e edges contains a bipartite subgraph with more than e/2 edges (see [11, p. xvii]).

Remark 5.2. Łuczak showed to us that (A.11) cannot be optimal if m and n are of very different order of magnitude. In particular, let p be a multiple of 4. Let e′ be the maximal number of edges of a graph on n vertices without circuit of length p/2. If m > pe′, he shows that (A.11) may be replaced by e < 3m.

We also get the following result, which enables us to conjecture a generalisation of the theorems of Erdős and Bondy and Simonovits.

Theorem 5.3. Let G be a nonempty graph on v vertices with e edges. Let s ⩾ 2 be an integer.

(i) If

e > 8D^{2} v^{1+1/s} with D > 9πs/4,

then one may choose two vertices v_{0} and v_{s} such that G contains more than D^{2s} /4 pairwise distinct trails from v_{0} to v_{s}, each of length s and with pairwise distinct edges.

(ii) One may draw the same conclusion if G is a bipartite graph whose two vertex classes have respectively m and n elements and

e > 4D^{2} min(m, n)^{1/s} max(m, n) with D > 9πs/4.

Proof. (i) According to [11, p. xvii], the graph G contains a bipartite subgraph with more than e/2 edges, so that we may apply (ii).

(ii) Combining inequalities (A.9) and (A.10), if D > 9πs/4, then there are vertices v_{0} and v_{s} such that the number c of pairwise distinct trails from v_{0} to v_{s}, each of length s and with pairwise distinct edges, satisfies (4c)^{1/2s} > D.

Two paths with equal endvertices are called independent if they have only their endvertices in common. Question 5.4. Let G be a graph on v vertices with e edges. Let s, l ⩾ 2 be integers. Is it so that there is a constant D such that if e > Dv^{1+1/s} , then G contains l pairwise independent paths of length s with equal endvertices? Remark 5.5. Note that by Theorem 4.2, the exponent 1 + 1/s is optimal in Theorem 5.3(i), whereas optimality of the exponent 1 + 2/p in Theorem 5.1 is an important open question in Graph Theory (see [49]).

One may also formulate Theorem 5.3(ii) in the following way.

Theorem 5.6. If a bipartite graph G_{2}(n, m) with n and m vertices in its two classes avoids any union of c pairwise distinct trails along s pairwise distinct edges between two given vertices as a subgraph, where the class of the first vertex is fixed, then the size e of the graph satisfies

e ⩽ 4 max((4c)^{1/2s} , 9πs/4)) min(m, n)^{1/s} max(m, n).

6 A random construction of graphs

Let us precise our construction of a random graph.

Random construction 6.1. Let C, R be two sets such that #C = m and #R = n. Let 0 ⩽ α ⩽ 1. A random bipartite graph on V = C ⨿ R is defined by selecting independently each edge in E = { {r, c} ⊆ V : (r, c) ∈ R × C } with the same probability α. The resulting random edge set is denoted by E′ ⊆ E and I′ ⊆ R × C denotes the associated random subset.

Our aim is to construct large sets while keeping down the Rudin number c_{s}.

Theorem 6.2. For each ε > 0 and for each integer s ⩾ 2, there is an α such that Random construction 6.1 yields subsets I′ ⊆ R × C with size

#I′ ∼ min(m, n)^{1/2+1/s} max(m, n)^{1/2−ε}

and with σ(2s) constant independent of m and n for mn → ∞.

Proof. Let us suppose without loss of generality that m ⩾ n. We want to estimate the Rudin number of trails in I′. Set C^{(∗)} = C for even s, C^{(∗)} = R for odd s and let (v_{0}, v_{s}) ∈ C × C^{(∗)} . Let l ⩾ 1 be a fixed integer. Then

ℙ[c_{s}(I′; v_{0}, v_{s}) ⩾ l]= ℙ[∃ l distinct trails (v_{0}^{j}, …, v_{s}^{j}) in 𝒞^{s}(I′; v_{0}, v_{s})]

= ℙ[ E′ ⊇ {{v_{i−1}^{j} , v_{i}^{j} }}_{i,j} : {(v_{0}^{j}, …, v_{s}^{j})}_{j=1}^{l} ⊆ 𝒞^{s}(R × C; v_{0}, v_{s}) ]

⩽ ∑_{k=⌈l^{1/s} ⌉}^{ls} #A_{k} · α^{k},

where A_{k} is the following set of l-element subsets of trails in 𝒞^{s}(R × C; v_{0}, v_{s}) built with k pairwise distinct edges

A_{k} = { {(v_{0}^{j}, …, v_{s}^{j})}_{j=1}^{l} ⊆ 𝒞^{s}(R × C; v_{0}, v_{s}) : #{{v_{i−1}^{j} , v_{i}^{j} }}_{i,j} = k };

the lower limit of summation is ⌈l^{1/s} ⌉ because one can build at most k^{s} pairwise distinct trails of length s with k pairwise distinct edges.

In order to estimate #A_{k}, we now have to bound the number of pairwise distinct vertices and the number of pairwise distinct column vertices in each set of l trails {(v_{0}^{j}, …, v_{s}^{j})}_{j=1}^{l} ∈ A_{k}. We claim that

#{ v_{i}^{j} : 1 ⩽ i ⩽ s − 1, 1 ⩽ j ⩽ l } ⩽ k(s − 1)/s,  (A.12)
#{ v_{2i}^{j} : 1 ⩽ i ⩽ ⌈s/2⌉ − 1, 1 ⩽ j ⩽ l } ⩽ k/2.  (A.13)

The second estimate is trivial, because each column vertex v_{2i}^{j} accounts for two distinct edges {v_{2i−1}^{j} , v_{2i}^{j}} and {v_{2i}^{j}, v_{2i+1}^{j} }. For the first estimate, note that each maximal sequence of h consecutive pairwise distinct vertices (v_{a+1}^{j} , …, v_{a+h}^{j} ) accounts for h + 1 pairwise distinct edges

{v_{a}^{j}, v_{a+1}^{j} }, {v_{a+1}^{j} , v_{a+2}^{j} }, …, {v_{a+h}^{j} , v_{a+h+1}^{j} };

as h ⩽ s − 1, h + 1 ⩾ hs/(s − 1). By (A.12) and (A.13),

#A_{k} ⩽ m^{k/2} n^{k/2−k/s} (k − k/s)^{ls−l} ⩽ (ls)^{ls} m^{k/2} n^{k/2−k/s} :

each element of A_{k} is obtained by a choice of at most k − k/s vertices, of which at most k/2 are column vertices, and the choice of an arrangement with repetitions of ls − l out of at most k − k/s vertices.

Put α = m^{−1/2} n^{−1/2+1/s} (#C · #C^{(∗)} )^{−ε} . Then

ℙ[ sup_{(v_{0} ,v_{s} )} c_{s}(I′; v_{0}, v_{s}) ⩾ l] ⩽ #C · #C^{(∗)} · (ls)^{ls} ∑_{k=⌈l^{1/s} ⌉}^{ls} (#C · #C^{(∗)} )^{−kε}

⩽ (ls)^{ls} \frac{(#C · #C^{(∗)} )^{1−⌈l^{1/s} ⌉ε}}{1 − (#C · #C^{(∗)} )^{−ε}} .

Choose l such that ⌈l^{1/s} ⌉ε > 1. Then this probability is little for mn large. On the other hand, #I′ is of order mnα with probability close to 1.

Remark 6.3. This construction yields much better results for s = 2. Keeping the notation of the proof above and m ⩾ n, we get k = 2l, A_{k} = \binom{n}{l} and

ℙ[ sup_{(v_{0} ,v_{2} )∈C×C} c_{2}(I′; v_{0}, v_{2}) ⩾ l] ⩽ m^{2} \binom{n}{l} α^{2l}.

Let l ⩾ 2 and α = m^{−1/l} n^{−1/2} . This yields sets I′ ⊆ R × C with size

#I′ ∼ n^{1/2} m^{1−1/l}

and with σ(4) constant independent of m and n. This case has been extensively studied in Graph theory as the “Zarankiewicz problem:” if c_{2}(I′; v_{0}, v_{2}) ⩽ l for all v_{0}, v_{2} ∈ C, then the graph I′ does not contain a complete bipartite subgraph on any two column vertices v_{0}, v_{2} and l + 1 row vertices. Reiman (see [11, Theorem VI.2.6]) showed that then

#I′ ⩽ (lnm(m − 1) + n^{2}/4)^{1/2} + n/2 ∼ l^{1/2} n^{1/2} m.

With use of finite projective geometries, he also showed that this bound is optimal for

n = l \frac{q^{r+1} − 1}{q^{2} − 1} \frac{q^{r} − 1}{q − 1} , m = \frac{q^{r+1} − 1}{q − 1}

with q a prime power and r ⩾ 2 an integer, and thus with m ⩽ n: there seems to be no constructive example of extremal graphs with c_{2}(I′; v_{0}, v_{2}) ⩽ l and m > n besides the trivial case of complete bipartite graphs with m > n = l − 1.

Remark 6.4. In the case s = 3, our result cannot be improved just by refining the estimation of #A_{k}. If we consider first l distinct paths that have their second vertex in common and then l independent paths, we get

#A_{2l+1} ⩾ \binom{m}{l} n , #A_{3l} ⩾ \binom{m}{l} \binom{n}{l} .

Therefore any choice of α as a monomial m^{−t} n^{−u} in the proof above must satisfy t ⩾ (l + 1)/(2l + 1), t + u ⩾ (2l + 2)/(3l) and this yields sets with

#I′ ≼ m^{1/2−1/2(4l+2)} n^{5/6−(7l+6)/(12l^{2} +6l)} .

Cycles and 1-unconditional matrices

Abstract

We characterise the 1-unconditional subsets (e_{rc} )_{(r,c)∈I} of the set of elementary matrices in the Schatten-von-Neumann class S^{p}. The set of couples I must be the set of edges of a bipartite graph without cycles of even length 4 ⩽ p if p is an even integer, and without cycles at all if p is a positive real number that is not an even integer. In the latter case, I is even a Varopoulos set of V-interpolation of constant 1. We also study the metric unconditional approximation property for the space S_{I}^{p} spanned by (e_{rc} )_{(r,c)∈I} in S^{p}.

1 Introduction

The starting point for this investigation has been the following isometric question on the Schattenvon-Neumann class S^{p}.

Question 1.1. Which matrix coefficients of an operator x ∈ S^{p} must vanish so that the norm of x does not depend on the argument, or on the sign, of the remaining nonzero matrix coefficients?

Let C be the set of columns and R be the set of rows for coordinates in the matrix. Let I ⊆ R×C be the set of matrix coordinates of the nonzero matrix coefficients of x (the pattern.) Question 1.1 describes the notion of a complex, or real, 1-unconditional basic sequence (e_{rc})_{(r,c)∈I} of elementary matrices in S^{p} (see Definition 4.1.)

By a convexity argument, Question 1.1 is equivalent to the following question on Schur multiplication.

Question 1.2. Which matrix coefficients of an operator x ∈ S^{p} must vanish so that for all matrices φ of complex, or real, numbers

∥φ ∗ x∥ ⩽ sup|φ_{rc}| ∥x∥,

where φ ∗ x is the Schur (or Hadamard or entrywise) product defined by

(φ ∗ x)_{rc} = φ_{rc}x_{rc}?

In the case p = ∞, Grothendieck’s inequality yields an estimation for the norm of Schur mul-

tiplication by φ in terms of the projective tensor product ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} : this norm is equivalent to the

supremum of the norm of those elements of ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} whose coefficient matrices are finite submatrices of φ. In the framework of tensor algebras over discrete spaces, Question 1.2 turns out to describe as well the isometric counterpart to Varopoulos’ V-Sidon sets as well as to his sets of V-interpolation. The following isometric question has however a different answer.

Question 1.3. Which coefficients of a tensor u ∈ ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} must vanish so that the norm of u is the maximal modulus of its coefficients?

In our answer to Question 1.2, S^{p} and Schur multiplication are treated as a noncommutative analogue to L^{p} and convolution. The main step is a careful study of the Schatten-von-Neumann norm ∥x∥ = (tr(x^{∗}x)^{p/2})^{1/p} for p an even integer. The rule of matrix multiplication provides an expression for this norm as a series in the matrix coefficients of x and their complex conjugate, indexed by the puples (v_{1}, v_{2}, …, v_{p}) satisfying (v_{2i−1} , v_{2i}), (v_{2i+1} , v_{2i}) ∈ I, where v_{p+1} = v_{1}: see the computation in Eq. (B.10). These are best understood as closed walks of length p on the bipartite graph G canonically associated to I: its vertex classes are C and R and its edges are given by the couples in I. A structure theorem for closed walks and a detailed study of the particular case in which G is a cycle yield the two following theorems that answer Questions 1.1 and 1.2.

Theorem 1.4. Let p ∈ (0, ∞] \ {2, 4, 6, …}. If the sequence of elementary matrices (e_{rc})_{(r,c)∈I} is a real 1-unconditional basic sequence in S^{p}, then the graph G associated to I contains no cycle. In this case, I is even a set of V-interpolation with constant 1: every sequence φ ∈ ℓ_{I}^{∞} may be interpolated

by a tensor u ∈ ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} such that ∥u∥ = ∥φ∥.

Theorem 1.5. Let p ∈ {2, 4, 6, …}. The sequence (e_{rc})_{(r,c)∈I} is a complex, or real, 1-unconditional basic sequence in S^{p} if and only if G contains no cycle of length 4, 6, …, p.

These theorems hold also for the complete counterparts to 1-unconditional basic sequences in the sense of Def. 4.1(c).

In particular, if we denote by U_{p} the property that (e_{rc})_{(r,c)∈I} is a 1-unconditional basic sequence in S^{p}, then we obtain the following hierarchy:

U_{p} for a p ∈ (0, ∞] \ {2, 4, 6, …} ⇒ ⋯⇒ U_{2n+2} ⇒ U_{2n} ⇒ ⋯⇒ U_{2} .

If C and R are finite, extremal graphs without cycles of given lengths remain an ongoing area of research in graph theory. Finite geometries seem to provide all known examples of such graphs when C and R become large. Proposition 11.6 and Remark 11.7 gather up known facts on this issue.

One may also avoid the terminology of graph theory and give an answer in terms of polygons drawn in a matrix by joining matrix coordinates with sides that follow alternately the row (horizontal) and the column (vertical) direction of the matrix:

– Suppose that p is not an even integer. If a pattern I contains the vertices of such a polygon, then there is an operator x ∈ S^{p} whose matrix coefficients vanish outside I and whose norm depends on the sign of its matrix coefficients. This condition is also necessary.

– If matrix coordinates of nonzero matrix coefficients of x are the vertices of such a polygon with n sides, then the norm of x in S^{p} depends on the argument of its matrix coefficients for every even integer p ⩾ n; if the matrix coefficients of x are real, then the norm of x even depends on the sign of its matrix coefficients. These conditions are also necessary.

An elementary example is given by the set

I = {(r, c) ∈ ℤ/7ℤ × ℤ/7ℤ : r + c ∈ {0, 1, 3}}.  (B.1)

The associated bipartite graph is known as the Heawood graph (Fig. B.1:) it is the incidence graph of the Fano plane (the finite projective plane PG(2, 2),) which is the smallest generalised triangle, and corresponds to the Steiner system S(2, 3; 7). It contains no cycle of length 4, but every pair of vertices is contained in a cycle of length 6.

Figure B.1: The Heawood graph Thus the p-trace norm of every matrix with pattern

( \begin{matrix}& 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 0 & ∗ & ∗ & 0 & ∗ & 0 & 0 & 0 \\ 1 & ∗ & 0 & ∗ & 0 & 0 & 0 & ∗ \\ 2 & 0 & ∗ & 0 & 0 & 0 & ∗ & ∗ \\ 3 & ∗ & 0 & 0 & 0 & ∗ & ∗ & 0 \\ 4 & 0 & 0 & 0 & ∗ & ∗ & 0 & ∗ \\ 5 & 0 & 0 & ∗ & ∗ & 0 & ∗ & 0 \\ 6 & 0 & ∗ & ∗ & 0 & ∗ & 0 & 0 \end{matrix} )

does not depend on the sign of its coefficients if and only if p ∈ {2, 4}.

These results give a complete description of the situation in which (e_{rc})_{(r,c)∈I} is a 1-uncondi-tional basis of the space S_{I}^{p} it spans in S^{p}. If this is not the case, S_{I}^{p} might still admit some other 1-unconditional basis. This leads to the following more general question.

Question 1.6. For which sets I does S_{I}^{p} admit some kind of almost 1-unconditional finite dimensional expansion of the identity?

The metric unconditional approximation property (muap) provides a formal definition for the object of Question 1.6: see Def. 10.1. We obtain the following results.

Theorem 1.7. Let p ∈ [1, ∞] \ {2, 4, 6, …}. If S_{I}^{p} has real (muap), then the distance of any two vertices that are not in the same vertex class is asymptotically infinite in G: their distance becomes arbitrarily large by deleting a finite number of edges from G.

Theorem 1.8. Let p ∈ {2, 4, 6, …}. The space S_{I}^{p} has complex, or real, (muap) if and only if any two vertices at distance 2j + 1 ⩽ p/2 are asymptotically at distance at least p − 2j + 1.

We now turn to a detailed description of this article. In Section 2, we provide tools for the computation of Schur multiplier norms. Section 3 characterises idempotent Schur multipliers and

0, 1-tensors in ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} of norm 1. In Section 4, we define the complex and real unconditional constants of basic sequences of elementary matrices and show that they are not equal in general. Section 5 looks back on Varopoulos’ results about tensor algebras over discrete spaces. Section 6 puts the connection between p-trace norm and closed walks of length p in the concrete form of closed walk relations. In Section 7, we compute the norm of relative Schur multipliers by signs in the case that G is a cycle, and estimate the corresponding unconditional constants. Section 8 is dedicated to a proof of Th. 1.4 and an answer to Question 1.3. Section 9 establishes Th. 1.5. In Section 10, we study the metric unconditional approximation property for spaces S_{I}^{p}. The final section provides four kinds of examples: sets obtained by a transfer of n-independent subsets of a discrete abelian group, Hankel sets, Steiner systems and Tits’ generalised polygons.

Terminology. C is the set of columns and R is the set of rows, both finite or countable and if necessary indexed by natural numbers. V, the set of vertices, is their disjoint union C ⨿ R: if there is a risk of confusion, an element n ∈ V that is a column (vs. a row) will be referred to as “col n” (vs. “row n”.) An edge on V is a pair {v, w} ⊆ V. A graph on V is given by a set of edges E. A bipartite graph on V with vertex classes C and R has only edges {r, c} such that c ∈ C and r ∈ R and may therefore be given alternatively by the set of couples I = {(r, c) ∈ R × C : {r, c} ∈ E}: this will be our custom throughout the article. A bipartite graph on V is complete if its set of couples I is the whole of R × C. Two graphs are disjoint if so are the sets of vertices of their edges. I is a column section if (r, c), (r′, c) ∈ I ⇒ r = r′, and a row section if (r, c), (r, c′) ∈ I ⇒ c = c′.

A walk of length s ⩾ 0 in a graph is a sequence (v_{0}, …, v_{s}) of s + 1 vertices such that {v_{0}, v_{1}}, …, {v_{s−1} , v_{s}} are edges of the graph. A walk is a path if its vertices are pairwise distinct. The distance of two vertices in a graph is the minimal length of a path in the graph that joins the two vertices; it is infinite if no such path exists. A closed walk of length p ⩾ 0 in a graph is a sequence (v_{1}, …, v_{p}) of p vertices such that {v_{1}, v_{2}}, …, {v_{p−1} , v_{p}}, {v_{p}, v_{1}} are edges of the graph. Note that p is necessarily even if the graph is bipartite. A closed walk is a cycle if its vertices are pairwise distinct. We take the convention that if a closed walk in a bipartite graph on V = C ⨿R is nonempty, then its first vertex is a column vertex: v_{1} ∈ C. We shall identify a path and a cycle with its set of edges {r, c} or the corresponding set of couples (r, c).

A bipartite graph on V is a tree if there is exactly one path between any two of its vertices. In this case, its vertices may be indexed by finite words over its set of vertices in the following way. Choose any row vertex r as root and index it by ∅. If v is a vertex and (r, c, …, v) is the unique path from r to v, let the word c^{⌢} ⋯^{⌢}v index v. Let W be the set of all words thus formed. Then

– ∅ ∈ W and every beginning of a word in W is also in W: if w ∈ W \ {∅}, then w is the concatenation w′^{⌢}v of a word w′ ∈ W with a letter v;

– words of even length index row vertices;

– words of odd length index column vertices;

– a pair of vertices is an edge exactly if their indices have the form w and w^{⌢}v, where w is a word and v is a letter.

A forest is a union of pairwise disjoint trees; equivalently, it is a cycle free graph.

Notation. Let 𝕋 = {z ∈ ℂ : |z| = 1}.

The unit ball of a Banach space X is denoted by B_{X}.

Given an index set I and q ∈ I, e_{q} is the sequence defined on I as the indicator function χ_{{q}} of the singleton {q}.

Let I = R × C and q = (r, c). Then e_{q} = e_{rc} is the elementary matrix identified with the operator from ℓ_{C}^{2} to ℓ_{R}^{2} that maps e_{c} on e_{r} and all other basis vectors on 0. The matrix coefficient at coordinate q of an operator x from ℓ_{C}^{2} to ℓ_{R}^{2} is x_{q} = tr e_{q}^{∗}x and its matrix representation is (x_{q})_{q∈R×C} = ∑ _{q∈R×C} x_{q}e_{q}. The support of x is {q ∈ R × C : x_{q} ≠ 0}.

The Schatten-von-Neumann class S^{p}, 0 < p < ∞, is the space of those compact operators x from ℓ_{C}^{2} to ℓ_{R}^{2} such that ∥x∥_{p}^{p} = tr|x|^{p} = tr(x^{∗}x)^{p/2} < ∞. S^{∞} is the space of compact operators with the operator norm. S^{p} is a quasi-normed space, and a Banach space if p ⩾ 1. Let (R_{n} × C_{n})_{n⩾0} be a sequence of finite sets that tends to R × C. Then the sequence of operators P_{n} : x ↦ ∑ _{q∈R_{n} ×C_{n}} x_{q}e_{q} tends pointwise to the identity on S^{p} if p ⩾ 1.

For I ⊆ R × C, the entry space S_{I}^{p} is the subspace of those x ∈ S^{p} whose support is a subset of I. S_{I}^{p} is also the closed subspace of S^{p} spanned by (e_{q})_{q∈I} .

The S^{p}-valued Schatten-von-Neumann class S^{p}(S^{p}) is the space of those compact operators x from ℓ_{C}^{2} to ℓ_{R}^{2}(S^{p}) such that ∥x∥_{p}^{p} = tr(tr|x|^{p}) < ∞, where the inner trace is the S^{p}-valued analogue of the usual trace: such operators have an S^{p}-valued matrix representation and their support is defined as in the scalar case. An element x ∈ S^{p}(S^{p}) can also be considered as a compact operator from ℓ_{C}^{2}(ℓ_{2}) = ℓ_{2} ⊗_{2} ℓ_{C}^{2} to ℓ_{R}^{2}(ℓ_{2}) = ℓ_{2} ⊗_{2} ℓ_{R}^{2} such that ∥x∥_{p}^{p} = tr⊗ tr|x|^{p} < ∞; the matrix coefficient of x at q is then x_{q} = (Id_{S^{p}} ⊗ tr)((Id_{ℓ_{2}} ⊗ e_{q}^{∗})x) and its matrix representation is ∑ _{q∈R×C} x_{q} ⊗ e_{q}. The entry space S_{I}^{p}(S^{p}) is defined in the same way as S_{I}^{p}.

A relative Schur multiplier on S_{I}^{p} is a sequence φ = (φ_{q})_{q∈I} ∈ ℂ^{I} such that the associated Schur multiplication operator M_{φ} defined by e_{q} ↦ φ_{q}e_{q} for q ∈ I is bounded on S_{I}^{p}. The Schur multiplier φ is furthermore completely bounded (c.b. for short) on S_{I}^{p} if Id_{S^{p}} ⊗ M_{φ}, the operator defined by x_{q}e_{q} ↦ φ_{q}x_{q}e_{q} for x_{q} ∈ S^{p} and q ∈ I, is bounded on S_{I}^{p}(S^{p}) (see [77, Lemma 1.7].) The norm of φ is the norm of M_{φ} and its complete norm is the norm of Id_{S^{p}} ⊗ M_{φ}. This norm is the supremum of the norm of its restrictions to finite rectangle sets R′× C′. Note that φ is a Schur multiplier on S^{∞} if and only if, for every bounded operator x: ℓ_{C}^{2} → ℓ_{R}^{2}, (φ_{q}x_{q}) is the matrix representation of a bounded operator; also φ is automatically c.b. on S^{∞} [79, Th. 5.1]. We used [77, 79] as a reference.

Let G be a compact abelian group endowed with its normalised Haar measure. Let Γ = Ĝ be the dual group of characters on G. The spectrum of an integrable function f on G is {γ ∈ Γ : f̂(γ) ≠ 0}. Let Λ ⊆ Γ. If X is a space of integrable functions on G, then X_{Λ} is the translation invariant subspace of those f ∈ X whose spectrum is a subset of Λ.

Let X be the space of continuous functions C(G) or the Lebesgue space L^{p}(G) with 0 < p < ∞. Then X_{Λ} is also the closed subspace of X spanned by Λ. A relative Fourier multiplier on X_{Λ} is a sequence μ = (μ_{γ})_{γ∈Λ} ∈ ℂ^{Λ} such that the associated convolution operator M_{μ} defined by γ ↦ μ_{γ}γ for γ ∈ Λ is bounded on X_{Λ} . The Fourier multiplier μ is furthermore c.b. if Id_{S^{p}} ⊗ M_{μ}, the operator defined by a_{γ}γ ↦ μ_{γ}a_{γ}γ for a_{γ} ∈ S^{p} and γ ∈ Λ, is bounded on the S^{p}-valued space X_{Λ} (S^{p}) (where p = ∞ if X = C(G).) The norm of μ is the norm of M_{μ} and its complete norm is the norm of Id_{S^{p}} ⊗ M_{μ}. Note that μ is a Fourier multiplier on C_{Λ} (G) if and only if, for every f ∈ L_{Λ}^{∞} (G), ∑ μ_{γ} f̂(γ)γ is the Fourier series of an element of L_{Λ}^{∞} (G): μ is a relative Fourier multiplier on L^{∞} (G); also μ is automatically c.b. on C_{Λ} (G) [79, Cor. 3.18].

Let X, Y be Banach spaces and u ∈ X ⊗ Y. Its projective tensor norm is

∥u∥_{X ⊗^{∧} Y} = inf{ ∑_{j=1}^{n} ∥x_{j} ∥ ∥y_{j} ∥ : u = ∑_{j=1}^{n} x_{j} ⊗ y_{j} }

and X ⊗^{∧} Y is the completion of X ⊗ Y with respect to this norm. Note that ℓ_{∞}^{n} ⊗^{∧} ℓ_{∞}^{m} ⊂ c_{0} ⊗^{∧} c_{0}

because ℓ_{∞}^{n} and ℓ_{∞}^{m} are 1-complemented in c_{0}, and that c_{0} ⊗^{∧} c_{0} ⊂ ℓ_{∞} ⊗^{∧} ℓ_{∞} because ℓ_{∞} is the bidual of c_{0}.

Let ∑ x_{j} ⊗ y_{j} be any representation of the tensor u. If ξ ⊗ η ∈ X^{∗} ⊗ Y^{∗}, we define ⟨ξ ⊗ η, u⟩ = ∑ ⟨ξ, x_{j} ⟩⟨η, y_{j} ⟩. The injective tensor norm of u is

∥u∥_{X ⊗^{∨} Y} = sup_{(ξ,η)∈B_{X^{∗}} ×B_{Y ^{∗}}} |⟨ξ ⊗ η, u⟩|

and X ⊗^{∨} Y is the completion of X ⊗ Y with respect to this norm.

If X and Y are both finite dimensional, then

(X ⊗^{∨} Y )^{∗} = X^{∗} ⊗^{∧} Y^{∗} and (X ⊗^{∧} Y )^{∗} = X^{∗} ⊗^{∨} Y^{∗}.

Further (c_{0} ⊗^{∧} c_{0})^{∗} = ℓ_{1} ⊗^{∨} ℓ_{1}: in fact, (c_{0} ⊗^{∧} c_{0})^{∗} may be identified with the space of bounded operators

from c_{0} to ℓ_{1} and ℓ_{1} ⊗^{∨} ℓ_{1} may be identified with the closure of finite rank operators in that space, and they are the same because every bounded operator from c_{0} to ℓ_{1} is compact and ℓ_{1} has the approximation property.

If X is a sequence space on C and Y is a sequence space on R, then the coefficient of the tensor u at (r, c) is ⟨e_{c} ⊗ e_{r}, u⟩. Its support is the set of coordinates (r, c) of its nonvanishing coefficients. One may use [90] as a reference.

2 Relative Schur multipliers

The following proposition is a straightforward consequence of [72].

Proposition 2.1. Let I ⊆ R × C and φ be a Schur multiplier on S_{I}^{∞} with norm D. Then φ is also a c.b. Schur multiplier on S_{I}^{p} for every p ∈ (0, ∞], with complete norm bounded by D.

Proof. We may assume that D = 1. Let R′× C′ be any finite subset of R × C. By [72, Th. 3.2], there exist vectors w_{c} and v_{r} of norm at most 1 in a Hilbert space H such that φ_{rc} = ⟨w_{c}, v_{r}⟩ for every (r, c) ∈ I ∩ R′× C′. If we define W: ℓ_{C′}^{2} → ℓ_{C′}^{2}(H) and V : ℓ_{R′}^{2} → ℓ_{R′}^{2}(H) by W ζ = (ζ_{c}w_{c})_{c∈C′} and V η = (η_{r}v_{r})_{r∈R′} , then V and W have norm at most 1, and the proposition follows from the factorisation

M_{φ} x = V^{∗}(x ⊗ Id_{H})W

for every x with support in I ∩ R′× C′.

Remark 2.2. Éric Ricard showed us an elementary proof that a Schur multiplier on S_{I}^{∞} is automatically c.b., included here by his kind permission. A Schur multiplier φ is bounded on S_{I}^{∞} by a constant D if and only if

∀ ξ ∈ B_{S_{I}^{∞}} ∀ η ∈ B_{ℓ_{R}^{2}} ∀ ζ ∈ B_{ℓ_{C}^{2}}  | ∑_{(r,c)∈I} η_{r}φ_{rc}ξ_{rc}ζ_{c} | ⩽ D.  (B.2)

It is furthermore completely bounded on S_{I}^{∞} by D if

∀ x ∈ B_{S_{I}^{∞} (S^{∞} )} ∀ y ∈ B_{ℓ_{R}^{2} (ℓ_{2} )} ∀ z ∈ B_{ℓ_{C}^{2} (ℓ_{2} )} | ∑_{(r,c)∈I} φ_{rc}⟨y_{r}, x_{rc}z_{c}⟩| ⩽ D.  (B.3)

Suppose that x, y, z are as quantified in Ineq. (B.3). Let

ξ_{rc} = ⟨y_{r}/∥y_{r}∥, x_{rc}z_{c}/∥z_{c}∥⟩, η_{r} = ∥y_{r}∥_{ℓ_{2}} and ζ_{c} = ∥z_{c}∥_{ℓ_{2}}.

Then ∥η∥_{ℓ_{R}^{2}} , ∥ζ∥_{ℓ_{C}^{2}} ⩽ 1 and

∥ξ∥ = sup{ | ∑_{(r,c)∈I} ⟨α_{r}y_{r}/∥y_{r}∥_{ℓ_{2}}, x_{rc}β_{c}z_{c}/∥z_{c}∥_{ℓ_{2}}⟩| : α ∈ B_{ℓ_{R}^{2}} , β ∈ B_{ℓ_{C}^{2}} }

⩽ ∥x∥ sup_{α∈B_{ℓ{2 \atop R}}} ∥(α_{r}y_{r}/∥y_{r}∥_{ℓ_{2}})∥_{ℓ_{R}^{2} (ℓ_{2} )} sup_{β∈B_{ℓ{2 \atop C}}} ∥(β_{c}z_{c}/∥z_{c}∥_{ℓ_{2}})∥_{ℓ_{C}^{2} (ℓ_{2} )} ⩽ 1,

so that Ineq. (B.2) implies Ineq. (B.3).

The fact that the canonical basis of an ℓ^{2} space is 1-unconditional yields that Schatten-von-Neumann norms are matrix unconditional in the terminology of [92]:

∀ ζ ∈ 𝕋^{C} ∀ η ∈ 𝕋^{R} ∥ ∑_{(r,c)∈R×C} ζ_{c}η_{r}a_{rc} e_{rc} ∥_{p} = ∥ ∑_{(r,c)∈R×C} a_{rc} e_{rc} ∥_{p}  (B.4)

for every finitely supported sequence of complex or S^{p}-valued coefficients a_{rc}. Let ζ ⊗ η denote the elementary Schur multiplier (ζ_{c}η_{r})_{(r,c)∈R×C} . Equation (B.4) shows that if ζ ∈ 𝕋^{C} and η ∈ 𝕋^{R}, then M_{ζ⊗η} is an isometry on every S^{p}. This yields that if ζ ∈ ℓ_{C}^{∞} , η ∈ ℓ_{R}^{∞} , then the complete norm of M_{ζ⊗η} is ∥ζ∥_{ℓ_{C}^{∞}} ∥η∥_{ℓ_{R}^{∞}} on every S^{p}.

Relative Schur multipliers also have a central place among operators on S_{I}^{p} because they appear as the range of a contractive projection defined by the following averaging scheme.

Definition 2.3. Let T: S_{J}^{p} → S_{I}^{p} be an operator. Let R′× C′ be a finite subset of R × C and let P_{R′×C′} be the contractive projection onto S_{R′×C′}^{p} defined by the Schur multiplier χ_{C′} ⊗ χ_{R′}. Then the average of T with respect to R′× C′ is given by

[T]_{R′×C′} (x) = ∫_{𝕋^{R}} dη ∫_{𝕋^{C}} dζ M_{ζ^{∗} ⊗η^{∗}} P_{R′×C′} T (M_{ζ⊗η} x),  (B.5)

where ζ^{∗} = (\overline{ζ_{c}})_{c∈C} and η^{∗} = (\overline{η_{r}})_{r∈R} .

Proposition 2.4. Let T: S_{J}^{p} → S_{I}^{p} be an operator and R′ × C′ a finite subset of R × C. Then [T]_{R′×C′} is a Schur multiplication operator from S_{J}^{p} to S_{I∩R′×C′}^{p} such that ∥[T]_{R′×C′} ∥ ⩽ ∥T ∥. In fact, [T]_{R′×C′} = M_{φ^{R′×C′}} with

φ_{rc}^{R′×C′} = \{ \begin{matrix}tr e_{rc}^{∗}T (e_{rc}) & if (r, c) ∈ J ∩ R′× C′ \\ 0 & if (r, c) ∈ J \ R′× C′. \end{matrix}

If T is a projection onto S_{I}^{p}, then φ^{R′×C′} = χ_{I∩R′×C′} , so that [T]_{R′×C′} is a projection onto S_{I∩R′×C′}^{p} . Let φ = (tr e_{q}^{∗}T (e_{q}))_{q∈J} . Then ∥M_{φ} ∥ ⩽ ∥T ∥ and we define the average of T by [T] = M_{φ}.

Proof. Formula (B.5) shows that ∥[T]_{R′×C′} (x)∥ ⩽ ∥T ∥ ∥x∥. We have

[T]_{R′×C′} (e_{rc})= ∫_{𝕋^{R}} dη ∫_{𝕋^{C}} dζ M_{ζ^{∗} ⊗η^{∗}} P_{R′×C′} T (ζ_{c}η_{r} e_{rc})

= ∫_{𝕋^{R}} dη ∫_{𝕋^{C}} dζ ζ_{c}η_{r}M_{ζ^{∗} ⊗η^{∗}} ∑_{(r′,c′)∈R′×C′} tr(e_{r′c′}^{∗} T (e_{rc}))e_{r′c′}

= ∑_{(r′,c′)∈R′×C′} ∫_{𝕋^{R}} dη ∫_{𝕋^{C}} dζ ζ_{c}η_{r} tr(e_{r′c′}^{∗} T (e_{rc}))ζ_{c′}^{−1} η_{r′}^{−1} e_{r′c′} = φ_{rc}^{R′×C′} e_{rc}.

As the norm of a Schur multiplier is the supremum of the norm of its restrictions to finite rectangle sets, this shows that φ is a Schur multiplier on S_{J}^{p} and ∥M_{φ} ∥ ⩽ ∥T ∥. If T is a projection onto S_{I}^{p}, note that tr e_{rc}^{∗}T (e_{rc}) = χ_{I} (r, c).

The following proposition relates Fourier multipliers to Herz-Schur multipliers in the fashion of [79, Th. 6.4] and will be very useful in the exact computation of the norm of certain relative Schur multipliers.

Proposition 2.5. Let Γ be a countable discrete abelian group and Λ ⊆ Γ. Let R and C be two copies of Γ and consider I = {(r, c) ∈ R × C : r − c ∈ Λ}. Let φ ∈ ℂ^{I} such that there is μ ∈ ℂ^{Λ} with φ(r, c) = μ(r − c) for all (r, c) ∈ I. Let G = Γ̂, so that Γ is the group of characters on the compact group G. Let p ∈ (0, ∞].

(a) The complete norm of the relative Schur multiplier φ on S_{I}^{p} is bounded by the complete norm of the relative Fourier multiplier μ on L_{Λ}^{p} (G).

(b) Suppose that Γ is finite. The norm of the relative Fourier multiplier μ on L_{Λ}^{p} (G) is bounded by the norm of the relative Schur multiplier φ on S_{I}^{p}. The same holds for complete norms.

Remark 2.6. Part (b) is just an abstract counterpart to [75, Chapter 6, Lemma 3.8], where the case of the finite cyclic group Γ = ℤ/nℤ is treated.

Proof. (a) is [77, Lemma 8.1.4]: for all a_{q} ∈ S^{p}, of which only a finite number are nonzero, and all g ∈ G, we have by matrix unconditionality (Eq. (B.4))

∥∑_{q∈I} a_{q}e_{q}∥_{S_{I}^{p} (S^{p})} = ∥ ∑_{(r,c)∈I} r(g)c(g)^{−1} a_{rc} e_{rc} ∥_{S_{I}^{p} (S^{p})}

= ∥∑_{γ∈Λ} ( ∑_{\begin{matrix}(r,c)∈I \\ r−c=γ \end{matrix}} a_{rc} e_{rc}) γ(g)∥_{S_{I}^{p} (S^{p})} = ∥∑_{γ∈Λ} ( ∑_{\begin{matrix}(r,c)∈I \\ r−c=γ \end{matrix}} a_{rc} e_{rc}) γ∥_{L_{Λ}^{p} (G,S^{p} (S^{p}))} . (B.6)

This yields an isometric embedding of S_{I}^{p}(S^{p}) in L_{Λ}^{p} (G, S_{I}^{p}(S^{p})). As S^{p}(S^{p}) may be identified with S^{p}(ℓ_{Γ}^{2}(ℓ^{2})),

∥∑_{q∈I} φ_{q}a_{q}e_{q}∥_{S_{I}^{p} (S^{p})} = ∥∑_{γ∈Λ} μ_{γ}( ∑_{\begin{matrix}(r,c)∈I \\ r−c=γ \end{matrix}} a_{rc} e_{rc}) γ∥_{L_{Λ}^{p} (G,S^{p} (S^{p}))} ⩽ ∥Id ⊗ M_{μ}∥∥∑_{q∈I} a_{q}e_{q}∥_{S_{I}^{p} (S^{p})} .

(b). Let us embed L_{Λ}^{p} (G) into S_{I}^{p} by f ↦ mf̂, where mf̂: ℓ_{C}^{2} → ℓ_{R}^{2} is the convolution operator defined by

mf̂e_{c} = f̂∗ e_{c} = ∑_{γ∈Λ} f̂(γ)e_{γ} ∗ e_{c} = ∑_{r−c∈Λ} f̂(r − c)e_{r} :

mf̂ has the matrix representation ∑ _{(r,c)∈I} f̂(r − c)e_{rc}. The characters g ∈ G form an orthonormal basis for ℓ_{C}^{2} such that mf̂g = f(g)g: therefore

∥mf̂∥_{p} =( ∑_{g∈G} |f(g)|^{p})^{1/p} = (# G)^{1/p}∥f∥_{L^{p} (G)} .

As M_{φ} mf̂ = m{ \atop M_{μ} f}, this shows that the norm of μ on L_{Λ}^{p} (G) is the norm of φ on the subspace of circulant matrices in S_{I}^{p}. The same holds for complete norms.

3 Idempotent Schur multipliers of norm 1

A Schur multiplier is idempotent if it is the indicator function χ_{I} of some set I ⊆ R × C; if χ_{I} is a Schur multiplier on S^{p}, then it is a projection of S^{p} onto S_{I}^{p}. Idempotent Schur multipliers on S^{p} and

tensors in ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} with 0, 1 coefficients of norm 1 may be characterised by the combinatorics of I.

Proposition 3.1. Let I ⊆ R × C be nonempty and 0 < p ≠ 2 < ∞. The following are equivalent.

(a) For every finite rectangle set R′× C′ intersecting I

∥ ∑_{(r,c)∈I∩R′×C′} e_{c} ⊗ e_{r} ∥_{ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞}} = 1.

(b) S_{I}^{p} is completely 1-complemented in S^{p}.

(c) S_{I}^{p} is 1-complemented in S^{p}.

(d) I is a union of pairwise disjoint complete bipartite graphs: there are pairwise disjoint sets R_{j} ⊆ R and pairwise disjoint sets C_{j} ⊆ C such that I = ⋃ R_{j} × C_{j}.

Property (d) means that the pattern I is, up to a permutation of columns and rows, block-diagonal:

( \begin{matrix}& C_{1} & C_{2} & C_{3} & ··· \\ R_{1} & ∗ & 0 & 0 & ⋯ \\ R_{2} & 0 & ∗ & 0 & ... \\ R_{3} & 0 & 0 & ∗ & ... \\ ⋮ & ⋮ & ... & ... & ... \end{matrix}) .

Proof. (b) ⇒ (c) is trivial.

(a) ⇒ (b). The complete norm of a Schur multiplier φ on S^{p} is the supremum of the complete norm of its restrictions φ′ = (φ_{q})_{q∈R′×C′} to finite rectangle sets R′× C′. Furthermore, the complete norm of an elementary Schur multiplier (η_{c}ζ_{r})_{(r,c)∈R×C} = η ⊗ ζ on S^{p} equals ∥η∥_{ℓ_{C}^{∞}} ∥ζ∥_{ℓ_{R}^{∞}} .

(c) ⇒ (d). If I is not a union of pairwise disjoint complete bipartite graphs, then there are r_{0}, r_{1} ∈ R and c_{0}, c_{1} ∈ C such that

I′ = I ∩ {r_{0}, r_{1}} × {c_{0}, c_{1}} = {(r_{0}, c_{0}), (r_{1}, c_{0}), (r_{0}, c_{1})}.

By Proposition 2.4, the average of a contractive projection of S^{p} onto S_{I}^{p} with respect to {r_{0}, r_{1}} × {c_{0}, c_{1}} would be the contractive projection associated to the Schur multiplier χ_{I′}. Let x(t), t ∈ ℝ, be the operator from ℓ_{C}^{2} to ℓ_{R}^{2} whose matrix coefficients vanish except for its {r_{0}, r_{1}}×{c_{0}, c_{1}} submatrix,

which is ( \begin{matrix}1 & \sqrt{2} \\ \sqrt{2} & t \end{matrix}) . Its eigenvalues are

\frac{1 + t + \sqrt{9 − 2t + t^{2}}}{2} = 2 + \frac{t}{3} + o(t), \frac{1 + t − \sqrt{9 − 2t + t^{2}}}{2} = −1 + \frac{2t}{3} + o(t),

so that

\{ \begin{matrix}∥x(t)∥_{∞} = 2 + t/3 + o(t) \\ ∥x(t)∥_{p}^{p} = 2^{p} + 1 + p(2^{p} − 4)t/6 + o(t) for 0 < p < ∞ \end{matrix}

and therefore ∥χ_{I′} ∗ x(t)∥_{p} = ∥x(0)∥_{p} > ∥x(t)∥_{p} for some t ≠ 0 if p ≠ 2.

(d) ⇒ (a). Suppose (d) and let R′× C′ intersect I. Then there are pairwise disjoint sets R′_{j} and pairwise disjoint sets C′_{j} such that I ∩ R′× C′ = R′_{1} × C′_{1} ∪ ⋯∪ R′_{n} × C′_{n} and

∑_{(r,c)∈I∩R′×C′} e_{c} ⊗ e_{r} = ∑_{j=1}^{n} χ_{C′_{j}} ⊗ χ_{R′_{j}} = average_{ϵ_{j} =±1}( ∑_{j=1}^{n} ϵ_{j} χ_{C′_{j}}) ⊗ ( ∑_{j=1}^{n} ϵ_{j} χ_{R′_{j}} )

which is an average of elementary tensors of norm 1, so that its projective tensor norm is bounded by 1, and actually is equal to 1.

Remark 3.2. Note that the proof of Prop. 3.1 shows that the norm of a projection M_{χ_{I}} : S^{∞} → S_{I}^{∞} is either 1 or at least 2/\sqrt{3}, as

∥( \begin{matrix}1 & \sqrt{2} \\ \sqrt{2} & −1 \end{matrix}) ∥_{∞} = \sqrt{3},  ∥( \begin{matrix}1 & \sqrt{2} \\ \sqrt{2} & 0 \end{matrix}) ∥_{∞} = 2.

4. UNCONDITIONAL BASIC SEQUENCES IN S^{p}  39

This is a noncommutative analogue to the fact that an idempotent measure on a locally compact abelian group G has either norm 1 or at least \sqrt{5}/2 [89, Th. 3.7.2]. The norm of M_{χ_{I}} actually equals 2/\sqrt{3} for I = \{(0, 0), (0, 1), (1, 0)\}, as shown in [53, Lemma 3]. In fact, the following decomposition holds:

e_{0} ⊗ e_{0} + e_{0} ⊗ e_{1} + e_{1} ⊗ e_{0} =

((e^{−iπ/12} , e^{iπ/4} ) ⊗ (e^{−iπ/12} , e^{iπ/4} ) + (e^{iπ/12} , e^{−iπ/4} ) ⊗ (e^{iπ/12} , e^{−iπ/4} ))/\sqrt{3}.

Remark 3.3. Results related to the equivalence of (c) with (d) have been obtained independently by Banks and Harcharras [4].

4 Unconditional basic sequences in S^{p}

Definition 4.1. Let 0 < p ⩽ ∞ and I ⊆ R × C. Let 𝕊 = 𝕋 (vs. 𝕊 = {−1, 1}.)

(a) I is an unconditional basic sequence in S^{p} if there is a constant D such that

∥∑_{q∈I} ϵ_{q}a_{q}e_{q}∥_{p} ⩽ D∥∑_{q∈I} a_{q}e_{q}∥_{p}  (B.7)

for every choice of signs ϵ_{q} ∈ 𝕊 and every finitely supported sequence of complex coefficients a_{q}. Its complex (vs. real) unconditional constant is the least such constant D.

(b) I is a completely unconditional basic sequence in S^{p} if there is a constant D such that (B.7) holds for every choice of signs ϵ_{q} ∈ 𝕊 and every finitely supported sequence of operator coefficients a_{q} ∈ S^{p}. Its complex (vs. real) complete unconditional constant is the least such constant D.

(c) I is a complex (vs. real, complex completely, real completely) 1-unconditional basic sequence in S^{p} if its complex (vs. real, complex complete, real complete) unconditional constant is 1: Inequality (B.7) turns into the equality

∥∑_{q∈I} ϵ_{q}a_{q}e_{q}∥_{p} = ∥∑_{q∈I} a_{q}e_{q}∥_{p}.

If Inequality (B.7) holds for every real choice of signs, then it also holds for every complex choice of signs at the cost of replacing D by Dπ/2 (see [93],) so that there is no need to distinguish between complex and real unconditional basic sequences.

The notions defined in (a) and (b) are called σ(p) sets and complete σ(p) sets in [37, §4] and [38] (see also the survey [80, § 9].) The notions defined in (c) are their isometric counterparts.

By [92, proof of Cor. 4], the real unconditional constant of any basis of S_{I}^{p} cannot be lower than a fourth of the real unconditional constant of I in S^{p}.

Example 4.2. A single column R × {c}, a single row {r} × C, the diagonal set {(rown, coln)}_{n∈ℕ} if R and C are copies of ℕ, are 1-unconditional basic sequences in all S^{p}. In fact, every column section and every row section (this is the terminology of [100, Def. 4.3]) is a 1-unconditional basic sequence; note that the length of every path in the corresponding graph is at most 2.

Note that the set I is a (completely) 1-unconditional basic sequence in S^{p} if and only if the relative Schur multipliers by signs on S_{I}^{p} define (complete) isometries. This yields by Prop. 2.1:

Proposition 4.3. Let I ⊆ R × C and 0 < p ⩽ ∞. If I is a real (vs. complex) 1-unconditional basic sequence in S^{∞} , then I is also a real (vs. complex) completely 1-unconditional basic sequence in S^{p}.

Example 4.4. If R = C = {0, …, n − 1}, 1 ⩽ p ⩽ ∞ and I = R × C, then the complex unconditional constant of the basis of elementary matrices in S^{p} is n^{|1/2−1/p|} and coincides with its complete unconditional constant (see [77, Lemma 8.1.5].) This is also the real unconditional constant if n = 2^{k}

is a power of 2 as the norm of Schur multiplication by the kth tensor power ( \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} ) ^{⊗k} (the kth Walsh

matrix) on S^{p} is (2^{|1/2−1/p|} )^{k} = n^{|1/2−1/p|} . Let us now show that if n = 3, the real unconditional constant of the basis of elementary matrices in S^{∞} is 5/3 and differs from its complex unconditional constant, \sqrt{3}. In fact, because the canonical bases of ℓ_{C}^{2} and ℓ_{R}^{2} are symmetric, the norm of a Schur multiplier by real signs turns out to equal the norm of one of the following three Schur multipliers:

( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}) , ( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & −1 \end{matrix} ) or ( \begin{matrix}−1 & 1 & 1 \\ 1 & −1 & 1 \\ 1 & 1 & −1 \end{matrix} ) .

The first one has norm 1: it defines the identity. The second one has the same norm as the Schur

multiplier ( \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} ) , which is \sqrt{2}, because the norm of that multiplier equals the norm of its tensor

product by Id_{ℓ_{2}^{2}}, which is ( \begin{matrix}1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & −1 & −1 \\ 1 & 1 & −1 & −1 \end{matrix} ) . By Prop. 2.5 for Γ = ℤ/3ℤ, the third one has the

same norm as the Fourier multiplier φ = (−1, 1, 1) on L^{∞} (G), where G = {z ∈ ℂ : z^{3} = 1}: as this multiplier acts by convolution with f = −1 + z + z^{2}, its norm is ∥f∥_{L^{1} (G)} , that is

(| − 1 + 1 + 1| + | − 1 + e^{2iπ/3} + e^{4iπ/3} | + | − 1 + e^{4iπ/3} + e^{2iπ/3} |)/3 = 5/3.

Complex interpolation yields that the real unconditional constant of the basis of elementary matrices is in fact strictly less than its complex counterpart in all S^{p} with p ≠ 2.

5 Varopoulos’ characterisation of unconditional matrices in S^{∞}

Our results may be seen as the isometric counterpart to results by Varopoulos [100] on tensor algebras over discrete spaces and their generalisation to S^{p}. He characterised unconditional basic sequences

of elementary matrices in S^{∞} in his study of the projective tensor product c_{0} ⊗^{∧} c_{0}. We gather up his results in the next theorem, as they are difficult to extract from the literature.

Theorem 5.1. Let I ⊆ R × C. The following are equivalent.

(a) I is an unconditional basic sequence in S^{∞} .

(b) I is an interpolation set for Schur multipliers on S^{∞} : every bounded sequence on I is the restriction of a Schur multiplier on S^{∞} .

(c) I is a V-Sidon set as defined in [100, Def. 4.1]: every null sequence on I is the restriction of

the sequence of coefficients of a tensor in c_{0}(C) ⊗^{∧} c_{0}(R).

(d) The coefficients of every tensor in ℓ_{C}^{1} ⊗^{∨} ℓ_{R}^{1} with support in I form an absolutely convergent series.

(e) (z_{c}z′_{r})_{(r,c)∈I} is a Sidon set in the dual of 𝕋^{C} × 𝕋^{R}, that is, an unconditional basic sequence in C(𝕋^{C} × 𝕋^{R}).

(f) There is a constant λ such that for all R′ ⊆ R and C′ ⊆ C with n elements # [I ∩R′×C′] ⩽ λn.

(g) I is a finite union of forests.

(h) I is a finite union of row sections and column sections.

(i) Every bounded sequence supported by I is a Schur multiplier on S^{∞} .

Sketch of proof. (a) ⇒ (b). If (a) holds, every sequence of signs ϵ ∈ {−1, 1}^{I} is a Schur multiplier on S_{I}^{∞} . By a convexity argument, this implies that every bounded sequence is a Schur multiplier on S_{I}^{∞} , which may be extended to a Schur multiplier on S^{∞} with the same norm by [72, Cor. 3.3].

(b) ⇒ (c) holds by Grothendieck’s inequality (see [79, § 5]) and an approximation argument.

(d) is but the formulation dual to (c) (see [99, § 6.2].)

(d) ⇒ (e). A computation yields

∥ ∑_{(r,c)∈I} a_{rc} e_{c} ⊗ e_{r} ∥_{ℓ_{C}^{1} ⊗^{∨} ℓ_{R}^{1}} = sup_{|z_{c} |,|z′_{r}|=1} | ∑_{(r,c)∈I} a_{rc}z_{c}z′_{r}|.  (B.8)

(e) ⇒ (f) is [100, Th. 4.2]. (The proof can be found in [99, § 6.3] and in [98, § 5].)

(f) ⇒ (g), (f) ⇒ (h) can be found in [98, Th. 6.1].

(g) ⇒ (h). In fact, a forest is the union of a column section I′ with a row section I′′(a bisection in the terminology of [100, Def. 4.3].) It suffices to prove this for a tree. Let its vertices be indexed by words as described in the Terminology. Then the set I′ of couples of the form (w, w^{⌢}c) with w a word and c a letter is a column section; the set I′′of couples of the form (w^{⌢}r, w) with w a word and r a letter is a row section.

(h) ⇒ (i) is [98, Th. 4.5]. Note that row sections and column sections form 1-unconditional basic sequences in S^{∞} and are 1-complemented in S^{∞} by Prop. 3.1.

(i) ⇒ (a) follows from the open mapping theorem.

6 Closed walk relations

We now introduce and study the combinatorial objects that we need in order to analyse the expansion of the function defined by

Φ_{I} (ϵ, a) = tr|∑_{q∈I} ϵ_{q}a_{q}e_{q}|^{p}  (B.9)

for I ⊆ R × C, a positive even integer p = 2k, signs ϵ_{q} ∈ 𝕋 and coefficients a_{q} ∈ ℂ, of which only a finite number are nonzero. In fact,

Φ_{I} (ϵ, a) = tr( ∑_{(r,c),(r′,c′)∈I} (ϵ_{rc}a_{rc} e_{rc})^{∗}(ϵ_{r′c′} a_{r′c′} e_{r′c′}))^{k}

= tr ∑_{\begin{matrix}(r_{1} ,c_{1} ),(r′_{1},c′_{1} ),..., \\ (r_{k} ,c_{k}),(r′_{k} ,c′_{k})∈I \end{matrix}} ∏_{i=1}^{k} (ϵ_{r_{i} c_{i}}^{−1} \overline{a_{r_{i} c_{i}}} e_{c_{i} r_{i}} )(ϵ_{r′_{i}c′_{i}} a_{r′_{i}c′_{i}} e_{r′_{i}c′_{i}} )

= ∑_{\begin{matrix}(r_{1} ,c_{1} ),(r_{1} ,c_{2} ),..., \\ (r_{k} ,c_{k}),(r_{k} ,c_{k+1} )∈I \end{matrix}}  ∏_{i=1}^{k} ϵ_{r_{i} c_{i}}^{−1} ϵ_{r_{i} c_{i+1}} \overline{a_{r_{i} c_{i}}} a_{r_{i} c_{i+1}} (where c_{k+1} = c_{1}.)

(B.10)

The latter sum runs over all closed walks (c_{1}, r_{1}, c_{2}, …, c_{k}, r_{k}) of length p in the graph I. With multinomial notation, its terms have the form ϵ^{β−α} \overline{a}^{α} a^{β} with |α| = |β| = k. The attempt to describe those couples (α, β) that effectively arise in this expansion yields the following definition.

Definition 6.1. Let p = 2k ⩾ 0 be an even integer and I ⊆ R × C.

(a) Let A_{k}^{I} = {α ∈ ℕ^{I} : ∑ _{q∈I} α_{q} = k} and set

B_{k}^{I} = {(α, β) ∈ A_{k}^{I} × A_{k}^{I} : ∀ r ∑ _{c} α_{rc} = ∑ _{c} β_{rc} and ∀ c ∑ _{r} α_{rc} = ∑ _{r} β_{rc}}.

(b) Two couples (α^{1}, β^{1}) ∈ B_{k_{1}}^{I}, (α^{2}, β^{2}) ∈ B_{k_{2}}^{I} are disjoint if k_{1}, k_{2} ⩾ 1 and

α_{rc}^{1} ⩾ 1 ⇒ ∀ (r′, c) ∈ I α_{r′c}^{2} = 0 and ∀ (r, c′) ∈ I α_{rc′}^{2} = 0.  (B.11)

(c) The set 𝒲_{k}^{I} of closed walk relations of length p in I is the subset of those (α, β) ∈ B_{k}^{I} that cannot be decomposed into the sum of two disjoint couples.

(d) Let W_{k}^{I} be the set of closed walks of length p in the graph I. To every closed walk P = (c_{1}, r_{1}, c_{2}, r_{2}, …, c_{k}, r_{k}) of length p we associate the couple (α, β) ∈ A_{k}^{I} × A_{k}^{I} defined by

α_{q} = # [i ∈ {1, …, k} : (r_{i}, c_{i}) = q]
β_{q} = # [i ∈ {1, …, k} : (r_{i}, c_{i+1} ) = q] (where c_{k+1} = c_{1}.)

We shall write P ∼ (α, β) and call n_{αβ} the number of elements of W_{k}^{I} mapped onto (α, β).

Note that the conditions in Eq. (B.11) is in fact symmetric and that it may be stated with β^{1} and β^{2} instead of α^{1} and α^{2}.

Example 6.2. Let R = C = {0, 1, 2, 3} and I = R × C. The couple (e_{00} + e_{11} + e_{22} + e_{33}, e_{01} + e_{10} + e_{23} + e_{32}) is an element of B_{4}^{I} \ 𝒲_{4}^{I}: it is the sum of the two disjoint closed walk relations (e_{00} + e_{11}, e_{01} + e_{10}) and (e_{22} + e_{33}, e_{23} + e_{32}).

Example 6.3. Let I = R × C = {0, 1} × {0, 1}. Two closed walks are associated with the closed walk relation (e_{00} + e_{11}, e_{01} + e_{10}) ∈ 𝒲_{2}^{I}: the two cycles (col0, row0, col1, row1) and (col1, row1, col0, row0). Six closed walks are mapped onto the closed walk relation (2e_{00} + 2e_{01}, 2e_{00} + 2e_{01}): the

\frac{4!}{2! 2!} concatenations of a permutation of (col1, row0), (col1, row0), (col0, row0), (col0, row0).

The next proposition shows that, for our purpose, closed walk relations describe entirely closed walks.

Proposition 6.4. Let p = 2k ⩾ 0 be an even integer and I ⊆ R × C. The image of the mapping in Def. 6.1(d) is 𝒲_{k}^{I}:

(a) if P ∈ W_{k}^{I} and P ∼ (α, β), then (α, β) ∈ 𝒲_{k}^{I};

(b) if (α, β) ∈ 𝒲_{k}^{I}, then there is a P ∈ W_{k}^{I} such that P ∼ (α, β), so that n_{αβ} ⩾ 1.

Proof. (a). Let P = (c_{1}, r_{1}, c_{2}, r_{2}, …, c_{k}, r_{k}). In fact,

∑ _{c} α_{rc} = # [i ∈ {1, …, k} : r_{i} = r] = ∑ _{c} β_{rc}
∑ _{r} α_{rc} = # [i ∈ {1, …, k} : c_{i} = c] = ∑ _{r} β_{rc}

and (α, β) ∈ B_{k}^{I}. If (α, β) = (α^{1}, β^{1}) + (α^{2}, β^{2}) with (α^{i}, β^{i}) ∈ B_{k_{i}}^{I} and k_{i} ⩾ 1, there is an i such that α_{r_{i} c_{i}}^{1} ⩾ 1 and α_{r_{i+1} c_{i+1}}^{2} ⩾ 1 (where (r_{i+1} , c_{i+1} ) = (r_{1}, c_{1}) if i = k.) If β_{r_{i} c_{i+1}}^{1} ⩾ 1, then ∑ _{r} α_{rc_{i+1}}^{1} ⩾ 1, so that there is an r such that α_{rc_{i+1}}^{1} ⩾ 1. Otherwise β_{r_{i} c_{i+1}}^{2} ⩾ 1, so that ∑ _{c} α_{r_{i} c}^{2} ⩾ 1 and there is a c such that α_{r_{i} c}^{2} ⩾ 1. Therefore (α^{1}, β^{1}) and (α^{2}, β^{2}) are not disjoint and (α, β) ∈ 𝒲_{k}^{I}.

(b). We have to find a closed walk of length p that is mapped onto (α, β). If k = 0, the empty closed walk suits. Suppose that k ⩾ 1; Consider a walk (c_{1}, r_{1}, c_{2}, r_{2}, …, c_{j} , r_{j} , c_{j+1} ) in I such that α_{q}^{1} = # [i : (r_{i}, c_{i}) = q] ⩽ α_{q} and β_{q}^{1} = # [i : (r_{i}, c_{i+1} ) = q] ⩽ β_{q} for every q ∈ R × C, and furthermore j is maximal. We claim (A) that c_{j+1} = c_{1} and (B) that j = k. Let (α^{2}, β^{2}) = (α, β) − (α^{1}, β^{1}).

(A). If c_{j+1} ≠ c_{1}, then

∑ _{r} α_{rc_{j+1}}^{1} = # [i ∈ {1, …, j} : c_{i} = c_{j+1} ]
∑ _{r} β_{rc_{j+1}}^{1} = # [i ∈ {1, …, j + 1} : c_{i} = c_{j+1} ] = 1 + ∑ _{r} α_{rc_{j+1}}^{1} ,

so that there must be r_{j+1} with α_{r_{j+1} c_{j+1}}^{2} ⩾ 1. But then

∑ _{c} β_{r_{j+1} c}^{2} = ∑ _{c} α_{r_{j+1} c}^{2} ⩾ 1

and there must be c_{j+2} such that β_{r_{j+1} c_{j+2}}^{2} ⩾ 1: j is not maximal.

(B). Suppose that j < k. Then (α^{1}, β^{1}) ∈ B_{j}^{I} and (α^{2}, β^{2}) ∈ B_{k−j}^{I} . By hypothesis, they are not disjoint: there are r, c, c′ such that α_{rc}^{1}α_{rc′}^{2} ⩾ 1 or r, r′, c such that α_{rc}^{1}α_{r′c}^{2} ⩾ 1. By interchanging R and C and by relabelling the vertices if necessary, we may suppose without loss of generality that for r′_{1} = r_{j} there is c′_{1} such that α_{r′_{1}c′_{1}}^{2} ⩾ 1. Then there is c′_{2} such that β_{r′_{1}c′_{2}}^{2} ⩾ 1. By the argument used in Claim (A), there is a closed walk (c′_{1}, r′_{1}, c′_{2}, …, c′_{j′}, r′_{j′}) such that # [i : (r′_{i}, c′_{i}) = q] ⩽ α_{q}^{2} and # [i : (r′_{i}, c′_{i+1} ) = q] ⩽ β_{q}^{2} (where c′_{j′+1} = c′_{1}.) Then the closed walk

(c_{1}, r_{1}, c_{2}, r_{2}, …, c_{j} , r_{j} , c′_{2}, r′_{2}, …, c′_{j′}, r′_{j′}, c′_{1}, r′_{1})

shows that j is not maximal.

We are now in position to state the following theorem, a matrix counterpart to the computation presented in [63, Prop. 2.5(ii)].

Theorem 6.5. Let p = 2k be a positive even integer and I ⊆ R × C.

(a) The function Φ_{I} in Eq. (B.9) has the expansion

Φ_{I} (ϵ, a) = ∑_{(α,β)∈𝒲_{k}^{I}} n_{αβ} ϵ^{β−α} \overline{a}^{α} a^{β},  (B.12)

where n_{αβ} ⩾ 1 for every (α, β) ∈ 𝒲_{k}^{I}.

(b) If ϵ ∈ 𝕋^{I} and a ∈ (S^{p})^{I} is finitely supported, then the function

Ψ_{I} (ϵ, a) = tr|∑_{q∈I} ϵ_{q}a_{q}e_{q}|^{p}  (B.13)

has the expansion

∑_{(α,β)∈𝒲_{k}^{I}} ϵ^{β−α} ∑_{(c_{1} ,r_{1} ,...,c_{k} ,r_{k})∼(α,β)} ∏_{i=1}^{k} a_{r_{i} c_{i}}^{∗} a_{r_{i} c_{i+1}} (with c_{k+1} = c_{1}.)  (B.14)

Proof. This follows from Def. 6.1 and Prop. 6.4.

Note that the edges of a closed walk P ∼ (α, β) are precisely those {r, c} such that α_{rc} + β_{rc} ⩾ 1. P is a cycle if and only if P does not have length 0 or 2 and ∑ _{r} α_{rc} ⩽ 1 for all c and ∑ _{c} α_{rc} ⩽ 1 for all r. We now show how to decompose closed walks into cycles.

Proposition 6.6. Let P = (c_{1}, r_{1}, c_{2}, r_{2}, …, c_{k}, r_{k}) ∼ (α, β) be a closed walk.

(a) If r_{i} = r_{j} (vs. c_{i} = c_{j}) for some i ≠ j, then P is the juxtaposition of two nonempty closed walks P_{1} ∼ (α^{1}, β^{1}) and P_{2} ∼ (α^{2}, β^{2}) such that (α, β) = (α^{1}, β^{1})+(α^{2}, β^{2}) and ∑ _{c} α_{r_{i} c}^{1} , ∑ _{c} α_{r_{i} c}^{2} ⩾ 1 (vs. ∑ _{r} α_{rc_{i}}^{1} , ∑ _{r} α_{rc_{i}}^{2} ⩾ 1.)

(b) P is the juxtaposition of nonempty closed walks P_{j} ∼ (α^{j} , β^{j}) such that ∑ _{r} α_{rc}^{j} ⩽ 1 for all c, ∑ _{c} α_{rc}^{j} ⩽ 1 for all r and (α, β) = ∑ (α^{j} , β^{j}).

(c) There are cycles P_{j} ∼ (α^{j} , β^{j}) and a γ such that (α, β) = (γ, γ) + ∑ (α^{j} , β^{j}).

Proof. (a). If r_{i} = r_{j} for i < j, we may suppose that j = k: consider the closed walks P_{1} = (c_{1}, r_{1}, …, c_{i}, r_{i}) and P_{2} = (c_{i+1} , r_{i+1} , …, c_{k}, r_{k}). If c_{i} = c_{j} for i < j, we may suppose that i = 1: consider then P_{1} = (c_{1}, r_{1}, …, c_{j−1} , r_{j−1} ) and P_{2} = (c_{j} , r_{j} , …, c_{k}, r_{k}).

(b). Use (a) in a maximality argument.

(c). Note that the closed walks P_{j} in (b) are either cycles or have length 2; in the latter case P_{j} = q ∼ (e_{q}, e_{q}) for some q ∈ I.

7 Schur multipliers on a cycle

We can realise a cycle of even length 2s, s ⩾ 2, in the following convenient way. Let Γ = ℤ/sℤ. Then the adjacency relation of integers modulo s turns Γ into the cycle (0, 1, …, s − 1) of length s. We double this cycle into the bipartite cycle (col0, row0, col1, row1, …, cols − 1 , rows − 1 ) on Γ ⨿ Γ, corresponding to the set of couples I = {(i, i), (i, i + 1) : i ∈ Γ} ⊆ Γ × Γ: I is the pattern

( \begin{matrix}& 0 & 1 & 2 & ··· & s−2 & s−1 \\ 0 & ∗ & ∗ & 0 & ... & 0 & 0 \\ 1 & 0 & ∗ & ∗ & ... & 0 & 0 \\ 2 & 0 & 0 & ∗ & ... & 0 & 0 \\ ⋮ & ... & ... & ... & ... & ... & ... \\ s−2 & 0 & 0 & 0 & ... & ∗ & ∗ \\ s−1 & ∗ & 0 & 0 & ... & 0 & ∗ \end{matrix} ) .

Γ is the group dual to G = Γ̂ = {z ∈ ℂ : z^{s} = 1}. We shall consider the space L_{Λ}^{p} (G) spanned by Λ = {1, z} in L^{p}(G), where z is the identical function on G: its norm is given by ∥a + bz∥_{L^{p} (G)} =

(s^{−1} ∑ _{z^{s} =1} |a + bz|^{p})^{1/p}.

Proposition 7.1. Let 0 < p ⩽ ∞, s ⩾ 2 and I = {(i, i), (i, i + 1) : i ∈ ℤ/sℤ}. Let ϵ ∈ 𝕋^{I} be a Schur multiplier by signs on S_{I}^{p}.

(a) The Schur multiplier ϵ has the same norm as the Schur multiplier ϵ̂ given by ϵ̂_{q} = 1 for q ≠ (s − 1 , 0) and ϵ̂_{s−1,0} = \overline{ϵ_{00}}ϵ_{01} … \overline{ϵ_{s−1 ,s−1}} ϵ_{s−1 ,0}.

(b) The Schur multiplier ϵ has the same norm as ϵ̌ given by ϵ̌_{ii} = 1 and ϵ̌_{i,i+1} = ϑ with ϑ any sth root of ϵ̂_{s−1 ,0} or its complex conjugate: without loss of generality, ϑ = e^{iα} with α ∈ [0, π/s].

(c) The norm of ϵ on S_{I}^{p} is bounded below by the norm of the relative Fourier multiplier μ : a+bz ↦ a + ϑbz on L_{Λ}^{p} (G); their complete norms are equal.

(d) The norm of ϵ on S_{I}^{1} and on S_{I}^{∞} is equal to the norm of μ on L_{Λ}^{1} (G) and on L_{Λ}^{∞} (G): this norm

is

\frac{cos(α/2 − π/2s)}{cosπ/2s} = \frac{max_{z^{s} =−1} |ϑ + z|}{|1 + e^{iπ/s} |} .

(e) The Schur multiplication operator M_{ϵ} is an isometry on S_{I}^{p} if and only if p/2 ∈ {1, 2, …, s − 1} or \overline{ϵ_{00}}ϵ_{01} … \overline{ϵ_{s−1 ,s−1}} ϵ_{s−1 ,0} = 1.

Proof. (a) and (b) follow from the matrix unconditionality of Schatten-von-Neumann norms (see Eq. (B.4)) and from the fact that the Schur multipliers ϵ and ϵ̄= (\overline{ϵ_{q}})_{q∈I} have the same norm on S_{I}^{p}.

(c) follows from Prop. 2.5.

(d). Let us compute f(β) = ∥1 + e^{iβ} z∥ _{L^{1} (G)} . As f(β) = f(β + 2π/s) = f(−β), we may suppose without loss of generality that β ∈ [0, π/s]. Then |β/2 + kπ/s| ⩽ π/2 if −⌊s/2⌋ ⩽ k ⩽ ⌈s/2⌉ − 1, so that

f(β) = \frac{1}{s} ∑_{k=−⌊s/2⌋}^{⌈s/2⌉−1} |1 + e^{iβ} e^{2ikπ/s} |

= \frac{2}{s} ∑_{k=−⌊s/2⌋}^{⌈s/2⌉−1} cos(β/2 + kπ/s)

= \frac{2}{s}ℜ( e^{iβ/2} \frac{e^{i⌈s/2⌉π/s} − e^{−i⌊s/2⌋/s}}{e^{iπ/s} − 1} )

= \frac{2}{s sin(π/2s)} · \{ \begin{matrix}cos(β/2 − π/2s) & if s is even \\ cos(β/2) & if s is odd. \end{matrix}

This shows in both cases that the norm of μ on L_{Λ}^{1} (G) is bounded below by cos(α/2 − π/2s)/ cos(π/2s). The complete norm of μ on L_{Λ}^{∞} (G) is equal to its norm and thus to the maximum of g(w) = ∥w + ϑz∥ _{L^{∞} (G)} /∥w + z∥ _{L^{∞} (G)} for w ∈ ℂ. Let w = re^{iβ} with r ⩾ 0 and β ∈ ℝ. Note that

∥w + z∥ _{L^{∞} (G)} = |r + e^{id(β,(2π/s)ℤ)} |

is a decreasing function of d(β, (2π/s)ℤ) and that

d(α − β, (2π/s)ℤ) < d(β, (2π/s)ℤ) ⇔ β ∈ ]α/2, π/s + α/2[ mod 2π/s.

As g(w) = g(wz) if z^{s} = 1, we may suppose without loss of generality that β ∈ ]α/2, π/s + α/2[. Therefore

g(w) = \{ \begin{matrix}| \frac{w + e^{iα}}{w + 1} | & if β ∈ ]α/2, π/s] \\ | \frac{w + e^{iα}}{w + e^{2iπ/s}} | & if β ∈ [π/s, π/s + α/2[. \end{matrix}

As g tends to 1 at infinity and g(w) = 1 if β ∈ {α/2, π/s + α/2}, the maximum principle shows that g attains its maximum with β = π/s. Finally,

g(re^{iπ/s} )^{2} = \frac{1 + 2r cos(π/s − α) + r^{2}}{1 + 2r cos(π/s) + r^{2}}

= 1 + \frac{cos(π/s − α) − cosπ/s}{cos(π/s) + (r + 1/r)/2} ⩽ g(e^{iπ/s} )^{2} = ( \frac{cos(π/2s − α/2)}{cosπ/2s} ) ^{2}.

8. 1-UNCONDITIONAL MATRICES IN S^{p}, p NOT AN EVEN INTEGER  45

(e). If p is not an even integer and ϑ^{s} ≠ 1, then μ is not an isometry on L_{Λ}^{p} (G): otherwise the functions z and ϑz would have the same distribution by the Plotkin-Rudin Equimeasurability Theorem (see [48, Th. 2]). If p ∈ {2, 4, …, 2s − 2}, then I contains no cycle of length 4, 6, …, p, so that by Prop. 6.6(c) every closed walk P ∼ (α, β) satisfies α = β. The function Φ_{I} (ϵ, a) in Eq. (B.9) is therefore constant in ϵ by Th. 6.5(a). If p ∈ {2s, 2s + 2, …}, the closed walk relation

(α, β) = ( ∑_{i∈Γ} e_{ii} , ∑_{i∈Γ} e_{i,i+1} ) + (p/2 − s)(e_{00}, e_{00})

satisfies n_{αβ} ⩾ 1 by Prop. 6.4. Then the coefficient of Φ_{I} (ϵ, a) in ā^{α} a^{β} equals

n_{αβ} \overline{ϵ_{00}}ϵ_{01} … \overline{ϵ_{s−1 ,s−1}} ϵ_{s−1 ,0}

and must equal the same quantity with ϵ replaced by 1 if ϵ defines an isometry on S_{I}^{p}.

Remark 7.2. See [50, p. 245] for a similar application of the Plotkin-Rudin Equimeasurability Theorem in (e).

The real unconditional constant of I is therefore the norm of ϵ̌ with α = π/s, and the complex unconditional constant is the maximum of the norm of ϵ̌ for α ∈ [0, π/s]. This yields

Corollary 7.3. Let 0 < p ⩽ ∞ and s ⩾ 2. Let I be the cycle of length 2s.

(a) I is a real 1-unconditional basic sequence in S^{p} if and only if p ∈ {2, 4, …, 2s − 2}.

(b) The real and complex unconditional constants of I in the spaces S^{1} and S^{∞} equal secπ/2s.

8 1-unconditional matrices in S^{p}, p not an even integer

We now state the announced isometric counterpart to Varopoulos’ characterisation of unconditional matrices in S^{∞} (Section 5) and its generalisation to S^{p} for p not an even integer.

Theorem 8.1. Let I ⊆ R × C be nonempty and p ∈ (0, ∞] \ 2ℕ. The following are equivalent.

(a) I is a complex completely 1-unconditional basic sequence in S^{p}.

(b) I is a complex 1-unconditional basic sequence in S^{p}.

(c) I is a real 1-unconditional basic sequence in S^{p}.

(d) I is a forest.

(e) For each ϵ ∈ 𝕋^{I} there are ζ ∈ 𝕋^{C} and η ∈ 𝕋^{R} such that ϵ_{rc} = ζ(c)η(r) for all (r, c) ∈ I.

(f) For each ϵ ∈ {−1, 1}^{I} there are ζ ∈ {−1, 1}^{C} and η ∈ {−1, 1}^{R} such that ϵ_{rc} = ζ(c)η(r) for all (r, c) ∈ I.

(g) I is a set of V-interpolation of constant 1: for all φ ∈ ℓ_{I}^{∞}

inf{ ∥ ∑_{(r,c)∈R×C} φ̃_{rc} e_{c} ⊗ e_{r} ∥_{ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞}} : φ̃|_{I} = φ} = sup_{q∈I}|φ_{q}|.  (B.15)

(h) I is a V-Sidon set of constant 1: for all φ ∈ c_{0}(I)

inf{ ∥ ∑_{(r,c)∈R×C} φ̃_{rc} e_{c} ⊗ e_{r} ∥_{c_{0} (C) ⊗^{∧} c_{0} (R)} : φ̃|_{I} = φ} = sup_{q∈I}|φ_{q}|.  (B.16)

(i) For every tensor u = ∑ _{(r,c)∈I} a_{rc} e_{c} ⊗ e_{r} in ℓ_{C}^{1} ⊗^{∨} ℓ_{R}^{1} with support in I we have ∥u∥_{ℓ_{C}^{1} ⊗^{∨} ℓ_{R}^{1}} =

∑ _{(r,c)∈I} |a_{rc}|.

(j) (z_{c}z′_{r})_{(r,c)∈I} is a Sidon set of constant 1 in the dual of 𝕋^{C} × 𝕋^{R}, that is, a 1-unconditional basic sequence in C(𝕋^{C} × 𝕋^{R}): if (a_{rc}) is finitely supported,

sup_{(z,z′)∈𝕋^{C} ×𝕋^{R}} | ∑_{(r,c)∈I} a_{rc}z_{c}z′_{r}| = ∑_{(r,c)∈I} |a_{rc}| .

(k) For all R′ ⊆ R and C′ ⊆ C with k ⩾ 1 elements # [I ∩ R′× C′] ⩽ 2k − 1.

(l) I is an isometric interpolation set for Schur multipliers on S^{∞} : every φ ∈ ℓ_{I}^{∞} is the restriction of a Schur multiplier on S^{∞} with norm ∥M_{φ} ∥ = ∥φ∥_{ℓ_{I}^{∞}} .

Proof. (a) ⇒ (b) ⇒ (c) is trivial.

(c) ⇒ (d). Suppose that I contains a cycle (c_{0}, r_{0}, …, c_{s−1} , r_{s−1} ) with s ⩾ 2. Cor. 7.3(a) shows that I is not a real 1-unconditional basic sequence in S^{p}.

(d) ⇔ (k). A tree on 2k vertices has exactly 2k−1 edges, so that a forest I satisfies (k). Conversely, a cycle of length 2s is a graph with s row vertices, s column vertices and 2s edges.

(d) ⇒ (e). Suppose first that I is a tree and index the vertices of its edges by words w ∈ W as described in the Terminology. Let us define η and ζ inductively. If r is the root of the tree, indexed by ∅, let η(r) = 1. Suppose that η and ζ have been defined for all vertices indexed by words of length at most 2n. If c is indexed by a word w of length 2n + 1, let r be the vertex indexed by the word of length 2n with which w begins and let ζ(c) = ϵ(r, c)/η(r). If r is indexed by a word w of length 2n + 2, let c be the vertex indexed by the word of length 2n + 1 with which w begins and let η(r) = ϵ(r, c)/ζ(c). If I is a union of pairwise disjoint trees, we may define η and ζ on each tree separately. We may finally extend η to R and ζ to C in an arbitrary manner.

(d) ⇒ (f) may be proved as (d) ⇒ (e).

(f) ⇒ (c). If (f) holds, then every Schur multiplier by signs ϵ ∈ {−1, 1}^{I} is elementary in the sense that ϵ = ζ ⊗ η. The complete norm of M_{ϵ} on any S_{I}^{p} is therefore ∥ζ∥_{ℓ_{C}^{∞}} ∥η∥_{ℓ_{R}^{∞}} = 1.

(e) ⇒ (g). If (e) holds, every φ ∈ 𝕋^{I} ⊆ ℓ_{I}^{∞} may be extended to an elementary tensor ζ ⊗ η of norm 1. (g) follows because every element of ℓ_{I}^{∞} with norm 1 is the half sum of two elements of 𝕋^{I}: note that e^{it} cosu = (e^{i(t+u)} + e^{i(t−u)} )/2.

(g) ⇒ (h). It suffices to check Equality (B.16) for φ with support contained in a finite rectangle

set R′× C′. As ℓ_{C′}^{∞} ⊗^{∧} ℓ_{R′}^{∞} is a subspace of ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} , Eq. (B.15) yields Eq. (B.16).

(h) ⇔ (i) because they are dual statements.

(i) ⇔ (j). Use Equality (B.8).

(h) ⇒ (l) may be deduced by the argument of Prop. 3.1(a) ⇒ (b).

(l) ⇒ (a). Taking sign sequences φ ∈ 𝕋^{I} in (l) shows that all relative Schur multipliers by signs on S_{I}^{∞} define isometries. Apply Prop. 4.3.

Remark 8.2. The equivalence of (e) with (j) may also be shown as a consequence of the characterisation of Sidon sets of constant 1 in [21].

Let us now answer Question 1.3.

Corollary 8.3. Let I ⊆ R × C. The following are equivalent.

(a) For all φ ∈ c_{0}(I) one has ∥∑ _{(r,c)∈I} φ_{rc} e_{c} ⊗ e_{r} ∥_{c_{0} (C) ⊗^{∧} c_{0} (R)} = sup_{q∈I} |φ_{q}|.

(b) There are pairwise disjoint sets R_{j} ⊆ R and pairwise disjoint sets C_{j} ⊆ C such that R_{j} or C_{j} is a singleton for each j and I = ⋃ R_{j} ×C_{j}: I is the union of the column section ⋃ _{# R_{j} =1} R_{j} ×C_{j} with the disjoint row section ⋃ _{# R_{j} >1} R_{j} × C_{j}.

(c) I is a union of pairwise disjoint star graphs: every path in I has length at most 2.

Proof. (a) ⇒ (b) follows from Prop. 3.1(a) ⇒ (d) and Th. 8.1(g) ⇒ (d).

(b) ⇔ (c). (b) holds if and only if (r, c), (r′, c), (r, c′) ∈ I ⇒ (r = r′ or c = c′) and therefore if and only if (c) holds.

(b) ⇒ (a). Suppose (b) and let φ ∈ c_{0}(I). Let α_{j} = sup_{(r,c)∈R_{j} ×C_{j}} |φ_{rc}|^{1/2} . If α_{j} = 0, let us define ϱ^{j} = 0 and γ^{j} = 0. Otherwise, if R_{j} is a singleton {r}, let us define ϱ^{j} = α_{j} e_{r} and γ^{j} by γ_{c}^{j} = φ_{rc}/α_{j} if c ∈ C_{j} and γ_{c}^{j} = 0 otherwise. Otherwise, C_{j} is a singleton {c} and we define γ^{j} = α_{j} e_{c} and ϱ^{j} by ϱ_{r}^{j} = φ_{rc}/α_{j} if r ∈ R_{j} and ϱ_{r}^{j} = 0 otherwise. Note that the γ^{j} have pairwise disjoint support and are null sequences, as well as the ϱ^{j}. Then

∑_{(r,c)∈I} φ_{rc} e_{c} ⊗ e_{r} = ∑_{j} γ^{j} ⊗ ϱ^{j} = average_{ϵ_{j} =±1}( ∑_{j} ϵ_{j} γ^{j} ) ⊗ ( ∑_{j} ϵ_{j} ϱ^{j} )

9. 1-UNCONDITIONAL MATRICES IN S^{p}, p AN EVEN INTEGER  47

is an average of elementary tensors in c_{0}(C) ⊗^{∧} c_{0}(R) of norm sup_{q∈I} |φ_{q}|, so that this average is also bounded by this norm, which obviously is a lower bound.

9 1-unconditional matrices in S^{p}, p an even integer

Let us now prove Theorem 1.5 as a consequence of Theorem 6.5 together with Proposition 6.6(c).

Theorem 9.1. Let I ⊆ R × C and p = 2k a positive even integer. The following assertions are equivalent.

(a) I is a complex completely 1-unconditional basic sequence in S^{p}.

(b) I is a complex 1-unconditional basic sequence in S^{p}.

(c) For every finite subset F ⊆ I there is an operator x ∈ S^{p}, whose support S contains F, such that ∥∑ ϵ_{q}x_{q}e_{q}∥_{p} does not depend on the complex choice of signs ϵ ∈ 𝕋^{S}.

(d) I is a real 1-unconditional basic sequence in S^{p}.

(e) For every finite subset F ⊆ I there is an operator x ∈ S^{p} with real matrix coefficients, whose support S contains F, such that ∥∑ ϵ_{q}x_{q}e_{q}∥_{p} does not depend on the real choice of signs ϵ ∈ {−1, 1}^{S}.

(f) Every closed walk P ∼ (α, β) of length 2s ⩽ 2k in I satisfies α = β.

(g) I does not contain any cycle of length 2s ⩽ 2k as a subgraph.

(h) For each v, w ∈ V there is at most one path in I of length l ⩽ k that joins v to w.

Proof. (a) ⇒ (b) ⇒ (c), (b) ⇒ (d) ⇒ (e) are trivial.

(c) ⇒ (g). Suppose that I contains a cycle P ∼ (γ, δ) of length 2s ⩽ 2k: the corresponding set of couples is F = {q : γ_{q} + δ_{q} = 1}. Let x be as in (c) and let (α, β) = (γ, δ) + (k − s)(e_{q}, e_{q}) for some arbitrary q ∈ F. Then (α, β) ∈ 𝒲_{k}^{S}. Consider f(ϵ) = ∥∑ ϵ_{q}x_{q}e_{q}∥_{p}^{p} as a function on the group 𝕋^{S}.

Then the Fourier coefficient f̂(ϵ^{β−α} ) of f at the Steinhaus character ϵ^{β−α} is, by Th. 6.5(a),

∑ \{n_{εζ} \overline{x}^{ε}x^{ζ} : (ε, ζ) ∈ 𝒲_{k}^{S} and ζ − ε = β − α\}

= \overline{x}^{γ}x^{δ} ∑ \{n_{εζ} \overline{x}^{ε−γ} x^{ζ−δ} : (ε, ζ) ∈ 𝒲_{k}^{S} and ζ − δ = ε − γ\}.

(Note that β − α = δ − γ.) As this last sum has only positive terms and contains at least the term corresponding to (α, β), f cannot be constant.

(e) ⇒ (g). Let P ∼ (γ, δ), F = {q : γ_{q} + δ_{q} = 1} and (α, β) be as in the proof of the implication (c) ⇒ (h). Let x be as in (e). Consider f(ϵ) = ∥∑ ϵ_{q}x_{q}e_{q}∥_{p}^{p} as a function on the group {−1, 1}^{S}.

Then the Fourier coefficient f̂(ϵ^{β−α} ) of f at the Walsh character ϵ^{β−α} is, by Th. 6.5(a),

∑ {n_{εζ}x^{ε+ζ} : (ε, ζ) ∈ 𝒲_{k}^{S} and ζ − ε ≡ β − α (mod 2)}

= x^{γ+δ} ∑ {n_{εζ}x^{ε+ζ−γ−δ} : (ε, ζ) ∈ 𝒲_{k}^{S} and ζ − ε ≡ δ − γ (mod 2)}.

As this last sum has only positive terms and contains at least the term corresponding to (α, β), f cannot be constant.

(f) ⇔ (g). Apply Prop. 6.6(c).

(g) ⇔ (h). If I contains a cycle (v_{0}, …, v_{2s−1} ), then I contains two distinct paths (v_{0}, …, v_{s}), (v_{0}, v_{2s−1} , …, v_{s}) of length s from v_{0} to v_{s}. If I contains two distinct paths (v_{0}, …, v_{l}), (v′_{0}, …, v′_{l′}) with v_{0} = v′_{0}, v_{l} = v′_{l′} and l, l′ ⩽ k, let a be minimal such that v_{a} ≠ v′_{a}, let b ⩾ a be minimal such that v_{b} ∈ {v′_{a}, …, v′_{l′}} and let d ⩾ a be minimal such that v′_{d} = v_{b}. Then (v_{a−1} , …, v_{b}, v′_{d−1} , …, v′_{a}) is a cycle in I of length 2s ⩽ 2k.

(f) ⇒ (a) holds by Theorem 6.5(b): If each (α, β) ∈ 𝒲_{k}^{I} satisfies α = β, then Eq. (B.14) shows that Ψ_{I} (ϵ, z) as defined in Eq. (B.13) is constant in ϵ.

Remark 9.2. The equivalence (b) ⇔ (g) is a noncommutative analogue to [63, Prop. 2.5(ii)].

Remark 9.3. In [64, Th. 2.7], the condition of Th. 9.1(f) is visualised in another way: a closed walk P = (c_{1}, r_{1}, …, c_{s}, r_{s}) ∼ (α, β) in ℕ × ℕ is considered as the polygonal closed curve γ in ℂ with sides parallel to the coordinate axes whose successive vertices are r_{1} + ic_{1}, r_{1} + ic_{2}, r_{2} + ic_{2}, …, r_{s−1} + ic_{s}, r_{s} + ic_{s}, r_{s} + ic_{1} and again r_{1} + ic_{1}. Then α = β if and only if the index with respect to γ of every point not on γ is zero, if and only if γ can be shrunk to a point inside of the set of its points.

Remark 9.4. One cannot drop the assumption that x has real matrix coefficients in Th. 9.1(e). Consider a 2 × 2 matrix x. Then trx^{∗}x = ∑ |x_{q}|^{2} and detx^{∗}x = |x_{00}x_{11} − x_{01}x_{10}|^{2}. This shows that

if ℜ(\overline{x_{00}x_{11}} x_{01}x_{10}) = 0, e.g. x = ( \begin{matrix}1 & 1 \\ 1 & i \end{matrix}) , then the singular values of x do not depend on the real

sign of the matrix coefficients of x, whereas (col0, row0, col1, row1) is a cycle of length 4.

Remark 9.5. Theorem 9.1(h) ⇒ (a) is the isometric counterpart to [38, Th. 3.1], which shows in particular that I is an unconditional basic sequence in S^{2k} if the number of walks in I between two given vertices of length k and with no edge repeated has a uniform bound. The following combinatorial problem arises naturally: if I satisfies this latter condition, is it so that I is the union of a finite number of sets I_{j} such that there is at most one path of length at most k in I_{j} between two given vertices? In the simplest case, k = 2, William Banks, Ilijas Farah, Asma Harcharras and Dominique Lecomte [5] have deduced from [87] that it is not so.

10 Metric unconditional approximation property for S_{I}^{p}

Let R, C be two copies of ℕ. It is well known that, apart from S^{2}, no S^{p} has an unconditional basis or just a local unconditional structure (see [80, § 4].) S^{1} and S^{∞} cannot even be embedded in a space with unconditional basis. If 1 < p < ∞, then S^{p} has the unconditional finite dimensional decomposition

⨁_{n∈ℕ} S_{{(r,c):r⩽n,c=n}}^{p} ⊕ S_{{(r,c):r=n+1,c⩽n}}^{p}

because the triangular projection associated to the idempotent Schur multiplier (χ_{r⩽c}) is bounded on S^{p}.

Definition 10.1. Let X be a separable Banach space and 𝕊 = 𝕋 (vs. 𝕊 = {−1, 1}.)

– A sequence (T_{k}) of operators on X is an approximating sequence if each T_{k} has finite rank and ∥T_{k}x − x∥ → 0 for every x ∈ X. An approximating sequence of commuting projections is a finite-dimensional decomposition.

– ([73].) The difference sequence (ΔT_{k}) of (T_{k}) is given by ΔT_{1} = T_{1} and ΔT_{k} = T_{k} − T_{k−1} for k ⩾ 2. X has the unconditional approximation property (uap) if there is an approximating sequence (T_{k}) such that for some constant D

∥∑_{k=1}^{n} ϵ_{k}ΔT_{k} ∥ ⩽ D for all n and ϵ_{k} ∈ 𝕊.

The complex (vs. real) unconditional constant of (T_{k}) is the least such constant D.

– ([22, § 3], [32, § 8].) X has the complex (vs. real) metric unconditional approximation property (muap) if, for every δ > 0, X has an approximating sequence with complex (vs. real) unconditional constant 1 + δ. By [22, Th. 3.8] and [32, Lemma 8.1], this is the case if and only if there is an approximating sequence (T_{k}) such that

sup_{ϵ∈𝕊}∥T_{k} + ϵ(Id − T_{k})∥ −→ 1.  (B.17)

X has (muap) if and only if, for every given δ > 0, X is isometric to a 1-complemented subspace of a space with a (1 + δ)-unconditional finite-dimensional decomposition [31, Cor. IV.4]. If X has (muap), then, for any given δ > 0, X is isometric to a subspace of a space with a (1+δ)-unconditional basis.

10. METRIC UNCONDITIONAL APPROXIMATION PROPERTY FOR S_{I}^{p}  49

Example 10.2. The simplest example is the subspace in S^{p} of operators with an upper triangular matrix. In fact, if I ⊆ R × C is such that all columns I ∩ R × {c} (vs. all rows I ∩ {r} × C) are finite, then S_{I}^{p} admits a 1-unconditional finite-dimensional decomposition in the corresponding finitely supported idempotent Schur multipliers χ_{I∩R×{c}} (vs. χ_{I∩{r}×C} .)

Our results on complete 1-unconditional basic sequences yield the following theorem.

Theorem 10.3. Let 1 ⩽ p ⩽ ∞. Let R_{r} ⊆ R, r ∈ ℕ, be pairwise disjoint and finite. Let C_{c} ⊆ C, c ∈ ℕ, be pairwise disjoint and finite. Let J ⊆ ℕ × ℕ and I = ⋃ _{(r,c)∈J} R_{r} × C_{c}. Then the sequence of Schur multipliers (χ_{R_{r} ×C_{c}} )_{(r,c)∈J} forms a complex 1-unconditional finite-dimensional decomposition for S_{I}^{p} if and only if J is a forest or p is an even integer and J contains no cycle of length 4, 6, …, p.

We may always suppose that approximating sequences on spaces S_{I}^{p} are associated to Schur multipliers. More precisely, we have

Proposition 10.4. Let 1 ⩽ p ⩽ ∞ and I ⊆ R × C. Let (T_{n}) be an approximating sequence on S_{I}^{p}. Then there is a sequence of Schur multipliers (φ_{n}) such that (M_{φ_{n}} ) is an approximating sequence on S_{I}^{p} and such that if (T_{n}) satisfies (B.17), then so does (M_{φ_{n}} ).

Proof. Let δ_{n} > 0 be such that δ_{n} → 0. As T_{n} has finite rank, there is a finite R_{n} × C_{n} ⊆ R × C such that the projection P_{R_{n} ×C_{n}} of S^{p} onto S_{R_{n} ×C_{n}}^{p} defined by the Schur multiplier χ_{C_{n}} ⊗ χ_{R_{n}} satisfies ∥P_{R_{n} ×C_{n}} T_{n} − T_{n}∥ < δ_{n}. Let φ_{n} be the Schur multiplier associated to [T_{n}]_{R_{n} ×C_{n}} . With the notation of Eq. (B.5),

M_{φ_{n}} (x) − x = ∫_{𝕋^{R}} dη ∫_{𝕋^{C}} dζ M_{ζ^{∗} ⊗η^{∗}} (P_{R_{n} ×C_{n}} T_{n} − Id)(M_{ζ⊗η} x).

As P_{R_{n} ×C_{n}} T_{n} tends to the identity uniformly on compact sets, this shows that M_{φ_{n}} is an approximating sequence. As

M_{φ_{n}} + ϵ(Id − M_{φ_{n}} ) = [P_{R_{n} ×C_{n}} T_{n} + ϵ(Id − P_{R_{n} ×C_{n}} T_{n})] ,

the norm of this operator is at most ∥T_{n} + ϵ(Id − T_{n})∥ + 2δ_{n}.

This proposition shows together with Prop. 2.1 the following results.

Corollary 10.5. Let 1 ⩽ p ⩽ ∞ and I ⊆ R × C.

– If S_{I}^{p} has (muap), then some sequence of Schur multipliers realises it.

– Let J ⊆ I. If S_{I}^{p} has (muap), then so does S_{J}^{p}.

– If S_{I}^{∞} has (muap), then so does S_{I}^{p}.

Let us define the following asymptotic properties.

Definition 10.6. Let 1 ⩽ p ⩽ ∞, I ⊆ R × C and 𝕊 = 𝕋 (vs. 𝕊 = {−1, 1}.)

– S_{I}^{p} is asymptotically unconditional if for every x ∈ S_{I}^{p} and for every bounded sequence (y_{n}) in S_{I}^{p} such that each matrix coefficient of y_{n} tends to 0

max_{ϵ∈𝕊}∥x + ϵy_{n}∥_{p} − min_{ϵ∈𝕊}∥x + ϵy_{n}∥_{p} −→ 0.

– I enjoys the property (𝒰 ) of block unconditionality in S^{p} if for each δ > 0 and finite F ⊆ I, there is a finite G ⊆ I such that

∀ x ∈ B_{S_{F}^{p}} ∀ y ∈ B_{S_{I\G}^{p}} max_{ϵ∈𝕊}∥x + ϵy∥_{p} − min_{ϵ∈𝕊}∥x + ϵy∥_{p} < δ.

The arguments of [63, § 6.2] show mutatis mutandis

Theorem 10.7. Let 1 ⩽ p ⩽ ∞, I ⊆ R × C and 𝕊 = 𝕋 (vs. 𝕊 = {−1, 1}.) Consider the following properties.

(a) S_{I}^{p} is asymptotically unconditional.

(b) I enjoys (𝒰 ) in S^{p}.

(c) S_{I}^{p} has (muap).

Then (c) ⇒ (a) ⇔ (b). If 1 < p < ∞, then (b) ⇔ (c). If p = 1, S_{I}^{1} has (muap) if and only if S_{I}^{1} has (uap) and I enjoys (𝒰 ) in S^{1}.

The case p = ∞ is extreme in the sense that the following properties are equivalent for S_{I}^{∞} : to be a dual space, to be reflexive, to have a finite cotype, not to contain c_{0}, because they are equivalent for I not to contain any sequence (r_{n}, c_{n}) with (r_{n}) and (c_{n}) injective, that is for I to be contained in the union of a finite set of lines and a finite set of columns, so that S_{I}^{∞} is isomorphic to ℓ_{I}^{2}.

Let us now introduce the asymptotic property on I that reflects the combinatorics imposed by (muap).

Definition 10.8. Let I ⊆ R × C and k ⩾ 1.

– I enjoys property 𝒥_{k} if for every path P = (c_{0}, r_{0}, …, c_{j} , r_{j}) of odd length 2j + 1 ⩽ k in I there is a finite set R′× C′ such that P cannot be completed with edges in I \ R′× C′ to a cycle of length 2s ∈ {4j + 2, …, 2k}.

– The asymptotic distance d_{∞} (r, c) of r ∈ R and c ∈ C in I is the supremum, over all finite rectangle sets R′× C′, of the distance from r to c in I \ R′× C′.

The asymptotic distance takes its values in {1, 3, 5, …, ∞}. Note that 𝒥_{1} is true and that 𝒥_{k} ⇒ 𝒥_{k−1} . This implication is strict: let R, C be two copies of ℕ and, given j ⩾ 1, consider the union I_{j} of all paths (col0, rownj + 1 , colnj + 1 , …, rownj + j , colnj + j , row0) of length 2j + 1. Then I_{j} contains no cycle of length 2s ∈ {4, …, 4j} and therefore enjoys 𝒥_{2j} , but fails 𝒥_{2j+1} ; I_{j} ∪ {(row0, col0)} contains no cycle of length 2s ∈ {4, …, 2j} and thus enjoys 𝒥_{j}, but fails 𝒥_{j+1} . In particular, the properties 𝒥_{k}, k ⩾ 2, are not stable under union with a singleton.

Let us now explicit the relationship between 𝒥_{k} and d_{∞} .

Proposition 10.9. Let I ⊆ R × C and k ⩾ 1.

(a) I enjoys 𝒥_{k} if and only if any two vertices r ∈ R and c ∈ C at distance 2j + 1 ⩽ k satisfy d_{∞} (r, c) ⩾ 2k − 2j + 1.

(b) If d_{∞} (r, c) ⩾ 2k + 1 for all (r, c) ∈ R × C, then I enjoys 𝒥_{k}.

(c) If d_{∞} (r, c) ⩽ k for some (r, c) ∈ R × C, then I fails 𝒥_{k}.

(d) I enjoys 𝒥_{k} for every k if and only if d_{∞} (r, c) = ∞ for every (r, c) ∈ R × C.

Proof. (a) is but a reformulation of the definition of𝒥_{k} and implies (b).

(d) is a consequence of (b) and (c).

(c). If d_{∞} (r, c) ⩽ k, then there is 0 ⩽ j ⩽ (k − 1)/2 such that there are infinitely many paths of length 2j + 1 from c to r: there is a path (c, r_{1}, c_{1}, …, r_{j} , c_{j} , r) that can be completed with edges outside any given finite set to a cycle of length 4j + 2 ⩽ 2k.

Theorem 10.10. Let I ⊆ R × C and 1 ⩽ p ⩽ ∞. If p is an even integer, then S_{I}^{p} has complex or real (muap) if and only if I enjoys 𝒥_{p/2}. If p = ∞ or if p is not an even integer, then S_{I}^{p} has real (muap) only if I enjoys 𝒥_{k} for every k.

Proof. Suppose that I enjoys (𝒰 ) in S^{p} and fails 𝒥_{k}. Then, for some s ⩽ k, I contains a sequence of cycles (c_{0}, r_{0}, …, c_{j−1} , r_{j−1} , c_{j}^{n}, r_{j}^{n}, …, c_{s−1}^{n} , r_{s−1}^{n} ) with the property that ∥x−y∥_{p} ⩽ (1+1/n)∥x+y∥_{p} for all x with support in {(r_{0}, c_{0}), (r_{0}, c_{1}), …, (r_{j−2} , c_{j−1} ), (r_{j−1} , c_{j−1} )} and all y with support in {(r_{j−1} , c_{j}^{n}), (r_{j}^{n}, c_{j}^{n}), …, (r_{s−1}^{n} , c_{s−1}^{n} ), (r_{s−1}^{n} , c_{0})}. With the notation of Section 7, this amounts to stating that the multiplier on I = {(i, i), (i, i + 1)} ⊆ ℤ/sℤ × ℤ/sℤ given by ϵ_{rc} = 1 if r, c ∈ \{0, …, j − 1\} and ϵ_{rc} = −1 otherwise actually is an isometry on S_{I}^{p}. As \overline{ϵ_{00}}ϵ_{01} … \overline{ϵ_{s−1 s−1}} ϵ_{s−1 0} = (−1)^{2s−2j+1} = −1, this implies by Prop. 7.1(e) that p/2 ∈ {1, 2, …, s − 1}.

Suppose that I enjoys 𝒥_{k}. We claim that for every finite F ⊆ I there is a finite G ⊆ I such that every closed walk P ∼ (α, β) of length 2k in I satisfies ∑ _{q∈I\G} β_{q} − α_{q} = 0. This signifies that given a closed walk (v_{0}, …, v_{2k−1} ) and 0 = a_{0} < b_{0} < ⋯< a_{m} < b_{m} < a_{m+1} = 2k such that v_{a_{i}} , …, v_{b_{i} −1} ∈ I \ G and v_{b_{i}}, …, v_{a_{i+1} −1} ∈ F,

{i ∈ {0, …, m} : a_{i}, b_{i} even} = {i ∈ {0, …, m} : a_{i}, b_{i} odd}.

Suppose that this is not true: then there is an s ⩽ k, there are 0 = a_{0} < b_{0} < ⋯< a_{m} < b_{m} < 2s and there are cycles (v_{a_{0}}^{n} , …, v_{b_{0} −1}^{n} , v_{b_{0}}, …, v_{a_{1} −1} , …, v_{a_{m}}^{n} , …, v_{b_{m} −1}^{n} , v_{b_{m}} , …, v_{2s−1} ) such that the (v_{i}^{n})_{n⩾0} are injective sequences of vertices and b_{i} − a_{i} is even for at least one index i: let us suppose so for i = 0. If b_{0} − a_{0} ⩾ s − 1, consider the path P = (v_{b_{0}}, …, v_{a_{0} −1} , v_{a_{0}}^{0} , …, v_{b_{m} −1}^{0} , v_{b_{m}} , …, v_{2s−1} ) of odd length 2s − 1 − (b_{0} − a_{0}); if b_{0} − a_{0} ⩽ s − 1, consider the path P = (v_{2s−1} , v_{a_{0}}^{0} , …, v_{b_{0} −1}^{0} , v_{b_{0}}) of odd length b_{0} − a_{0} + 1. Then P can be completed with vertices outside any given finite set to a cycle of length at most 2s because (v_{2s−1} , v_{a_{0}}^{n} , …, v_{b_{0} −1}^{n} , v_{b_{0}}) is a path of length b_{0} − a_{0} + 1 in I for every n. This proves that I fails 𝒥_{s}.

The claim shows that I enjoys (𝒰 ) in S^{p} for p = 2k. In fact, if ϵ̃∈ 𝕋^{F ∪(I\G)} is defined by ϵ̃_{q} = 1 for q ∈ F and ϵ̃_{q} = ϵ ∈ 𝕋 for q ∈ I \ G, then, with the notation of Th. 6.5,

Φ_{F ∪(I\G)} (ϵ̃, a) = ∑_{(α,β)∈𝒲_{k}^{F ∪(I\G)}} n_{αβ} ϵ^{∑ _{q∈I\G} β_{q}−α_{q}} \overline{a}^{α} a^{β}

does not depend on ϵ, so that ∥x + ϵy∥_{2k} = ∥x + y∥_{2k} if x ∈ S_{F}^{2k} and y ∈ S_{I\G}^{2k} , and S_{I}^{2k} has complex (muap) by Th. 10.7(b) ⇒ (c).

Remark 10.11. This theorem is a noncommutative analogue to [63, Th. 7.5].

11 Examples

One of Varopoulos’ motivations for the study of the projective tensor product ℓ_{∞} ⊗^{∧} ℓ_{∞} are lacunary sets in a locally compact abelian group.

Let Γ be a discrete abelian group and Λ ⊆ Γ. Let us say that Λ is n-independent if every element of Γ admits at most one representation as the sum of n terms in Λ, up to a permutation. For example, the geometric sequence {j^{k}}_{k⩾0} with j ∈ {2, 3, …} is n-independent in ℤ if and only if j ⩾ n [63, § 3]. If Λ is n-independent for all n, then Λ is independent. Let

Z_{n} = {ζ ∈ ℤ^{Λ} : ∑_{γ∈Λ} ζ_{γ} = 0 and ∑_{γ∈Λ} |ζ_{γ}| ⩽ 2n]

and Z = ⋃ Z_{n}. Then Λ is n-independent if and only if, for every ζ ∈ Z_{n},

∑_{γ∈Λ} ζ_{γ}γ = 0 ⇒ ζ = 0

and Λ is independent if and only if this holds for every ζ ∈ Z.

Let us say that Λ is n-independent modulo 2 if in every representation of an element of Γ as the sum of n terms in Λ, each element of Λ appears the same number of times modulo 2. In other words, for every ζ ∈ Z_{n},

∑_{γ∈Λ} ζ_{γ}γ = 0 ⟹ ∀ γ ∈ Λ ζ_{γ} = 0 (mod 2) ;

Λ is independent modulo 2 if this holds for every ζ ∈ Z. If Γ contains no element of order 2, then one may always suppose that at least one coefficient ζ_{γ} of a nontrivial relation ∑ ζ_{γ}γ = 0 is odd, so that these two latter notions “modulo 2” coincide with the two former ones.

Let G = Γ̂, so that Γ is the group of characters on G. Then the computation presented in [63, Prop. 2.5(ii)] for the case Γ = ℤ shows that Λ is a complex (vs. real) 1-unconditional basic sequence in L^{p}(G) with p ∈ 2ℕ^{∗} if and only if Λ is p/2-independent (vs. modulo 2). Furthermore Λ is a complex (vs. real) 1-unconditional basic sequence in L^{p}(G) with p ∈ (0, ∞]\2ℕ^{∗} if and only if Λ is independent (vs. modulo 2). If Γ contains no element of order 2, then a real 1-unconditional basic sequence in L^{p}(G) is also complex 1-unconditional. All these results hold also for the complete counterparts to 1-unconditional basic sequences.

Results on lacunary sets in a discrete abelian group transfer to lacunary matrices in the following way, as in [98, Th. 4.2].

Proposition 11.1. Let Γ be a discrete abelian group and R, C be countable subsets of Γ. To every Λ ⊆ R + C associate I_{Λ} = {(r, c) ∈ R × C : r + c ∈ Λ}. Let G = Γ̂.

(a) If Λ is a complex 1-unconditional basic sequence in L^{4}(G), then I_{Λ} is a 1-unconditional basic sequence in S^{4}.

(b) Suppose that each element of Γ admits at most one representation as the sum of an element of R with an element of C. Then every I ⊆ R×C has the form I = I_{Λ} with Λ = {r + c : (r, c) ∈ I} . If Λ is a real 1-unconditional basic sequence in L^{p}(G), then I_{Λ} is a 1-unconditional basic sequence in S^{p}.

(c) Let p = 2k be a positive even integer. Suppose that R ∩ C = ∅ and R ∪ C is k-independent modulo 2. I_{Λ} is a 1-unconditional basic sequence in S^{p} if and only if Λ is a real 1-unconditional basic sequence in L^{p}(G).

Proof. (a). Let P = (c, r, c′, r′) be a closed walk in I_{Λ} . Then r + c, r′+ c′, r + c′ and r′+ c are in Λ while (r + c) + (r′+ c′) = (r + c′) + (r′+ c): if Λ is 2-independent, then r + c ∈ {r + c′, r′+ c}, so that c = c′ or r = r′ and P is not a cycle.

(b). For each γ ∈ Λ, let q_{γ} = (r_{γ}, c_{γ}) be the unique element of I such that r_{γ} + c_{γ} = γ. If Λ is a real 1-unconditional basic sequence in L^{p}(G), then it is also a complete real 1-unconditional basic sequence in L^{p}(G). Let φ ∈ {−1, 1}^{I_{Λ}} , so that φ_{q_{γ}} ∈ {−1, 1} for all γ ∈ Λ. Then, as in Eq. (B.6),

∥ ∑_{q∈I_{Λ}} a_{q}e_{q}∥_{S_{I_{Λ}}^{p} (S^{p})} = ∥ ∑_{(r,c)∈I_{Λ}} r(g)c(g)a_{rc} e_{rc} ∥_{S_{I_{Λ}}^{p} (S^{p})}

= ∥∑_{γ∈Λ} a_{q_{γ}} e_{q_{γ}} γ(g)∥_{S_{I_{Λ}}^{p} (S^{p})} = ∥∑_{γ∈Λ} a_{q_{γ}} e_{q_{γ}} γ∥_{L_{Λ}^{p} (G,S^{p} (S^{p}))}

so that as in Eq. (2), by complete real 1-unconditionality of Λ in L^{p}(G),

∥ ∑_{q∈I_{Λ}} φ_{q}a_{q}e_{q}∥_{S_{I_{Λ}}^{p} (S^{p})} = ∥∑_{γ∈Λ} φ_{q_{γ}} a_{q_{γ}} e_{q_{γ}} γ∥_{L_{Λ}^{p} (G,S^{p} (S^{p}))} = ∥ ∑_{q∈I_{Λ}} a_{q}e_{q}∥_{S_{I_{Λ}}^{p} (S^{p})} .

(c). Each element of Γ admits at most one representation as the sum of an element of R with an element of C, so that (b) yields sufficiency. Suppose that Λ is not a real 1-unconditional basic sequence in L^{p}(G) and let ζ ∈ Z_{k} such that ∑ _{γ∈Λ} ζ_{γ}γ = 0 and J = {(r, c) ∈ I_{Λ} : ζ_{r+c} ≠ 0 (mod 2)} is nonempty; J has at most 2k elements. Let P = (v_{1}, …, v_{j}) be a path in J of maximal length. Then ζ_{v_{j−1} +v_{j}} is odd and ∑ {ζ_{v_{j} +v} : v_{j} + v ∈ Λ} is even because it is the coefficient of v_{j} in the relation ∑ _{γ∈Λ} ζ_{γ}γ = 0 and R ∪ C is k-independent modulo 2. There is therefore v_{j+1} distinct from v_{j−1} such that ζ_{v_{j} +v_{j+1}} is odd. As j is maximal and R ∩ C = ∅, v_{j+1} = v_{j+1−2i} for some 2 ⩽ i ⩽ k, so that (v_{j+1−2i} , …, v_{j}) is a cycle of length 2i in J: I_{Λ} is not a 1-unconditional basic sequence in S^{p}.

Let R and C be any countable sets. Consider G = {−1, 1}^{C} ×{−1, 1}^{R}. If we denote by ((ϵ_{c})_{c∈C} , (ϵ′_{r})_{r∈R} ) a generic point in G, then the set of Rademacher functions {ϵ_{c}}_{c∈C} ∪ {ϵ′_{r}}_{r∈R} is a real 1-unconditional basic sequence in C(G), so that it is independent modulo 2 in Ĝ. Similarly, the set of Steinhaus functions {z_{c}}_{c∈C} ∪ {z′_{r}}_{r∈R} is independent in the dual of 𝕋^{C} × 𝕋^{R}. This yields:

Corollary 11.2. Let I ⊆ R × C and p ∈ (0, ∞]. The following are equivalent:

– I is a 1-unconditional basic sequence in S^{p}.

– {ϵ_{c}ϵ′_{r} : (r, c) ∈ I} is a real 1-unconditional basic sequence in L^{p}(G).

– {z_{c}z′_{r} : (r, c) ∈ I} is a 1-unconditional basic sequence in L^{p}(𝕋^{C} × 𝕋^{R}).

Remark 11.3. The isomorphic counterpart is also true: I is a completely unconditional basic sequence in S^{p} (i.e., a complete σ(p) set) if and only if {ϵ_{c}ϵ′_{r} : (r, c) ∈ I} is a completely unconditional basic sequence in L^{p}(G) (a Λ(p)_{cb} set in Ĝ, see [37] and [77, § 8.1],) if and only if {z_{c}z′_{r} : (r, c) ∈ I} is a completely unconditional basic sequence in L^{p}(𝕋^{C} × 𝕋^{R}). This follows e.g. from the proof of Prop. 11.1(b) and the iterated noncommutative Khinchin inequality [77, Eq. (8.4.11)].

Harcharras [37] used Peller’s discovery [74] of the link between Fourier and Hankel Schur multipliers to produce unconditional basic sequences in S^{p} that are unions of antidiagonals in ℕ × ℕ. We have in our context the rather disappointing

Proposition 11.4. Let Λ ⊆ ℕ ⊆ ℤ and I = {(r, c) ∈ ℕ × ℕ : r + c ∈ Λ}.

(a) I is a 1-unconditional basic sequence in S^{4} if and only if {z^{λ}}_{λ∈Λ} is a 1-unconditional basic sequence in L^{4}(𝕋).

(b) If Λ contains three elements λ < μ < ν such that λ + μ ⩾ ν, then I is not a 1-unconditional basic sequence in S^{p} if p ∈ (0, ∞] \ {2, 4}.

(c) If Λ = {λ_{k}} with λ_{k+1} > 2λ_{k} for all k, then I is a 1-unconditional basic sequence in S^{p} for every p.

Proof. (a). Sufficiency follows from Prop. 11.1(a) with R = C = ℕ. Conversely, if Λ contains a solution to λ + μ = λ′ + μ′ with λ < λ′ ⩽ μ′ < μ, then I contains the cycle (col0, rowλ, colλ′ − λ, rowμ′).

(b). Consider the cycle (col0, rowλ, colν − λ, rowμ − ν + λ, colν − μ, rowμ).

(c). In fact, I is a forest. Let P = (c_{1}, r_{1}, …, c_{k}, r_{k}) be a closed walk in I. We may suppose without loss of generality that r_{1} + c_{2} is a maximal element of {r_{1} + c_{1}, r_{1} + c_{2}, …, r_{k} + c_{k}, r_{k} + c_{1}}. Then r_{1}+c_{1} ⩽ r_{1}+c_{2} and r_{2}+c_{2} ⩽ r_{1}+c_{2}. One of these inequalities must be an equality and P is not a cycle: for otherwise 2(r_{1} + c_{1}) < r_{1} + c_{2} and 2(r_{2} + c_{2}) < r_{1} + c_{2} because r_{1} + c_{1}, r_{1} + c_{2}, r_{2} + c_{2} ∈ Λ, so that 2(r_{1} + c_{1} + r_{2} + c_{2}) < 2(r_{1} + c_{2}) and c_{1} + r_{2} < 0.

Remark 11.5. Further computations yield the following result. If {z^{λ}}_{λ∈Λ} is a 1-unconditional basic sequence in L^{6}(𝕋) and if {λ < μ < ν} ⊆ Λ ⇒ λ + μ < ν, then I is a 1-unconditional basic sequence in S^{6}; the converse does not hold.

Let us now give an overview of the known extremal bipartite graphs without cycle of length 4, 6, …, 2k and their size. Look up [9, Def. I.3.1] for the definition of a Steiner system and [56, Def. 1.3.1] for the definition of a generalised polygon. An elementary example is given in the introduction with (B.1).

Proposition 11.6. Let 2 ⩽ n ⩽ m, I ⊆ R × C with # C = n and # R = m, and e = # I.

(a) If I is a 1-unconditional basic sequence in S^{4}, then

n ⩾ 1 + ( \frac{e}{m} − 1) + ( \frac{e}{m} − 1) ( \frac{e}{n} − 1) ,

that is e^{2} − me − mn(n − 1) ⩽ 0. Equality holds if and only if I is the incidence graph of a Steiner system S(2, e/m; n) on n points and m blocks.

(b) If I is a 1-unconditional basic sequence in S^{6}, then

n ⩾ 1 + ( \frac{e}{m} − 1) + ( \frac{e}{m} − 1) ( \frac{e}{n} − 1) + ( \frac{e}{m} − 1)^{2}( \frac{e}{n} − 1) ,

that is e^{3} − (m + n)e^{2} + 2mne − m^{2}n^{2} ⩽ 0. Equality holds if and only if I is the incidence graph of the quadrangle (the cycle of length 8) or of a generalised quadrangle with n points and m lines.

(c) If I is a 1-unconditional basic sequence in S^{2k} with k ⩾ 1 an integer, then

n ⩾ ∑_{i=0}^{k} ( \frac{e}{m} − 1)^{⌈ \frac{i}{2} ⌉}( \frac{e}{n} − 1)^{⌊ \frac{i}{2} ⌋}.  (B.18)

Equality holds if I is the incidence graph of the (k + 1)-gon (the cycle of length 2k + 2) or of a generalised (k + 1)-gon with n points and m lines.

Proof. By Theorem 9.1(b) ⇒ (g), I is a 1-unconditional basic sequence in S^{2k}, with k ⩾ 1 an integer, if and only if I is a graph of girth 2k + 2 in the sense of [44]. Therefore (a) and (b) are shown in [65, Prop. 4, Th. 8, Rem. 10]. Inequality (B.18) is [44, Eq. (1)] and the sufficient condition for equality follows from [56, Lemma 1.5.4].

Consult [9, Tables A1.1, A5.1] for examples of Steiner systems and [56, Table 2.1] for examples of generalised polygons. In both cases, the corresponding incidence graph is biregular: every vertex in R has same degree s + 1 and every vertex in C has same degree t + 1. Arbitrarily large generalised (k + 1)-gons exist only if 2k ∈ {4, 6, 10, 14} [56, Lemma 1.7.1]; for 2k ∈ {6, 10, 14}, it follows from [56, Lemma 1.5.4] that

n = (s + 1)\frac{(st)^{(k+1)/2} − 1}{st − 1} , m = (t + 1)\frac{(st)^{(k+1)/2} − 1}{st − 1} .

Remark 11.7. Let I ⊆ R × C with # C = # R = n. Inequality (B.18) shows that if I is a 1-uncon-ditional basic sequence in S^{2k}, then # I ⩽ n^{1+1/k} + (s − 1)n/s. If p ∉ {4, 6, 10}, the existence of 1-unconditional basic sequences in S^{2k} such that # I ≽ n^{1+1/k} is in fact an important open problem in graph theory: extremal graphs cannot correspond to generalised polygons and necessarily have less structure.

Acknowledgement. I would like to thank Éric Ricard for his comments and suggestions.

Keywords. Schatten-von-Neumann class, Schur product, graph with a given girth, 1-unconditional basic sequence, metric unconditional approximation property, V-Sidon set, lacunary set.

2000 Mathematics Subject Classification. 47B10, 46B15, 46B04, 43A46, 05C38, 46B28.

Matrix inequalities with applications to the theory of iterated kernels

with William Banks, Asma Harcharras and Éric Ricard.

Abstract

For an m × n matrix A with nonnegative real entries, Atkinson, Moran and Watterson proved the inequality s(A)^{3} ⩽ mns(AA^{t}A), where A^{t} is the transpose of A, and s(·) is the sum of the entries. We extend this result to finite products of the form AA^{t}AA^{t} …A or AA^{t}AA^{t} …A^{t} and give some applications to the theory of iterated kernels.

1 Introduction

For any matrix A, let s(A) denote the sum of its entries. For any integer k ≥ 1, we define

A^{(2k)} = (AA^{t})^{k}, A^{(2k+1)} = (AA^{t})^{k}A,

where A^{t} denotes the transpose of A. In Section 2, we prove the following sharp inequalities:

Theorem 1.1. Let A be an m×n matrix with nonnegative real entries. Then for every integer k ⩾ 1, the following matrix inequalities hold:

s(A)^{2k} ⩽ m^{k−1} n^{k}s(A^{(2k)} ), s(A)^{2k+1} ⩽ m^{k}n^{k}s(A^{(2k+1)} ).

For the special case of symmetric matrices, this theorem was proved in 1959 by Mulholland and Smith [60], thus settling an earlier conjecture of Mandel and Hughes [57] that had been based on the study of certain genetical models. For arbitrary matrices (with nonnegative entries), Theorem 1.1 also generalises the matrix inequality

s(A)^{3} ⩽ mns(AA^{t}A),

which was first proved in 1960 by Atkinson, Moran and Watterson [3] using methods of perturbation theory.

Theorem 1.1 has a graph theoretic interpretation when applied to matrices with entries in {0, 1}. Let G be a graph with red vertices labeled 1, …, m and blue vertices labeled 1, …, n such that every edge connects only vertices of distinct colours: G is a bipartite graph. Its reduced incidence matrix is an m × n matrix A such that a_{i,j} = 1 if red vertex i is adjacent to blue vertex j, and a_{i,j} = 0 otherwise. Then s(A) is the size of G, while s(A^{(ℓ)}) is the number of walks on G of length ℓ starting from a red vertex, i.e., the number of sequences (v_{0}, …, v_{ℓ}) such that v_{0} is a red vertex and every pair {v_{i}, v_{i+1} } is an edge in G. Theorem 1.1 then yields the optimal lower bound of the number of walks in terms of the size of G. We do not know of a corresponding lower bound for the number of trails (walks with no edge repeated) or paths (walks with no vertex repeated).

Recall that an m × n matrix A is said to be bistochastic if every row sum of A is equal to s(A)/m, and every column sum of A is equal to s(A)/n. In Section 3 we prove the following asymptotic form of Theorem 1.1:

Theorem 1.2. Let A be an m × n matrix with nonnegative real entries. If A is bistochastic, then for all k ⩾ 1,

s(A)^{2k} = m^{k−1} n^{k}s(A^{(2k)} ), s(A)^{2k+1} = m^{k}n^{k}s(A^{(2k+1)} ).

If A is not bistochastic, then there exist constants c > 0 and γ > 1 (depending only on A) such that for all ℓ ⩾ 1,

s(A)^{ℓ} < cγ^{−ℓ} (mn)^{ℓ/2}s(A^{(ℓ)}).

As we show in Sections 2 and 3, both of the above theorems, though stated for arbitrary rectangular matrices with nonnegative entries, follow from the special case of square matrices.

Theorem 1.2 has an immediate application. Atkinson, Moran and Watterson [3] conjectured that for a nonnegative symmetric kernel function K(x, y) that is integrable (in a suitable sense) over the square 0 ⩽ x, y ⩽ a, the inequality

∫_{0}^{a} ∫_{0}^{a} K_{ℓ}(x, y) dx dy ⩾ \frac{1}{a^{ℓ−1}} ( ∫_{0}^{a} ∫_{0}^{a} K(x, y) dx dy) ^{ℓ}  (C.1)

holds for all ℓ ⩾ 1. Here K_{ℓ}(x, y) denotes the ℓ-th order iterate of K(x, y), which is defined recursively by

K_{1}(x, y) = K(x, y), K_{ℓ}(x, y) = ∫_{0}^{a} K_{ℓ−1} (x, t)K(t, y) dt.

Beesack [7] showed that the Atkinson-Moran-Watterson conjecture follows from the matrix identities of Mulholland and Smith described above. Using Beesack’s ideas together with Theorem 1.2, we prove in Section 4 the following asymptotic form of the Atkinson-Moran-Watterson inequality (C.1):

Theorem 1.3. Let K(x, y) be a nonnegative symmetric kernel function that is integrable over the

square 0 ⩽ x, y ⩽ a, and consider the function f(x) = ∫_{0}^{a} K(x, y) dy defined on the interval 0 ⩽ x ⩽ a.

If f(x) is constant almost everywhere, then for all ℓ ⩾ 1

∫_{0}^{a} ∫_{0}^{a} K_{ℓ}(x, y) dx dy = \frac{1}{a^{ℓ−1}} ( ∫_{0}^{a} ∫_{0}^{a} K(x, y) dx dy) ^{ℓ}.

If not, there exist constants c > 0 and γ > 1 (depending only on K) such that for all ℓ ⩾ 1

∫_{0}^{a} ∫_{0}^{a} K_{ℓ}(x, y) dx dy > \frac{cγ^{ℓ}}{a^{ℓ−1}} ( ∫_{0}^{a} ∫_{0}^{a} K(x, y) dx dy) ^{ℓ}.

Remark 1.4. Using an approximation argument as in the proof of Theorem 1.3, Theorem 1.1 can be also applied to establish an analogue to inequalities (C.1) and Theorem 1.3 in the case of nonsymmetric kernel functions. Let K(x, y) be any nonnegative kernel function that is integrable on the rectangle 0 ⩽ x ⩽ a, 0 ⩽ y ⩽ b and let K_{ℓ} be the ℓ-th order iterate of K defined by K_{1}(x, y) = K(x, y) and for each integer k ≥ 1,

K_{2k}(x, x′) = ∫_{0}^{b} K_{2k−1} (x, y)K(x′, y) dy,

K_{2k+1} (x, y) = ∫_{0}^{a} K_{2k}(x, x′)K(x′, y) dx′.

In this case, inequalities (C.1) become

∫_{0}^{a} ∫_{0}^{b} K_{2k+1} (x, y) dx dy ≥ \frac{1}{a^{k}b^{k}}( ∫_{0}^{a} ∫_{0}^{b} K(x, y) dx dy) ^{2k+1}

∫_{0}^{a} ∫_{0}^{a} K_{2k}(x, x′) dx dx′ ≥ \frac{1}{a^{k−1} b^{k}}( ∫_{0}^{a} ∫_{0}^{b} K(x, y) dx dy) ^{2k}.

The analogue of Theorem 1.3 is then obvious.

2 Matrix inequality

Given a matrix A = (a_{i,j} ) and an integer ℓ ⩾ 0, we denote by a_{i,j}^{(ℓ)} the (i, j)-th entry of A^{(ℓ)}, so that A^{(ℓ)} = (a_{i,j}^{(ℓ)}). This notation will be used often in the sequel.

Lemma 2.1. Let B = (b_{i,j} ) be a d × d matrix with nonnegative real entries. For any two sequences {α_{i}} and {β_{i}} of nonnegative real numbers, the following inequality holds:

(I′_{2}) :  ∑_{i,j=1}^{d} α_{i}β_{i}b_{i,j} ⩽ d^{\frac{1}{2}} ( ∑_{i,j=1}^{d} α_{i}^{2}β_{j}^{2}b_{i,j}^{(2)}) ^{\frac{1}{2}} .

Proof. To prove the lemma, we apply the Cauchy-Schwarz inequality twice as follows:

∑_{i,j=1}^{d} α_{i}β_{i}b_{i,j} = ∑_{i,k=1}^{d} α_{i}β_{i}b_{i,k} ⩽ d^{\frac{1}{2}} ( ∑_{k=1}^{d} ( ∑_{i=1}^{d} α_{i}β_{i}b_{i,k} )^{2}) ^{\frac{1}{2}} .  (C.2)

∑_{i,j=1}^{d} α_{i}β_{i}b_{i,j} ≤ d^{\frac{1}{2}} ( ∑_{i,j,k=1}^{d} α_{i}α_{j} β_{i}β_{j} b_{i,k} b_{j,k} ) ^{\frac{1}{2}}

= d^{\frac{1}{2}} ( ∑_{i,j=1}^{d} α_{i}α_{j} β_{i}β_{j} b_{i,j}^{(2)}) ^{\frac{1}{2}}

= d^{\frac{1}{2}} ( ∑_{i,j=1}^{d} α_{i}β_{j} (b_{i,j}^{(2)})^{\frac{1}{2}} · α_{j} β_{i}(b_{j,i}^{(2)})^{\frac{1}{2}} ) ^{\frac{1}{2}}

⩽ d^{\frac{1}{2}} ( ∑_{i,j=1}^{d} α_{i}^{2}β_{j}^{2}b_{i,j}^{(2)}) ^{\frac{1}{2}} .

Here we have used the fact that B^{(2)} = BB^{t} is a symmetric matrix.

Theorem 2.2. Let B = (b_{i,j} ) be a square d × d matrix with nonnegative real entries, and let {α_{i}} be any sequence of nonnegative real numbers. Then for each integer ℓ ⩾ 1, we have

(I_{ℓ}) :  ∑_{i,j=1}^{d} α_{i}b_{i,j} ⩽ d^{\frac{ℓ−1}{ℓ}} ( ∑_{i,j=1}^{d} α_{i}^{ℓ}b_{i,j}^{(ℓ)}) ^{\frac{1}{ℓ}} .

Proof of Theorem 2.2. The case ℓ = 1 is trivial while the case ℓ = 2 is a consequence of the lemma above. We prove the general case by induction. Suppose that p ⩾ 2, and the inequalities (I_{1}), (I_{2}), …, (I_{p}) hold for all square matrices with nonnegative real entries. If p = 2k − 1 is an odd integer, then the inequality (I_{p+1} ) follows immediately from (I_{2}) and (I_{k}). Indeed, since B^{(2k)} = B^{(2)(k)} , we have

∑_{i,j=1}^{d} α_{i}b_{i,j} ⩽ d^{\frac{1}{2}} ( ∑_{i,j=1}^{d} α_{i}^{2}b_{i,j}^{(2)}) ^{\frac{1}{2}} ⩽ d^{\frac{1}{2}} ( d^{\frac{k−1}{k}} ( ∑_{i,j=1}^{d} α_{i}^{2k}b_{i,j}^{(2)(k)} ) ^{\frac{1}{k}} ) ^{\frac{1}{2}} .  (C.3)

Thus

∑_{i,j=1}^{d} α_{i}b_{i,j} ⩽ d^{\frac{2k−1}{2k}} ( ∑_{i,j=1}^{d} α_{i}^{2k}b_{i,j}^{(2k)} ) ^{\frac{1}{2k}} .

If p = 2k is an even integer, then the inequality (I_{p+1} ) follows from Hölder’s inequality, and the inequalities (I_{k}) and (I′_{2}). Indeed, by Hölder’s inequality, we have

∑_{i,j=1}^{d} α_{i}b_{i,j} ⩽ d^{\frac{1}{2k+1}} ( ∑_{i=1}^{d} αi^{\frac{2k+1}{2k}} ( ∑_{j=1}^{d} b_{i,j} ) ^{\frac{2k+1}{2k}} ) ^{\frac{2k}{2k+1}} .  (C.4)

Let I denote the term between parentheses, and set β_{i} = ∑_{j=1}^{d} b_{i,j} for each i. Then

I = ∑_{i=1}^{d} αi^{\frac{2k+1}{2k}} ( ∑_{j=1}^{d} b_{i,j} ) ^{\frac{2k+1}{2k}} = ∑_{i,j=1}^{d} αi^{\frac{2k+1}{2k}} βi^{\frac{1}{2k}} b_{i,j} .

Applying (I_{k}), it follows that

I ⩽ d^{\frac{k−1}{k}} ( ∑_{i,j=1}^{d} αi^{\frac{2k+1}{2}} βi^{\frac{1}{2}} b_{i,j}^{(k)}) ^{\frac{1}{k}} .

Applying the lemma to the sequences \{αi^{\frac{2k+1}{2}} \} and \{βi^{\frac{1}{2}} \}, and using the fact that B^{(k)(2)} = B^{(2k)} , we see that

I ⩽ d^{\frac{k−1}{k}} ( d^{\frac{1}{2}} ( ∑_{i,j=1}^{d} α_{i}^{2k+1} β_{j} b_{i,j}^{(k)(2)} ) ^{\frac{1}{2}} ) ^{\frac{1}{k}} = d^{\frac{2k−1}{2k}} ( ∑_{i,j=1}^{d} α_{i}^{2k+1} β_{j} b_{i,j}^{(2k)} ) ^{\frac{1}{2k}} .

Putting everything together, we have therefore shown that

∑_{i,j=1}^{d} α_{i}b_{i,j} ⩽ d^{\frac{2k}{2k+1}} ( ∑_{i,j=1}^{d} α_{i}^{2k+1} β_{j} b_{i,j}^{(2k)} ) ^{\frac{1}{2k+1}} .

Finally, note that

∑_{j=1}^{d} β_{j} b_{i,j}^{(2k)} = ∑_{ℓ=1}^{d} b_{i,ℓ}^{(2k)} β_{ℓ} = ∑_{j,ℓ=1}^{d} b_{i,ℓ}^{(2k)} b_{ℓ,j} = ∑_{j=1}^{d} b_{i,j}^{(2k+1)}

since B^{(2k+1)} = B^{(2k)} B. Consequently,

∑_{i,j=1}^{d} α_{i}b_{i,j} ⩽ d^{\frac{2k}{2k+1}} ( ∑_{i,j=1}^{d} α_{i}^{2k+1} b_{i,j}^{(2k+1)} ) ^{\frac{1}{2k+1}}  (C.5)

and (I_{p+1} ) holds for the case p = 2k. Theorem 2.2 now follows by induction.

Proof of Theorem 1.1. For the case of square matrices, Theorem 1.1 follows immediately from Theorem 2.2. Indeed, taking α_{i} = 1 for each i, the inequality (I_{ℓ}) yields the corresponding inequality in Theorem 1.1.

Now, let A be an m × n matrix with nonnegative real entries, put d = mn, and let B be the d × d matrix with nonnegative real entries defined as the tensor product B = A ⊗ J_{n,m} , where J_{n,m} is the n × m matrix with every entry equal to 1. For any integers ℓ, k ⩾ 0, the relations

B^{(ℓ)} = A^{(ℓ)} ⊗ J_{n,m}^{(ℓ)} , s(B^{(ℓ)}) = s(A^{(ℓ)})s(J_{n,m}^{(ℓ)} ),
s(J_{n,m}^{(2k)} ) = m^{k}n^{k+1} , s(J_{n,m}^{(2k+1)} ) = m^{k+1} n^{k+1} .

are easily checked. In particular, s(B) = mns(A). Applying Theorem 1.1 to the matrix B and using these identities, the inequalities of Theorem 1.1 follow for the matrix A.

3 Asymptotic matrix inequality

As will be shown below, Theorem 1.2 is a consequence of the following more precise theorem for square matrices:

Theorem 3.1. Let B be a square d × d matrix with nonnegative real entries and s(B) ≠ 0. Let λ be the largest eigenvalue of B^{(2)} = BB^{t}, and put γ = λd^{2}/s(B)^{2}. Then γ ⩾ 1, and there exists a constant c > 0 (depending only on B) such that for all integers ℓ ⩾ 0,

s(B)^{ℓ} < cγ^{− \frac{ℓ}{2}} d^{ℓ−1} s(B^{(ℓ)}).  (C.6)

Moreover, the following assertions are equivalent:

(a) γ = 1,

(b) s(B)^{ℓ} = d^{ℓ−1} s(B^{(ℓ)}) for every integer ℓ ⩾ 0,

(c) s(B)^{ℓ} = d^{ℓ−1} s(B^{(ℓ)}) for some integer ℓ ⩾ 3,

(d) B is bistochastic.

Proof. We express B^{(2)} = BB^{t} in the form B^{(2)} = U^{t}DU, where U = (u_{i,j} ) is an orthogonal matrix, and D is a diagonal matrix diag(λ_{1}, …, λ_{d}) with λ_{1} ⩾ … ⩾ λ_{d} ⩾ 0. Here λ = λ_{1}. For each ν = 1, …, d, let E_{ν} be the projection matrix whose (ν, ν)-th entry is 1, and all other entries are equal to 0. Put A_{ν} = U^{t}E_{ν}U for each ν. Then for all integers k ⩾ 0,

B^{(2k)} = ∑_{ν=1}^{d} λ_{ν}^{k}A_{ν}, B^{(2k+1)} = ∑_{ν=1}^{d} λ_{ν}^{k}A_{ν}B.

By a straightforward calculation, we see that for each ν

s(A_{ν}) = ( ∑_{i=1}^{d} u_{ν,i} )^{2},  s(A_{ν}B) = ( ∑_{i=1}^{d} u_{ν,i} ) ( ∑_{j,k=1}^{d} u_{ν,k} b_{k,j} ) .  (C.7)

In particular, s(A_{ν}) ⩾ 0. By Theorem 2.2, it follows that

\frac{s(B)^{2}}{d} ⩽ s(B^{(2)}) = ∑_{ν=1}^{d} λ_{ν}s(A_{ν}) ⩽ λ ∑_{ν=1}^{d} s(A_{ν}) = λd.  (C.8)

Therefore, γ = \frac{λd^{2}}{s(B)^{2}} ⩾ 1. Now, from the definition of γ, we have

\frac{γ^{\frac{ℓ}{2}} s(B)^{ℓ}}{d^{ℓ−1} s(B^{(ℓ)})} = d\frac{λ^{\frac{ℓ}{2}}}{s(B^{(ℓ)})} ·

Then, in order to show inequality (C.6), we will show that the λ^{\frac{ℓ}{2}} /s(B^{(ℓ)}) are bounded above by a constant that is independent of ℓ. Indeed, let C_{ℓ} = B^{(ℓ)}/s(B^{(ℓ)}) for every ℓ ⩾ 0. Since each C_{ℓ} has nonnegative real entries, and s(C_{ℓ}) = 1, the entries of C_{ℓ} all lie in the closed interval [0, 1]. Thus the entries of the matrices UC_{2k}U^{t} and UC_{2k+1} B^{t}U^{t} are bounded by a constant that depends only on B. Noting that for each nonnegative integer k, we have

UC_{2k}U^{t} = \frac{D^{k}}{s(B^{(2k)} )} , UC_{2k+1} B^{t}U^{t} = \frac{D^{k+1}}{s(B^{(2k+1)} )} ,

and on examining the (1, 1)th entry for each of these matrices, we see that λ^{k} /s(B^{(2k)} ) and λ^{k+1} / s(B^{(2k+1)} ) are both bounded above by a constant that is independent of k. Consequently, inequality (C.6) holds.

(a)⇒ (b): If γ = 1, then λd = s(B)^{2}/d, hence from (C.8) we see that s(A_{ν}) = 0 whenever λ_{ν} ≠ λ. By (C.7), we also have that s(A_{ν}B) = 0 whenever λ_{ν} ≠ λ. Thus

s(B^{(2k)} ) = ∑_{ν=1}^{d} λ_{ν}^{k}s(A_{ν}) = λ^{k} ∑_{ν:λ_{ν} =λ} s(A_{ν})

= λ^{k} ∑_{ν=1}^{d} s(A_{ν}) = λ^{k}d = \frac{s(B)^{2k}}{d^{2k−1}} ,

s(B^{(2k+1)} ) = ∑_{ν=1}^{d} λ_{ν}^{k}s(A_{ν}B) = λ^{k} ∑_{ν:λ_{ν} =λ} s(A_{ν}B)

= λ^{k} ∑_{ν=1}^{d} s(A_{ν}B) = λ^{k}s(B) = \frac{s(B)^{2k+1}}{d^{2k}} ·

(b)⇒ (a): If (b) holds, then inequality (C.6) implies 1 < cγ^{− \frac{ℓ}{2}} for some γ ≥ 1 and all integers ℓ ≥ 0. This forces γ = 1.

(b)⇒ (c): Trivial.

(c)⇒ (d): Suppose that ℓ = 2k + 1 ⩾ 3 is an odd integer such that s(B)^{ℓ} = d^{ℓ−1} s(B^{(ℓ)}). Taking every α_{i} = 1 in the proof of Theorem 2.2, our hypothesis means that equality holds in (C.5), hence (C.4) must also hold with equality:

∑_{i,j=1}^{d} b_{i,j} = d^{\frac{1}{2k+1}} ( ∑_{i=1}^{d} ( ∑_{j=1}^{d} b_{i,j} ) ^{\frac{2k+1}{2k}} ) ^{\frac{2k}{2k+1}} .

By Hölder’s inequality, this is only possible if all of the row sums of B are equal. Since ℓ is odd and s is transpose-invariant, we also have

s(B^{t})^{ℓ} = d^{ℓ−1} s((B^{(ℓ)})^{t}) = d^{ℓ−1} s((B^{t})^{(ℓ)}).

Thus all of the row sums of B^{t} are equal as well, and B is bistochastic.

Now suppose that ℓ = 2k ⩾ 4 is an even integer such that s(B)^{ℓ} = d^{ℓ−1} s(B^{(ℓ)}). By taking every α_{i} = 1 in (C.3), we see that s(B)^{2} = ds(B^{(2)}). Then, taking every α_{i} = β_{i} = 1 in the proof of the lemma, we see that equality holds in (C.2) which is only possible if all of the column sums of B are equal. Therefore s(BA) = βs(A) for every d × d matrix A, where β = s(B)/d is the sum of each column of B. In particular,

\begin{aligned}s(B)^{ℓ} & = d^{ℓ−1} s(B^{(ℓ)}) = d^{ℓ−1} βs((B^{t})^{(ℓ−1)} ) \\ & = d^{ℓ−1} βs((B^{(ℓ−1)} )^{t}) = d^{ℓ−1} βs(B^{(ℓ−1)} ), \end{aligned}

thus s(B)^{ℓ−1} = d^{ℓ−2} s(B^{(ℓ−1)} ). Since ℓ − 1 is odd, we can apply the previous result to conclude that B is bistochastic.

(d)⇒ (b): Suppose B is bistochastic, with every row or column sum equal to β = s(B)/d. For any d × d matrix A, one has s(AB) = βs(A) and s(AB^{t}) = βs(A). In particular, s(B^{(2k+1)} ) = βs(B^{(2k)} ) and s(B^{(2k+2)} ) = βs(B^{(2k+1)} ) for all k ⩾ 0. Consequently,

s(B^{(ℓ)}) = β^{ℓ−1} s(B) = \frac{s(B)^{ℓ}}{d^{ℓ−1}} , ℓ ⩾ 0.

This completes the proof.

Corollary 3.2. Let B be a square d × d matrix with nonnegative real entries and s(B) ≠ 0. Let β_{j} be the j-th column sum of B for each j, and put

δ = 1 + \frac{1}{2s(B)^{2}} ∑_{i,j=1}^{d} (β_{i} − β_{j})^{2}.

Then there exists a constant c > 0 (depending only on B) such that for all ℓ ⩾ 0, we have

s(B)^{ℓ} < cδ^{− \frac{ℓ}{2}} d^{ℓ−1} s(B^{(ℓ)}).

Proof. Note first that for any d × d matrix B, if β_{j} denotes the j-th column sum of B, then it is easily seen that

s(B^{(2)}) = \frac{s(B)^{2}}{d} + \frac{1}{2d} ∑_{i,j=1}^{d} (β_{i} − β_{j})^{2}.  (C.9)

Using the notation of Theorem 3.1 and applying the relations (C.8) and (C.9), we have

γ = \frac{λd^{2}}{s(B)^{2}} ⩾ \frac{ds(B^{(2)})}{s(B)^{2}} = 1 + \frac{1}{2s(B)^{2}} ∑_{i,j=1}^{d} (β_{i} − β_{j})^{2} = δ.

The corollary therefore follows from (C.6).

Proof of Theorem 1.2. Given an m × n matrix A with nonnegative real entries, we proceed as in the proof of Theorem 1.1: put d = mn, and let B = A⊗J_{n,m} . Note that A is bistochastic if and only if B is bistochastic. Applying the corollary above to B, Theorem 1.2 follows immediately for the matrix A. The details are left to the reader.

4 Asymptotic kernel inequality

Proof of Theorem 1.3. By changing variables if necessary, we can assume that a = 1. For simplicity, we will also assume that K(x, y) is continuous. Consider the function f(x) defined by

f(x) = ∫_{0}^{1} K(x, y) dy, x ∈ [0, 1].

If f(x) is a constant function, then since K(x, y) is symmetric, the equality

∫_{0}^{1} ∫_{0}^{1} K_{ℓ}(x, y) dx dy = ( ∫_{0}^{1} ∫_{0}^{1} K(x, y) dx dy) ^{ℓ}

for all ℓ ⩾ 1 follows from an easy inductive argument.

Now suppose that f(x) is not constant, and let m and M denote respectively the minimum and maximum value of f(x) on [0, 1]. Choose ε > 0 such that 4ε < M − m. For every integer d ⩾ 1, let 𝒰_{i}^{[d]} be the open interval

𝒰_{i}^{[d]} = ( \frac{i − 1}{d} , \frac{i}{d}) , 1 ⩽ i ⩽ d,

and let 𝒰_{i,j}^{[d]} be the rectangle 𝒰_{i}^{[d]} × 𝒰_{j}^{[d]} for 1 ⩽ i, j ⩽ d. Let K^{[d]}(x, y) be defined on [0, 1] × [0, 1] as follows:

min\{K(s, t) : (s, t) ∈ \overline{𝒰_{i,j}^{[d]}} \} if (x, y) ∈ 𝒰_{i,j}^{[d]} for some 1 ⩽ i, j ⩽ d
K(x, y)  otherwise.

Here \overline{𝒰_{i,j}^{[d]}} denotes the closure of 𝒰_{i,j}^{[d]}. Noting that K^{[d]}(x, y) is constant on each rectangle 𝒰_{i,j}^{[d]}, let B_{[d]} be the d × d matrix whose (i, j)-th entry is equal to K^{[d]}(𝒰_{i,j}^{[d]}). Let K_{ℓ}^{[d]}(x, y) denote the ℓ-th order iterate of K^{[d]}(x, y) for each ℓ ⩾ 1. Then

K_{ℓ}^{[d]}(x, y) = ∫_{0}^{1} K_{ℓ−1}^{[d]} (x, t)K^{[d]}(t, y) dt = ∑_{k=1}^{d} ∫_{𝒰_{k}^{[d]}} K_{ℓ−1}^{[d]} (x, t)K^{[d]}(t, y) dt.

It follows by induction that K_{ℓ}^{[d]}(x, y) is also constant on each rectangle 𝒰_{i,j}^{[d]}, and

K_{ℓ}^{[d]}(𝒰_{i,j}^{[d]}) = \frac{1}{d} ∑_{k=1}^{d} K_{ℓ−1}^{[d]} (𝒰_{i,k}^{[d]})K^{[d]}(𝒰_{k,j}^{[d]});

by induction, this is the (i, j)-th entry of the matrix \frac{1}{d^{ℓ−1}} B_{[d]}^{(ℓ)}. In other words,

( K_{ℓ}^{[d]}(𝒰_{i,j}^{[d]})) = \frac{1}{d^{ℓ−1}} B_{[d]}^{(ℓ)},  for all ℓ, d ⩾ 1.  (C.10)

Now since f(x) is continuous, we can choose d sufficiently large such that for some integers 1 ⩽ i_{m}, i_{M} ⩽ d, we have

f(x) < m + ε, for all x ∈ 𝒰_{i_{m}}^{[d]},
f(x) > M − ε, for all x ∈ 𝒰_{i_{M}}^{[d]}.

Taking d larger if necessary, we can further assume that

0 ⩽ K(x, y) − K^{[d]}(x, y) < ε

for all 0 ⩽ x, y ⩽ 1. Fixing this value of d, we define

γ = 1 + \frac{ε^{2}}{2d^{2}( ∫_{0}^{1} ∫_{0}^{1} K(x, y) dx dy)^{2}} .

Finally, since γ^{− \frac{1}{4}} < 1, we can choose e sufficiently large so that K^{[de]}(x, y) > γ^{− \frac{1}{4}} K(x, y) for all 0 ⩽ x, y ⩽ 1. For this value of e, we therefore have

∫_{0}^{1} ∫_{0}^{1} K^{[de]}(x, y) dx dy > γ^{− \frac{1}{4}} ∫_{0}^{1} ∫_{0}^{1} K(x, y) dx dy.

By the corollary to Theorem 3.1 applied to the matrix B_{[de]}, there exists a constant c > 0, which is independent of ℓ, such that

s(B_{[de]})^{ℓ} < cδ^{− \frac{ℓ}{2}} (de)^{ℓ−1} s(B_{[de]}^{(ℓ)})

for all integers ℓ ⩾ 0, where

δ = 1 + \frac{1}{2s(B_{[de]})^{2}} ∑_{i,j=1}^{de} (β_{[de],i} − β_{[de],j} )^{2}.

Here β_{[de],j} denotes the j-th column sum of B_{[de]} for each j. We now claim that δ > γ. Granting this fact for the moment, we apply (C.10) to K^{[de]}(x, y) and obtain:

∫_{0}^{1} ∫_{0}^{1} K_{ℓ}(x, y) dx dy ⩾ ∫_{0}^{1} ∫_{0}^{1} K_{ℓ}^{[de]}(x, y) dx dy = \frac{1}{(de)^{2}} ∑_{i,j=1}^{de} K_{ℓ}^{[de]}( 𝒰_{i,j}^{[de]})

= \frac{1}{(de)^{ℓ+1}} s(B_{[de]}^{(ℓ)}) > c^{−1} δ^{\frac{ℓ}{2}} (de)^{−2ℓ} s(B_{[de]})^{ℓ}

= c^{−1} δ^{\frac{ℓ}{2}} ( \frac{1}{(de)^{2}} ∑_{i,j=1}^{de} K^{[de]}( 𝒰_{i,j}^{[de]}) ) ^{ℓ}= c^{−1} δ^{\frac{ℓ}{2}} ( ∫_{0}^{1} ∫_{0}^{1} K^{[de]}(x, y) dx dy) ^{ℓ}

> c^{−1} δ^{\frac{ℓ}{2}} γ^{− \frac{ℓ}{4}} ( ∫_{0}^{1} ∫_{0}^{1} K(x, y) dx dy) ^{ℓ}> c^{−1} γ^{\frac{ℓ}{4}} ( ∫_{0}^{1} ∫_{0}^{1} K(x, y) dx dy) ^{ℓ}.

This completes the proof of the theorem modulo our claim that δ > γ. To see this, let 𝒱 be any interval of the form 𝒰_{i}^{[de]} such that 𝒱 ⊂ 𝒰_{i_{m}}^{[d]}. Note that there are e such intervals. Since B^{[de]} is a symmetric matrix, the column sum β_{[de],𝒱} of B_{[de]} corresponding to the interval 𝒱 is equal to the “𝒱 -th” row sum, which can be bounded as follows:

β_{[de],𝒱} = ∑_{j=1}^{de} K^{[de]}(𝒱 , 𝒰_{j}^{[de]}) = (de)^{2} ∫_{𝒱} ∫_{0}^{1} K^{[de]}(x, y) dy dx

⩽ (de)^{2} ∫_{𝒱} ∫_{0}^{1} K(x, y) dy dx = (de)^{2} ∫_{𝒱} f(x) dx < de(m + ε).

Similarly, let 𝒲 be any interval of the form 𝒰_{i}^{[de]} such that 𝒲 ⊂ 𝒰_{i_{M}}^{[d]}. Again, there are e such intervals, and by a similar calculation, the column sum β_{[de],𝒲} satisfies the bound

β_{[de],𝒲} = ∑_{j=1}^{de} K^{[de]}(𝒲 , 𝒰_{j}^{[de]}) > de(M − 2ε).

Thus

∑_{i,j=1}^{de} (β_{[de],i} − β_{[de],j} )^{2} ⩾ ∑_{𝒱 ,𝒲} (β_{[de],𝒲} − β_{[de],𝒱} )^{2} > d^{2}e^{4}(M − m − 3ε)^{2} > d^{2}e^{4}ε^{2}.

On the other hand, we have

s(B_{[de]}) = (de)^{2} ∫_{0}^{1} ∫_{0}^{1} K^{[de]}(x, y) dx dy ⩽ (de)^{2} ∫_{0}^{1} ∫_{0}^{1} K(x, y) dx dy,

and the claim follows.

Keywords. Matrix inequality, iterated kernel.

2000 Mathematics Subject Classification. 15A45.

The size of bipartite graphs with girth eight

Abstract

Reiman’s inequality for the size of bipartite graphs of girth six is generalised to girth eight. It is optimal in as far as it admits the algebraic structure of generalised quadrangles as case of equality. This enables us to obtain the optimal estimate e ∼ v^{4/3} for balanced bipartite graphs. We also get an optimal estimate for very unbalanced graphs.

1 Introduction

De Caen and Székely recently proposed a new bound for the size of a bipartite graph of girth eight, that is a bipartite graph without cycle of length four and six. We adapt their method to obtain the following cubic inequality.

Theorem 1.1. Let G be a bipartite graph on v + w vertices.

(i) If G contains no cycle of length 4 and 6, then its size e satisfies

e^{3} − (v + w)e^{2} + 2vwe − v^{2}w^{2} ⩽ 0.

(ii) If v ⩾ ⌊w^{2}/4⌋, then furthermore e ⩽ v + ⌊w^{2}/4⌋.

Part (i) is the right generalisation of Reiman’s inequality for bipartite graphs of girth 6 (see Prop. 3.1) to girth 8. It is optimal in the sense that it is an equality for all known extremal graphs constructed via finite fields. Part (ii) describes the case of very unbalanced bipartite graphs and is optimal: there is a graph, constructed by hand, for which it is an equality.

Let us give a brief description of this article. Section 2 describes a way to translate uncoloured graphs into bipartite graphs and its converse. This permits to get two propositions on very unbalanced graphs.

Section 3 summarises facts about bipartite graphs of girth six that should be folklore and well known although I did not see them printed.

Section 4 is the core of the paper. We adapt an inequality of Atkinson et al. to get an optimal lower bound on the number of paths of length 3 in a bipartite graph (Cor. 4.6). This enables us to bypass the final step in the proof of [20, Th. 1] and to get our theorem.

2 Uncoloured graphs and bipartite graphs

Expanding a graph to a bipartite graph

We propose the following construction of a bipartite graph out of an uncoloured graph. Let G′ be an uncoloured graph with set of vertices V. Then the bipartite graph G is defined as follows:

– the first class of vertices of G is V;

– the second class W of vertices of G is the set of edges of G′;

– the set of edges of G is { {x, y} : y is an edge of G′ with endpoint x }.

Thus every vertex of W has degree 2 and the size of G is twice the size of G′.

Contracting a bipartite graph to an uncoloured graph

Let us describe an inverse construction. Let G be a bipartite graph with colour classes V and W. Let G′ be the following graph:

– its set of vertices is V;

– its set of edges is { {x, z} ⊆ V : ∃y {x, y} and {z, y} are edges of G }.

The size of G′ is at most half the size of G. If G contains no cycle of length 4, then, given {x, z}, there is at most one y such that {x, y} and {z, y} are edges of G, so that the size of G′ is exactly

∑_{y∈W} \binom{d(y)}{2} ⩽ \binom{#V}{2} .  (D.1)

(We recognise here [11, Inequality (2), p. 310] for s = t = 2.) Thus each vertex y ∈ W of degree at least 2 contributes at least 1 to sum (D.1). This yields

Proposition 2.1. Let G be a bipartite graph on v + w vertices that contains no cycle of length 4. (i) If w > \binom{v}{2}, then there are at least w − \binom{v}{2} vertices in W of degree 0 or 1.

(ii) If its minimal degree is at least 2, then w ⩽ \binom{v}{2} and v ⩽ \binom{w}{2}.

If G contains no cycle of length 4 nor 6, then G′ contains no triangle and its size is at most ⌊v^{2}/4⌋. This argument proves

Proposition 2.2. Let G be a bipartite graph on v + w vertices that contains no cycle of length 4 or 6.

(i) If w > ⌊v^{2}/4⌋, then there are at least ⌈w − v^{2}/4⌉ vertices in W with degree 0 or 1. (ii) If its minimal degree is at least 2, then w ⩽ ⌊v^{2}/4⌋ and v ⩽ ⌊w^{2}/4⌋.

3 Bipartite graphs of girth six

The following estimate is well known as Reiman’s inequality, but its cases of equality were not written down explicitly. Reading the proof of [11, Th. VI.2.6], one gets with [9, Def. I.3.1]

Proposition 3.1. Let v ⩽ w. A graph of girth at least 6 on v + w vertices with e edges satisfies

O(v, w, e) = e^{2} − we − vw(v − 1) ⩽ 0
e ⩽ \sqrt{vw(v − 1) + w^{2}/4} + w/2.

We have equality if and only if it is the incidence graph of a Steiner system S(2, k; v) on v points with block degree k given by wk(k − 1) = v(v − 1).

Note that by symmetry, we also get O(w, v, e) ⩽ 0, but this is superfluous by

Lemma 3.2. Let v ⩽ w. Let e be the positive root of X^{2} − vX − vw(w − 1). Then O(v, w, e) ⩾ 0.

Proof. As (vw)^{2} − vvw − vw(w − 1) = vw(vw − v − w + 1) ⩾ 0, we have e ⩽ vw. Therefore

e^{2} − we − vw(v − 1) = e^{2} − ve − vw(w − 1) + (v − w)e + vw(w − v)
= (vw − e)(w − v) ⩾ 0 .

Remark 3.3. The case of equality in Prop. 3.1 may be described further as follows. By [9, Cor. I.2.11], every vertex in V has same degree r and every vertex in W has same degree k with

k − 1 | v − 1 and k(k − 1) | v(v − 1),  (D.2)

so that v = 1 + r(k − 1) and k | r(r − 1). For given k, this set of conditions is in fact sufficient for the existence of an extremal graph for large r: this is Wilson’s Theorem [9, Th. XI.3.8]. For example, we have the following complete sets of parameters (v, w, r, k):

(1 + r(k − 1), r(1 + r(k − 1))/k, r, k) for 1 ⩽ k ⩽ 5 and k | r(r − 1).

The first set of parameters satisfying (D.2) for which an extremal graph does not exist is (36, 42, 7, 6). Consult [9, Table A1.1] for all known block designs with r ⩽ 17. [9, Table A5.1] provides the following sets of parameters (v, w, r, k) for block designs: given any prime power q and natural number n, given t ⩽ s,

( q^{n}, q^{n−1} \frac{q^{n} − 1}{q − 1} , \frac{q^{n} − 1}{q − 1} , q) , ( \frac{q^{n+1} − 1}{q − 1} , \frac{q^{n+1} − 1}{q^{2} − 1} \frac{q^{n} − 1}{q − 1} , \frac{q^{n} − 1}{q − 1} , q + 1) ,

(q^{3} + 1, q^{2}(q^{2} − q + 1), q^{2}, q + 1) , (2^{t+s} − 2^{s} + 2^{t}, (2^{s} + 1)(2^{s} − 2^{s−t} + 1), 2^{s} + 1, 2^{t}) .

The following proposition provides a simpler but coarser bound.

Proposition 3.4. Let G be a bipartite graph on vertex classes V and W with #V = v and #W = w without cycles of length 4. Its size satisfies

e ⩽ \{ \begin{matrix}\sqrt{2vw(v − 1)} & if w ⩽ v(v − 1)/2 \\ v(v − 1)/2 + w & otherwise. \end{matrix}

We have optimality in the second alternative for the bipartite expansion of a complete graph on V as described in Section 2, on which we add w − v(v − 1)/2 new edges by connecting any vertex of V to w − v(v − 1)/2 new vertices in colour class W.

Proof. By Proposition 2.1, if w > v(v − 1)/2, then w − v(v − 1)/2 vertices in W have degree 0 or 1. If we remove them, we remove at most w − v(v − 1)/2 edges and the remaining graph has at most v(v − 1) edges because O(v, v(v − 1)/2, v(v − 1)) = 0. The first alternative follows from

O(v, w, \sqrt{2vw(v − 1)}) = w\sqrt{v(v − 1)}(\sqrt{v(v − 1)} − \sqrt{2w}).

4 Bipartite graphs of girth eight

Statement of the theorem

Consult [56, Def. 1.3,1] for the definition of generalised polygons.

Theorem 4.1. Let G be a bipartite graph on vertex classes V and W with #V = v and #W = w. If G contains no cycle of length 4 or 6, then its size e satisfies

P (v, w, e) = e^{3} − (v + w)e^{2} + 2vwe − v^{2}w^{2} ⩽ 0.  (D.3)

We have equality exactly in two cases:

(i) if G is the complete bipartite graph and v = 1 or w = 1;

(ii) if G is the incidence graph of a generalised quadrangle.

Remark 4.2. Let us first note that this polynomial has exactly one positive root in e for positive v, w. It suffices to this purpose to show that its discriminant is negative. This is −v^{2}w^{2}D with

D = 27p^{2} + 4s^{3} − 36sp − 4s^{2} + 32p, s = v + w, p = vw.

Let us study this quantity for s ⩾ 2, p ⩾ s − 1. We have

\frac{dD}{dp} = 54p − 36s + 32 ⩾ 54p − 36(p + 1) + 32 = 18p − 4 > 0,

so that its minimum satisfies p = s − 1, which implies D = (4s − 5)(s − 1)^{2} ⩾ 3. Therefore Inequality (D.3) is equivalent to an inequality of form e ⩽ e(v, w).

Remark 4.3. The case of equality in Th. 4.1 implies the following: every vertex in V has same degree s + 1 and every vertex in W has same degree t + 1. By [56, Cor. 1.5.5, Th. 1.7.1], s + t | st(1 + st) and

v = (t + 1)(1 + st) , w = (s + 1)(1 + st) , e = (s + 1)(t + 1)(1 + st).

Let us suppose, by symmetry, that s ⩽ t. If s = 0, we get case (i). If s = 1, we obtain exactly the examples of extremal graphs produced by de Caen and Székely: W consists of t + 1 horizontal lines and as much vertical lines and V is the set of (t + 1)^{2} intersection points and G is the point-line incidence graph of this grid (this is also the bipartite expansion of a complete bipartite graph on (t + 1) + (t + 1) vertices.) Otherwise s, t ⩾ 2 and G is in fact the incidence graph of a generalised quadrangle, so that by [56, Th. 1.7.2], t ⩽ s^{2}. Let q be a prime power. Then there are generalised quadrangles with set of parameters (s, t) any of (q, q), (q, q^{2}), (q^{2}, q^{3}), (q − 1, q + 1); all known ones fit in this list. In particular, by [56, Th. 1.7.9], if t ⩾ s = 2, then t = 2 or t = 4 and in each case there is exactly one extremal graph. By [56, Sec. 1.7.11], if t ⩾ s = 3, then there is a (unique) extremal graph exactly if t = 3, 5, 9. There is a unique extremal graph with s = t = 4. It is open whether there exists a generalised quadrangle with s = 4 and t ∈ {11, 12}.

A generalisation of an inequality of Atkinson et al.

We first need an optimal lower bound on the number of paths of length 3. Let us prove the following inequality.

Theorem 4.4. Let (a_{ij} )_{1⩽i⩽v,1⩽j⩽w} be a matrix of nonnegative coefficients and ρ, γ ⩾ 0. Let

a_{i⋆} = ∑_{j=1}^{w} a_{ij} , a_{⋆j} = ∑_{i=1}^{v} a_{ij} , e = ∑_{i=1}^{n} ∑_{j=1}^{v} a_{ij} .  (D.4)

If a_{i⋆} ⩾ 2ρ and a_{⋆j} ⩾ 2γ, then

ϕ = ∑_{i=1}^{v} ∑_{j=1}^{w} a_{ij} (a_{i⋆} − ρ)(a_{⋆j} − γ) ⩾ e(e/v − ρ)(e/w − γ),  (D.5)

equality holding exactly if a_{i⋆} and a_{⋆j} are constant.

This refines the inequality in [3], which states

ψ = ∑_{i=1}^{v} ∑_{j=1}^{w} a_{ij} a_{i⋆}a_{⋆j} ⩾ e^{3}/vw  (D.6)

as, by the Arithmetic-Quadratic Mean Inequality,

ϕ − ψ = −γ ∑_{i=1}^{v} a_{i⋆}^{2} − ρ ∑_{j=1}^{w} a_{⋆j}^{2} + ργe ⩽ e(−γe/v − ρe/w + ργ)  (D.7)

Remark 4.5. If v = w and a is diagonal, Inequality (D.6) is the Arithmetic-Cubic Mean Inequality and Inequality (D.5) becomes

\frac{1}{v} ∑_{i=1}^{v} a_{ii} (a_{ii} − ρ)(a_{ii} − γ) ⩾ \frac{e}{v} \frac{e − vρ}{v} \frac{e − vγ}{v} ,

which is true by Chebyshev’s Inequality [39, Th. 43] if a_{ii} ⩾ ρ and a_{ii} ⩾ γ. For our “non commutative Chebyshev Inequality”, the conditions a_{i⋆} ⩾ 2ρ and a_{⋆j} ⩾ 2γ cannot be weakened to a_{i⋆} ⩾ ρ and a_{⋆j} ⩾ γ, as we have the following counterexamples:

( \begin{matrix}2 & 5 \\ 4 & 0 \end{matrix}) , ( \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}) .

Proof. If (D.5) is an equality, then so are (D.7) and (D.6) and our case of equality follows from the identical case of equality in [3], whose proof we now imitate. We shall suppose that a_{i⋆} > 2ρ or a_{⋆j} > 2γ, so that the whole inequality follows by continuity. Fix e and suppose that under this condition the a_{ij} are chosen so to minimise ϕ. We may suppose that the rows and the columns have been permuted such that the sequences (a_{i⋆}) and (a_{⋆j} ) are nondecreasing:

a_{1⋆} ⩽ ⋯⩽ a_{v⋆}, a_{⋆1} ⩽ ⋯⩽ a_{⋆w} .  (D.8)

If one of these sequences is constant, the inequality follows by the Arithmetic-Quadratic inequality (and the case of equality is easy). Let us suppose that this is not so.

One can suppose that a_{1w} and a_{v1} are positive. Let us show the argument for a_{1w}. If a_{1w} = 0, there are k, l such that a_{1k}, a_{lw} > 0. Make a perturbation by adding α to a_{1w} and to a_{lk} and subtracting α to a_{1k} and to a_{lw}. The row and column sums a_{i⋆} and a_{⋆j} are unaltered and ϕ increases of

Δϕ = α((a_{1⋆} − ρ)(a_{⋆w} − γ) + (a_{l⋆} − ρ)(a_{⋆k} − γ)
− (a_{1⋆} − ρ)(a_{⋆k} − γ) − (a_{l⋆} − ρ)(a_{⋆m} − γ))
= α(a_{1⋆}a_{⋆w} + a_{l⋆}a_{⋆k} − a_{1⋆}a_{⋆k} − a_{l⋆}a_{⋆w})
= α(a_{1⋆} − a_{l⋆})(a_{⋆w} − a_{⋆k}),

so that ϕ does not increase.

Now make the following perturbation: add 2α to a_{11} and subtract α to a_{1w} and to a_{v1}. Let us compute the differential of ϕ: as

\frac{dϕ}{da_{rc}} = (a_{r⋆} − ρ)(a_{⋆c} − γ) + ∑_{i=1}^{v} a_{ic}(a_{i⋆} − ρ) + ∑_{j=1}^{w} a_{rj} (a_{⋆j} − γ),

dϕ = dα( 2\frac{dϕ}{da_{11}} − \frac{dϕ}{da_{1w}} − \frac{dϕ}{da_{v1}} )
= dα( (a_{1⋆} − ρ)(a_{⋆1} − a_{⋆w}) + (a_{1⋆} − a_{v⋆})(a_{⋆1} − γ) + ∑_{i=1}^{v} a_{i1}(a_{i⋆} − ρ)

+ ∑_{j=1}^{w} a_{1j} (a_{⋆j} − γ) − ∑_{i=1}^{v} a_{iw} (a_{i⋆} − ρ) − ∑_{j=1}^{w} a_{vj} (a_{⋆j} − γ))

For positive dα, we have by (D.8)

dϕ ⩽ dα((a_{1⋆} − ρ)(a_{⋆1} − a_{⋆w}) + (a_{1⋆} − a_{v⋆})(a_{⋆1} − γ) + a_{⋆1}(a_{v⋆} − ρ)
+ a_{1⋆}(a_{⋆w} − γ) − a_{⋆w} (a_{1⋆} − ρ) − a_{v⋆}(a_{⋆1} − γ))
= dα((a_{1⋆} − 2ρ)(a_{⋆1} − a_{⋆w}) + (a_{⋆1} − 2γ)(a_{1⋆} − a_{v⋆}))
< 0 ,

which contradicts the minimum hypothesis.

Corollary 4.6. Let G be a bipartite graph on v + w vertices and of minimal degree 2. Then the number of paths of length 3 in G is at least e(e/v − 1)(e/w − 1). This bound is achieved exactly if the graph is regular for each of its two colours.

Proof. A path of length 3 is a sequence of 4 vertices (x, y, z, t) with no repetition such that

{x, y}, {y, z}, {z, t} ∈ G.

Given two adjacent vertices y and z, the number of paths (x, y, z, t) makes (d(y) − 1)(d(z) − 1), where d denotes the degree of a vertex. Therefore the number of all paths of length 3 is

∑_{{y,z}∈G} (d(y) − 1)(d(z) − 1).

Let (a_{ij} )_{1⩽i⩽v,1⩽j⩽w} be the reduced incidence matrix of G: a_{ij} = 1 if the ith vertex of the first class is adjacent to the jth vertex of the second class; otherwise a_{ij} = 0. Then this sum is

∑_{i=1}^{v} ∑_{j=1}^{w} a_{ij} (a_{i⋆} − 1)(a_{⋆j} − 1),  (D.9)

so that it suffices to take ρ = γ = 1 in Th. 4.4.

Proof of Theorem 4.1

The case of equality follows from [56, Lemma 1.4.1] because its axiom (i) is exactly what makes Bound (D.10) an equality.

I now follow the proof of [20, Th. 1]. If G contains no cycle of length 4, there is no path of length 3 between two adjacent vertices; if G contain no cycle of length 6, there is at most one path of length 3 between non-adjacent vertices of different colour. Therefore the sum (D.9) is bounded by

vw − e with e = ∑_{i=1}^{n} ∑_{j=1}^{v} a_{ij} .  (D.10)

By Corollary 4.6, if all the vertices of G have degree at least two, one has

vw − e ⩾ e^{3}/vw − (1/v + 1/w)e^{2} + e

and therefore (D.3). In order to get rid of this degree condition, we have to do an induction on the sum s = v + w of the number of columns and the number of rows of the incidence matrix. If v = 1, then P (v, w, e) = (e − w)(e^{2} − e + w), so that the inequality states e ⩽ w, which is trivial; symmetrically for w = 1. Suppose the result is true for all v × w incidence matrices with v + w = s. Consider now a v × w incidence matrix with v + w = s + 1 and v, w ⩾ 2. If each vertex has degree at least two, the result is true; otherwise there is a column or a row containing only zeroes or exactly one “1”. Apply the induction hypothesis on the matrix without this row or column: we get P (v − 1, w, e − 1) ⩽ 0 or P (v, w − 1, e − 1) ⩽ 0 and we may apply the following growth lemma to conclude.

Lemma 4.7. Let v, w ⩾ 1. If P (v, w, e) ⩽ 0, then P (v + 1, w, e + 1) ⩽ 0.

Proof. In fact, one has

P (v + 1, w, e + 1) − P (v, w, e) = 2e^{2} + (1 − 2v)e + (w − w^{2})(2v + 1) − v,

which is negative as long as

0 ⩽ e ⩽ e_{0} =(2v − 1 + \sqrt{(2v + 1) (2v + 8w^{2} − 8w + 1)} )/4 = (2v − 1 + Δ)/4.

Let us use that P (v, w, e) has a unique root in e and compute P (v, w, e_{0}). This makes

(4vw^{2} + 2w^{2} + 1)Δ/16 + (−16vw^{3} − 8v^{2}w^{2} − 8w^{3} + 8vw^{2} + 2w^{2} − 2v + 4w − 1)/16.

Then either the second term in this sum is positive and P (v, w, e_{0}) is positive, or the conjugate expression of this sum is positive, and the product of the sum with this conjugate expression is

(w − 1)^{2}w^{2}(8v^{3}w^{2} + 4v^{2}w^{2} − 2vw^{2} + 2v^{2} − w^{2} − 4vw + 2v − 2w)/8,

which is positive if v, w ⩾ 1.

Further remarks

Remark 4.8. Theorem 4.1 does not always give the right order of magnitude for the maximal size of a graph of girth 8: as

P (v, w, (vw)^{2/3} ) = 2(vw)^{5/3} − (vw)^{4/3} (v + w)) ⩽ 2(vw)^{5/3} − 2(vw)^{4/3+1/2} ⩽ 0,

we expect to find maximal graphs of size (vw)^{2/3} : De Caen and Székely [20, Th. 4] find a counterexample to this expectation if v “lies in an interval just slightly below” w^{2}. They conjecture [19] that this is the case as soon as v ≫ w^{5/4} and v ≪ w^{2}.

In the case of v = w, let us give the following approximation for the real root of the polynomial. For

e = v^{4/3} + \frac{2}{3}v − \frac{2}{9}v^{2/3} − \frac{20}{81}v^{1/3} ,

P (v, v, e) = \frac{40}{243}v^{7/3} + \frac{376}{2187}v^{2} − \frac{80}{2187}v^{5/3} − \frac{800}{19683}v^{4/3} − \frac{8000}{531441}v ⩾ \frac{129808}{531441},

P (v, v, e − 16/81) = −\frac{8}{531441}(v^{1/3} − 1)(39366v^{7/3} + 28431v^{2} + 8262v^{5/3}
− 8748v^{4/3} − 11880v − 6560v^{2/3} − 2432v^{1/3} − 512)
⩽ 0.

In particular,

Corollary 4.9. Let G be a bipartite graph of size e with v vertices in each vertex class. If the girth of G is at least 8, then

e < v^{4/3} + \frac{2}{3}v − \frac{2}{9}v^{2/3} − \frac{20}{81}v^{1/3} .

Let us now show that we generalise the following estimations for the size of bipartite graphs of girth 8 in [20, Th. 1]:

(i) if the minimal degree of G is at least 2, then e ⩽ 2^{1/3} (vw)^{2/3} ; (ii) if v ≼ w^{2} or w ≼ v^{2}, then e ≼ (vw)^{2/3} . In fact,

P (v, w, 2^{1/3} (vw)^{2/3} ) = (vw)^{4/3} (w^{2/3} − 2^{2/3} v^{1/3} )(v^{2/3} − 2^{2/3} w^{1/3} ),

which is nonnegative exactly if v ⩽ w^{2}/4 and w ⩽ v^{2}/4 or if (v, w) is among {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)}, and this is the case by Prop. 2.2 if the minimal degree is at least 2.

Furthermore, by Prop. 2.2, if w > ⌊v^{2}/4⌋, then ⌈w − v^{2}/4⌉ vertices in W have degree 0 or 1. If we remove them, we remove at most ⌈w − v^{2}/4⌉ edges and the remaining graph has at most ⌊v^{2}/2⌋ edges because P (v, ⌊v^{2}/4⌋, ⌊v^{2}/2⌋ + 1) > 0. This yields

Proposition 4.10. Let G be a bipartite graph on vertex classes V and W with #V = v and #W = w without cycles of length 4 and 6. Its size satisfies

e ⩽ \{ \begin{matrix}2^{1/3} (vw)^{2/3} & if max(v, w) ⩽ ⌊min(v, w)^{2}/4⌋ \\ ⌊min(v, w)^{2}/4⌋ + max(v, w) & otherwise. \end{matrix}

We have optimality in the second alternative: make a bipartition V = V_{1} ∪ V_{2} with V_{1} = ⌈v/2⌉ and V_{2} = ⌊v/2⌋, let G′ be the complete bipartite graph on the colour classes V_{1} and V_{2}, which has ⌊v^{2}/4⌋ edges. Now consider the bipartite expansion of G′, add ⌈w − v^{2}/4⌉ new vertices to colour class W, and connect each of them to some vertex of V.

Note that this estimate yields another proof of [35, Th. 1] by means of [35, Th. 3]. Remark 4.11. Our inequality condenses the following facts about the behaviour of e for fixed w and large v. If w ⩽ 3, then extremal graphs of girth 8 do not contain any cycle at all, so that their size is e = v + w − 1; if v ⩾ w = 4 and if v = w = 5, then extremal graphs of girth 8 contain exactly one cycle, so that their size is e = v + w; if v > w = 5, then extremal graphs of girth 8 contain exactly one “θ-graph”, so that their size is e = v + w + 1.

Ordering simultaneously the columns and lines of 0,1 matrices

with Nikolai Kosmatov.

Let M be a 0, 1 matrix with m rows and n columns:

M = (a_{ij} ) = ( \begin{matrix}a_{11} & a_{12} & … & a_{1n} \\ a_{21} & a_{22} & … & a_{2n} \\ ⋮ & ⋮ & & ⋮ \\ a_{m1} & a_{m2} & … & a_{mn} \end{matrix} ) with a_{ij} ∈ \{0, 1\}.

There are 2^{m+n} such matrices. We would like to study these matrices up to a permutation of the rows and a permutation of the columns, and therefore propose an order on the set of rows and on the set of columns.

Definition 1. Let L_{i} = L_{i}(M) = (a_{i1}, a_{i2}, …, a_{in}), 1 ⩽ i ⩽ m, be the lines of M and let C_{j} = C_{j} (M) = (a_{1j} , a_{2j} , …, a_{mj} ), 1 ⩽ j ⩽ n, be its columns. The rows (vs. columns) of M are ordered if they form an increasing sequence with respect to the lexicographic order.

Note that if the rows and columns are considered as digit sequences of binary numbers, the lexicographic order is just the natural order on ℕ.

Theorem 2. There is a permutation of rows and columns that orders them simultaneously.

Definition 3. The numbers r , s ⩾ 1, 0 = i_{0} < i_{1} < i_{2} < ⋯< i_{r} = m, 0 = j_{0} < j_{1} < j_{2} < ⋯< j_{s} = n define a grid on m × n that divides M in r × s block matrices

B_{t,u} = B_{t,u} (M) = ( \begin{matrix}a_{i_{t−1} +1,j_{u−1} +1} & … & a_{i_{t−1} +1,j_{u}} \\ ⋮ & & ⋮ \\ a_{i_{t},j_{u−1} +1} & … & a_{i_{t},j_{u}} \end{matrix} ) , 1 ⩽ t ⩽ r, 1 ⩽ u ⩽ s,

so that

M = ( \begin{matrix}B_{1,1} & … & B_{1,s} \\ ⋮ & & ⋮ \\ B_{r,1} & … & B_{r,s} \end{matrix} ) = ( \begin{matrix}a_{i_{0} +1,j_{0} +1} & … & a_{i_{0} +1,j_{1}} & … & a_{i_{0} +1,j_{s−1} +1} & … & a_{i_{0} +1,j_{s}} \\ ⋮ & & ⋮ & & ⋮ & & ⋮ \\ a_{i_{1} ,j_{0} +1} & … & a_{i_{1} ,j_{1}} & … & a_{i_{1} ,j_{s−1} +1} & … & a_{i_{1} ,j_{s}} \\ a_{i_{1} +1,j_{0} +1} & … & a_{i_{1} +1,j_{1}} & … & a_{i_{1} +1,j_{s−1} +1} & … & a_{i_{1} +1,j_{s}} \\ ⋮ & & ⋮ & & ⋮ & & ⋮ \\ a_{i_{2} ,j_{0} +1} & … & a_{i_{2} ,j_{1}} & … & a_{i_{2} ,j_{s−1} +1} & … & a_{i_{2} ,j_{s}} \\ ⋮ & & ⋮ & & ⋮ & & ⋮ \\ a_{i_{r−1} +1,j_{0} +1} & … & a_{i_{r−1} +1,j_{1}} & … & a_{i_{r−1} +1,j_{s−1} +1} & … & a_{i_{r−1} +1,j_{s}} \\ ⋮ & & ⋮ & & ⋮ & & ⋮ \\ a_{i_{r} ,j_{0} +1} & … & a_{i_{r} ,j_{1}} & … & a_{i_{r} ,j_{s−1} +1} & … & a_{i_{r} ,j_{s}} \end{matrix}  ) .

A permutation of lines L_{p_{1}} and L_{p_{2}} is admissible if i_{t−1} < p_{1} ⩽ i_{t} and i_{t−1} < p_{2} ⩽ i_{t} for some t ∈ {1, 2, …, r}; otherwise, it is inadmissible. A permutation of columns C_{q_{1}} and C_{q_{2}} is admissible if j_{u−1} < q_{1} ⩽ j_{u} and j_{u−1} < q_{2} ⩽ j_{u} for some u ∈ {1, 2, …, s}; otherwise, it is inadmissible. In other words, admissible permutations of rows and columns only permute matrix coefficients inside the block matrices defined by the grid. Matrix M is ordered with respect to the grid if

L_{i_{0} +1} ⩽ L_{i_{0} +2} ⩽ ⋯⩽ L_{i_{1}} , L_{i_{1} +1} ⩽ ⋯⩽ L_{i_{2}} , …, L_{i_{r−1} +1} ⩽ ⋯⩽ L_{i_{r}} ;
C_{j_{0} +1} ⩽ C_{j_{0} +2} ⩽ ⋯⩽ C_{j_{1}} , C_{j_{1} +1} ⩽ ⋯⩽ C_{j_{2}} , …, C_{j_{s−1} +1} ⩽ ⋯⩽ C_{j_{s}} ,

where ⩽ is the lexicographic order.

Theorem 2 is an immediate consequence of Lemma 4: it suffices to consider the trivial grid given by r = s = 1, 0 = i_{0} < i_{1} = m, 0 = j_{0} < j_{1} = n.

Lemma 4. The rows and columns of a 0, 1 matrix may be simultaneously ordered with respect to a given grid by admissible permutations of its rows and columns.

Proof by induction. If m = 1 or n = 1, this is trivial. Let us suppose m, n > 1 and let us construct the first line and then the first column of the reordered matrix. Let

A = ( \begin{matrix}∑ _{q=j_{0} +1}^{j_{1}} a_{1q} & ∑ _{q=j_{1} +1}^{j_{2}} a_{1q} & … & ∑ _{q=j_{s−1} +1}^{j_{s}} a_{1q} \\ ∑ _{q=j_{0} +1}^{j_{1}} a_{2q} & ∑ _{q=j_{1} +1}^{j_{2}} a_{2q} & … & ∑ _{q=j_{s−1} +1}^{j_{s}} a_{2q} \\ ⋮ & ⋮ & & ⋮ \\ ∑ _{q=j_{0} +1}^{j_{1}} a_{i_{1} q} & ∑ _{q=j_{1} +1}^{j_{2}} a_{i_{1} q} & … & ∑ _{q=j_{s−1} +1}^{j_{s}} a_{i_{1} q} \end{matrix} )

be the matrix of numbers of ones in the lines of the block matrices B_{1,u}, 1 ⩽ u ⩽ s. Let L_{v}(A) be the minimal line of A: we permute L_{1}(M) with L_{v}(M) and still call M the resulting matrix. Then

∑_{q=j_{0} +1}^{j_{1}} a_{1q} ⩽ ∑_{q=j_{0} +1}^{j_{1}} a_{iq},

for each i ∈ {1, 2, …, i_{1}}, so that either some a_{1j} , 1 ⩽ j ⩽ j_{1}, vanishes, either B_{1,1} is a matrix of ones. Let us permute the columns of M in such a way that that the first lines of all the resulting block matrices B_{1,u}, 1 ⩽ u ⩽ s, consist in a block of zeroes followed by a block of ones:

B_{1,u} = ( \begin{matrix}0 & 0 & … & 0 & 0 & 1 & 1 & … & 1 & 1 \\ ∗ & ∗ & … & ∗ & ∗ & ∗ & ∗ & … & ∗ & ∗ \\ ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ \\ ∗ & ∗ & … & ∗ & ∗ & ∗ & ∗ & … & ∗ & ∗ \end{matrix}) .

We still call M the resulting matrix. We now refine the grid on m × n in such a way that the first lines of the block matrices of M with respect to the new grid either consist only of zeroes or consist only of ones, as follows:

( \begin{matrix}0 & … & 0 & 1 & … & 1 & … & 0 & … & 0 & 1 & … & 1 \\ ∗ & … & ∗ & ∗ & … & ∗ & … & ∗ & … & ∗ & ∗ & … & ∗ \\ ⋮ & & ⋮ & ⋮ & & ⋮ & & ⋮ & & ⋮ & ⋮ & & ⋮ \\ ∗ & … & ∗ & ∗ & … & ∗ & … & ∗ & … & ∗ & ∗ & … & ∗ \\ ∗ & … & ∗ & ∗ & … & ∗ & … & ∗ & … & ∗ & ∗ & … & ∗ \\ ⋮ & & ⋮ & ⋮ & & ⋮ & & ⋮ & & ⋮ & ⋮ & & ⋮ \end{matrix} ) ,  (E.1)

where the double lines belong to the original grid and the simple ones have been added now. Let this new grid be defined by the numbers r, s′ ⩾ 1, 0 = i_{0} < i_{1} < i_{2} < ⋯< i_{r} = m, 0 = j′_{0} < j′_{1} < j′_{2} < ⋯< j′_{s′} = n, and let B′_{t,u}, t ∈ {1, 2, …, r}, u ∈ {1, 2, …, s′}, be the block matrices with respect to it. The block matrix B′_{1,1} has one of the two following forms:

( \begin{matrix}0 & 0 & … & 0 & 0 \\ ∗ & ∗ & … & ∗ & ∗ \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ ∗ & ∗ & … & ∗ & ∗ \end{matrix}) or ( \begin{matrix}1 & 1 & … & 1 & 1 \\ 1 & 1 & … & 1 & 1 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 1 & 1 & … & 1 & 1 \end{matrix}) .

We now proceed similarly with the first column. Let

B = ( \begin{matrix}∑ _{p=i_{0} +1}^{i_{1}} a_{p1} & ∑ _{p=i_{0} +1}^{i_{1}} a_{p2} & … & ∑ _{p=i_{0} +1}^{i_{1}} a_{pj′_{1}} \\ ∑ _{p=i_{1} +1}^{i_{2}} a_{p1} & ∑ _{p=i_{1} +1}^{i_{2}} a_{p2} & … & ∑ _{p=i_{1} +1}^{i_{2}} a_{pj′_{1}} \\ ⋮ & ⋮ & & ⋮ \\ ∑ _{p=i_{r−1} +1}^{i_{r}} a_{p1} & ∑ _{p=i_{r−1} +1}^{i_{r}} a_{p2} & … & ∑ _{p=i_{r−1} +1}^{i_{r}} a_{pj′_{1}} \end{matrix} )

be the matrix of numbers of ones in the columns of the block matrices B′_{t,1}, 1 ⩽ t ⩽ r. Let C_{v}(B) be the minimal column of B: we permute C_{1}(M) with C_{v}(M) and still call M the resulting matrix. Let us permute the rows of M in such a way that the first columns of all the block matrices B′_{t,1}, 1 ⩽ t ⩽ r, consist in a block of zeroes followed by a block of ones:

B′_{t,1}(M) = ( \begin{matrix}0 & ∗ & … & ∗ \\ ⋮ & ⋮ & & ⋮ \\ 0 & ∗ & … & ∗ \\ 1 & ∗ & … & ∗ \\ ⋮ & ⋮ & & ⋮ \\ 1 & ∗ & … & ∗ \end{matrix})

The form of B′_{1,1} shows that this can be done without permuting the first row, so we do not. We still call the resulting matrix M. We now refine the grid on m × n in such a way that the first columns of the block matrices with respect to the new grid either consist only of zeroes or consist only of ones, so that also the tranpose of M has a form as in (E.1). Let the new grid be defined by the numbers r′, s′ ⩾ 1, 0 = i′_{0} < i′_{1} < i′_{2} < ⋯< i′_{r′} = m, 0 = j′_{0} < j′_{1} < j′_{2} < ⋯< j′_{s′} = n.

Let N be the m − 1 × n − 1 submatrix in the lower right corner of M. By induction hypothesis, N may be ordered by admissible permutations with respect to the grid on m − 1 × n − 1 induced by the grid just defined. These permutations are also admissible with respect to the latter grid and do not permute the first line and the first column of M. Let us again call the resulting matrices M and N.

As the rows and columns of N are ordered, we have for M

L_{i_{0} +2} ⩽ ⋯⩽ L_{i_{1}} , L_{i_{1} +1} ⩽ ⋯⩽ L_{i_{2}} , …, L_{i_{r−1} +1} ⩽ ⋯⩽ L_{i_{r}} ;
C_{j_{0} +2} ⩽ ⋯⩽ C_{j_{1}} , C_{j_{1} +1} ⩽ ⋯⩽ C_{j_{2}} , , …, C_{j_{s−1} +1} ⩽ ⋯⩽ C_{j_{s}} ,

and our choice for the first line and the first column of M ensures that

L_{i_{0} +1} ⩽ L_{i_{0} +2} , C_{j_{0} +1} ⩽ C_{j_{0} +2} .

Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group

with Éric Ricard

Abstract

We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group Γ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets; unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum Λ ⊆ Γ; the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2; the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1.

1 Introduction

Let Λ be a subset of ℤ and let x be a bounded measurable function on the circle 𝕋 with Fourier spectrum in Λ: we write x ∈ L_{Λ}^{∞} , x ∼ ∑ _{k∈Λ} x_{k}z^{k}. The matrix of the associated operator y ↦ xy on L^{2} with respect to its trigonometric basis is the Toeplitz matrix

(x_{r−c} )_{(r,c)∈ℤ×ℤ} = ( \begin{matrix}& ··· & 1 & 0 & −1 & ··· \\ ⋮ & ... & ... & ... & ... & ... \\ 1 & ... & x_{0} & x_{1} & x_{2} & ... \\ 0 & ... & x_{−1} & x_{0} & x_{1} & ... \\ −1 & ... & x_{−2} & x_{−1} & x_{0} & ... \\ ⋮ & ... & ... & ... & ... & ... \end{matrix})

with support in Λ̋ = {(r, c) : r − c ∈ Λ}.

This is an entry point to the interplay between harmonic analysis and operator theory. In the general case of a discrete group Γ, the counterpart to a bounded measurable function is defined as a bounded operator on ℓ_{Γ}^{2} whose matrix has the form (x_{rc^{−1}} )_{(r,c)∈Γ ×Γ} for some sequence (x_{γ})_{γ∈Γ} . This will be the framework of the body of this article, while the introduction sticks to the case Γ = ℤ.

We are concerned with two kinds of multipliers. A sequence φ = (φ_{k})_{k∈Λ} defines

– the relative Fourier multiplication operator on trigonometric polynomials with spectrum in Λ by

∑_{k∈Λ} x_{k}z^{k} ↦ ∑_{k∈Λ} φ_{k}x_{k}z^{k};  (F.1)

– the relative Schur multiplication operator on finite matrices with support in Λ̋ by

(x_{r,c})_{(r,c)∈ℤ×ℤ} ↦ (φ̋_{r,c} x_{r,c})_{(r,c)∈ℤ×ℤ}  (F.2)

where φ̋_{r,c} = φ_{r−c} .

Marek Bożejko and Gero Fendler proved that these two multipliers have the same norm. The operator (F.1) is nothing but the restriction of (F.2) to Toeplitz matrices. They noted that it is automatically completely bounded: it has the same norm when acting on trigonometric series with operator coefficients x_{k}, and this permits to remove this restriction. Schur multiplication is also automatically completely bounded.

This observation has been extended by Gilles Pisier to multipliers acting on a translation invariant Lebesgue space L_{Λ}^{p} and on the subspace Sp ; Λ̋ of elements of a Schatten-von-Neumann class supported by Λ̋, respectively: it yields that the complete norm of a relative Schur multiplier (F.2) remains bounded by the complete norm of the relative Fourier multiplier (F.1).

But L_{Λ}^{p} is not a subspace of Sp ; Λ̋, so that a relative Fourier multiplier may not be viewed anymore as the restriction of a relative Schur multiplier to Toeplitz matrices. We point out that this difficulty may be overcome by using Szegő’s limit theorem: a bounded measurable real function is the weak^{∗} limit of the normalised counting measure of eigenvalues of finite truncates of its Toeplitz matrix. Note that other types of approximation are also available, as the completely positive approximation property and Reiter sequences combined with complex interpolation: they are compared in Section 3 in terms of local embeddings of L^{p} into S^{p}. They are more canonical than Szegő’s limit theorem, but give no access to general Orlicz norms.

Theorem 1.1. Let ψ: ℝ^{+} → ℝ^{+} be a continuous nondecreasing function vanishing only at 0. The norm of the following operators is equal:

– the relative Fourier multiplication operator (F.1) on the Lebesgue-Orlicz space L_{Λ}^{ψ} (S^{ψ} ) of S^{ψ} - valued trigonometric series with spectrum in Λ;

– the relative Schur multiplication operator (F.2) on the Schatten-von-Neumann-Orlicz class Sψ ; Λ̋ (S^{ψ} ) of S^{ψ} -valued matrices with support in Λ̋.

Look at Theorem 2.6 for the precise statement in the general case of an amenable group Γ, for which a block matrix variant of Szegő’s limit theorem in the style of Erik Bédos [6], Theorem 2.1, is available.

An application of this theorem to the class of all unimodular Fourier multipliers yields a transfer of lacunary subsets into lacunary matrix patterns. Call Λ unconditional in L^{p} if (z^{k})_{k∈Λ} is an unconditional basis of L_{Λ}^{p}, and Λ̋ unconditional in S^{p} if the sequence (e_{q})_{q∈Λ̋} of elementary matrices is an unconditional basis of Sp ; Λ̋. These properties are also known as Λ(max(2, p)) and σ(p), respectively; they have natural “complete” counterparts that are also known as Λ(p)_{cb} (K(p)_{cb} if p ⩽ 2) and σ(p)_{cb}, respectively: see Definitions 4.1 and 4.2.

Corollary 1.2. If Λ̋ is unconditional in S^{p}, then Λ is unconditional in L^{p}. Λ̋ is completely unconditional in S^{p} if and only if Λ is completely unconditional in L^{p}.

Look at Proposition 4.3 for the precise statement in the general case of a discrete group Γ.

The two most prominent multipliers are the Riesz projection and the Hilbert transform. The first consists in letting φ be the indicator function of nonnegative integers and transfers into the upper triangular truncation of matrices. The second corresponds to the sign function and transfers into the Hilbert matrix transform. We obtain the following partial results.

Theorem 1.3. The norm of the matrix Riesz projection and of the matrix Hilbert transform on S^{ψ} (S^{ψ} ) coincide with their norm on S^{ψ} .

– If p is a power of 2, the norm of the matrix Hilbert transform on S^{p} is cot(π/2p).

– The norm of the matrix Riesz projection on S^{4} is \sqrt{2}.

The transfer technique lends itself naturally to the case where Λ contains a sumset R + C: if subsets R′ and C′ are extracted so that the r + c with r ∈ R′ and c ∈ C′ are pairwise distinct, they may play the rôle of rows and columns. Here are consequences of the conditionality of the sequence of elementary matrices e_{r,c} in S^{p} for p ≠ 2 and of the unboundedness of the Riesz transform on S^{1} and S^{∞} , respectively.

Theorem 1.4. If (z^{k})_{k∈Λ} is a completely unconditional basis of L_{Λ}^{p} with p ≠ 2, then Λ does not contain sumsets R + C of arbitrarily large sets.

– If L_{Λ}^{1} admits some completely unconditional approximating sequence, or

– if the space C_{Λ} of continuous functions with spectrum in Λ admits some unconditional approximating sequence,

then Λ does not contain the sumset R + C of two infinite sets.

The proof of the second part of this theorem consists in constructing infinite subsets R′ and C′, and skipped block sums ∑ (T_{k_{j+1}} −T_{k_{j}} ) of a given approximating sequence that act like the projection on the “upper triangular” part of R′+ C′. Look at Proposition 4.8 and Theorem 7.4 for the precise statement in the general case of a discrete group Γ.

In the case of quasi-normed Schatten-von-Neumann classes S^{p} with p < 1, the transfer technique yields a new proof for the following result of Alexey Alexandrov and Vladimir Peller.

Theorem 1.5. Let 0 < p < 1. The Fourier multiplier φ is contractive on L^{p} or on L^{p}(S^{p}) if and only if the Schur multiplier φ̋ is contractive on S^{p} or on S^{p}(S^{p}), if and only if the sequence φ is the Fourier transform of an atomic measure of the form ∑ a_{g}δ_{g} on 𝕋 with ∑ |a_{g}|^{p} ⩽ 1.

The emphasis put on relative Schur multipliers motivates the natural question of how the norm of an elementary Schur multiplier, that is a rank 1 matrix (ϱ_{r,c}) = (x_{r}y_{c}), gets affected when the action of ϱ is restricted to matrices with a given support. The surprising answer is the following.

Theorem 1.6. Let I ⊆ R × C and consider (x_{r})_{r∈R} and (y_{c})_{c∈C} . The relative Schur multiplier on S_{I}^{∞} given by (x_{r}y_{c})_{(r,c)∈I} has norm sup_{(r,c)∈I} |x_{r}y_{c}|.

Let us finally describe the content of this article. Section 2 is devoted to transfer techniques for Fourier and Schur multipliers provided by a block matrix Szegő limit theorem. This theorem provides local embeddings of L^{ψ} into S^{ψ} ; Section 3 shows how interpolation may be used to define such embeddings for the scale of L^{p} spaces. Section 4 is devoted to the transfer of lacunary sets into lacunary matrix patterns; the unconditional constant of a set Λ is related to the size of the sumsets it contains. Section 5 deals with Toeplitz Schur multipliers for p < 1 and comments on the case p ⩾ 1. The Riesz projection and the Hilbert transform are studied in Section 6. In Section 7, the presence of sumsets in a spectrum Λ is shown to be an obstruction for the existence of completely unconditional bases for L_{Λ}^{p}. The last section provides a norm-preserving extension for partially specified rank 1 Schur multipliers.

Notation and terminology. Let 𝕋 = {z ∈ ℂ : |z| = 1}.

Given an index set C and c ∈ C, e_{c} is the sequence defined on C as the indicator function χ_{{c}} of the singleton {c}, so that (e_{c})_{c∈C} is the canonical Schauder basis of the Hilbert space of square summable sequences indexed by C, denoted by ℓ_{C}^{2}. We will use the notation ℓ_{n}^{2} = ℓ_{{1,2,...,n}}^{2} and ℓ^{2} = ℓ_{ℕ}^{2}.

Given a product set I = R × C and q = (r, c), the indicator function e_{q} = e_{r,c} is the elementary matrix identified with the linear operator from ℓ_{C}^{2} to ℓ_{R}^{2} that maps e_{c} on e_{r} and all other basis vectors on 0. The matrix coefficient at coordinate q of a linear operator x from ℓ_{C}^{2} to ℓ_{R}^{2} is x_{q} = tr e_{q}^{∗}x and its matrix representation is (x_{q})_{q∈R×C} = ∑ _{q∈R×C} x_{q}e_{q}. The support or pattern of x is {q ∈ R × C : x_{q} ≠ 0}.

The space of all bounded operators from ℓ_{C}^{2} to ℓ_{R}^{2} is denoted by 𝔹(ℓ_{C}^{2}, ℓ_{R}^{2}), and its subspace of compact operators is denoted by S^{∞} .

Let ψ: ℝ^{+} → ℝ^{+} be a continuous nondecreasing function vanishing only at 0. The Schattenvon-Neumann-Orlicz class S^{ψ} is the space of those compact operators x from ℓ_{C}^{2} to ℓ_{R}^{2} such that trψ(|x|/a) < ∞ for some a > 0. If ψ is convex, then S^{ψ} is a Banach space for the norm given by ∥x∥_{S^{ψ}} = inf{a > 0 : trψ(|x|/a) ⩽ 1}. Otherwise, S^{ψ} is a Fréchet space for the F-norm given by ∥x∥_{S^{ψ}} = inf{a > 0 : trψ(|x|/a) ⩽ a} (see [70, Chapter 3].) This space may also be constructed as the noncommutative Lebesgue-Orlicz space L^{ψ} (tr) associated to the von Neumann algebra 𝔹(ℓ_{C}^{2}, ℓ_{R}^{2}) endowed with the normal faithful semifinite trace tr. If ψ is the power function t ↦ t^{p}, this space is denoted S^{p}: if p ⩾ 1, then ∥x∥_{S^{p}} = (tr|x|^{p})^{1/p}; if p < 1, then ∥x∥_{S^{p}} = (tr|x|^{p})^{1/(1+p)} .

If # C = # R = n, then 𝔹(ℓ_{C}^{2}, ℓ_{R}^{2}) identifies with the space of n × n matrices denoted S_{n}^{∞} , and we write S_{n}^{ψ} for S^{ψ} . Let (R_{n} × C_{n}) be a sequence of finite sets such that each element of R × C eventually is in R_{n} × C_{n}. Then the sequence of operators P_{n} : x ↦ ∑ _{q∈R_{n} ×C_{n}} x_{q}e_{q} tends pointwise to the identity on S^{ψ} .

For I ⊆ R × C, we define the space S_{I}^{ψ} as the closed subspace of S^{ψ} spanned by (e_{q})_{q∈I} : this coincides with the subspace of those x ∈ S^{ψ} whose support is a subset of I.

A relative Schur multiplier on S_{I}^{ψ} is a sequence ϱ = (ϱ_{q})_{q∈I} ∈ ℂ^{I} such that the associated Schur multiplication operator M_{ϱ} defined by e_{q} ↦ ϱ_{q}e_{q} for q ∈ I is bounded on S_{I}^{ψ} . The norm ∥ϱ∥_{M(S_{I}^{ψ} )} of ϱ is defined as the norm of M_{ϱ}. This norm is the supremum of the norm of its restrictions to finite rectangle sets R′× C′. We used [77, 79] as a reference.

Let Γ be a discrete group with identity ϵ. The reduced C^{∗}-algebra of Γ is the closed subspace spanned by the left translations λ_{γ} (the linear operators defined on ℓ_{Γ}^{2} by λ_{γ} e_{β} = e_{γβ} ) in 𝔹(ℓ_{Γ}^{2} ); we denote it by C, set in roman type. The von Neumann algebra of Γ is its weak^{∗} closure, endowed with the normal faithful normalised finite trace τ defined by τ(x) = x_{ϵ,ϵ}; we denote it by L^{∞} . Let ψ: ℝ^{+} → ℝ^{+} be a continuous nondecreasing function vanishing only at 0: we define the noncommutative Lebesgue-Orlicz space L^{ψ} of Γ as the completion of L^{∞} with respect to the norm given by ∥x∥_{L^{ψ}} = inf{a > 0 : τ(ψ(|x|/a)) ⩽ 1} if ψ is convex and with respect to the F -norm given by∥x∥_{L^{ψ}} = inf{a > 0 : τ(ψ(|x|/a)) ⩽ a} otherwise. If ψ is the power function t ↦ t^{p}, this space is denoted L^{p}: if p ⩾ 1, then ∥x∥_{L^{p}} = τ(|x|^{p})^{1/p}; if p < 1, then ∥x∥_{L^{p}} = τ(|x|^{p})^{1/(1+p)} . The Fourier coefficient of x at γ is x_{γ} = τ(λ_{γ}^{∗}x) = x_{γ,ϵ} and its Fourier series is ∑ _{γ∈Γ} x_{γ}λ_{γ}. The spectrum of an element x is {γ ∈ Γ : x_{γ} ≠ 0}. Let X be the C^{∗}-algebra C or the space L^{ψ} and let Λ ⊆ Γ: then we define X_{Λ} as the closed subspace of X spanned by the λ_{γ} with γ ∈ Λ. We skip the general question for which spaces X this coincides with the subspace of those x ∈ X whose spectrum is a subset of Λ, but note that this is the case if Γ is an amenable group (or if L^{∞} has the QWEP by [45, Theorem 4.4]) and ψ is the power function t ↦ t^{p}; note also that our definition of X_{Λ} makes it a subspace of the heart of X: if x ∈ X_{Λ} , then τ(ψ(|x|/a)) is finite for all a > 0.

A relative Fourier multiplier on X_{Λ} is a sequence φ = (φ_{γ})_{γ∈Λ} ∈ ℂ^{Λ} such that the associated Fourier multiplication operator M_{φ} defined by λ_{γ} ↦ φ_{γ}λ_{γ} for γ ∈ Λ is bounded on X_{Λ} . The norm ∥φ∥_{M(X_{Λ} )} of φ is defined as the norm of M_{φ}. Fourier multipliers on the whole of the C^{∗}-algebra C are also called multipliers of the Fourier algebra A(Γ) (which may be identified with L^{1}); they form the set M(A(Γ)).

The space S^{ψ} (S^{ψ} ) is the space of those compact operators x from ℓ^{2} ⊗ ℓ_{C}^{2} to ℓ^{2} ⊗ ℓ_{R}^{2} such that ∥x∥_{S^{ψ} (S^{ψ} )} = inf{a : tr⊗ trψ(|x|/a) ⩽ 1}: it is the noncommutative Lebesgue-Orlicz space L^{ψ} (tr⊗ tr) associated to the von Neumann algebra 𝔹(ℓ^{2}) ⊗ 𝔹(ℓ_{C}^{2}, ℓ_{R}^{2}). One may think of S^{ψ} (S^{ψ} ) as the S^{ψ} - valued Schatten-von-Neumann class: we define the matrix coefficient of x at q by x_{q} = (Id_{S^{ψ}} ⊗ tr) ((Id_{ℓ^{2}} ⊗ e_{q}^{∗})x) ∈ S^{ψ} and its matrix representation by ∑ _{q∈R×C} x_{q} ⊗ e_{q}. The support of x and the subspace S_{I}^{ψ} (S^{ψ} ) are defined in the same way as S_{I}^{ψ} .

Similarly, the space L^{ψ} (tr⊗τ) is the noncommutative Lebesgue-Orlicz space associated to the von Neumann algebra 𝔹(ℓ^{2}) ⊗ L^{∞} = L^{∞} (tr⊗τ). One may think of L^{ψ} (tr⊗τ) as the S^{ψ} -valued noncommutative Lebesgue space: we define the Fourier coefficient of x at γ by x_{γ} = (Id_{S^{ψ}} ⊗τ)((Id_{ℓ^{2}} ⊗ λ_{γ}^{∗})x) ∈ S^{ψ} and its Fourier series by ∑ _{γ∈Γ} x_{γ} ⊗ λ_{γ}; the spectrum of x is defined accordingly. The subspace L_{Λ}^{ψ} (tr⊗τ) is the closed subspace of L^{ψ} (tr⊗τ) spanned by the x⊗λ_{γ} with x ∈ S^{ψ} and γ ∈ Λ.

An operator T on S_{I}^{ψ} is bounded on S_{I}^{ψ} (S^{ψ} ) if the linear operator Id_{S^{ψ}} ⊗ T defined by x ⊗ y ↦ x ⊗ T (y) for x ∈ S^{ψ} and y in S_{I}^{ψ} on finite tensors extends to a bounded operator Id_{S^{ψ}} ⊗ T on S_{I}^{ψ} (S^{ψ} ). The norm of a Schur multiplier ϱ on S_{I}^{ψ} (S^{ψ} ) is defined as the norm of Id_{S^{ψ}} ⊗ M_{ϱ}. Similar definitions hold for an operator T on L_{Λ}^{ψ} ; the norm of a Fourier multiplier φ on L_{Λ}^{ψ} (tr⊗τ) is the norm of Id_{S}^{ψ} ⊗M_{φ} on L_{Λ}^{ψ} (tr⊗τ).

Let ψ be the power function t ↦ t^{p} with p ⩾ 1: the norms on S^{p}(S^{p}) and L^{p}(tr⊗τ) describe the canonical operator space structure on S^{p} and L^{p}, respectively: see [77, Corollary 1.4]; we should rather use the notation S^{p}[S^{p}] and S^{p}[L^{p}]. This explains the following terminology. An operator T on S_{I}^{p} is completely bounded (c.b. for short) if Id_{S^{p}} ⊗ T is bounded on S_{I}^{p}(S^{p}); the norm of Id_{S^{p}} ⊗ T is the complete norm of T (compare [77, Lemma 1.7].) The complete norm ∥ϱ∥_{M_{cb} (S_{I}^{p})} of a Schur multiplier ϱ is defined as the complete norm of M_{ϱ}. Note that the complete norm of a Schur multiplier ϱ on S_{I}^{∞} is equal to its norm [72, Theorem 3.2]: ∥ϱ∥_{M_{cb} (S_{I}^{∞} )} = ∥ϱ∥_{M(S_{I}^{∞} )} . The complete norm ∥φ∥_{M_{cb} (L_{Λ}^{p} )} of a Fourier multiplier φ is defined as the complete norm of M_{φ}. The complete norm of an operator T on C_{Λ} is the norm of Id_{S^{∞}} ⊗ T on the subspace of S^{∞} ⊗ C spanned by the x ⊗ λ_{γ} with x ∈ S^{∞} and γ ∈ Λ; in the case Λ = Γ, φ is also called a c.b. multiplier of the Fourier algebra A(Γ) and one writes φ ∈ M_{cb}(A(Γ)); if Γ is amenable, the complete norm of a Fourier multiplier φ on C_{Λ} is equal to its norm [25, Corollary 1.8]: ∥φ∥_{M_{cb} (C_{Λ} )} = ∥φ∥_{M(C_{Λ} )} .

An element whose norm is at most 1 is contractive, and if its complete norm is at most 1, it is completely contractive.

If Γ is abelian, let G be its dual group and endow it with its unique normalised Haar measure m: then the Fourier transform identifies the C^{∗}-algebra C as the space of continuous functions on G, L^{∞} as the space of classes of bounded measurable functions on (G, m), L^{ψ} as the Lebesgue-Orlicz space of classes of ψ-integrable functions on (G, m), τ(x) as ∫_{G} x(g) dm(g), L^{ψ} (tr⊗τ) as the S^{ψ} -valued Lebesgue-Orlicz space L^{ψ} (S^{ψ} ) and x_{γ} as x̂(γ).

2 Transfer between Fourier and Schur multipliers

Let Λ be a subset of a discrete group Γ and let φ be a relative Fourier multiplier on C_{Λ} , the closed subspace spanned by (λ_{γ})_{γ∈Λ} in the reduced C^{∗}-algebra of Γ. Let x ∈ C_{Λ} : the matrix of x is constant down the diagonals in the sense that for every (r, c) ∈ Γ × Γ, x_{r,c} = x_{rc^{−1} ,ϵ} = x_{rc^{−1}} ; we say that x is a Toeplitz operator on ℓ_{Γ}^{2} . Furthermore, the matrix of the Fourier product M_{φ} x of φ with x is given by (M_{φ} x)_{r,c} = φ_{rc^{−1}} x_{r,c}. This shows that if we set Λ̋ = {(r, c) ∈ Γ × Γ : rc^{−1} ∈ Λ} and φ̋_{r,c} = φ_{rc^{−1}} , then M_{φ} x is the Schur product Mφ̋x of φ̋ with x. We have transferred the Fourier multiplier φ into the Schur multiplier φ̋: this shows at once that the norm of the Fourier multiplier φ on C_{Λ} is the norm of the Schur multiplier φ̋ on the subspace of Toeplitz elements of 𝔹(ℓ_{Γ}^{2} ) with support in Λ̋, and that the same holds for complete norms.

We shall now give us the means to generalise this identification to the setting of Lebesgue-Orlicz spaces L^{ψ} : we shall bypass the main obstacle, that L^{ψ} may not be considered as a subspace of S^{ψ} , by the Szegő limit theorem as stated by Erik Bédos [6, Theorem 10].

As we want to compute complete norms of multipliers, we shall generalise the Szegő limit theorem to the block matrix case, not considered in [6]. Let us first recall the scalar case. Consider a discrete amenable group Γ: it admits a Følner averaging net of sets (Γ_{ι}), that is,

– each Γ_{ι} is a finite subset of Γ;

– # (γΓ_{ι}ΔΓ_{ι}) = o(# Γ_{ι}) for each γ ∈ Γ.

Each set Γ_{ι} corresponds to the orthogonal projection p_{ι} of ℓ_{Γ}^{2} onto its (# Γ_{ι})-dimensional subspace of sequences supported by Γ_{ι}. The truncate of a selfadjoint operator y ∈ 𝔹(ℓ_{Γ}^{2} ) with respect to Γ_{ι} is y_{ι} = p_{ι}yp_{ι}: it has # Γ_{ι} eigenvalues α_{j}, counted with multiplicities, and its normalised counting measure of eigenvalues is

μ_{ι} = \frac{1}{# Γ_{ι}} ∑_{j=1}^{# Γ_{ι}} δ_{α_{j}} .

If y is a Toeplitz operator, that is, if y ∈ L^{∞} , Erik Bédos [6, Theorem 10] proves that (μ_{ι}) converges weak^{∗} to the spectral measure of y with respect to τ, which is the unique Borel probability measure μ on ℝ such that

τ(ψ(y)) = ∫_{ℝ} ψ(α)dμ(α)

for every continuous function ψ on ℝ that tends to zero at infinity. If Γ is abelian, then y may be identified as the class of a real-valued bounded measurable function on the group G dual to Γ and μ is the distribution of y.

The matrix Szegő limit theorem is the analogue of this result for selfadjoint elements y ∈ S_{n}^{∞} ⊗L^{∞} , whose S_{n}^{∞} -valued spectral measure μ is defined by

∫_{ℝ} ψ(α)dμ(α) = Id_{ℓ_{n}^{2}} ⊗ τ(ψ(y)).

The orthogonal projection p̃_{ι} = Id_{ℓ_{n}^{2}} ⊗ p_{ι} defines the truncate y_{ι} = p̃_{ι}yp̃_{ι} ∈ S_{n}^{∞} ⊗ 𝔹(ℓ_{Γ_{ι}}^{2} ), and the S_{n}^{∞} -valued normalised counting measure of eigenvalues μ_{ι} by

∫_{ℝ} ψ(α)dμ_{ι}(α) = Id_{ℓ_{n}^{2}} ⊗ \frac{tr}{# Γ_{ι}} (ψ(y_{ι})).

Theorem 2.1 (Matrix Szegő limit theorem). Let Γ be a discrete amenable group and let (Γ_{ι}) be a Følner averaging net for Γ. Let y be a selfadjoint element of S_{n}^{∞} ⊗ L^{∞} . The net (μ_{ι}) of S_{n}^{∞} -valued normalised counting measures of eigenvalues of the truncates of y with respect to Γ_{ι} converges in the weak^{∗} topology to the spectral measure of y:

∫_{ℝ} ψ(α)dμ_{ι}(α) → Id_{ℓ_{n}^{2}} ⊗ τ(ψ(y))

for every continuous function ψ on ℝ that tends to zero at infinity.

Sketch of proof. Let us first suppose that y ∈ S_{n}^{∞} ⊗ C: we may suppose that y = ∑ _{γ∈Γ} y_{γ} ⊗ λ_{γ} with only a finite number of the y_{γ} ∈ S_{n}^{∞} nonzero: the S_{n}^{∞} -valued matrix of y_{ι} for the canonical basis of ℓ_{Γ_{ι}}^{2} is (y_{rc^{−1}} )_{(r,c)∈Γ_{ι}×Γ_{ι}} . It suffices to prove that

Id ⊗ \frac{tr}{#Γ_{ι}} (y_{ι}^{k}) → Id ⊗ τ(y^{k})  (F.3)

for every k. This is trivial if k = 0. If k = 1, then

Id ⊗ \frac{tr}{#Γ_{ι}} (y_{ι}) = \frac{1}{#Γ_{ι}} ∑_{c∈Γ_{ι}} y_{c,c} = Id ⊗ τ(y)

as y_{c,c} = y_{cc^{−1}} = y_{ϵ}. If k ⩾ 2, the same formula holds with y^{k} instead of y:

Id ⊗ τ(y^{k}) = Id ⊗ \frac{tr}{#Γ_{ι}} (p̃_{ι}y^{k} p̃_{ι}),

so that we wish to prove

Id ⊗ tr(p̃_{ι}y^{k} p̃_{ι}− (p̃_{ι}yp̃_{ι})^{k}) = o(# Γ_{ι}).

Note that

∥Id ⊗ tr(p̃_{ι}y^{k} p̃_{ι}− (p̃_{ι}yp̃_{ι})^{k})∥_{S_{n}^{1}} ⩽ ∥p̃_{ι}y^{k} p̃_{ι}− (p̃_{ι}yp̃_{ι})^{k}∥_{S^{1} (S_{n}^{1} )} .

Lemma 5 in [6] provides the following estimate: as

p̃_{ι}y^{k} p̃_{ι}− (p̃_{ι}yp̃_{ι})^{k} = p̃_{ι}y^{k−1} (yp̃_{ι}− p̃_{ι}yp̃_{ι}) + (p̃_{ι}y^{k−1} p̃_{ι}− (p̃_{ι}yp̃_{ι})^{k−1} )yp̃_{ι},

an induction yields

∥p̃_{ι}y^{k} p̃_{ι}− (p̃_{ι}yp̃_{ι})^{k}∥_{S^{1} (S_{n}^{1} )} ⩽ (k − 1)∥y∥_{S_{n}^{∞} ⊗L^{∞}}^{k−1} ∥yp̃_{ι}− p̃_{ι}yp̃_{ι}∥_{S^{1} (S_{n}^{1} )} .

It suffices to consider the very last norm for each term y_{γ} ⊗ λ_{γ} of y: let h ∈ ℓ_{n}^{2} and β ∈ Γ; as

((y_{γ} ⊗ λ_{γ})p̃_{ι}− p̃_{ι}(y_{γ} ⊗ λ_{γ})p̃_{ι})(h ⊗ e_{β}) = \{ \begin{matrix}y_{γ}(h)e_{γβ} & if β ∈ Γ_{ι} and γβ ∉ Γ_{ι} \\ 0 & otherwise, \end{matrix}

the definition of a Følner averaging net yields

∥(y_{γ} ⊗ λ_{γ})p̃_{ι}− p̃_{ι}(y_{γ} ⊗ λ_{γ})p̃_{ι}∥_{S^{1} (S_{n}^{1} )} ⩽ #(Γ_{ι} \ γ^{−1} Γ_{ι})∥y_{γ}∥_{S_{n}^{1}} = o(# Γ_{ι}).

An approximation argument as in [6, proof of Proposition 4] permits to conclude for y ∈ L^{∞} .

Let us now describe and prove the L^{ψ} version of the transfer described at the beginning of this section.

Lemma 2.2. Let Γ be a discrete amenable group and p > 0. Let Λ ⊆ Γ and φ ∈ ℂ^{Λ} . Consider the associated Toeplitz set Λ̋ = {(r, c) ∈ Γ ×Γ : rc^{−1} ∈ Λ} and the Toeplitz matrix defined by φ̋_{r,c} = φ_{rc^{−1}} . Let ψ: ℝ^{+} → ℝ^{+} be a continuous nondecreasing function vanishing only at 0.

(a) The norm of the relative Fourier multiplier φ on L_{Λ}^{ψ} is bounded by the norm of the relative Schur multiplier φ̋ on Sψ ; Λ̋.

(b) The norm of the relative Fourier multiplier φ on L_{Λ}^{ψ} (tr⊗τ) is bounded by the norm of the relative Schur multiplier φ̋ on Sψ ; Λ̋ (S^{ψ} ).

Theorem 2.6 (a) below will provide the full picture of case (b): both norms are in fact equal. This is not so in case (a): see Remark 5.2.

Proof. A Toeplitz matrix has the form (x_{rc^{−1}} )_{(r,c)∈Λ̋} . Our definition of the space L_{Λ}^{ψ} (cf. section on Notation and terminology) ensures that we may suppose that only a finite number of the x_{γ} are nonzero for the computation of the norm of φ. Then (x_{rc^{−1}} )_{(r,c)∈Λ̋} is the matrix of the operator x = ∑ _{γ∈Λ} x_{γ}λ_{γ} for the canonical basis of ℓ_{Γ}^{2} .

Let y = x^{∗}x and let us use the notation of Theorem 2.1. Let ψ̃ be a continuous function with compact support such that ψ̃(t) = ψ(t) on [0, ∥x∥^{2}].

(a). By Szegő’s limit theorem (as given in [6, Theorem 1]),

\frac{1}{#Γ_{ι}} trψ(p_{ι}yp_{ι}) = \frac{1}{#Γ_{ι}} tr ψ̃(p_{ι}yp_{ι}) → τ(ψ̃(y)) = τ(ψ(y))

Let us describe how φ̋ acts on xp_{ι}. Schur multiplication with φ̋ transforms the matrix of xp_{ι}, that is the truncated Toeplitz matrix (x_{rc^{−1}} )_{(r,c)∈Λ̋∩Γ ×Γ_{ι}} , into the matrix (φ̋_{rc^{−1}} x_{rc^{−1}} )_{(r,c)∈Λ̋∩Γ ×Γ_{ι}} , so that it transforms xp_{ι} into (M_{φ} x)p_{ι}.

(b). Combine the argument in (a) with the matrix Szegő limit theorem.

In the case of a finite abelian group, no limit theorem is needed: this case has been considered in [66, Proposition 2.5 (b)]. Remark 2.3. Our technique proves in fact that the norm of a Fourier multiplier is the upper limit of the norm of the corresponding relative Schur multipliers on subspaces of truncated Toeplitz matrices. We ignore whether it is actually their supremum.

The following well-known argument has been used (first in [17], see [18, Proposition D.6]) to show that the complete norm of the Fourier multiplier φ on L_{Λ}^{∞} bounds the complete norm of the Schur multiplier φ̋ on S∞ ; Λ̋ , so that we have in full generality ∥φ∥_{M_{cb} (C_{Λ} )} = ∥φ̋∥_{M_{cb} (S∞ ; Λ̋ )} . This argument permits to strengthen Lemma 2.2 (b).

Lemma 2.4. Let Γ be a discrete group and let R and C be subsets of Γ. To Λ ⊆ Γ associate Λ̋ = {(r, c) ∈ R × C : rc^{−1} ∈ Λ}; given φ ∈ ℂ^{Λ} define φ̋ ∈ ℂΛ̋ by φ̋_{r,c} = φ_{rc^{−1}} . Let ψ: ℝ^{+} → ℝ^{+} be a continuous nondecreasing function vanishing only at 0. The norm of the relative Schur multiplier φ̋ on Sψ ; Λ̋ (S^{ψ} ) is bounded by the norm of the relative Fourier multiplier φ on L_{Λ}^{ψ} (tr⊗τ).

Proof. We adapt the argument in [77, Lemma 8.1.4]. Let x_{q} ∈ S^{ψ} , of which only a finite number are nonzero. The space L^{ψ} (tr⊗ tr⊗τ) is a left and right L^{∞} (tr⊗ tr⊗τ)-module and ∑ _{γ∈Γ} e_{γγ} ⊗ λ_{γ} is a unitary in L^{∞} (tr⊗τ), so that

∥∑_{q∈Λ̋} x_{q} ⊗ e_{q}∥_{Sψ ; Λ̋ (S^{ψ} )} = ∥( Id ⊗ ∑_{r∈R} e_{r,r} ⊗ λ_{r}) ( ∑_{q∈Λ̋} x_{q} ⊗ e_{q} ⊗ λ_{ϵ}) ( Id ⊗ ∑_{c∈C} e_{c,c} ⊗ λ_{c}^{∗}) ∥_{L^{ψ} (tr ⊗ tr ⊗τ)}

= ∥ ∑_{(r,c)∈Λ̋} x_{r,c} ⊗ e_{r,c} ⊗ λ_{rc^{−1}} ∥_{L^{ψ} (tr ⊗ tr ⊗τ)}

= ∥∑_{γ∈Λ} ( ∑_{rc^{−1} =γ} x_{r,c} ⊗ e_{r,c} ) ⊗ λ_{γ} ∥_{L_{Λ}^{ψ} (tr ⊗ tr ⊗τ)} .

3. LOCAL EMBEDDINGS OF L^{p} INTO S^{p}  81

This yields an isometric embedding of Sψ ; Λ̋ (S^{ψ} ) in L_{Λ}^{ψ} (tr⊗ tr⊗τ). As S^{ψ} (S^{ψ} ) is the Schatten-von-Neumann-Orlicz class for the Hilbert space ℓ^{2} ⊗ ℓ_{Γ}^{2} which may be identified with ℓ^{2},

∥∑_{q∈Λ̋} x_{q} ⊗ φ̋_{q} e_{q}∥_{Sψ ; Λ̋ (S^{ψ} )} = ∥∑_{γ∈Λ} ( ∑_{rc^{−1} =γ} x_{r,c} ⊗ e_{r,c} ) ⊗ φ_{γ}λ_{γ} ∥_{L_{Λ}^{ψ} (tr ⊗ tr ⊗τ)}

⩽ ∥Id_{S^{ψ}} ⊗ M_{φ} ∥∥∑_{q∈Λ̋} x_{q} ⊗ e_{q}∥_{Sψ ; Λ̋ (S^{ψ} )} .

Remark 2.5. The proof of Lemma 2.4 shows also the following transfer. Let (r_{i}) and (c_{j}) be sequences in Γ, consider Λ̆ = {(i, j) ∈ ℕ × ℕ : r_{i}c_{j} ∈ Λ} and define φ̆ ∈ ℂΛ̆ by φ̆(i, j) = φ(r_{i}c_{j}). Then the norm of the relative Schur multiplier φ̆ on Sψ ; Λ̆(S^{ψ} ) is bounded by the norm of the relative Fourier multiplier Id_{S^{ψ}} ⊗ M_{φ} on L_{Λ}^{ψ} (tr⊗τ) (compare with [79, Theorem 6.4].) In particular, if the r_{i}c_{j} are pairwise distinct, this permits to transfer every Schur multiplier, not just the Toeplitz ones. See [66, Section 11] for applications of this transfer.

Here is the announced strengthening of Lemma 2.2.

Theorem 2.6. Let Γ be a discrete amenable group. Let Λ ⊆ Γ and φ ∈ ℂ^{Λ} . Consider the associated Toeplitz set Λ̋ = {(r, c) ∈ Γ × Γ : rc^{−1} ∈ Λ} and the Toeplitz matrix defined by φ̋_{r,c} = φ_{rc^{−1}} .

(a) Let ψ: ℝ^{+} → ℝ^{+} be a continuous nondecreasing function vanishing only at 0. The norm of the relative Fourier multiplier φ on L_{Λ}^{ψ} (tr⊗τ) and the norm of the relative Schur multiplier φ̋ on Sψ ; Λ̋ (S^{ψ} ) are equal.

(b) Let p ⩾ 1. The complete norm of the relative Fourier multiplier φ on L_{Λ}^{p} and the complete norm of the relative Schur multiplier φ̋ on Sp ; Λ̋ are equal:

∥φ∥_{M_{cb} (L_{Λ}^{p} )} = ∥φ̋∥_{M_{cb} (Sp ; Λ̋)} .

(c) The norm of the relative Fourier multiplier φ on C_{Λ} , its complete norm, the norm of the relative Schur multiplier φ̋ on S∞ ; Λ̋ and its complete norm are equal:

∥φ∥_{M(C_{Λ} )} = ∥φ∥_{M_{cb} (C_{Λ} )} = ∥φ̋∥_{M_{cb} (S∞ ; Λ̋ )} = ∥φ̋∥_{M(S∞ ; Λ̋ )} .

(d) Suppose that Λ = Γ. The norm of the Fourier algebra multiplier φ, its complete norm, the norm of the Schur multiplier φ̋ on S^{∞} and its complete norm are equal:

∥φ∥_{M(A(Γ))} = ∥φ∥_{M_{cb} (A(Γ))} = ∥φ̋∥_{M_{cb} (S^{∞} )} = ∥φ̋∥_{M(S^{∞} )} .

Proof. Combine Proposition 2.2 (b) with Lemma 2.4. Recall that if Γ is amenable, the norm of a Fourier multiplier φ on C_{Λ} is equal to its complete norm [25, Corollary 1.8] and that the complete norm of a Schur multiplier φ̋ on S∞ ; Λ̋ is equal to its norm [72, Theorem 3.2].

3 Local embeddings of L^{p} into S^{p}

The proof of Lemma 2.2 can be interpreted as an embedding of L^{ψ} into an ultraproduct of finitedimensional spaces S_{n}^{ψ} that intertwines Fourier and Toeplitz Schur multipliers. If we restrict ourselves to power functions ψ: t ↦ t^{p} with p ⩾ 1, such embeddings are well known and the proof of Lemma 2.2 does not need the full strength of the Matrix Szegő limit theorem but only the existence of such embeddings. In this section, we explain two ways to obtain them by interpolation.

The first way is to extend the classical result that the reduced C^{∗}-algebra C of a discrete group Γ has the completely contractive approximation property if Γ is amenable. We follow the approach of [18, Theorem 2.6.8]. Let Γ be a discrete amenable group and Γ_{ι} be a Følner averaging net of sets. As above, we denote by p_{ι} the orthogonal projection from ℓ_{Γ}^{2} to ℓ_{Γ_{ι}}^{2} . Define the compression φ_{ι} and the embedding ψ_{ι} by

\begin{matrix}φ_{ι}: C & → 𝔹(ℓ_{Γ_{ι}}^{2} ) \\ x & ↦ p_{ι}xp_{ι} \end{matrix} and \begin{matrix}ψ_{ι}: 𝔹(ℓ_{Γ_{ι}}^{2} ) & → C \\ e_{r,c} & ↦ (1/ # Γ_{ι})λ_{r}λ_{c^{−1}} . \end{matrix}  (F.4)

If we endow 𝔹(ℓ_{Γ_{ι}}^{2} ) with the normalised trace, these maps are unital completely positive, trace preserving (and normal) and the net (ψ_{ι}φ_{ι}) converges pointwise to the identity of C. One can therefore extend them by interpolation to completely positive contractions on the respective noncommutative Lebesgue spaces: recall that L^{p}(𝔹(ℓ_{Γ_{ι}}^{2} ), (1/ # Γ_{ι}) tr) is (# Γ_{ι})^{−1/p} S_{# Γ_{ι}}^{p} . We get a net of complete contractions

φ̃_{ι} : L^{p} → (# Γ_{ι})^{−1/p} S_{# Γ_{ι}}^{p} and ψ̃_{ι} : (# Γ_{ι})^{−1/p} S_{# Γ_{ι}}^{p} → L^{p},

such that (ψ̃_{ι} φ̃_{ι}) converges pointwise to the identity of L^{p}. Moreover, the definitions (F.4) show that these maps also intertwine Fourier and Toeplitz Schur multipliers.

Remark 3.1. This approach is more canonical as it allows to extend the transfer to the vector-valued spaces in the sense of [77, Chapter 3]. Recall that for any hyperfinite semifinite von Neumann algebra M and any operator space E, one can define L^{p}(M, E): for p = ∞, this space is defined as M ⊗_{min} E; for p = 1, this space is defined as M_{∗}^{op} ˆ⊗E; these spaces form an interpolation scale for the complex method when 1 ⩽ p ⩽ ∞. For us, M will be 𝔹(ℓ^{2}) or the group von Neumann algebra L^{∞} . As the maps ψ_{ι} and φ_{ι} are unital completely positive and trace preserving and normal, they define simultaneously complete contractions on M and M_{∗}. By interpolation, the maps ψ_{ι}⊗ Id_{E} and φ_{ι} ⊗ Id_{E} are still complete contractions on the spaces L_{p}(E) and S^{p}[E]. Let φ: Γ → ℂ: the transfer shows that the norm of Id_{E} ⊗ M_{φ} on L^{p}(E) is bounded by the norm of Id_{E} ⊗ Mφ̋ on S^{p}[E], and that their complete norms coincide. In formulas,

∥Id_{E} ⊗ M_{φ} ∥_{𝔹(L^{p} (E))} ⩽ ∥Id_{E} ⊗ Mφ̋∥_{𝔹(S^{p} [E])} ,
∥Id_{E} ⊗ M_{φ} ∥_{cb(L^{p} (E))} = ∥Id_{E} ⊗ Mφ̋∥_{cb(S^{p} [E])} .

This approximation is two-sided whereas the proof of Lemma 2.2 uses only a one-sided approximation. This subtlety makes a difference if one tries to give a direct proof by complex interpolation, as we shall do now.

Proposition 3.2. Let Γ be a discrete amenable group and let (μ_{ι}) be a Reiter net of means for Γ:

– each μ_{ι} is a positive sequence summing to 1 with finite support Γ_{ι} ⊆ Γ and viewed as a diagonal operator from ℓ_{Γ_{ι}}^{2} to ℓ_{Γ}^{2} , so that

∥μ_{ι}∥_{S^{1}} = ∑_{γ∈Γ_{ι}} (μ_{ι})_{γ} = 1;

– the net (μ_{ι}) satisfies, for each γ ∈ Γ, Reiter’s Property P_{1}:

∑_{β∈Γ} |(μ_{ι})_{γ^{−1} β} − (μ_{ι})_{β} | → 0.  (F.5)

Let x ∈ S_{n}^{∞} ⊗ L^{∞} = L^{∞} (tr⊗τ) and p ⩾ 1. Then

lim sup∥xμ_{ι}^{1/p}∥_{S^{p} (S_{n}^{p} )} = ∥x∥_{L^{p} (tr ⊗τ)} .

Proof. Consider x = ∑ _{γ∈Γ} x_{γ} ⊗ λ_{γ} with only a finite number of the x_{γ} ∈ S_{n}^{∞} nonzero. As

∑_{β∈Γ} |(μ_{ι})_{γ^{−1} β}^{1/2} − (μ_{ι})_{β}^{1/2} | ^{2} ⩽ ∑_{β∈Γ} |(μ_{ι})_{γ^{−1} β} − (μ_{ι})_{β} | ,

Property P_{1} implies Property P_{2}:

∥λ_{γ}μ_{ι}^{1/2} − μ_{ι}^{1/2} λ_{γ}∥_{S^{2}} → 0,

so that

∥xμ_{ι}^{1/2} − μ_{ι}^{1/2} x∥_{S^{2} (S_{n}^{2} )} → 0.

As the S_{n}^{∞} -valued matrix of x for the canonical basis of ℓ_{Γ}^{2} is (x_{rc^{−1}} )_{(r,c)∈Γ ×Γ} ,

∥xμ_{ι}^{1/2} ∥_{S^{2} (S_{n}^{2} )}^{2} = ∑_{(r,c)∈Γ ×Γ} ∥x_{rc^{−1}} ∥_{S_{2}^{n}}^{2} (μ_{ι})_{c}

= ∑_{c∈Γ} (μ_{ι})_{c} ∑_{r∈Γ} ∥x_{rc^{−1}} ∥_{S_{2}^{n}}^{2}

= ∑_{c∈Γ} (μ_{ι})_{c}∥x∥_{L^{2} (tr ⊗τ)}^{2} = ∥x∥_{L^{2} (tr ⊗τ)}^{2} .

By density and continuity, the result extends to all x ∈ L^{2}(tr⊗τ).

Let us prove now that for x ∈ L^{∞} (tr⊗τ)

lim sup∥xμ_{ι}∥_{S^{1} (S_{n}^{1} )} ⩽ ∥x∥_{L^{1} (tr ⊗τ)} .

The polar decomposition x = u|x| yields a factorisation x = ab with a = u|x|^{1/2} and b = |x|^{1/2} in L^{∞} (tr⊗τ) such that

∥a∥_{L^{2} (tr ⊗τ)} = ∥b∥_{L^{2} (tr ⊗τ)} = ∥x∥_{L^{1} (tr ⊗τ)}^{1/2}
∥a∥_{L^{∞} (tr ⊗τ)} = ∥x∥_{L^{∞} (tr ⊗τ)}^{1/2} .

Then xμ_{ι} = a(bμ_{ι}^{1/2} − μ_{ι}^{1/2} b)μ_{ι}^{1/2} + aμ_{ι}^{1/2} bμ_{ι}^{1/2} , so that the Cauchy-Schwarz inequality yields

∥xμ_{ι}∥_{S^{1} (S_{n}^{1} )} ⩽ ∥a∥_{L^{∞} (tr ⊗τ)} ∥(bμ_{ι}^{1/2} − μ_{ι}^{1/2} b)μ_{ι}^{1/2} ∥_{S^{1} (S_{n}^{1} )} + ∥aμ_{ι}^{1/2} bμ_{ι}^{1/2} ∥_{S^{1} (S_{n}^{1} )}
⩽ ∥a∥_{L^{∞} (tr ⊗τ)} ∥bμ_{ι}^{1/2} − μ_{ι}^{1/2} b∥_{S^{2} (S_{n}^{2} )} + ∥a∥_{L^{2} (tr ⊗τ)} ∥b∥_{L^{2} (tr ⊗τ)}

and therefore our claim. Now complex interpolation yields

lim sup∥xμ_{ι}^{1/p}∥_{S^{p} (S_{n}^{p} )} ⩽ ∥x∥_{L^{p} (tr ⊗τ)}  (F.6)

for x ∈ L^{∞} (tr⊗τ) and p ∈ [1, ∞]. Indeed, let u be the unitary appearing in the polar decomposition of x. Consider the function f(z) = u|x|^{pz}μ_{ι}^{z} analytic in the strip 0 < ℑz < 1 and continuous on its closure: then f(it) is a product of unitaries for t ∈ ℝ, so that

∥f(it)∥_{L^{∞} (tr ⊗τ)} = 1.

Also

∥f(1 + it)∥_{S^{1} (S_{n}^{1} )} = ∥|x|^{p}μ_{ι}∥_{S^{1} (S_{n}^{1} )}

As S^{p}(S_{n}^{p}) is the complex interpolation space (S^{∞} (S_{n}^{∞} ), S^{1}(S_{n}^{1}))_{1/p},

∥xμ_{ι}^{1/p}∥_{S^{p} (S_{n}^{p} )} = ∥f(1/p)∥_{S^{p} (S_{n}^{p} )} ⩽ ∥|x|^{p}μ_{ι}∥_{S^{1} (S_{n}^{1} )}^{1/p} .

Then, taking the upper limit and using the estimate on S^{1}(S_{n}^{1})

lim sup∥xμ_{ι}^{1/p}∥_{S^{p} (S_{n}^{p} )} ⩽ lim sup∥|x|^{p}μ_{ι}∥_{S^{1} (S_{n}^{1} )}^{1/p}
⩽ ∥|x|^{p}∥_{L^{1} (tr ⊗τ)}^{1/p} = ∥x∥_{L^{p} (tr ⊗τ)} .

The reverse inequality is obtained by duality; first note that for y ∈ L^{∞} (tr⊗τ) ,

limtryμ_{i} = τ(y).

With the above notation and the inequality for p′,

∥x∥_{L^{p}}^{p} = τ(|x|^{p}) = limtr|x|^{p}μ_{i} = limtrμ_{i}^{1−1/p} |x|^{1−p} u^{∗}xμ_{i}^{1/p}
⩽ lim sup∥μ_{i}^{1−1/p} |x|^{1−p} ∥_{S^{p′} (S_{n}^{p′})} ∥xμ_{i}^{1/p}∥_{S^{p} (S_{n}^{p} )}
= lim sup∥|x|^{1−p} μ_{i}^{1−1/p} ∥_{S^{p′} (S_{n}^{p′})} ∥xμ_{i}^{1/p}∥_{S^{p} (S_{n}^{p} )}
⩽ ∥|x|^{1−p} ∥_{L^{p′}} lim sup∥xμ_{i}^{1/p}∥_{S^{p} (S_{n}^{p} )}

so that

lim sup∥xμ_{ι}^{1/p}∥_{S^{p} (S_{n}^{p} )} = ∥x∥_{L^{p} (tr ⊗τ)}^{p} .

Remark 3.3. Let μ be any positive diagonal operator with trμ = 1 and p ⩾ 2: then ∥xμ^{1/p}∥_{S^{p} (S_{n}^{p} )} ⩽ ∥x∥_{L^{p}} for all x ∈ L^{∞} (tr⊗τ). The Reiter condition is only necessary to go below exponent 2.

In the same way, using interpolation, we can come back to approximation on both sides using Reiter means, that is

∥μ_{ι}^{1/2p} xμ_{ι}^{1/2p} ∥_{S^{p} (S_{n}^{p} )} ⩽ ∥x∥_{L^{p}}

and we have

lim sup∥μ_{ι}^{1/2p} xμ_{ι}^{1/2p} ∥_{S^{p} (S_{n}^{p} )} = ∥x∥_{L^{p} (tr ⊗τ)} .

Note that this formula is in the same spirit as the first approach of this section.

4 Transfer of lacunary sets into lacunary matrix patterns

As a first application of Theorem 2.6, let us mention that it provides a shortcut for some arguments in [37] as it permits to transfer lacunary subsets of discrete group Γ into lacunary matrix patterns in Γ × Γ. Let us first introduce the following terminology.

Definition 4.1. Let Γ be a discrete group and Λ ⊆ Γ. Let X be the reduced C^{∗}-algebra C of Γ or its noncommutative Lebesgue space L^{p} for p ∈ [1, ∞[.

(a) The set Λ is unconditional in X if the Fourier series of every x ∈ X_{Λ} converges unconditionally: there is a constant D such that

∥ ∑_{γ∈Λ′} x_{γ}ε_{γ}λ_{γ} ∥_{X} ⩽ D∥x∥_{X}

finite Λ′ ⊆ Λ and ε_{γ} ∈ 𝕋. The minimal constant D is the unconditional constant of Λ in X.

(b) If X = C, let X̃ = S^{∞} ⊗ C; if X = L^{p}, let X̃ = L^{p}(tr⊗τ). The set Λ is completely unconditional in X if the Fourier series of every x ∈ X̃_{Λ} converges unconditionally: there is a constant D such that

∥ ∑_{γ∈Λ′} x_{γ} ⊗ ε_{γ}λ_{γ} ∥X̃ ⩽ D∥x∥X̃

for finite Λ′ ⊆ Λ and ε_{γ} ∈ 𝕋. The minimal constant D is the complete unconditional constant of Λ in X.

Unconditional sets in L^{p} have been introduced as “Λ(p) sets” in [37, Definition 1.1] for p > 2: if Γ is abelian, they are Walter Rudin’s Λ(p) sets if p > 2 and his Λ(2) sets if p < 2: see [88, 15]. Asma Harcharras [37, Definition 1.5, Comments 1.9] termed completely unconditional sets in L^{p} “Λ(p)_{cb} sets” if p ∈ ]2, ∞[, and “K(p)_{cb} sets” if p ∈ ]1, 2]; her definitions are equivalent to ours by the noncommutative Khinchin inequality.

Sets that are unconditional in C have been introduced as “unconditional Sidon sets” in [16]. If Γ is amenable, Fourier multipliers are automatically c.b. on C_{Λ} , so that such sets are automatically completely unconditional in C, and there are at least three more equivalent definitions for the counterpart of Sidon sets in an abelian group. If Γ is nonamenable, these definitions are not all equivalent anymore and our notion of completely unconditional sets in C corresponds to Marek Bożejko’s “c.b. Sidon sets.”

Definition 4.2. Let 1 ⩽ p ⩽ ∞ and I ⊆ R × C.

(a) The set I is unconditional in S^{p} if the matrix representation of every x ∈ S_{I}^{p} converges unconditionally: there is a constant D such that

∥∑_{q∈I′} x_{q}ε_{q}e_{q}∥_{p} ⩽ D∥x∥_{p}

for finite I′ ⊆ I and ε_{q} ∈ 𝕋. The minimal constant D is the unconditional constant of I in S^{p}.

(b) The set I is completely unconditional in S^{p} if the matrix representation of every x ∈ S_{I}^{p}(S^{p}) converges unconditionally: there is a constant D such that

∥∑_{q∈I′} x_{q} ⊗ ε_{q}e_{q}∥_{p} ⩽ D∥x∥_{p}

for finite I′ ⊆ I and ε_{q} ∈ 𝕋. The minimal constant D is the complete unconditional constant of I in S^{p}.

Harcharras [37, Definitions 4.1 and 4.4, Remarks 4.6 (iv)] termed unconditional and completely unconditional sets in S^{p} “σ(p) sets” and “σ(p)_{cb} sets,” respectively; her definitions are equivalent by the noncommutative Khinchin inequality.

Proposition 4.3. Let Γ be a discrete group. Let Λ ⊆ Γ and consider the associated Toeplitz set Λ̋ = {(r, c) ∈ Γ × Γ : rc^{−1} ∈ Λ}. Let p ∈ [1, ∞[.

(a) If Γ is amenable, then Λ is unconditional in L^{p} if Λ̋ is unconditional in S^{p}.

(b) If Λ is completely unconditional in L^{p}, then Λ̋ is completely unconditional in S^{p}. The converse holds if Γ is amenable.

Proof. The first part of (b) follows by the argument of the proof of [37, Proposition 4.7]: let us sketch it. Consider the isometric embedding of the space Sp ; Λ̋(S^{p}) in L_{Λ}^{p} (tr⊗ tr⊗τ) that is given in the proof of Lemma 2.4 and apply the equivalent Definition 1.5 in [37] of the complete unconditionality of Λ: this gives the complete unconditionality of Λ̋ in the equivalent Definition 4.4 in [37].

Unconditionality in L^{p} expresses the uniform boundedness of relative unimodular Fourier multipliers on L_{Λ}^{p}; complete unconditionality expresses their uniform complete boundedness. Unconditionality in S^{p} expresses the uniform boundedness of relative unimodular Schur multipliers on Sp ; Λ̋; complete unconditionality expresses their uniform complete boundedness. The second part of (b) follows therefore from Theorem 2.6 (b) and (a) follows from Lemma 2.2 (a).

Remark 4.4. This transfer does not pass to the limit p = ∞: Nicholas Varopoulos ([100, Theorem 4.2], see [66, § 5] for a reader’s guide) proved that unconditional sets in S^{∞} are finite unions of patterns whose rows or whose columns contain at most one element, and this excludes sets of the form Λ̋ for any infinite Λ.

Remark 4.5. See [66, Remark 11.3] for an illustration of Proposition 4.3 (b) in a particular context.

Remark 4.6. Let p be an even integer greater or equal to 4. The existence of a σ(p)_{cb} set that is not a σ(q) set for any q > p [37, Theorem 4.9] becomes a direct consequence of Walter Rudin’s construction [88, Theorem 4.8] of a Λ(p) set that is not a Λ(q) set for any q > p, because this set has property B(p/2) [37, Definition 2.4] and is therefore Λ(p)_{cb} by [37, Theorem 1.13] (in fact, it is even “1-unconditional” in L^{p} because B(p/2) is “p/2-independence” [66, § 11].)

Remark 4.7. In the same way, Theorem 5.2 in [37] becomes a mere reformulation of [37, Proposition 3.6] if one remembers that the Toeplitz Schur multipliers are 1-complemented in the Schur multipliers for an amenable discrete group and for all classical norms. Basically results on Λ(p)_{cb} sets produce results on σ(p)_{cb} sets.

Let us now estimate the complete unconditional constant of sumsets. In the case Γ = ℤ, Harcharras [37, Prop. 2.8] proved that a completely unconditional set in L^{p} cannot contain the sumset of characters A + A for arbitrary large finite sets A: in particular, if Λ ⊇ A + A with A infinite, then Λ is not a completely unconditional set. Her proof provided thus examples of Λ(p) sets that are not Λ(p)_{cb} sets.

We generalise Harcharras’ result in two directions. Compare [54, § 1.4].

Proposition 4.8. Let Γ be a discrete group and p ≠ 2. A completely unconditional set in L^{p} cannot contain the sumset of two arbitrarily large sets. More precisely, let R and C be subsets of Γ with # R ⩾ n and # C ⩾ n^{3}. Then, for any p ⩾ 1, the complete unconditional constant of the sumset RC in L^{p} is at least n^{|1/2−1/p|} .

Proof. Let r_{1}, …, r_{n} be pairwise distinct elements in R. We shall select inducively elements c_{1}, …, c_{n} in C such that the r_{i}c_{j} are pairwise distinct. Assume there are c_{1}, …, c_{m−1} such that the induction hypothesis

∀ i, k ⩽ n ∀ j, l ⩽ m − 1 (i, j) ≠ (k, l) ⇒ r_{i}c_{j} ≠ r_{k}c_{l}.

holds. We are looking for an element c_{m} ∈ C such that

∀ i, k ⩽ n ∀ l ⩽ m − 1 r_{i}c_{m} ≠ r_{k}c_{l}.

Such an element exists as long as m ⩽ n because the set {r_{i}^{−1} r_{k}c_{l} : i, k ⩽ n, l ⩽ m − 1} has at most (n(n − 1) + 1)(m − 1) < n^{3} elements.

The end of the proof is the same as Harcharras’. The unconditional constant of the canonical basis of elementary matrices in S_{n}^{p} is n^{|1/2−1/p|} ; in particular, there is an unimodular Schur multiplier φ̆ on S_{n}^{p} of norm n^{|1/2−1/p|} (which is also its complete norm, by the way): see [77, Lemma 8.1.5]. Let Λ be the sumset {r_{i}c_{j} : i, j ⩽ n}; as the r_{i}c_{j} are pairwise distinct, we may define a sequence φ ∈ ℂ^{Λ} by φ_{r_{i} c_{j}} = φ̆_{i,j} . By Remark 2.5, the complete norm of the Fourier multiplier φ on L_{Λ}^{p} is bounded below by the complete norm of the Schur multiplier φ̆ on S_{I}^{p}.

Example 4.9. Λ = {2^{i}−2^{j} : i > j} is not a complete Λ(p) set for any p ≠ 2. Indeed, {2^{i}−2^{j} } = Λ∪−Λ does not and if Λ did, then also −Λ and Λ ∪ −Λ.

5 Toeplitz Schur multipliers on S^{p} for p < 1

When 0 < p < 1, a complete characterisation of bounded Schur multiplier of Toeplitz type has been obtained by Alexey Alexandrov and Vladimir Peller in [1, Theorem 5.1]. This result was an easy consequence of their deep results on Hankel Schur multipliers. The transfer approach provides a direct proof.

Corollary 5.1. Let 0 < p < 1. Let Γ be a discrete abelian group with dual group G. Let φ be a sequence indexed by Γ and define the associated Toeplitz matrix φ̋ ∈ ℂΛ̋ by φ̋(r, c) = φ(rc^{−1} ) for (r, c) ∈ Γ × Γ. Then the following are equivalent:

(a) The sequence φ is the Fourier transform of an atomic measure μ = ∑ a_{g}δ_{g} on G with ∑ |a_{g}|^{p} ⩽ 1;

(b) The Fourier multiplier φ is contractive on L^{p};

(c) The Fourier multiplier φ is contractive on L^{p}(S^{p});

(d) The Schur multiplier φ̋ is contractive on S^{p};

(e) The Schur multiplier φ̋ is contractive on S^{p}(S^{p}).

Proof. The implication (d) ⇒ (b) follows from Lemma 2.2 (a). The equivalence (c) ⇔ (e) follows from Theorem 2.6 (a). The characterisation (a) ⇔ (b) is an old result of Daniel Oberlin [67]. It is plain that (e) ⇒ (d). At last, (a) ⇒ (c) is obvious by the p-triangular inequality.

Remark 5.2. As a consequence, we get that the norm of a Toeplitz Schur multiplier on S^{p}(S^{p}) coincides with its norm on S^{p} when p < 1. If p ∈ {1, 2, ∞}, this holds for every Schur multiplier. Let p ∈ ]1, 2[ ∪ ]2, ∞[. Then we still do not know whether Schur multipliers are automatically c.b. on S^{p}. But from [77, Proposition 8.1.3], we know that (b) and (c) are not equivalent: if Γ is an infinite abelian group, there is a bounded Fourier multiplier on L^{p} that is not c.b. This example is easy to describe: if an infinite set A ⊆ Γ is lacunary enough, the sumset A + A is unconditional in L^{p} (see [54, Theorem 5.13]); by Proposition 4.8, it cannot be completely unconditional. In particular, this shows that in Lemma 2.2 (a) we cannot remove the restriction to truncated Toeplitz matrices in the computation of the Schur multiplier norm, that is, (b) ⇒ (d) does not hold.

Remark 5.3. Our questions may also be addressed in the case of a compact group: a measurable function φ on 𝕋 defines

– the Fourier multiplier on measurable functions on 𝕋 by x ↦ φx;

– the Schur multiplier on integral operators on L^{2}(𝕋) with kernel a measurable function x on 𝕋 × 𝕋 by x ↦ φ̋x, where φ̋(z, w) = φ(zw^{−1} ).

Victor Olevskii [69] constructed a continuous function φ that defines a bounded Fourier multiplier on the space of functions with p-summable Fourier series endowed with the norm given by ∥x∥ = (∑ |x̂(n)|^{p})^{1/p} for every p ∈ ]1, ∞[, while the corresponding Schur multiplier is not bounded on the Schatten-von-Neumann class S^{p} of operators on L^{2}(𝕋) for any p ∈ ]1, 2[ ∪]2, ∞[.

6 The Riesz projection and the Hilbert transform

In this section, we concentrate on Γ = ℤ, the dual group of 𝕋.

Proposition 6.1. Let ϱ be a linear combination of the identity and the upper triangular projection of ℕ × ℕ: there are z, w ∈ ℂ so that ϱ_{i,j} = z if i ⩽ j and ϱ_{i,j} = w if i > j. Then the norm of the Schur multiplier ϱ on S^{ψ} coincides with the norm of the Schur multiplier ϱ on S^{ψ} (S^{ψ} ).

Proof. Let a ∈ S_{m}^{ψ}(S_{n}^{ψ}): a may be considered as an m × m matrix (a_{ij} ) whose entries a_{ij} are n × n matrices, and a may be identified with the block matrix

ã = ( \begin{matrix}0 & a_{11} & 0 & a_{12} & ⋯ \\ 0 & 0 & 0 & 0 & ⋯ \\ 0 & a_{21} & 0 & a_{22} & ⋯ \\ 0 & 0 & 0 & 0 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} )

In this identification, Id_{S_{n}^{ψ}} ⊗ M_{ϱ}(a) is M_{ϱ}(ã).

The Hilbert transform ℋ is the Schur multiplier obtained by choosing z = −1 and w = 1. The upper triangular operators in S^{p} can be seen as a noncommutative H^{p} space, and ℋ corresponds exactly to the Hilbert transform in this setting (see [83, 58]). Using classical results on H^{p} spaces, all Hilbert transforms are c.b. for 1 < p < ∞ (see [101, 83, 58]).

On the torus 𝕋, the classical Hilbert transform H corresponds to the Fourier multiplier given by the sign function (with the convention sgn(0) = 1) and its norm on L^{p} is cot(π/2 max(p, p′)) = csc(π/p) + cot(π/p) for 1 < p < ∞. The story of the computation of this norm starts with a paper by Israel Gohberg and Naum Krupnik [34] for p a power of 2. The remaining cases were handled by Stylianos Pichorides [76] and Brian Cole (see [30]) independently. The most achieved results are those of Brian Hollenbeck, Nigel Kalton and Igor Verbitsky [42], but they rely on complex variable methods that are not available in the operator-valued case. When p is a power of 2 (or its conjugate), a combination of arguments of Gohberg and Krupnik [33] with some of László Zsidó [101] yields the following result.

Theorem 6.2. Let p ∈ ]1, ∞[. The norm and the complete norm of the Hilbert transform ℋ on S^{p} coincide with the complete norm of the Hilbert transform H on L^{p}: if s̋gn(i, j) = sgn(i−j) for i, j ⩾ 1,

∥ s̋gn∥_{M(S^{p})} = ∥ s̋gn∥_{M_{cb} (S^{p})} = ∥ sgn∥_{M_{cb} (L^{p})} .

If p is a power of 2, then these norms coincide with the norm of H on L^{p}:

∥ s̋gn∥_{M(S^{p})} = ∥ s̋gn∥_{M_{cb} (S^{p})} = ∥ sgn∥_{M_{cb} (L^{p})} = ∥ sgn∥_{M(L^{p})} = cot(π/2p).

Proof. Let p ⩾ 2. The norm of H on L^{p} is cot(π/2p) and the three other norms are equal by the transfer theorem 2.6 and the above proposition: we only need to compute the complete norm of H. Let H̃ = Id_{S^{p}} ⊗ H be the Hilbert transform on L^{p}(tr⊗τ). We shall use Mischa Cotlar’s trick to go from L^{p} to L^{2p}: the equality sgni sgnj + 1 = sgn(i + j)(sgni + sgnj) shows that

(H̃f)(H̃g) + fg = H̃((H̃f)g + f(H̃g)).  (F.7)

First step. The function sgn is not odd because of its value in 0: this can be fixed in the following way. Let Λ = 2ℤ + 1. The norm of H̃ on L^{p}(tr⊗τ) is equal to its norm on L_{Λ}^{p} (tr⊗τ). In fact, let D be defined by Df(z) = zf(z^{2}): D is a complete isometry on L^{p} with range L_{Λ}^{p} that commutes with H.

Second step. Let S be the real subspace of L_{Λ}^{p} (tr⊗τ) consisting of functions with values in S^{p} so that f(z) is selfadjoint for almost all z ∈ 𝕋. Let us apply Vern Paulsen’s off-diagonal trick [71, Lemma 8.1] to show that the norm of H̃ on L^{p} is equal to its norm on S. Let f ∈ L_{Λ}^{p} (tr⊗τ): identifying S_{2}^{p}(S^{p}) with S^{p},

g(z) = ( \begin{matrix}0 & f(z) \\ f(z)^{∗} & 0 \end{matrix} )

defines an element of S. As adjoining is isometric on S^{p},

∥g∥_{S} = 2^{1/p}∥f∥_{L^{p} (tr ⊗τ)} .

Let us now consider

H̃g = ( \begin{matrix}0 & H̃f \\ H̃(f^{∗}) & 0 \end{matrix} ) .

As 0 ∉ Λ by step 1, the equality sgn(−i) = − sgni holds for i ∈ Λ: this yields that H̃(f^{∗}) = −(H̃f)^{∗}. Therefore

∥H̃g∥_{S} = 2^{1/p}∥H̃f∥_{L^{p} (tr ⊗τ)} .

Third step. Let u_{p} be the norm of H̃ on L^{p}(tr⊗τ): then u_{2p} ⩽ u_{p} + \sqrt{1 + u_{p}} . It suffices to prove this estimate for f ∈ S, and by approximation we may suppose that f is a finite linear combination of terms a_{i} ⊗ z^{i} + a_{i}^{∗} ⊗ z^{−i} with a_{i} finite matrices. Note that H̃f = −(H̃f)^{∗}. Formula (F.7) with f = g yields, combined with Hölder’s inequality,

∥(H̃f)^{2}∥_{L^{p} (tr ⊗τ)} ⩽ ∥f^{2}∥_{L^{p} (tr ⊗τ)} + 2u_{p}∥f∥_{L^{2p} (tr ⊗τ)} ∥H̃f∥_{L^{2p} (tr ⊗τ)} .

Since f and H̃f take normal values,

∥f^{2}∥_{L^{p} (tr ⊗τ)} = ∥f∥_{L^{2p} (tr ⊗τ)}^{2}
∥(H̃f)^{2}∥_{L^{p} (tr ⊗τ)} = ∥H̃f∥_{L^{2p} (tr ⊗τ)}^{2} .

Therefore, if ∥f∥_{L^{2p} (tr ⊗τ)} = 1, ∥H̃f∥_{L^{2p} (tr ⊗τ)} must be smaller than the bigger root of t^{2} − 2u_{p}t − 1, that is

∥H̃f∥_{L^{2p} (tr ⊗τ)}^{2} ⩽ u_{p} + \sqrt{u_{p}^{2} + 1}

and u_{2p} ⩽ u_{p} + \sqrt{u_{p}^{2} + 1} .

Fourth step. The multiplier H is an isometry on L^{2}(tr⊗τ), so that u_{2} = 1 = cot(π/4). As cot(ϑ/2) = cotϑ + \sqrt{cot^{2} ϑ + 1} for ϑ ∈ ]0, π[, we conclude by an induction.

Unfortunately, we cannot deal with other values of p > 2 by this method.

The Riesz projection 𝒯 is the Schur multiplier obtained by choosing z = 0 and w = 1 in Proposition 6.1: it is the projection on the upper triangular part. On the torus, the classical Riesz projection T, that is the projection onto the analytic part, corresponds to the Fourier multiplier given by the indicator function χ_{ℤ^{+}} of nonnegative integers; its norm on L^{p} has been computed by Hollenbeck and Verbitsky [43]: it is csc(π/p). As for the Hilbert transform, we know that the norm and the complete norm of 𝒯 on S^{p} are equal and coincide with the complete norm of T on L^{p}; but, to the best of our knowledge, there is no simple formula like (F.7) to go from exponent p to 2p. We only obtained the following computation.

Proposition 6.3. Let p ∈ ]1, ∞[. The norm and the complete norm of the Riesz projection 𝒯 on S^{p} coincide with the complete norm of the Riesz projection T on L^{p}: if χ̋_{ℤ^{+}} (i, j) = χ_{ℤ^{+}} (i−j) for i, j ⩾ 1,

∥χ̋_{ℤ^{+}} ∥_{M(S^{p})} = ∥χ̋_{ℤ^{+}} ∥_{M_{cb} (S^{p})} = ∥χ_{ℤ^{+}} ∥_{M_{cb} (L^{p})} .

If p = 4, then these norms coincide with the norm of T on L^{p}:

∥χ̋_{ℤ^{+}} ∥_{M(S^{4} )} = ∥χ̋_{ℤ^{+}} ∥_{M_{cb} (S^{4} )} = ∥χ_{ℤ^{+}} ∥_{M_{cb} (L^{4} )} = ∥χ_{ℤ^{+}} ∥_{M(L^{4} )} = \sqrt{2}.

Proof. We shall compute the norm of 𝒯 on S^{4}. Let x be a finite upper triangular matrix and let y be a finite strictly lower triangular matrix. We have to prove that

\sqrt{2}∥x + y∥_{S^{4}} ⩾ ∥x∥_{S^{4}} .

Let us make the obvious estimates on S^{2} and use the fact that adjoining is isometric:

∥𝒯 (xx^{∗})∥_{S^{2}} = ∥𝒯 ((x + y)x^{∗})∥_{S^{2}} ⩽ ∥x + y∥_{S^{4}} ∥x∥_{S^{4}}

and similarly,

∥(Id − 𝒯 )(xx^{∗})∥_{S^{2}} = ∥(Id − 𝒯 )(x(x + y)^{∗})∥_{S^{2}} ⩽ ∥x∥_{S^{4}} ∥x + y∥_{S^{4}} .

As 𝒯 and Id − 𝒯 have orthogonal ranges,

∥x∥_{S^{4}}^{4} = ∥xx^{∗}∥_{S^{2}}^{2} = ∥(Id − 𝒯 )(xx^{∗})∥_{S^{2}}^{2} + ∥𝒯 (xx^{∗})∥_{S^{2}}^{2} ⩽ 2∥x∥_{S^{4}}^{2} ∥x + y∥_{S^{4}}^{2} .

7 Unconditional approximating sequences

The following definition makes sense for general operator spaces, but we chose to state it only in our specific context.

Definition 7.1. Let Γ be a discrete group and Λ ⊆ Γ. Let X be the reduced C^{∗}-algebra of Γ or its noncommutative Lebesgue space L^{p} for p ∈ [1, ∞[.

(a) A sequence (T_{k}) of operators on X_{Λ} is an approximating sequence if each T_{k} has finite rank and T_{k}x → x for every x ∈ X_{Λ} . It is a complete approximating sequence if the T_{k} are uniformly c.b. If X admits a complete approximating sequence, then X_{Λ} enjoys the c.b. approximation property.

(b) The difference sequence (ΔT_{k}) of a sequence (T_{k}) is given by ΔT_{1} = T_{1} and ΔT_{k} = T_{k} − T_{k−1} for k ⩾ 2. An approximating sequence (T_{k}) is unconditional if the operators

∑_{k=1}^{n} ε_{k}ΔT_{k} with n ⩾ 1 and ε_{k} ∈ {−1, 1}  (F.8)

are uniformly bounded on X_{Λ} : then X_{Λ} enjoys the unconditional approximation property.

(c) An approximating sequence (T_{k}) is completely unconditional if the operators in (F.8) are uniformly c.b. on X_{Λ} : then X_{Λ} enjoys the complete unconditional approximation property. The minimal uniform bound of these operators is the complete unconditional constant of X_{Λ} .

We may always suppose that a complete approximating sequence on C_{Λ} is a Fourier multiplier sequence: see [36, Theorem 2.1]. We may also do so on L_{Λ}^{p} if L^{∞} has the so-called QWEP: see [45, Theorem 4.4]. More precisely, the following proposition holds.

Proposition 7.2. Let Γ be a discrete group and Λ ⊆ Γ. Let X either be its reduced C^{∗}-algebra or its noncommutative Lebesgue space L^{p}, where p ∈ [1, ∞[ and L^{∞} has the QWEP. If X_{Λ} enjoys the completely unconditional approximation property with constant D, then, for every D′ > D, there is a complete approximating sequence of Fourier multipliers (φ_{k}) that realises the completely unconditional approximation property with constant D′: the Fourier multipliers ∑ _{k=1}^{n} ε_{k}Δφ_{k} are uniformly completely bounded by D′ on X_{Λ} .

Let us now describe how to skip blocks in an approximating sequence in order to construct an operator that acts like the Riesz projection on the sumset of two infinite sets. The following trick will be used in the induction below: compare [61, proof of Theorem 4.2]:

( \begin{matrix}1 & 1 0 \\ 0 & 1 0 \\ 0 & 0 0 \end{matrix}) − ( \begin{matrix}1 & 1 0 \\ 1 & 1 0 \\ 1 & 1 0 \end{matrix}) + ( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}) = ( \begin{matrix}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix})

Lemma 7.3. Let Γ be a discrete group and Λ ⊆ Γ. Suppose that Λ contains the sumset RC of two infinite sets R and C. Let (T_{k}) be either an approximating sequence on L_{Λ}^{p} with p ∈ [1, ∞[, or an approximating sequence of Fourier multipliers on C_{Λ} . Let ε > 0. There is a sequence (r_{i}) in R, a sequence (c_{i}) in C and there are indices l_{1} < k_{2} < l_{2} < k_{3} < … such that, for every n, the skipped block sum

U_{n} = T_{l_{1}} + (T_{l_{2}} − T_{k_{2}} ) + ⋯+ (T_{l_{n}} − T_{k_{n}} )  (F.9)

acts, up to ε, as the Riesz projection on the sumset {r_{i}c_{j} }_{i,j⩽n} :

\{ \begin{matrix}∥U_{n}(λ_{r_{i} c_{j}} ) − λ_{r_{i} c_{j}} ∥ < ε & if i ⩽ j ⩽ n , \\ ∥U_{n}(λ_{r_{i} c_{j}} )∥ < ε & if j < i ⩽ n . \end{matrix}  (F.10)

Proof. Let us construct the sequences and indices by induction. If n = 1, let r_{1} and c_{1} be arbitrary; there is l_{1} such that ∥T_{l_{1}} (λ_{r_{1} c_{1}} ) − λ_{r_{1} c_{1}} ∥ < ε. Suppose that r_{1}, …, r_{n}, c_{1}, …, c_{n}, l_{1}, …, k_{n}, l_{n} have been constructed. Let δ > 0 to be chosen later.

– The operator U_{n} defined by Equation (F.9) has finite rank. If it is a Fourier multiplier, one can choose an element r_{n+1} ∈ R such that U_{n}(λ_{r_{n+1} c_{j}} ) = 0 for j ⩽ n. If it acts on L_{Λ}^{p} with p ∈ [1, ∞[, one can choose an element r_{n+1} ∈ R such that ∥U_{n}(λ_{r_{n+1} c_{j}} )∥ < δ for j ⩽ n because (λ_{γ})_{γ∈Γ} is weakly null in L^{p}.

– There is k_{n+1} > l_{n} such that ∥T_{k_{n+1}} (λ_{γ}) − λ_{γ}∥ < δ for γ ∈ {r_{i}c_{j} : 1 ⩽ i ⩽ n + 1, 1 ⩽ j ⩽ n}.

– Again, choose c_{n+1} ∈ C such that ∥(U_{n} − T_{k_{n+1}} )(λ_{r_{i} c_{n+1}} )∥ < δ for i ⩽ n + 1.

– Again, choose l_{n+1} > k_{n+1} such that ∥T_{l_{n+1}} (λ_{γ}) − λ_{γ}∥ < δ for γ ∈ {r_{i}c_{j} : 1 ⩽ i, j ⩽ n + 1}. Let U_{n+1} = U_{n} + (T_{l_{n+1}} − T_{k_{n+1}} ). If i ⩽ n + 1 and j ⩽ n, then

∥ΔU_{n+1} (λ_{r_{i} c_{j}} )∥ ⩽ ∥T_{l_{n+1}} (λ_{r_{i} c_{j}} ) − λ_{r_{i} c_{j}} ∥ + ∥λ_{r_{i} c_{j}} − T_{k_{n+1}} (λ_{r_{i} c_{j}} )∥ < 2δ,

so that

∥U_{n+1} (λ_{r_{i} c_{j}} ) − λ_{r_{i} c_{j}} ∥ < ε + 2δ if i ⩽ j ⩽ n
∥U_{n+1} (λ_{r_{i} c_{j}} )∥ < ε + 2δ if j < i ⩽ n
∥U_{n+1} (λ_{r_{n+1} c_{j}} )∥ < 3δ if j ⩽ n.

If i ⩽ n + 1, then

∥U_{n+1} (λ_{r_{i} c_{n+1}} ) − λ_{r_{i} c_{n+1}} ∥ ⩽ ∥(U_{n} − T_{k_{n+1}} )(λ_{r_{i} c_{n+1}} )∥ + ∥T_{l_{n+1}} (λ_{r_{i} c_{n+1}} ) − λ_{r_{i} c_{n+1}} ∥ < 2δ.

This shows that our choice of r_{n+1} , c_{n+1} , k_{n+1} and l_{n+1} is adequate if δ is small enough.

This construction will provide an obstacle to the unconditionality of sumsets.

Theorem 7.4. Let Γ be a discrete group and Λ ⊆ Γ. Suppose that Λ contains the sumset RC of two infinite sets R and C.

(a) Let 1 < p < ∞. The complete unconditional constant of any approximating sequence for L^{p} is bounded below by the norm of the Riesz projection on S^{p}, and thus by cscπ/p.

(b) The spaces L_{Λ}^{1} and C_{Λ} do not enjoy the complete unconditional approximation property.

(c) If Γ is amenable, then the space C_{Λ} does not enjoy the unconditional approximation property.

Proof. Let (T_{k}) be an approximating sequence on L_{Λ}^{p}. By Lemma 7.3, for every ε > 0 and every n, there are elements r_{1}, …, r_{n} ∈ R, c_{1}, …, c_{n} ∈ C such that the Fourier multiplier φ given by the indicator function of {r_{i}c_{j} }_{i⩽j} is near to a skipped block sum U_{n} of (T_{k}) in the sense that ∥U_{n}(λ_{r_{i} c_{j}} )− φ_{r_{i} c_{j}} λ_{r_{i} c_{j}} ∥ < ε. But U_{n} is the mean of two operators of the form (F.8): its complete norm will provide a lower bound for the complete unconditional constant of X_{Λ} . Let us repeat the argument of Lemma 2.4 with x ∈ S_{n}^{p}: as

∥ ∑_{i,j=1}^{n} x_{i,j} e_{i,j} ∥_{S_{n}^{p}} = ∥( ∑_{i=1}^{n} e_{i,i} ⊗ λ_{r_{i}} ) ( ∑_{i,j=1}^{n} x_{i,j} e_{i,j} ⊗ λ_{ϵ}) ( ∑_{j=1}^{n} e_{j,j} ⊗ λ_{c_{j}} ) ∥_{L^{p} (tr ⊗τ)}

= ∥∑_{i=1}^{n} x_{i,j} e_{i,j} ⊗ λ_{r_{i} c_{j}} ∥_{L^{p} (tr ⊗τ)}

and

∥∑_{i=1}^{n} x_{i,j} e_{i,j} ⊗ (U_{n}(λ_{r_{i} c_{j}} ) − φ_{r_{i} c_{j}} λ_{r_{i} c_{j}} )∥_{L^{p} (tr ⊗τ)} < n^{2}ε∥x∥_{S_{n}^{p}} ,

the complete norm of U_{n} is nearly bounded below by the norm of the Riesz projection on S_{n}^{p}:

∥∑_{i=1}^{n} x_{i,j} e_{i,j} ⊗ U_{n}(λ_{r_{i} c_{j}} )∥_{L^{p} (tr ⊗τ)} > ∥∑_{i⩽j} x_{i,j} e_{i,j} ⊗ λ_{r_{i} c_{j}} ∥_{L^{p} (tr ⊗τ)} − n^{2}ε∥x∥_{S_{n}^{p}}
= ∥𝒯 (x)∥_{S_{n}^{p}} − n^{2}ε∥x∥_{S_{n}^{p}} .

This proves as well (a) as the first assertion in (b), because the Riesz projection is unbounded on S^{1}. Let (T_{k}) be an approximating sequence on C_{Λ} : by Lemma 7.2, we may suppose that (T_{k}) is a sequence of Fourier multipliers. Thus the second assertion in (b) follows from Lemma 7.3 combined with the preceding argument (where S_{n}^{p} is replaced by S_{n}^{∞} and L^{p}(tr⊗τ) by S_{n}^{∞} ⊗ C) and the unboundedness of the Riesz projection on S^{∞} . For (c), note that the Fourier multipliers T_{k} are automatically c.b. on C_{Λ} if Γ is amenable [25, Corollary 1.8].

Theorem 7.4 (b) has been devised originally to prove that the Hardy space H^{1}, corresponding to the case Λ = ℕ ⊆ ℤ and p = 1, admits no completely unconditional basis: see [85, 86]. Theorem 7.4 (c) both generalises the fact that a sumset cannot be a Sidon set (see [54, §§ 1.4, 6.6] for two proofs and historical remarks, or [51, Proposition IV.7]) and Daniel Li’s result [50, Corollary 13] that the space C_{Λ} does not have the “metric” unconditional approximation property if Γ is abelian and Λ contains a sumset. Li [50, Theorem 10] also constructed a set Λ ⊆ ℤ such that C_{Λ} has this property while Λ contains the sumset of arbitrarily large sets. This theorem also provides a new proof that the disc algebra has no unconditional basis and answers [64, Question 6.1.6].

Example 7.5. Neither the span of products {r_{i}r_{j} } of two Rademacher functions in the space of continuous functions on {−1, 1}^{∞} nor the span of products {s_{i}s_{j} } of two Steinhaus functions in the space of continuous functions on 𝕋^{∞} have an unconditional basis.

8 Relative Schur multipliers of rank one

Let ϱ be an elementary Schur multiplier on S^{∞} , that is,

ϱ = x ⊗ y = (x_{r}y_{c})_{(r,c)∈R×C} :

then its norm is sup_{r∈R} |x_{r}| sup_{c∈C} |y_{c}|. How is this norm affected if ϱ is only partially specified, that is, if the action of ϱ is restricted to matrices with a given support?

Theorem 8.1. Let I ⊆ R × C and consider (x_{r})_{r∈R} and (y_{c})_{c∈C} . The relative Schur multiplier on S_{I}^{∞} given by (x_{r}y_{c})_{(r,c)∈I} has norm sup_{(r,c)∈I} |x_{r}y_{c}|.

Note that the norm of the Schur multiplier (x_{r}y_{c})_{(r,c)∈I} is bounded by sup_{r∈R} |x_{r}| sup_{c∈C} |y_{c}| because the matrix (x_{r}y_{c})_{(r,c)∈R×C} trivially extends (x_{r}y_{c})_{(r,c)∈I} ; the proof below provides a constructive nontrivial extension of this Schur multiplier that is a composition of ampliations of the Schur multiplier in the following lemma.

Lemma 8.2. The Schur multiplier ( \begin{matrix}\overline{z} & w \\ \overline{w} & z \end{matrix}) has norm max(|z|, |w|) on S_{2}^{∞} .

Proof. This follows from the decomposition

( \begin{matrix}\overline{z} & w \\ \overline{w} & z \end{matrix}) = \frac{|z| + |w|}{2} ( \begin{matrix}t̄u \\ tū \end{matrix}) ⊗ (\overline{tu} tu) + \frac{|z| − |w|}{2} ( \begin{matrix}t̄u \\ −tū \end{matrix} ) ⊗ (\overline{tu} −tu ) ,

where t, u ∈ 𝕋 are chosen so that z = |z|t^{2} and w = |w|u^{2}.

Proof of Theorem 8.1. We may suppose that C is the finite set {1, …, m} and that R is the finite set {1, …, n}, that each y_{c} is nonzero and that each row in R contains an element of I. We may also suppose that (|x_{r}|)_{r∈R} and (|y_{c}|)_{c∈C} are nonincreasing sequences. For each r ∈ R let c_{r} be the least column index of elements of I in or above row r: in other words,

c_{r} = min_{r′⩽r}min{c : (r′, c) ∈ I}.

The sequence (c_{r})_{r∈R} is nonincreasing. Let us define its inverse (r_{c})_{c∈C} in the sense that r_{c} ⩽ r ⇔ c_{r} ⩽ c: for each c ∈ C let r_{c} = min{r : c_{r} ⩽ c}. Given r, let r′ ⩽ r be such that (r′, c_{r}) ∈ I: then |x_{r}y_{c_{r}} | ⩽ |x_{r′}y_{c_{r}} |, so that sup_{r∈R} |x_{r}y_{c_{r}} | ⩽ sup_{(r,c)∈I} |x_{r}y_{c}| and the rank 1 Schur multiplier

ϱ_{0} = (x_{r}y_{c_{r}} )_{(r,c)∈R×C}

with pairwise equal columns is bounded by sup_{(r,c)∈I} |x_{r}y_{c}| on S_{n}^{∞} . We will now “correct” ϱ_{0} without increasing its norm so as to make it an extension of (x_{r}y_{c})_{(r,c)∈I} . Let r ∈ R and c′ ⩾ c_{r}: then

x_{r}y_{c′} = x_{r}y_{c_{r}} \frac{y_{c_{r} +1}}{y_{c_{r}}} ⋯\frac{y_{c′}}{y_{c′−1}} = x_{r}y_{c_{r}} ∏_{c_{r} ⩽c⩽c′−1} \frac{y_{c+1}}{y_{c}}

= x_{r}y_{c_{r}} ∏_{\begin{matrix}r⩾r_{c} \\ c′⩾c+1 \end{matrix}} \frac{y_{c+1}}{y_{c}} .

This shows that it suffices to compose the Schur multiplier ϱ_{0} with the m−1 rank 2 Schur multipliers with block matrix

ϱ_{c} =  ( \begin{matrix}& 1 ···c & c+1 ···m \\ 1 ; ⋮ ; r_{c} −1 & \overline{( \frac{y_{c+1}}{y_{c}} )} & 1 \\ r_{c} ; ⋮ ; n & 1 & \frac{y_{c+1}}{y_{c}} \end{matrix} ) ,

each of which has norm 1 on S_{n}^{∞} by Lemma 8.2.

Remark 8.3. We learnt after submitting this article that Timur Oikhberg proved independently Theorem 8.1 and gave some applications to it: compare [68].

Remark 8.4. As an illustration, let C = R = {1, ..., n} and I = {(r, c) : r ⩾ c}, and let a_{i} be an increasing sequence of positive numbers. Take x_{r} = a_{r} and y_{c} = 1/a_{c}. Then the relative Schur multiplier (a_{r}/a_{c})_{r⩽c} has norm 1. The above proof actually constructs the norm 1 extension (min(a_{r}/a_{c}, a_{c}/a_{r}))_{(r,c)} . If we put a_{i} = e^{x_{i}} , we recover that (e^{−|x_{r} −x_{c} |} )_{(r,c)} is positive definite, that is, |·| is a conditionally negative function on ℝ.

2000 Mathematics subject classification: Primary 47B49; Secondary 43A22, 43A46, 46B28.

Key words and phrases: Fourier multiplier, Toeplitz Schur multiplier, lacunary set, unconditional approximation property, Hilbert transform, Riesz projection.

The Sidon constant of sets with three elements

Abstract

We solve an elementary extremal problem on trigonometric polynomials and obtain the exact value of the Sidon constant for sets with three elements {n_{0}, n_{1}, n_{2}}: it is

sec (π gcd(n_{1} − n_{0}, n_{2} − n_{0})/2max |n_{i} − n_{j} | ) .

1 Introduction

Let Λ = {λ_{0}, λ_{1}, λ_{2}} be a set of three frequencies and ϱ_{0}, ϱ_{1}, ϱ_{2} three positive intensities. We solve the following extremal problem:

(†)

To find ϑ_{0}, ϑ_{1}, ϑ_{2} three phases such that, putting c_{j} = ϱ_{j} e^{iϑ_{j}} , the maximum max_{t} |c_{0}e^{iλ_{0} t} + c_{1}e^{iλ_{1} t} + c_{2}e^{iλ_{2} t} | is minimal.

This enables us to generalise a result of D. J. Newman. He solved the following extremal problem for Λ = {0, 1, 2}:

(‡)

To find f(t) = c_{0}e^{iλ_{0} t} + c_{1}e^{iλ_{1} t} + c_{2}e^{iλ_{2} t} with ∥f∥_{∞} = max_{t} |f(t)| ⩽ 1 such that ∥f̂∥_{1} = |c_{0}| + |c_{1}| + |c_{2}| is maximal.

Note that for such an f, ∥f̂∥_{1} is the Sidon constant of Λ. Newman’s argument is the following (see [94, Chapter 3]): by the parallelogram law,

max_{t}|f(t)|^{2} = max_{t}|f(t)|^{2} ∨ |f(t + π)|^{2}
⩾ max_{t}(|f(t)|^{2} + |f(t + π)|^{2})/2
= max_{t}(|c_{0} + c_{1}e^{it} + c_{2}e^{i2t}|^{2} + |c_{0} − c_{1}e^{it} + c_{2}e^{i2t}|^{2})/2
= max_{t}|c_{0} + c_{2}e^{i2t}|^{2} + |c_{1}|^{2} = (|c_{0}| + |c_{2}|)^{2} + |c_{1}|^{2}
⩾ (|c_{0}| + |c_{1}| + |c_{2}|)^{2}/2

and equality holds exactly for multiples and translates of f(t) = 1 + 2ie^{it} + e^{i2t}.

Let us describe this paper briefly. We use a real-variable approach: Problem (†) reduces to studying a function of form

Φ(t, ϑ) = |1 + re^{iϑ} e^{ikt} + se^{ilt}|^{2} for r, s > 0, k ≠ l ∈ ℤ^{∗}

and more precisely Φ^{∗}(ϑ) = max_{t} Φ(t, ϑ). We obtain the variations of Φ^{∗}: the point is that we find “by hand” a local minimum of Φ^{∗} and that any two minima of Φ^{∗} are separated by a maximum of Φ^{∗}, which corresponds to an extremal point of Φ and therefore has a handy description. The solution to Problem (‡) then turns out to derive easily from this.

The initial motivation was twofold. In the first place, we wanted to decide whether sets Λ = {λ_{n}} such that λ_{n+1} /λ_{n} is bounded by some q may have a Sidon constant arbitrarily close to 1 and to find evidence among sets with three elements. That there are such sets, arbitrarily large but finite, may in fact be proven by the method of Riesz products in [47, Appendix V, §1.II]. In the second place, we wished to show that the real and complex unconditionality constants are distinct for basic sequences of characters e^{int}; we prove however that they coincide in the space 𝒞 (𝕋) for sequences with three terms.

Notation. 𝕋 = { z ∈ ℂ : |z| = 1 } and e_{λ}(z) = z^{λ} for z ∈ 𝕋 and λ ∈ ℤ.

2 Definitions

Definition 2.1. (1) Let Λ ⊆ ℤ. Λ is a Sidon set if there is a constant C such that for all trigonometric polynomials f(t) = ∑ _{λ∈Λ} c_{λ} e^{iλt} with spectrum in Λ we have

∥f̂∥_{1} = ∑_{λ∈Λ} |c_{λ}| ⩽ C max_{t}|f(t)| = ∥f∥_{∞} .

The optimal C is called the Sidon constant of Λ.

(2) Let X be a Banach space. The sequence (x_{n}) ⊆ X is a real (vs. complex) unconditional basic sequence in X if there is a constant C such that

∥∑ ϑ_{n}c_{n}x_{n} ∥_{X} ⩽ C∥∑ c_{n}x_{n} ∥_{X}

for every real (vs. complex) choice of signs ϑ_{n} ∈ {−1, 1} (vs. ϑ_{n} ∈ 𝕋) and every finitely supported family of coefficients (c_{n}). The optimal C is the real (vs. complex) unconditionality constant of (x_{n}) in X.

Let us state the two following well known facts.

Proposition 2.2. (1) The Sidon constant of Λ is the complex unconditionality constant of the sequence of functions (e_{λ})_{λ∈Λ} in the space 𝒞 (𝕋).

(2) The complex unconditionality constant is at most π/2 times the real unconditionality constant.

Proof. (1) holds because ∥∑ ϑ_{λ}c_{λ} e_{λ} ∥_{∞} = ∑ |c_{λ}| for ϑ_{λ} = \overline{c_{λ}}/|c_{λ}|.

(2) Because the complex unconditionality constant of the sequence (ϵ_{n}) of Rademacher functions in 𝒞 ({−1, 1}^{∞} ) is π/2 (see [93]),

sup_{ϑ_{n} ∈𝕋} ∥∑ ϑ_{n}c_{n}x_{n} ∥_{X} = sup_{x^{∗} ∈B_{X^{∗}}} sup_{ϑ_{n} ∈𝕋} sup_{ϵ_{n} =±1} |∑ ϑ_{n}c_{n}⟨x^{∗}, x_{n}⟩ϵ_{n} |
⩽ π/2 sup_{x^{∗} ∈B_{X^{∗}}} sup_{ϵ_{n} =±1} |∑ c_{n}⟨x^{∗}, x_{n}⟩ϵ_{n} |
= π/2 sup_{ϵ_{n} =±1} ∥∑ ϵ_{n}c_{n}x_{n} ∥_{X} .

Furthermore the real unconditionality constant of (ϵ_{n}) in 𝒞 ({−1, 1}^{∞} ) is 1: therefore the factor π/2 is optimal.

Let us straighten out the expression of the Sidon constant. For

f(t) = c_{0}e^{iλ_{0} t} + c_{1}e^{iλ_{1} t} + c_{2}e^{iλ_{2} t} , c_{j} = ϱ_{j} e^{iϑ_{j}} ,

the supremum norm ∥f∥_{∞} of f is equal to

∥ϱ_{0} + ϱ_{1}e^{iϑ} e_{λ_{1} −λ_{0}} + ϱ_{2}e_{λ_{2} −λ_{0}} ∥_{∞} , ϑ = \frac{λ_{1} − λ_{2}}{λ_{2} − λ_{0}} ϑ_{0} + ϑ_{1} + \frac{λ_{0} − λ_{1}}{λ_{2} − λ_{0}} ϑ_{2}  (G.1)

and therefore the Sidon constant C of Λ = {λ_{0}, λ_{1}, λ_{2}} may be written

C = max_{r,s>0,ϑ} (1 + r + s)/∥1 + re^{iϑ} e_{k} + se_{l}∥_{∞} with \{ \begin{matrix}k = λ_{1} − λ_{0} \\ l = λ_{2} − λ_{0}. \end{matrix}  (G.2)

By change of variables, we may suppose w.l.o.g. that k and l are coprime.

3. A SOLUTION TO EXTREMAL PROBLEM (†)  95

3 A solution to Extremal problem (†)

Let us first establish

Lemma 3.1. Let λ_{1}, …, λ_{k} ∈ ℤ^{∗} and ϱ_{1}, …, ϱ_{k} > 0. Let

f(t, ϑ) = 1 + ϱ_{1}e^{i(λ_{1} t+ϑ_{1} )} + ⋯+ ϱ_{k−1} e^{i(λ_{k−1} t+ϑ_{k−1} )} + ϱ_{k}e^{iλ_{k} t}

and Φ(t, ϑ) = |f(t, ϑ)|^{2}. The critical points (t, ϑ) such that ∇Φ(t, ϑ) = 0 satisfy either f(t, ϑ) = 0 or λ_{1}t + ϑ_{1} ≡ ⋯≡ λ_{k−1} t + ϑ_{k−1} ≡ λ_{k}t ≡ 0 mod π.

Proof. As Φ = (ℜf)^{2} + (ℑf)^{2}, the critical points (t, ϑ) satisfy

\{ \begin{matrix}ℜ\frac{∂f}{∂t} (t, ϑ) ℜf(t, ϑ) + & ℑ\frac{∂f}{∂t} (t, ϑ) ℑf(t, ϑ) = 0 \\ − sin(λ_{i}t + ϑ_{i}) ℜf(t, ϑ) + & cos(λ_{i}t + ϑ_{i}) ℑf(t, ϑ) = 0 & (1 ⩽ i ⩽ k − 1), \end{matrix}

which simplifies to

− sin(λ_{i}t + ϑ_{i}) ℜf(t, ϑ) + cos(λ_{i}t + ϑ_{i}) ℑf(t, ϑ) = 0 (1 ⩽ i ⩽ k, ϑ_{k} = 0).

Suppose that f(t, ϑ) ≠ 0: then the system above implies that

− sin(λ_{i}t + ϑ_{i}) cos(λ_{j} t + ϑ_{j}) + cos(λ_{i}t + ϑ_{i}) sin(λ_{j} t + ϑ_{j}) = 0 (1 ⩽ i, j ⩽ k, ϑ_{k} = 0)

and it simplifies therefore to

sin(λ_{i}t + ϑ_{i}) = 0 (1 ⩽ i ⩽ k, ϑ_{k} = 0).

The following result is the core of the paper.

Lemma 3.2. Let r, s > 0, k, l ∈ ℤ^{∗} distinct and coprime. Let

Φ(t, ϑ) = |1 + re^{iϑ} e^{ikt} + se^{ilt}|^{2}
= 1 + r^{2} + s^{2} + 2r cos(kt + ϑ) + 2s coslt + 2rs cos((l − k)t − ϑ).

Let Φ^{∗}(ϑ) = max_{t} Φ(t, ϑ). Then Φ^{∗} is an even function with period 2π/|l| that decreases on [0, π/|l|]. Therefore min_{ϑ} Φ^{∗}(ϑ) = Φ^{∗}(π/l).

Proof. Φ^{∗} is continuous (see [81, Chapter 5.4]) and even, as Φ(t, −ϑ) = Φ(−t, ϑ). Φ^{∗} is (2π/|l|)- periodical: let j ∈ ℤ be such that jk ≡ 1 mod. l. Then

Φ(t + 2jπ/l, ϑ) = |1 + re^{i(ϑ+2πjk/l)} e^{ikt} + se^{ilt}|^{2} = Φ(t, ϑ + 2π/l).

Thus Φ^{∗} attains its minimum on [0, π/|l|]. Furthermore, we have

Φ(−t − 2jπ/l, π/l − ϑ) = Φ(t + 2jπ/l, −π/l + ϑ) = Φ(t, π/l + ϑ),

so that Φ^{∗} has an extremum at π/l. Now

Φ^{∗}(π/l + ϑ) = Φ^{∗}(π/l) + |ϑ| max_{Φ(t,π/l)=Φ^{∗} (π/l)}  | \frac{∂Φ}{∂ϑ} (t, π/l)| + o(ϑ).

Choose a t such that Φ(t, π/l) = Φ^{∗}(π/l). If ∂Φ/∂ϑ(t, π/l) ≠ 0, then this shows that Φ^{∗} has a local minimum and a cusp at π/l. Let us now suppose that ∂Φ/∂ϑ(t, π/l) = 0. If Φ^{∗} had a local maximum at π/l, then (t, π/l) would be a critical point of Φ, so that by Lemma 3.1 cos(kt+π/l) = δ, coslt = ϵ, cos((l − k)t − π/l) = δϵ for some δ, ϵ ∈ {−1, 1}. One necessarily would have (δ, ϵ) ≠ (1, 1). Furthermore,

\frac{∂^{2}Φ}{∂ϑ^{2}} (t, π/l) = −2rδ(1 + sϵ) ⩽ 0
|\begin{matrix}∂^{2}Φ/∂t^{2} & ∂^{2}Φ/∂t∂ϑ \\ ∂^{2}Φ/∂ϑ∂t & ∂^{2}Φ/∂ϑ^{2} \end{matrix} | (t, π/l) = 4rsl^{2}(δϵ + rϵ + sδ) ⩾ 0,

which would imply ϵ = −1, r = 0, s = 1. Therefore Φ^{∗} has a local minimum at π/l. Let us show that then Φ^{∗} must decrease on [0, π/|l|]. Otherwise there are 0 ⩽ ϑ_{0} < ϑ_{1} ⩽ π/|l| such that Φ^{∗}(ϑ_{1}) > Φ^{∗}(ϑ_{0}). As π/|l| is a local minimum, there is a ϑ_{0} < ϑ^{∗} < π/|l| such that

Φ^{∗}(ϑ^{∗}) = max_{ϑ_{0} ⩽ϑ⩽π/|l|} Φ^{∗}(ϑ) = max_{\begin{matrix}0⩽t<2π \\ ϑ_{0} ⩽ϑ⩽π/|l| \end{matrix}} Φ(t, ϑ),

i.e., there further is some t^{∗} such that Φ has a local maximum at (t^{∗}, ϑ^{∗}). But then kt^{∗} +ϑ^{∗} ≡ lt^{∗} ≡ 0 mod π and ϑ^{∗} ≡ 0 mod π/l and this is false.

By Computation (G.1) and Lemma 3.2, we obtain

Theorem 3.3. Let λ_{0}, λ_{1}, λ_{2} ∈ ℝ and ϱ_{0}, ϱ_{1}, ϱ_{2} > 0. The solution to Extremal problem (†) is the following.

– If the smallest additive group containing λ_{1} − λ_{0} and λ_{2} − λ_{0} is dense in ℝ, then the maximum is independent of the phases ϑ_{0}, ϑ_{1}, ϑ_{2} and makes ϱ_{0} + ϱ_{1} + ϱ_{2}.

– Otherwise let d = gcd(λ_{1} − λ_{0}, λ_{2} − λ_{0}) be a generator of this group. Then the sought phases ϑ_{0}, ϑ_{1}, ϑ_{2} are given by

ϑ_{0}(λ_{2} − λ_{1}) + ϑ_{1}(λ_{0} − λ_{2}) + ϑ_{2}(λ_{1} − λ_{0}) ≡ dπ mod 2dπ.

In particular, these phases may be chosen among 0 and π.

4 A solution to Extremal problem (‡)

There are two cases where one can make explicit computations by Lemma 3.2.

Example 4.1. The real and complex unconditionality constant of \{0, 1, 2\} in 𝒞 (𝕋) is \sqrt{2}. Indeed, a case study shows that

∥1 + ire_{1} + se_{2}∥_{∞} = \{ \begin{matrix}r + |s − 1| & if r|s − 1| ⩾ 4s \\ (1 + s)(1 + r^{2}/4s)^{1/2} & if r|s − 1| ⩽ 4s \end{matrix}

and this permits to compute the maximum (G.2), which is obtained for r = 2, s = 1. This yields another proof to Newman’s result presented in the Introduction.

Example 4.2. The real and complex unconditionality constant of \{0, 1, 3\} in 𝒞 (𝕋) is 2/\sqrt{3}. Indeed, a case study shows that ∥1 + re^{iπ/3} e_{1} + se_{3}∥_{∞} makes

\{ \begin{matrix}1 + r − s & if s ⩽ r/(4r + 9) \\ ( \frac{2}{27} s(r^{2} + 9 + 3r/s)^{3/2} − \frac{2}{27} r^{3}s + \frac{2}{3}r^{2} + rs + s^{2} + 1)^{1/2} & if s ⩾ r/(4r + 9) \end{matrix}

and this permits to compute the maximum (G.2), which is obtained exactly at r = 3/2, s = 1/2.

These examples are particular cases of the following theorem.

Theorem 4.3. Let λ_{0}, λ_{1}, λ_{2} ∈ ℤ be distinct. Then the Sidon constant of Λ = {λ_{0}, λ_{1}, λ_{2}} is sec(π/2n), where n = max|λ_{i} − λ_{j} |/ gcd(λ_{1} − λ_{0}, λ_{2} − λ_{0}).

Proof. We may suppose λ_{0} < λ_{1} < λ_{2}. Let k = (λ_{1} − λ_{0})/ gcd(λ_{1} − λ_{0}, λ_{2} − λ_{0}) and l = (λ_{2} − λ_{0})/ gcd(λ_{1} − λ_{0}, λ_{2} − λ_{0}). By Lemma 3.2, the Arithmetic-Geometric Mean Inequality bounds the Sidon constant C of {0, k, l} in the following way:

C = max_{r,s>0} \frac{1 + r + s}{∥1 + re^{iπ/l} e_{k} + se_{l}∥_{∞}}  ⩽ max_{r,s>0} \frac{1 + r + s}{|1 + re^{iπ/l} + s|}

= max_{r,s>0}( 1 − sin^{2} \frac{π}{2l} \frac{4r(1 + s)}{(1 + r + s)^{2}} ) ^{−1/2}
⩽ (1 − sin^{2}(π/2l))^{−1/2} = sec(π/2l).

This inequality is sharp: we have equality for s = k/(l − k) and r = 1 + s. In fact the derivative of |1 + re^{iπ/l} e^{ikt} + se^{ilt}|^{2} is then

\frac{8kl}{k − l} cos \frac{kt + π/l}{2} sin \frac{lt}{2} cos \frac{(l − k)t − π/l}{2} ,

so that its critical points are

\frac{2j + 1}{k} π − \frac{π}{kl}, \frac{2j}{l} π, \frac{2j + 1}{l − k} π + \frac{π}{l(l − k)} : j ∈ ℤ,

where it makes

4s^{2} sin^{2} \frac{2j + 1 + l}{2k} π, 4r^{2} cos^{2} \frac{2j + 1}{2l} π, 4 cos^{2} \frac{2j + 1 + k}{2(l − k)} π : j ∈ ℤ.

Therefore the maximum of |1 + re^{iπ/l} e^{ikt} + se^{ilt}| is 2r cos(π/2l).

This proof and (G.1) yield also the more precise

Proposition 4.4. Let Λ = {λ_{0}, λ_{1}, λ_{2}} ⊆ ℤ. The solution to Extremal problem (‡) is a multiple of

f(t) = ϵ_{0} |λ_{1} − λ_{2}| e^{iλ_{0} t} + ϵ_{1} |λ_{0} − λ_{2}| e^{iλ_{1} t} + ϵ_{2} |λ_{0} − λ_{1}| e^{iλ_{2} t}

with ϵ_{0}, ϵ_{1}, ϵ_{2} ∈ {−1, 1} real signs such that

– ϵ_{0}ϵ_{1} = −1 if 2^{j} | λ_{1} − λ_{0} and 2^{j} ∤ λ_{2} − λ_{0} for some j;

– ϵ_{0}ϵ_{2} = −1 if 2^{j} ∤ λ_{1} − λ_{0} and 2^{j} | λ_{2} − λ_{0} for some j;

– ϵ_{1}ϵ_{2} = −1 otherwise.

The Sidon constant of Λ is attained for this f. Therefore the complex and real unconditionality constants of {e_{λ}}_{λ∈Λ} in 𝒞 (𝕋) coincide for sets Λ with three elements.

5 Some consequences

Let us underline the following consequences of our computation.

Corollary 5.1. (1) The Sidon constant of sets with three elements is at most \sqrt{2}.

(2) The Sidon constant of \{0, n, 2n\} is \sqrt{2}, while the Sidon constant of \{0, n + 1, 2n\} is at most sec(π/2n) = 1 + π^{2}/8n^{2} + o(n^{−2} ) and thus arbitrarily close to 1.

(3) The Sidon constant of {λ_{0} < λ_{1} < λ_{2}} does not depend on λ_{1} but on the g.c.d. of λ_{1} − λ_{0} and λ_{2} − λ_{0}.

Theorem 4.3 also shows anew that no set of integers with more than two elements has Sidon constant 1 (see [94, p. 21] or [21]). Recall now that Λ = {λ_{n}} ⊆ ℤ is a Hadamard set if there is a q > 1 such that |λ_{n+1} /λ_{n}| ⩾ q for all n. By [63, Cor. 9.4], the Sidon constant of Λ is at most 1 + π^{2}/(2q^{2} − 2 − π^{2}) if q > \sqrt{π^{2}/2 + 1} ≈ 2.44. On the other hand Theorem 4.3 shows

Corollary 5.2. (1) If there is an integer q ⩾ 2 such that Λ ⊇ {λ, λ + μ, λ + qμ} for some integers λ and μ, then the Sidon constant of Λ is at least

sec(π/2q) > 1 + π^{2}/(8q^{2}).

(2) In particular, we have the following bounds for the Sidon constant C of the set Λ = {q^{k}}, q ∈ ℤ \ {0, ±1, ±2}:

1 + \frac{π^{2}}{8 max(−q, q + 1)^{2}} < C ⩽ 1 + \frac{π^{2}}{2q^{2} − 2 − π^{2}} .

6 Three questions

(a) Is there a set Λ for which the real and complex unconditionality constants of {e_{λ}}_{λ∈Λ} in 𝒞 (𝕋) differ? The same question is open in spaces L^{p}(𝕋), 1 ⩽ p < ∞, and even for the case of three element sets if p is not a small even integer, and especially for the set {0, 1, 2, 3} in any space but L^{2}(𝕋).

(b) Let q > 1. Are there infinite sets Λ = {λ_{n}} such that |λ_{n+1} /λ_{n}| ⩽ q with Sidon constant arbitrarily close to 1? What about the sequence of integer parts of the powers of a transcendental number σ > 1 (see [63, Cor. 2.10, Prop. 3.2])?

(c) The only set with more than three elements with known Sidon constant is {0, 1, 2, 3, 4}, for which it makes 2 (see [94, Chapter 3]). Can one compute the Sidon constant of sets with four elements? I conjecture that the Sidon constant of {0, 1, 2, 3} is 5/3.

The maximum modulus of a trigonometric trinomial

Abstract

Let Λ be a set of three integers and let C_{Λ} be the space of 2π-periodic functions with spectrum in Λ endowed with the maximum modulus norm. We isolate the maximum modulus points x of trigonometric trinomials T ∈ C_{Λ} and prove that x is unique unless |T | has an axis of symmetry. This enables us to compute the exposed and the extreme points of the unit ball of C_{Λ} , to describe how the maximum modulus of T varies with respect to the arguments of its Fourier coefficients and to compute the norm of unimodular relative Fourier multipliers on C_{Λ} . We obtain in particular the Sidon constant of Λ.

1 Introduction

Let λ_{1}, λ_{2} and λ_{3} be three pairwise distinct integers. Let r_{1}, r_{2} and r_{3} be three positive real numbers. Given three real numbers t_{1}, t_{2} and t_{3}, let us consider the trigonometric trinomial

T (x) = r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + r_{3}e^{i(t_{3} +λ_{3} x)}  (H.1)

for x ∈ ℝ. The λ’s are the frequencies of the trigonometric trinomial T, the r’s are the moduli or intensities and the t’s the arguments or phases of its Fourier coefficients r_{1}e^{it_{1}} , r_{2}e^{it_{2}} and r_{3}e^{it_{3}} .

\begin{matrix}−1 \\ −e^{iπ/3} \end{matrix}

Figure H.1: The unit circle, the hypotrochoid H with equation z = 4e^{−i2x} + e^{ix}, the segment from −1 to the unique point on H at maximum distance and the segments from −e^{iπ/3} to the two points on H at maximum distance.

The maximum modulus of a trigonometric trinomial has an interpretation in plane geometry. Without loss of generality, we may assume that λ_{2} is between λ_{1} and λ_{3}. Let H be the curve with complex equation

z = r_{1}e^{i(t_{1} −(λ_{2} −λ_{1} )x)} + r_{3}e^{i(t_{3} +(λ_{3} −λ_{2} )x)}  (−π < x ⩽ π).  (H.2)

H is a hypotrochoid: it is drawn by a point at distance r_{3} to the centre of a circle with radius r_{1}|λ_{2} − λ_{1}|/|λ_{3} − λ_{2}| that rolls inside another circle with radius r_{1}|λ_{3} − λ_{1}|/|λ_{3} − λ_{2}|. The maximum modulus of (H.1) is the maximum distance of points z ∈ H to a given point −r_{2}e^{it_{2}} of the complex plane. If λ_{3} − λ_{2} = λ_{2} − λ_{1}, then H is the ellipse with centre 0, half major axis r_{1} + r_{3} and half minor axis |r_{1} − r_{3}|. Note that an epitrochoid (Ptolemy’s epicycle) amounts also to a hypotrochoid. Figure H.1 illustrates the particular case T (x) = 4e^{−i2x} + e^{it} + e^{ix}.

We deduce an interval on which T attains its maximum modulus independently of the moduli of its Fourier coefficients (see Theorem 7.1 (a) for a detailed answer.) We prove in particular the following result.

Theorem 1.1. Let d = gcd(λ_{2} − λ_{1}, λ_{3} − λ_{2}) and let τ be the distance of

\frac{λ_{2} − λ_{3}}{d} t_{1} + \frac{λ_{3} − λ_{1}}{d} t_{2} + \frac{λ_{1} − λ_{2}}{d} t_{3}  (H.3)

to 2πℤ. The trigonometric trinomial T attains its maximum modulus at a unique point modulo 2π/d, with multiplicity 2, unless τ = π.

Theorem 1.1 shows that if there are two points of the hypotrochoid H at maximum distance to −r_{2}e^{it_{2}} , it is so only because −r_{2}e^{it_{2}} lies on an axis of symmetry of H.

We obtain a precise description of those trigonometric trinomials that attain their maximum modulus twice modulo 2π/d: see Theorem 7.1 (c). Their rôle becomes clear by the following result in convex geometry: they yield the exposed points of the unit ball of the ambient normed space. Let us first put up the proper functional analytic framework. Let Λ = {λ_{1}, λ_{2}, λ_{3}} be the spectrum of the trigonometric trinomial T and write e_{λ} : x ↦ e^{iλx} . Let C_{Λ} be the space of functions spanned by {e_{λ} : λ ∈ Λ}, endowed with the maximum modulus norm. Recall that a point P of a convex set K is exposed if there is a hyperplane that meets K only in P; P is extreme if it is not the midpoint of any two other points of K.

Theorem 1.2. Let K be the unit ball of the space C_{Λ} and let P ∈ K.

(a) The point P is an exposed point of K if and only if P is either a trigonometric monomial e^{iα} e_{λ} with α ∈ ℝ and λ ∈ Λ or a trigonometric trinomial that attains its maximum modulus, 1, at two points modulo 2π/d. Every linear functional on C_{Λ} attains its norm on an exposed point of K.

(b) The point P is an extreme point of K if and only if P is either a trigonometric monomial e^{iα} e_{λ} with α ∈ ℝ and λ ∈ Λ or a trigonometric trinomial such that 1 − |P |^{2} has four zeroes modulo 2π/d, counted with multiplicities.

We describe the dependence of the maximum modulus of the trigonometric trinomial T on the arguments. The general issue has been studied for a long time; [52, 91] are two early references. In particular, the following problem has been addressed: see [26, page 2 and Supplement].

Extremal problem 1.3 (Complex Mandel′shtam problem). To find the minimum of the maximum modulus of a trigonometric polynomial with given Fourier coefficient moduli.

It appeared originally in electrical circuit theory: “L. I. Mandel′shtam communicated to me a problem on the phase choice of electric currents with different frequencies such that the capacity of the resulting current to blow [the circuit] is minimal”, tells N. G. Chebotarëv in [24, p. 396], where he discusses applications of a formula given in Section 9 that we would like to advertise.

Our main theorem solves an elementary case of the complex Mandel′shtam problem.

Theorem 1.4. The maximum modulus of T as defined in (H.1) is a strictly decreasing function of τ as defined in Theorem 1.1. In particular,

min_{t_{1} ,t_{2} ,t_{3}} max_{x}|r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + r_{3}e^{i(t_{3} +λ_{3} x)} | = max_{x}|ϵ_{1}r_{1}e^{iλ_{1} x} + ϵ_{2}r_{2}e^{iλ_{2} x} + ϵ_{3}r_{3}e^{iλ_{3} x} |

if ϵ_{1}, ϵ_{2} and ϵ_{3} are real signs +1 or −1 such that ϵ_{i}ϵ_{j} = −1, where i, j, k is a permutation of 1, 2, 3 such that the power of 2 in λ_{i} − λ_{j} is greater than the power of 2 in λ_{i} − λ_{k} and in λ_{k} − λ_{j}.

This shows that the maximum modulus is minimal when the phases are chosen in opposition, independently of the intensities r_{1}, r_{2} and r_{3}.

The decrease of the maximum modulus of (H.1) may be bounded as shown in the next result.

Theorem 1.5. Let d and τ be defined as in Theorem 1.1. Suppose that λ_{2} is between λ_{1} and λ_{3}. The quotient of the maximum modulus of T by |r_{1} + r_{2}e^{iτd/|λ_{3} −λ_{1} |} + r_{3} | is a strictly increasing function of τ unless r_{1} : r_{3} = |λ_{3} − λ_{2}| : |λ_{2} − λ_{1}|, in which case it is constantly equal to 1.

When r_{1} : r_{3} = |λ_{3} −λ_{2}| : |λ_{2} −λ_{1}|, the hypotrochoid H with equation (H.2) is a hypocycloid with |λ_{3} − λ_{1}|/d cusps: the rolling point is on the rolling circle. Note that an epicycloid amounts also to an hypocycloid. Figure H.2 illustrates the particular case T (x) = (1/3)e^{−i2x} + e^{it} + (2/3)e^{ix}.

−1 −e^{iπ/3}

Figure H.2: The unit circle, the deltoid H with equation z = (1/3)e^{−i2x} +(2/3)e^{ix}, the segment from −1 to the unique point on H at maximum distance and the segments from −e^{iπ/3} to the two points on H at maximum distance.

We may deduce from Theorem 1.5 a less precise but handier inequality.

Theorem 1.6. Let d and τ be defined as in Theorem 1.1. Let

D = \frac{max(|λ_{2} − λ_{1}|, |λ_{3} − λ_{2}|, |λ_{3} − λ_{1}|)}{gcd(λ_{2} − λ_{1}, λ_{3} − λ_{1})}  (H.4)

be the quotient of the diameter of Λ by d. Let t′_{1}, t′_{2} and t′_{3} be another three real numbers and define correspondingly τ′. If τ > τ′, then

max_{x}|r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + r_{3}e^{i(t_{3} +λ_{3} x)} |

⩾ \frac{cos(τ/2D)}{cos(τ′/2D)} max_{x}|r_{1}e^{i(t′_{1} +λ_{1} x)} + r_{2}e^{i(t′_{2} +λ_{2} x)} + r_{3}e^{i(t′_{3} +λ_{3} x)} |

with equality if and only if r_{1} : r_{2} : r_{3} = |λ_{3} − λ_{2}| : |λ_{3} − λ_{1}| : |λ_{2} − λ_{1}|.

Figure H.3 illustrates the inequalities obtained in Theorems 1.5 and 1.6 for the particular case T (x) = 4e^{−i2x} + e^{it} + e^{ix}, as in Figure H.1.

If we choose τ′ = 0 in Theorem 1.6, we get the solution to an elementary case of the following extremal problem.

Extremal problem 1.7. To find the minimum of the maximum modulus of a trigonometric polynomial with given spectrum, Fourier coefficient arguments and moduli sum.

Theorem 1.8. Let τ be defined as in Theorem 1.1 and D be given by (H.4). Then

\frac{max_{x} |r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + r_{3}e^{i(t_{3} +λ_{3} x)} |}{r_{1} + r_{2} + r_{3}} ⩾ cos(τ/2D)

with equality if and only if τ = 0 or r_{1} : r_{2} : r_{3} = |λ_{3} − λ_{2}| : |λ_{3} − λ_{1}| : |λ_{2} − λ_{1}|.

0 m 5  t π/3 m = max_{x} |4e^{−i2x} + e^{it} + e^{ix} |  m = |4 + e^{it} + 1| m = (4 + 1 + 1) cos(t/2)

Figure H.3: Let H be the hypotrochoid with equation z = 4e^{−i2x} + e^{ix}. This plot shows the maximum distance m of points z ∈ H to the point −e^{it} and the two estimates of this maximum distance provided by Theorems 1.5 and 1.6 for t ∈ [0, π/3].

The dependence of the maximum modulus of (H.1) on the arguments may also be expressed as properties of relative multipliers. Given three real numbers t_{1}, t_{2} and t_{3}, the linear operator on C_{Λ} defined by e_{λ_{j}} ↦ e^{it_{j}} e_{λ_{j}} is a unimodular relative Fourier multiplier: it multiplies each Fourier coefficient of elements of C_{Λ} by a fixed unimodular number; let us denote it by (t_{1}, t_{2}, t_{3}). Consult [40] for general background on relative multipliers.

Theorem 1.9. The unimodular relative Fourier multiplier (t_{1}, t_{2}, t_{3}) has norm

cos((π − τ)/2D)/ cos(π/2D),

where τ is defined as in Theorem 1.1 and D is given by (H.4), and attains its norm exactly at functions of the form

r_{1}e^{i(u_{1} +λ_{1} x)} + r_{2}e^{i(u_{2} +λ_{2} x)} + r_{3}e^{i(u_{3} +λ_{3} x)}

with r_{1} : r_{2} : r_{3} = |λ_{3} − λ_{2}| : |λ_{3} − λ_{1}| : |λ_{2} − λ_{1}| and

\frac{λ_{2} − λ_{3}}{d} u_{1} + \frac{λ_{3} − λ_{1}}{d} u_{2} + \frac{λ_{1} − λ_{2}}{d} u_{3} = π mod 2π.

The maximum of the norm of unimodular relative Fourier multipliers is the complex unconditional constant of the canonical basis (e_{λ_{1}} , e_{λ_{2}} , e_{λ_{3}} ) of C_{Λ} . As

r_{1} + r_{2} + r_{3} = max_{x}|r_{1}e^{iλ_{1} x} + r_{2}e^{iλ_{2} x} + r_{3}e^{iλ_{3} x} |,

this constant is the minimal constant C such that

r_{1} + r_{2} + r_{3} ⩽ C max_{x}|r_{1}e^{i(u_{1} +λ_{1} x)} + r_{2}e^{i(u_{2} +λ_{2} x)} + r_{3}e^{i(u_{3} +λ_{3} x)} |;

it is therefore the Sidon constant of Λ. It is also the solution to the following extremal problem.

Extremal problem 1.10 (Sidon constant problem). To find the minimum of the maximum modulus of a trigonometric polynomial with given spectrum and Fourier coefficient moduli sum.

Setting τ = π in Theorem 1.9, we obtain the following result.

Corollary 1.11. The Sidon constant of Λ is sec(π/2D), where D is given by (H.4). It is attained exactly at functions of the form given in Theorem 1.9.

Finally, we would like to stress that each of the above results gives rise to open questions if the set Λ is replaced by any set of four integers.

Let us now give a brief description of this article. In Sections 2 and 3, we use carefully the invariance of the maximum modulus under rotation, translation and conjugation to reduce the arguments t_{1}, t_{2} and t_{3} of the Fourier coefficients of the trigonometric trinomial T to the variable τ. Section 4 shows how to further reduce this study to the trigonometric trinomial

r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx}  (H.5)

with k and l positive coprime integers and t ∈ [0, π/(k + l)]. In Section 5, we prove that (H.5) attains its maximum modulus for x ∈ [−t/k, t/l]. Section 6 studies the variations of the modulus of (H.5) for x ∈ [−t/k, t/l]: it turns out that it attains its absolute maximum only once on that interval. This yields Theorem 1.1. Section 7 restates the results of the two previous sections for a general trigonometric trinomial T. Section 8 is dedicated to the proof of Theorem 1.2. In Section 9, we compute the directional derivative of the maximum modulus of (H.5) with respect to the argument t and prove Theorems 1.4, 1.5, 1.6 and 1.8. In Section 10, we prove Theorem 1.9 and show how to lift unimodular relative Fourier multipliers to operators of convolution with a linear combination of two Dirac measures. Section 11 replaces our computation of the Sidon constant in a general context; it describes the initial motivation for this research.

Part of these results appeared previously, with a different proof, in [64, Chapter II.10] and in [62].

Notation. Throughout this article, λ_{1}, λ_{2} and λ_{3} are three pairwise distinct integers, Λ is the set {λ_{1}, λ_{2}, λ_{3}} and d = gcd(λ_{2} − λ_{1}, λ_{3} − λ_{2}). If λ is an integer, e_{λ} is the function x ↦ e^{iλx} of the real variable x. A trigonometric polynomial is a linear combination of functions e_{λ}; it is a monomial, binomial or trinomial if this linear combination has one, two or three nonzero coefficients, respectively. The normed space C_{Λ} is the three-dimensional space of complex functions spanned by e_{λ} with λ ∈ Λ, endowed with the maximum modulus norm. The Dirac measure δ_{x} is the linear functional T ↦ T (x) of evaluation at x on the space of continuous functions. Given three real numbers t_{1}, t_{2} and t_{3}, the linear operator on C_{Λ} defined by e_{λ_{j}} ↦ e^{it_{j}} e_{λ_{j}} is a unimodular relative Fourier multiplier denoted by (t_{1}, t_{2}, t_{3}).

2 Isometric relative Fourier multipliers

The rôle of Quantity (H.3) is explained by the following lemma.

Lemma 2.1. Let t_{1}, t_{2} and t_{3} be real numbers. The unimodular relative Fourier multiplier M = (t_{1}, t_{2}, t_{3}) is an isometry on C_{Λ} if and only if

\frac{λ_{2} − λ_{3}}{d} t_{1} + \frac{λ_{3} − λ_{1}}{d} t_{2} + \frac{λ_{1} − λ_{2}}{d} t_{3} ∈ 2πℤ.  (H.6)

Then it is a unimodular multiple of a translation: there are real numbers α and v such that Mf(x) = e^{iα} f(x − v) for all f ∈ C_{Λ} and all x ∈ ℝ.

Proof. If M is a unimodular multiple of a translation by a real number v, then

|r_{1}e^{i(t_{1} +λ_{1} v)} + r_{2}e^{i(t_{2} +λ_{2} v)} + r_{3}e^{i(t_{3} +λ_{3} v)} | = r_{1} + r_{2} + r_{3},

which holds if and only if

t_{1} + λ_{1}v = t_{2} + λ_{2}v = t_{3} + λ_{3}v modulo 2π.  (H.7)

There is a v satisfying (H.7) if and only if Equation (H.6) holds as (H.7) means that there exist integers a_{1} and a_{3} such that

v = \frac{t_{2} − t_{1} + 2πa_{1}}{λ_{1} − λ_{2}} = \frac{t_{2} − t_{3} + 2πa_{3}}{λ_{3} − λ_{2}} .

If t_{1}, t_{2} and t_{3} are three real numbers satisfying (H.6), let v be such that (H.7) holds. Then

r_{1}e^{i(t_{1} +u_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +u_{2} +λ_{2} x)} + r_{3}e^{i(t_{3} +u_{3} +λ_{3} x)}

= e^{i(t_{2} +λ_{2} v)} (r_{1}e^{i(u_{1} +λ_{1} (x−v))} + r_{2}e^{i(u_{2} +λ_{2} (x−v))} + r_{3}e^{i(u_{3} +λ_{3} (x−v))} )

for all real numbers u_{1}, u_{2}, u_{3} and x.

3 The arguments of the Fourier coefficients of a trigonometric trinomial

We have used a translation and a rotation to reduce the three arguments of the Fourier coefficients of a trigonometric trinomial to just one variable. Use of the involution \overline{f(−x)} of C_{Λ} allows us to restrict even further the domain of that variable.

Lemma 3.1. Let t_{1}, t_{2} and t_{3} be real numbers and let t̃_{2} be the representative of

\frac{λ_{2} − λ_{3}}{λ_{3} − λ_{1}} t_{1} + t_{2} + \frac{λ_{1} − λ_{2}}{λ_{3} − λ_{1}} t_{3}  (H.8)

modulo 2π/|λ_{3} − λ_{1}| in [−πd/|λ_{3} − λ_{1}|, πd/|λ_{3} − λ_{1}|[.

(a) There are real numbers α and v such that

r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + r_{3}e^{i(t_{3} +λ_{3} x)} = e^{iα} (r_{1}e^{iλ_{1} (x−v)} + r_{2}e^{i(t̃_{2} +λ_{2} (x−v))} + r_{3}e^{iλ_{3} (x−v))} )

(H.9)

for all x.

(b) Let t = |t̃_{2}| be the distance of (H.8) to (2πd/|λ_{3} − λ_{1}|)ℤ. There is a sign ε ∈ {+1, −1} such that

|r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + r_{3}e^{i(t_{3} +λ_{3} x)} | = |r_{1}e^{iλ_{1} ε(x−v)} + r_{2}e^{i(t+λ_{2} ε(x−v))} + r_{3}e^{iλ_{3} ε(x−v))} |

for all x.

Proof. (a). The argument t̃_{2} is chosen so that the relative multiplier (t_{1}, t_{2} − t̃_{2}, t_{3}) is an isometry. (b). If t̃_{2} is negative, take the conjugate under the modulus of the right hand side in (H.9).

Remark 3.2. This proves the following periodicity formula:

|r_{1}e^{iλ_{1} x} + r_{2}e^{i(t+2πd/(λ_{3} −λ_{1} )+λ_{2} x)} + r_{3}e^{iλ_{3} x} | = |r_{1}e^{iλ_{1} (x−v)} + r_{2}e^{i(t+λ_{2} (x−v))} + r_{3}e^{iλ_{3} (x−v)} |

for all x and t, where v satisfies λ_{1}v = 2πd/(λ_{3} − λ_{1}) + λ_{2}v = λ_{3}v modulo 2π, that is

v = \frac{2mπ}{λ_{3} − λ_{1}} with m an inverse of \frac{λ_{3} − λ_{2}}{d} modulo \frac{λ_{3} − λ_{1}}{d} .

4 The frequencies of a trigonometric trinomial

We may suppose without loss of generality that λ_{1} < λ_{2} < λ_{3}. Let k = (λ_{2}−λ_{1})/d and l = (λ_{3}−λ_{2})/d. Then

r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + r_{3}e^{i(t_{3} +λ_{3} x)} = e^{iλ_{2} x} (r_{1}e^{i(t_{1} −k(dx))} + r_{2}e^{it_{2}} + r_{3}e^{i(t_{3} +l(dx))} )

This defines an isometry between C_{Λ} and C_{{−k,0,l}} and shows that C_{Λ} is normed by the maximum modulus norm on [0, 2π/d[. With Lemma 3.1 (b), this shows that a homothety by d^{−1} allows us to restrict our study to the function

f(t, x) = |r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} | ^{2}

for x ∈ ℝ with k and l two positive coprime numbers and t ∈ [0, π/(k + l)]. We have

f(−t, x) = f(t, −x)  (H.10)

and Remark 3.2 shows that

f(t + 2π/(k + l), x) = f(t, x − 2mπ/(k + l))  (H.11)

for all x and t, where m is the inverse of l modulo k + l. In particular, if t = π/(k + l), we have the symmetry relation

f(π/(k + l), x) = f(π/(k + l), 2mπ/(k + l) − x) .  (H.12)

5 Location of the maximum point

The purpose of our first proposition is to deduce the existence of a small interval on which a trigonometric trinomial attains its maximum modulus. Note that a trigonometric binomial attains its maximum modulus at a point that depends only on the phase of its coefficients:

– |r_{1}e^{−ikx} + r_{2}e^{it} | attains its maximum at −t/k independently of r_{1} and r_{2},

– |r_{1}e^{−ikx} + r_{3}e^{ilx} | attains its maximum at 0 independently of r_{1} and r_{3},

– |r_{2}e^{it} + r_{3}e^{ilx} | attains its maximum at t/l independently of r_{2} and r_{3}. The next proposition shows that if the point at which a trigonometric trinomial attains its maximum modulus changes with the modulus of its coefficients, it changes very little; we get bounds for this point that are independent of the intensities.

Proposition 5.1. Let k and l be positive coprime integers. Let r_{1}, r_{2} and r_{3} be three positive real numbers. Let t ∈ [0, π/(k + l)]. Let

f(x) = |r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} | ^{2}

for x ∈ ℝ.

(a) The function f attains its absolute maximum in the interval [−t/k, t/l].

(b) If f attains its absolute maximum at a point y outside of [−t/k, t/l] modulo 2π, then t = π/(k+l) and 2mπ/(k + l) − y lies in [−t/k, t/l] modulo 2π, where m is the inverse of l modulo k + l.

Proof. (a). We have

f(x) = r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + 2 · (r_{1}r_{2} cos(t + kx) + r_{1}r_{3} cos((k + l)x) + r_{2}r_{3} cos(t − lx)).  (H.13)

Let us prove that f attains its absolute maximum on [−t/k, t/l]. Let y be outside of [−t/k, t/l] modulo 2π. Let I be the set of all x ∈ [−t/k, t/l] such that

cos(t + kx) ⩾ cos(t + ky)
cos((k + l)x) ⩾ cos((k + l)y)
cos(t − lx) ⩾ cos(t − ly).

Note that if x ∈ [−t/k, t/l], then

\{ \begin{matrix}t + kx & ∈ [0, (k + l)t/l] \\ (k + l)x & ∈ [−(k + l)t/k, (k + l)t/l] \\ t − lx & ∈ [0, (k + l)t/k], \end{matrix}

and that (k + l)t/k, (k + l)t/l ∈ [0, π]. Let – α be the distance of t/k + y to (2π/k)ℤ,

– β be the distance of y to (2π/(k + l))ℤ,

– γ be the distance of t/l − y to (2π/l)ℤ. Then

I = [−t/k, t/l] ∩ [−t/k − α, −t/k + α] ∩ [−β, β] ∩ [t/l − γ, t/l + γ].  (H.14)

Let us check that I is the nonempty interval

I = [max(−t/k, −β, t/l − γ), min(t/l, −t/k + α, β)].  (H.15)

In fact, we have the following triangular inequalities:

– −β ⩽ −t/k + α because t/k is the distance of (t/k + y) − y to (2π/k(k + l))ℤ;

– t/l − γ ⩽ −t/k + α because t/l + t/k is the distance of (t/k + y) + (t/l − y) to (2π/kl)ℤ;

– t/l − γ ⩽ β because t/l is the distance of (t/l − y) + y to (2π/l(k + l))ℤ.

The other six inequalities that are necessary to deduce (H.15) from (H.14) are obvious.

(b). We have proved in (a) that there is an x ∈ [−t/k, t/l] such that cos(t + kx) ⩾ cos(t + ky), cos((k + l)x) ⩾ cos((k + l)y) and cos(t − lx) ⩾ cos(t − ly). In fact, at least one of these inequalities is strict unless there are signs δ, ε, η ∈ {−1, 1} such that t + kx = δ(t + ky), t − lx = ε(t − ly) and (k + l)x = η(k + l)y modulo 2π. Two out of these three signs are equal and the corresponding two equations imply the third one with the same sign. This system is therefore equivalent to

\{ \begin{matrix}k(x − y) & = 0 \\ l(x − y) & = 0 \end{matrix} or \{ \begin{matrix}k(x + y) & = −2t \\ l(x + y) & = 2t \end{matrix}

modulo 2π. The first pair of equations yields x = y modulo 2π because k and l are coprime. Let m be an inverse of l modulo k + l; then the second pair of equations is equivalent to

\{ \begin{matrix}2(k + l)t & = 0 \\ x + y & = 2mt \end{matrix}

modulo 2π. Therefore g does not attain its absolute maximum at y unless t = π/(k + l) and 2mπ/(k + l) − y ∈ [−t/k, t/l].

Remark 5.2. This proposition is a complex counterpart to Lemma 2.1. i) in [84], where cosine trinomials are investigated.

6 Uniqueness of the maximum point

Note that

r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} = r_{3}e^{−il(−x)} + r_{2}e^{it} + r_{1}e^{ik(−x)}
= r′_{1}e^{−ik′x′} + r_{2}e^{it} + r′_{3}e^{il′x′}

with r′_{1} = r_{3}, r′_{3} = r_{1}, k′ = l, l′ = k and x′ = −x. We may therefore suppose without loss of generality that kr_{1} ⩽ lr_{3}.

Our second proposition studies the points at which a trigonometric trinomial attains its maximum modulus. Note that if k = l = 1, the derivative of |f|^{2} has at most 4 zeroes, so that the modulus of f has at most two maxima and attains its absolute maximum in at most two points. Proposition 6.1 shows that this is true in general, and that if it may attain its absolute maximum in two points, it is so only because of the symmetry given by (H.12).

Proposition 6.1. Let k and l be positive coprime integers. Let r_{1}, r_{2} and r_{3} be three positive real numbers such that kr_{1} ⩽ lr_{3}. Let t ∈ ]0, π/(k + l)]. Let

f(x) = |r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} | ^{2}

for x ∈ [−t/k, t/l].

(a) There is a point x^{∗} ∈ [0, t/l] such that df/dx > 0 on ]−t/k, x^{∗}[ and df/dx < 0 on ]x^{∗}, t/l[.

(b) There are three cases:

1. f attains its absolute maximum at 0 if and only if kr_{1} = lr_{3};

2. f attains its absolute maximum at t/l if and only if l = 1, t = π/(k + 1) and k^{2}r_{1}r_{2} + (k + 1)^{2}r_{1}r_{3} − r_{2}r_{3} ⩽ 0;

3. otherwise, f attains its absolute maximum in ]0, t/l[.

(c) The function f attains its absolute maximum with multiplicity 2 unless l = 1, t = π/(k +1) and k^{2}r_{1}r_{2} + (k + 1)^{2}r_{1}r_{3} − r_{2}r_{3} = 0, in which case it attains its absolute maximum at π/(k + 1) with multiplicity 4.

Proof. (a). By Proposition 5.1, the derivative of f has a zero in [−t/k, t/l]. Let us study the sign of this derivative. Equation (H.13) yields

\frac{1}{2} \frac{df}{dx}(x) = −kr_{1}r_{2} sin(t + kx) − (k + l)r_{1}r_{3} sin((k + l)x) + lr_{2}r_{3} sin(t − lx).  (H.16)

We wish to compare sin(t + kx) with sin(t − lx): note that

sin(t + kx) − sin(t − lx) = 2 sin((k + l)x/2) cos(t + (k − l)x/2),

and that if x ∈ [−t/k, t/l], then

−π/2 ⩽ −π/2k ⩽ −(k + l)t/2k ⩽ (k + l)x/2 ⩽ (k + l)t/2l ⩽ π/2l ⩽ π/2

0 ⩽ t + (k − l)x/2 ⩽ \{ \begin{matrix}t + (l − k)t/2k = (k + l)t/2k & if k ⩽ l \\ t + (k − l)t/2l = (k + l)t/2l & if l ⩽ k \end{matrix} ⩽ π/2.

Suppose that x ∈ [−t/k, 0[: then it follows that sin(t+kx) ⩽ sin(t−lx) and also sin((k +l)x) ⩽ 0, with equality if and only if k = 1 and −x = t = π/(1 + l). This yields with kr_{1} ⩽ lr_{3} that

\frac{1}{2} \frac{df}{dx}(x) ⩾ −(k + l)r_{1}r_{3} sin((k + l)x) ⩾ 0  (H.17)

with equality if and only if k = 1 and −x = t = π/(1 + l).

Suppose that x ∈ [0, t/l]. If l ⩾ 2, then

\{ \begin{matrix}t + kx & ∈ [t, (k + l)t/l] ⊂ [t, π/2] \\ (k + l)x & ∈ [0, (k + l)t/l] ⊂ [t, π/2] \\ t − lx & ∈ [0, t] ⊂ [0, π/3], \end{matrix}

so that the second derivative of f is strictly negative on [0, t/l]: its derivative is strictly decreasing on this interval and (a) is proved. If l = 1, let g(x) = f(t − x) for x ∈ [0, t]: we have to prove that there is a point x^{∗} such that dg/dx > 0 on ]0, x^{∗}[ and dg/dx < 0 on ]x^{∗}, t[. We already know that (dg/dx)(0) ⩾ 0 and that dg/dx has a zero on [0, t]. Put α = (k + 1)t: then

\frac{1}{2} \frac{dg}{dx}(x) = kr_{1}r_{2} sin(α − kx) + (k + 1)r_{1}r_{3} sin(α − (k + 1)x) − r_{2}r_{3} sinx

and it suffices to prove that

\frac{1}{2 sinx} \frac{dg}{dx}(x) = kr_{1}r_{2} \frac{sin(α − kx)}{sinx} + (k + 1)r_{1}r_{3} \frac{sin(α − (k + 1)x)}{sinx} − r_{2}r_{3}  (H.18)

is a strictly decreasing function of x on ]0, α/(k + 1)]. Let us study the sign of

\frac{d}{dx} \frac{sin(α − kx)}{sinx} = \frac{−k cos(α − kx) sinx − sin(α − kx) cosx}{sin^{2} x}

for α ∈ ]0, π] and x ∈ ]0, α/k]. If k = 1, then

−k cos(α − kx) sinx − sin(α − kx) cosx = − sinα ⩽ 0

and the inequality is strict unless α = π. Let us prove by induction on k that

k cos(α − kx) sinx + sin(α − kx) cosx > 0

for all k ⩾ 2, α ∈ ]0, π] and x ∈ ]0, α/k]. This will complete the proof of (a). Let k ⩾ 1 and x ∈ ]0, α/(k + 1)]. Then

(k + 1) cos(α − (k + 1)x) sinx + sin(α − (k + 1)x) cosx

= (k + 1) cos(α − kx) cosx sinx + (k + 1) sin(α − kx) sin^{2} x + sin(α − kx) cos^{2} x − cos(α − kx) sinx cosx

= (k cos(α − kx) sinx + sin(α − kx) cosx) cosx +(k + 1) sin(α − kx) sin^{2} x

⩾ (k + 1) sin(α − kx) sin^{2} x > 0

(b). 1. By Proposition 5.1 and (a), f attains its absolute maximum at 0 if and only if 0 is a critical point for f. We have

\frac{1}{2} \frac{df}{dx}(0) = (lr_{3} − kr_{1})r_{2} sint ⩾ 0

and equality holds if and only if kr_{1} = lr_{3}.

2. We have

\frac{1}{2} \frac{df}{dx}(t/l) = (−kr_{1}r_{2} − (k + l)r_{1}r_{3}) sin((k + l)t/l) ⩽ 0

and equality holds if and only if l = 1 and t = π/(k + 1). Let l = 1 and t = π/(k + 1) and let us use the notation introduced in the last part of the proof of (a): we need to characterise the case that g has a maximum at 0. As α = π and

\frac{1}{2 sinx} \frac{dg}{dx}(x) = k^{2}r_{1}r_{2} + (k + 1)^{2}r_{1}r_{3} − r_{2}r_{3} + o(x)  (H.19)

is a strictly decreasing function of x on ]0, π/(k + 1)], g has a maximum at 0 if and only if k^{2}r_{1}r_{2} + (k + 1)^{2}r_{1}r_{3} − r_{2}r_{3} ⩽ 0.

(c). If l ⩾ 2, then the second derivative of f is strictly negative on [0, t/l]. If l = 1, then the derivative of (H.18) is strictly negative on ]0, α/(k + 1)]: this yields that the second derivative of g can only vanish at 0. By (b) 2., g has a maximum at 0 only if t = π/(k + 1); then

\frac{1}{2} \frac{d^{2}g}{dx^{2}}(0) = k^{2}r_{1}r_{2} + (k + 1)^{2}r_{1}r_{3} − r_{2}r_{3}  (H.20)
\frac{1}{2} \frac{d^{4}g}{dx^{4}}(0) = −k^{4}r_{1}r_{2} − (k + 1)^{4}r_{1}r_{3} + r_{2}r_{3}  (H.21)

If (H.20) vanishes, then the sum of (H.21) with (H.20) yields

\frac{1}{2} \frac{d^{4}g}{dx^{4}}(0) = −k(k + 1)r_{1}((k − 1)kr_{2} + (k + 1)(k + 2)r_{3}) < 0.

Remark 6.2. We were able to prove directly that the system

f(x) = f(y)

\frac{df}{dx}(x) = \frac{df}{dx}(y) = 0

\frac{d^{2}f}{dx^{2}} (x), \frac{d^{2}f}{dx^{2}} (y) ⩽ 0

implies x = y modulo 2π or t = π/(k + l) and x + y = 2mπ/(k + l), but our computations are very involved and opaque.

Remark 6.3. This proposition is a complex counterpart to [84, Lemma 2.1. ii)].

Remark 6.4. Suppose that l = k = 1. If t ∈ ]0, π/2[, it is necessary to solve a generally irreducible quartic equation in order to compute the maximum of f. If t = π/2, it suffices to solve a linear equation and one gets the following expression for max_{x} |r_{1}e^{−ix} + ir_{2} + r_{3}e^{ix} | :

\{ \begin{matrix}(r_{1} + r_{3})\sqrt{1 + r_{2}^{2}/4r_{1}r_{3}} & if |r_{1}^{−1} − r_{3}^{−1} | < 4r_{2}^{−1} \\ r_{2} + |r_{3} − r_{1}| & otherwise. \end{matrix}

This formula appears in [2, (3.1)]. In the first case, the maximum is attained at the two points x^{∗} such that sinx^{∗} = r_{2}(r_{3} − r_{1})/4r_{1}r_{3}.

Remark 6.5. Suppose that l = 1 and k = 2. If t ∈ ]0, π/3[, it is necessary to solve a generally irreducible sextic equation in order to compute the maximum of f. If t = π/3, it suffices to solve a quadratic equation and one gets the following expression for max_{x} |r_{1}e^{−i2x} + r_{2}e^{iπ/3} + r_{3}e^{ix} | : if r_{1}^{−1} − 4r_{3}^{−1} < 9r_{2}^{−1} , then its square makes

r_{1}^{2} + \frac{2}{3}r_{2}^{2} + r_{3}^{2} + r_{1}r_{2} + 2r_{1}r_{3}[( ( \frac{r_{2}}{3r_{3}} )^{2} + \frac{r_{2}}{3r_{1}} + 1) ^{3/2} − ( \frac{r_{2}}{3r_{3}} )^{3}]

and the maximum is attained at the two points x^{∗} such that

2 cos(π/3 − x^{∗}) = ( ( \frac{r_{2}}{3r_{3}} )^{2} + \frac{r_{2}}{3r_{1}} + 1) ^{1/2} − \frac{r_{2}}{3r_{3}} ;

otherwise, it makes −r_{1} + r_{2} + r_{3}.

7 The maximum modulus points of a trigonometric trinomial

If we undo all the reductions made in Sections 3 and 4 and at the beginning of Section 6, we get the following theorem.

Theorem 7.1. Let λ_{1}, λ_{2} and λ_{3} be three pairwise distinct integers such that λ_{2} is between λ_{1} and λ_{3}. Let r_{1}, r_{2} and r_{3} be three positive real numbers. Given three real numbers t_{1}, t_{2} and t_{3}, consider the trigonometric trinomial

T (x) = r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} + r_{3}e^{i(t_{3} +λ_{3} x)}

for x ∈ ℝ. Let d = gcd(λ_{2} − λ_{1}, λ_{3} − λ_{2}) and choose integers a_{1} and a_{3} such that

τ = \frac{λ_{2} − λ_{3}}{d} (t_{1} − 2πa_{1}) + \frac{λ_{3} − λ_{1}}{d} t_{2} + \frac{λ_{1} − λ_{2}}{d} (t_{3} − 2πa_{3}),

satisfies |τ| ⩽ π. Let t̃_{1} = t_{1} − 2πa_{1} and t̃_{3} = t_{3} − 2πa_{3}.

(a) The trigonometric trinomial T attains its maximum modulus at a unique point of the interval

bounded by (t̃_{1} − t_{2})/(λ_{2} − λ_{1}) and (t_{2} − t̃_{3})/(λ_{3} − λ_{2}). More precisely,

– if r_{1}|λ_{2} −λ_{1}| ⩽ r_{3}|λ_{3} −λ_{2}|, then this point is between (t̃_{1}−t̃_{3})/(λ_{3} −λ_{1}) and (t_{2} −t̃_{3})/(λ_{3} − λ_{2});

– if r_{1}|λ_{2} −λ_{1}| ⩾ r_{3}|λ_{3} −λ_{2}|, then this point is between (t̃_{1}−t̃_{3})/(λ_{3} −λ_{1}) and (t̃_{1}−t_{2})/(λ_{2} − λ_{1});

– T attains its maximum modulus at (t̃_{1}−t̃_{3})/(λ_{3}−λ_{1}) if and only if r_{1}|λ_{2}−λ_{1}| = r_{3}|λ_{3}−λ_{2}|

or τ = 0.

(b) The function T attains its maximum modulus at a unique point modulo 2π/d, and with multiplicity 2, unless |τ| = π.

(c) Suppose that |τ| = π, i.e.,

\frac{λ_{2} − λ_{3}}{d} t_{1} + \frac{λ_{3} − λ_{1}}{d} t_{2} + \frac{λ_{1} − λ_{2}}{d} t_{3} = π mod 2π.  (H.22)

Let s be a solution to 2t_{1} + λ_{1}s = 2t_{2} + λ_{2}s = 2t_{3} + λ_{3}s modulo 2π: s is unique modulo 2π/d. Then T (s − x) = e^{i(2t_{2} +λ_{2} s)} \overline{T (x)} for all x. Suppose that |λ_{3} − λ_{2}| ⩽ |λ_{2} − λ_{1}|. There are three cases.

1. If λ_{2} − λ_{1} = k(λ_{3} − λ_{2}) with k ⩾ 2 integer and

r_{1}^{−1} − k^{2}r_{3}^{−1} ⩾ (k + 1)^{2}r_{2}^{−1} ,

then T attains its maximum modulus, −r_{1} + r_{2} + r_{3}, only at x = (t_{2} − t̃_{3})/(λ_{3} − λ_{2}) modulo 2π/d, with multiplicity 2 if the inequality is strict and with multiplicity 4 if there is equality;

2. if λ_{2} − λ_{1} = λ_{3} − λ_{2} and

|r_{1}^{−1} − r_{3}^{−1} | ⩾ 4r_{2}^{−1} ,

then T attains its maximum modulus, r_{2} + |r_{3} − r_{1}|, at a unique point x modulo 2π/d, with multiplicity 2 if the inequality is strict and with multiplicity 4 if there is equality. This point is (t_{2} − t̃_{3})/(λ_{3} − λ_{2}) if r_{1} < r_{3}, and (t̃_{1} − t_{2})/(λ_{2} − λ_{1}) if r_{3} < r_{1};

3. otherwise T attains its maximum modulus at exactly two points x and y modulo 2π/d, with multiplicity 2, where x is strictly between (t̃_{1} − t_{2})/(λ_{2} − λ_{1}) and (t_{2} − t̃_{3})/(λ_{3} − λ_{2}), and x + y = s modulo 2π/d.

Note that s − x = x modulo 2π/d in Cases 1 and 2.

8 Exposed and extreme points of the unit ball of C_{Λ}

The characterisation of the maximum modulus points of a trigonometric trinomial enables us to compute the exposed and the extreme points of the unit ball of C_{Λ} . We begin with a lemma.

Lemma 8.1. (a) A trigonometric trinomial with a given spectrum that attains its maximum mo-

dulus at two given points modulo 2π/d is determined by its value at these points.

(b) The trigonometric trinomials with a given spectrum that attain their maximum modulus with multiplicity 4 at a given point and have a given value at this point lie on a parabola.

Proof. We will use the notation of Theorem 7.1. Without loss of generality, we may suppose that λ_{1} = −k, λ_{2} = 0 and λ_{3} = l with k and l positive coprime integers. Let x and y be two real numbers that are different modulo 2π/d, let ϑ and ζ be real numbers and let ϱ be a positive real number.

(a). Let us prove that at most one trigonometric trinomial T attains its maximum modulus at x and y and satisfies T (x) = ϱe^{iϑ} and T (y) = ϱe^{iζ}. Let us translate T by (x+y)/2: we may suppose that x + y = 0. Let us divide T by e^{i(ϑ+ζ)/2} : we may suppose that ϑ + ζ = 0. As T attains its maximum modulus at the two points x and y, we have s = x + y = 0 and 2t_{1} − ks = 2t_{2} = 2t_{3} + ls = ϑ + ζ = 0 modulo 2π. Therefore t_{1} = t_{2} = t_{3} = 0 modulo π. Let p_{j} = e^{it_{j}} r_{j}: the p_{j} are nonzero real numbers. We have

T (x) = p_{1}e^{−ikx} + p_{2} + p_{3}e^{ilx} = ϱe^{iϑ},

so that, multiplying by e^{−iϑ} and taking real and imaginary parts,

p_{1} cos(ϑ + kx) + p_{3} cos(ϑ − lx) = ϱ − p_{2} cosϑ  (H.23)
p_{1} sin(ϑ + kx) + p_{3} sin(ϑ − lx) = −p_{2} sinϑ.  (H.24)

The computation

\frac{1}{2} \frac{d|T |^{2}}{dx} (x) = ℜ( \overline{T (x)}\frac{dT}{dx}(x)) = ℜ( \overline{T (x)}(−ikp_{1}e^{−ikx} + ilp_{3}e^{ilx} )) ,

yields

kp_{1} sin(ϑ + kx) − lp_{3} sin(ϑ − lx) = 0.  (H.25)

Equations (H.24) and (H.25) yield p_{1} and p_{3} as linear functions of p_{2} because sin(ϑ+kx) sin(ϑ−lx) ≠ 0: otherwise both factors would vanish, so that ϑ = x = 0 modulo π and x = y modulo 2π. As ϱ ≠ 0, Equation (H.23) has at most one solution in p_{2}.

(b). We are necessarily in Case 1 or 2 of Theorem 7.1(c), so that we may suppose that l = 1. Let us determine all trigonometric trinomials T that attain their maximum modulus at x with multiplicity 4 and satisfy T (x) = ϱe^{iϑ}. Let us translate T by x: we may suppose that x = 0. Let us divide T by e^{iϑ}: we may suppose that ϑ = 0. As T attains its maximum modulus at 0 with multiplicity 4, we have s − 0 = 0 and 2t_{1} − ks = 2t_{2} = 2t_{3} + s = 2ϑ = 0 modulo 2π. Therefore t_{1} = t_{2} = t_{3} = 0 modulo π. Let p_{j} = e^{it_{j}} r_{j}: the p_{j} are nonzero real numbers and satisfy the system

\{ \begin{matrix}p_{1} + p_{2} + p_{3} = ϱ \\ k^{2}p_{1}p_{2} + (k + 1) ^{2}p_{1}p_{3} + p_{2}p_{3} = 0, \end{matrix}

that is

\{ \begin{matrix}p_{2} = ϱ − p_{1} − p_{3} \\ (kp_{1} − p_{3})^{2} = ϱ(k^{2}p_{1} + p_{3}). \end{matrix}

This is the equation of a parabola.

Remark 8.2. The equality

max_{x}|r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} | = r_{1} + r_{2}  (H.26)

shows that the exposed points of the unit ball of the space C_{{λ_{1} ,λ_{2} }} are the trigonometric monomials e^{iα} e_{λ_{1}} and e^{iα} e_{λ_{2}} with α ∈ ℝ and that no trigonometric binomial is an extreme point of the unit ball of C_{Λ} .

8. EXPOSED AND EXTREME POINTS OF THE UNIT BALL OF C_{Λ}  111

Proof of Theorem 1.2 (a). Necessity. Let f ∈ K. Let us make four remarks:

– if f is an exposed point, then ∥f∥ = 1;

– if l is a nonzero linear functional on C_{Λ} and z is a complex number such that l^{−1} (z) ∩ K = {f}, then |l(f)| = |z| = ∥l∥, so that l attains its norm at f;

– f is exposed if and only if ∥f∥ = 1 and there is a nonzero linear functional l on C_{Λ} that attains its norm only at multiples of f;

– if l is a linear functional on C_{Λ} that attains its norm at f and μ is the measure on [0, 2π/d[ identified with the Hahn-Banach extension of l to the space of continuous functions on [0, 2π/d[, then the support of μ must be a subset of the maximum modulus points of f on [0, 2π/d[.

Theorem 7.1 (b, c) tells that a trigonometric trinomial attains its maximum modulus at one or two points modulo 2π/d. We have therefore to show that trigonometric binomials and trigonometric trinomials with only one maximum modulus point modulo 2π/d are not exposed.

– A linear functional that attains its modulus at a trigonometric binomial attains its norm at a trigonometric monomial because this trigonometric binomial is a convex combination of two trigonometric monomials with same norm by Equation (H.26).

– If f attains its maximum modulus at a unique point x ∈ [0, 2π/d[, then l must be a multiple of the Dirac measure δ_{x} at x, so that l attains also its norm at the monomials in C_{Λ} .

These arguments show also that every linear functional on C_{Λ} attains its norm at monomials or at trigonometric trinomials with two maximum modulus points.

Sufficiency. Conversely, the trigonometric monomial e^{iα} e_{λ} is exposed to the linear form

P ↦ \frac{1}{2π} ∫_{0}^{2π} P (x)e^{−i(α+λx)} dx.

A trigonometric trinomial T that attains its maximum modulus, 1, at two points x_{1}^{∗} and x_{2}^{∗} modulo 2π/d is exposed, by Lemma 8.1 (a), to any nontrivial convex combination of the unimodular multiples of Dirac measures \overline{T (x_{1}^{∗})}δ_{x_{1}^{∗}} and \overline{T (x_{2}^{∗})}δ_{x_{2}^{∗}} .

Remark 8.3. This is a complex counterpart to Lemma 2.3 in [84], dealing with the exposed points of the unit ball of the three-dimensional space spanned by the functions 1, cosx and coskx in the space of continuous functions.

Proof of Theorem 1.2 (b). Let K be the unit ball of C_{Λ} . Straszewicz’s Theorem [97] tells that the exposed points of K are dense in the set of its extreme points. Let P be a limit point of exposed points of K. If P is a trigonometric monomial, P is exposed. If P is a trigonometric binomial, P is not an extreme point of K by Remark 8.2. If P is a trigonometric trinomial, it is the limit point of trigonometric trinomials that attain their maximum modulus twice modulo 2π/d, so that either P also attains its maximum modulus twice modulo 2π/d or, by Rolle’s Theorem, P attains its maximum modulus with multiplicity 4. Let us prove that if a trigonometric trinomial T attains its maximum modulus with multiplicity 4 at a point x, then T is an extreme point of K. Suppose that T is the midpoint of two points A and B in K. Then |A(x)| ⩽ 1, |B(x)| ⩽ 1 and (A(x) + B(x))/2 = T (x), so that A(x) = B(x) = T (x). Furthermore

|T (x + h)| ⩽ \frac{|A(x + h)| + |B(x + h)|}{2} = 1 + \frac{h^{2}}{4} ( \frac{d^{2}|A|}{dx^{2}}(x) + \frac{d^{2}|B|}{dx^{2}}(x)) + o(h^{2})

so that, as |T (x + h)| = 1 + o(h^{3}),

\frac{d^{2}|A|}{dx^{2}}(x) + \frac{d^{2}|B|}{dx^{2}}(x) ⩾ 0 while \frac{d^{2}|A|}{dx^{2}}(x), \frac{d^{2}|B|}{dx^{2}}(x) ⩽ 0

and therefore A and B also attain their maximum modulus with multiplicity 4 at x. As this implies that A and B are trigonometric trinomials, Lemma 8.1 (b) yields that T, A and B lie on a parabola: this implies A = B = T.

Remark 8.4. The set of extreme points of the unit ball of C_{Λ} is not closed: for example, if λ_{2} is between λ_{1} and λ_{3}, every absolutely convex combination of e_{λ_{1}} and e_{λ_{3}} is a limit point of exposed points. Remark 8.5. If λ_{2} is between λ_{1} and λ_{3}, and λ_{1} − λ_{2} is not a multiple of λ_{3} − λ_{2} nor vice versa, then we obtain that every extreme point of the unit ball of C_{Λ} is exposed. Remark 8.6. In particular, compare our description of the extreme points of the unit ball of C_{{0,1,2}} with the characterisation given by K. M. Dyakonov in [27, Theorem 1]. He shows in his Example 1 that it is false that “in order to recognize the extreme points”, “one only needs to know ‘how often’ [the modulus of a trigonometric polynomial] takes the extremal value 1”. We show that with the exception of trigonometric binomials, the extreme points of the unit ball of C_{{0,1,2}} are characterised by the number of zeroes of 1 − |P |^{2}. Remark 8.7. This is a complex counterpart to [59], dealing with the extreme points of the unit ball of the three-dimensional space spanned by the functions 1, x^{n} and x^{m} in the space of real valued continuous functions on [−1, 1].

9 Dependence of the maximum modulus on the arguments

We wish to study how the maximum modulus of a trigonometric trinomial depends on the phase of its coefficients. We shall use the following formula that gives an expression for the directional derivative of a maximum function. It was established in [23]. Elementary properties of maximum functions are addressed in [81, Part Two, Problems 223–226].

N. G. Chebotarëv’s formula ([26, Theorem VI.3.2, (3.6)]). Let I ⊂ ℝ be an open interval and let

K be a compact space. Let f(t, x) be a function on I × K that is continuous along with \frac{∂f}{∂t} (t, x). Let

f^{∗}(t) = max_{x∈K}f(t, x).

Then f^{∗}(t) admits the following expansion at every t ∈ I:

f^{∗}(t + h) = f^{∗}(t) + max_{f(t,x)=f^{∗} (t)} ( h\frac{∂f}{∂t} (t, x)) + o(h).  (H.27)

Proposition 9.1. Let k and l be positive coprime integers. Let r_{1}, r_{2} and r_{3} be three positive real numbers. Then

max_{x}|r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} |

is an even 2π/(k + l)-periodic function of t that decreases strictly on [0, π/(k + l)]: in particular

min_{t} max_{x}|r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} | = max_{x}|r_{1}e^{−ikx} + r_{2}e^{iπ/(k+l)} + r_{3}e^{ilx} |.

Proof. Let

f(t, x) = |r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} | ^{2}.  (H.28)

By (H.10) and (H.11), f^{∗} is an even 2π/(k + l)-periodic function.

Let t ∈ ]0, π/(k + l)[ and choose x^{∗} such that f(t, x^{∗}) = f^{∗}(t): then x^{∗} ∈ [−t/k, t/l] by Proposition 5.1, so that

\frac{1}{2r_{2}} \frac{∂f}{∂t} (t, x^{∗}) = −r_{1} sin(t + kx^{∗}) − r_{3} sin(t − lx^{∗}) < 0

because t + kx^{∗} ∈ [0, (k + l)t/l] and t − lx^{∗} ∈ [0, (k + l)t/k] do not vanish simultaneously. By Formula (H.27), f^{∗} decreases strictly on [0, π/(k + l)].

Proposition 9.2. Let k and l be positive coprime integers. Let r_{1}, r_{2} and r_{3} be three positive real numbers. Then

\frac{max_{x} |r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} |}{|r_{1} + r_{2}e^{it} + r_{3} |}  (H.29)

is an increasing function of t ∈ [0, π/(k + l)]. If kr_{1} = lr_{3}, it is constantly equal to 1; otherwise it is strictly increasing.

Proof. Let f(t, x) be as in (H.28): then the expression (H.29) is g^{∗}(t)^{1/2} with

g(t, x) = \frac{f(t, x)}{f(t, 0)} .

If kr_{1} = lr_{3}, then f(t, 0) = f^{∗}(t), so that g^{∗}(t) = 1. As shown in the beginning of Section 6, we may suppose without loss of generality that kr_{1} < lr_{3}. Let t ∈ ]0, π/(k + l)[ and choose x^{∗} such that f(t, x^{∗}) = f^{∗}(t): then x^{∗} ∈ ]0, t/l[ by Propositions 5.1 and 6.1 and

\frac{f(t, 0)^{2}}{2r_{2}} \frac{∂g}{∂t}(t, x^{∗}) = \frac{1}{2r_{2}} ( \frac{∂f}{∂t} (t, x^{∗})f(t, 0) − f(t, x^{∗})\frac{∂f}{∂t} (t, 0))
= (−r_{1} sin(t + kx^{∗}) − r_{3} sin(t − lx^{∗}))f(t, 0) + f^{∗}(t)(r_{1} + r_{3}) sint
= h(0)f^{∗}(t) − h(x^{∗})f(t, 0)

with

h(x) = r_{1} sin(t + kx) + r_{3} sin(t − lx).

Let us show that h is strictly decreasing on [0, t/l]: in fact, if x ∈ ]0, t/l[,

\frac{dh}{dx}(x) = kr_{1} cos(t + kx) − lr_{3} cos(t − lx) < (kr_{1} − lr_{3}) cos(t − lx) < 0.

As f^{∗}(t) > f(t, 0) and h(0) > h(x^{∗}), (∂g/∂t)(t, x^{∗}) > 0. By N. G. Chebotarëv’s formula, g^{∗} increases strictly on [0, π/(k + l)].

It is possible to describe the decrease of the maximum modulus of a trigonometric trinomial independently of the r’s as follows.

Proposition 9.3. Let k and l be two positive coprime integers. Let r_{1}, r_{2} and r_{3} be three positive real numbers. Let 0 ⩽ t′ < t ⩽ π/(k + l). Then

max_{x}|r_{1}e^{−ikx} + r_{2}e^{it′} + r_{3}e^{ilx} | ⩽ \frac{cos(t′/2)}{cos(t/2)} max_{x}|r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} |  (H.30)

with equality if and only if r_{1} : r_{2} : r_{3} = l : k + l : k.

Proof. Let us apply Proposition 9.2. We have

\begin{matrix}|r_{1} + r_{2}e^{it′} + r_{3} | & 2 \\ |r_{1} + r_{2}e^{it} + r_{3} | & 2 \end{matrix} = 1 + \frac{2r_{2}(r_{1} + r_{3})(cost′− cost)}{(r_{1} + r_{3})^{2} + 2r_{2}(r_{1} + r_{3}) cost + r_{2}^{2}}

= 1 + \frac{cost′− cost}{cost + (r_{2}^{2} + (r_{1} + r_{3})^{2})/2r_{2}(r_{1} + r_{3})}

⩽ 1 + \frac{cost′− cost}{cost + 1} = \frac{cost′+ 1}{cost + 1}

by the arithmetic-geometric inequality, with equality if and only if r_{2} = r_{1} + r_{3}. Therefore Inequality (H.30) holds, with equality if and only if kr_{1} = lr_{3} and r_{2} = r_{1} + r_{3}.

We may now find the minimum of the maximum modulus of a trigonometric trinomial with given spectrum, Fourier coefficient arguments and moduli sum. Proposition 9.3 yields with t′ = 0

Corollary 9.4. Let k and l be two positive coprime integers. Let r_{1}, r_{2} and r_{3} be three positive real numbers. Let t ∈ ]0, π/(k + l)]. Then

\frac{max_{x} |r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} |}{r_{1} + r_{2} + r_{3}} ⩾ cos(t/2)

with equality if and only if r_{1} : r_{2} : r_{3} = l : k + l : k.

Remark 9.5. There is a shortcut proof to Corollary 9.4:

\frac{max_{x} |r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} |}{r_{1} + r_{2} + r_{3}} ⩾ \frac{|r_{1} + r_{2}e^{it} + r_{3} |}{r_{1} + r_{2} + r_{3}}

= \sqrt{1 − \frac{4(r_{1} + r_{3})r_{2}}{(r_{1} + r_{2} + r_{3}) ^{2}} sin^{2}(t/2)}

⩾ \sqrt{1 − sin^{2}(t/2)} = cos(t/2)

and equality holds if and only if |r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} | is maximal for x = 0 and r_{1} + r_{3} = r_{2}.

10 The norm of unimodular relative Fourier multipliers

We may now compute the norm of unimodular relative Fourier multipliers.

Corollary 10.1. Let k and l be two positive coprime integers. Let t ∈ [0, π/(k + l)]. Let M be the relative Fourier multiplier (0, t, 0) that maps the element

r_{1}e^{iu_{1}} e_{−k} + r_{2}e^{iu_{2}} e_{0} + r_{3}e^{iu_{3}} e_{l}  (H.31)

of the normed space C_{{−k,0,l}} on

r_{1}e^{iu_{1}} e_{−k} + r_{2}e^{i(t+u_{2} )} e_{0} + r_{3}e^{iu_{3}} e_{l}.

Then M has norm cos(π/2(k + l) − t/2)/ cos(π/2(k + l)) and attains its norm exactly at elements of form (H.31) with r_{1} : r_{2} : r_{3} = l : k + l : k and

−lu_{1} + (k + l)u_{2} − ku_{3} = π mod 2π.

Proof. This follows from Proposition 9.3 and the concavity of cos on [0, π/2].

Remark 10.2. This corollary enables us to guess how to lift M to an operator that acts by convolution with a measure μ. Note that μ is a Hahn-Banach extension of the linear form f ↦ Mf(0). The relative multiplier M is an isometry if and only if t = 0 and μ is the Dirac measure in 0. Otherwise, t ≠ 0; the proof of Theorem 1.2 (a) in Section 8 shows that μ is a linear combination αδ_{y} + βδ_{w} of two Dirac measures such that the norm of M is |α| + |β|. Let f(x) = le^{−ikx} + (k + l)e^{iπ/(k+l)} + ke^{ilx}: M attains its norm at f, f attains its maximum modulus at 0 and 2mπ/(k + l), and Mf attains its maximum modulus at 2mπ/(k + l), where m is the inverse of l modulo k + l. As

(|α| + |β|) max_{x}|f(x)| = max_{x}|Mf(x)|
= |μ ∗ f(2mπ/(k + l))|
= |αf(2mπ/(k + l) − y) + βf(2mπ/(k + l) − w)|,

we must choose {y, w} = {0, 2mπ/(k + l)}. A computation yields then

μ = e^{it/2} \frac{sin(π/(k + l) − t/2)}{sin(π/(k + l))} δ_{0} + e^{i(t/2+π/(k+l))} \frac{sin(t/2)}{sin(π/(k + l))} δ_{2mπ/(k+l)} .

If k = l = 1, this is a special case of the formula appearing in [41, proof of Prop. 1]. Consult [95] on this issue.

11 The Sidon constant of integer sets

Let us study the maximum modulus of a trigonometric trinomial with given spectrum and Fourier coefficient moduli sum. We get the following result as an immediate consequence of Corollary 9.4.

Proposition 11.1. Let k and l be two positive coprime integers. Let r_{1}, r_{2} and r_{3} be three positive real numbers. Let t ∈ [0, π/(k + l)]. Then

max_{x}|r_{1}e^{−ikx} + r_{2}e^{it} + r_{3}e^{ilx} | ⩾ cos(π/2(k + l)) · (r_{1} + r_{2} + r_{3})

with equality if and only if r_{1} : r_{2} : r_{3} = l : k + l : k and t = π/(k + l).

This means that the Sidon constant of {−k, 0, l} equals sec(π/2(k + l)).

The Sidon constant of integer sets was previously known only in the following three instances:

– The equality

max_{x}|r_{1}e^{i(t_{1} +λ_{1} x)} + r_{2}e^{i(t_{2} +λ_{2} x)} | = r_{1} + r_{2}

shows that the Sidon constant of sets with one or two elements is 1.

– The Sidon constant of \{−1, 0, 1\} is \sqrt{2} and it is attained for e_{−1} + 2i + e_{1}. Let us give the original argument: if f(x) = |r_{1}e^{−ix} + r_{2}e^{it} + r_{3}e^{ix} |^{2}, the parallelogram identity and the arithmetic-quadratic inequality yield

max_{x}f(x) ⩾ max_{x} \frac{f(x) + f(x + π)}{2}

= max_{x} \frac{|r_{1}e^{−ix} + r_{3}e^{ix} + r_{2}e^{it} | ^{2} + |r_{1}e^{−ix} + r_{3}e^{ix} − r_{2}e^{it} | ^{2}}{2}

= max_{x} |r_{1}e^{−ix} + r_{3}e^{ix} | ^{2} + |r_{2}e^{it} |^{2}

= (r_{1} + r_{3})^{2} + r_{2}^{2} ⩾ \frac{(r_{1} + r_{2} + r_{3})^{2}}{2} .

– The Sidon constant of {0, 1, 2, 3, 4} is 2 and it is attained for 1 + 2e_{1} + 2e_{2} − 2e_{3} + e_{4}. These results were obtained by D. J. Newman (see [94].) The fact that the Sidon constant of sets of three integers cannot be 1 had been noted with pairwise different proofs in [94, 21, 50].

Remark 11.2. The real algebraic counterpart is better understood: the maximal absolute value of a real algebraic polynomial of degree at most n with given coefficient absolute value sum is minimal for multiples of the nth Chebyshev polynomial T_{n} (look up the last paragraph of [28], and [82, Theorem 16.3.3] for a proof). As the sum of the absolute values of T_{n}’s coefficients is the integer t_{n} nearest to (1 + \sqrt{2}) ^{n}/2, we have for real a_{0}, a_{1}, …, a_{n}

max_{x∈[−1,1]} |a_{0} + a_{1}x + ⋯+ a_{n}x^{n} | ⩾ t_{n}^{−1} (|a_{0}| + |a_{1}| + ⋯+ |a_{n}|).

The following estimates for the Sidon constant of large integer sets are known.

– E. Beller and D. J. Newman [8] showed that the Sidon constant of {0, 1, …, n} is equivalent to \sqrt{n}.

– (Hadamard sets.) Let q > 1 and suppose that the sequence (λ_{j})_{j⩾1} grows with geometric ratio q: |λ_{j+1} | ⩾ q|λ_{j} | for every j. Then the Sidon constant of {λ_{1}, λ_{2}, …} is finite; it is at most 4.27 if q ⩾ 2 (see [46]), at most 2 if q ⩾ 3 (see [54]), and at most 1 + π^{2}/ (2q^{2} − 2 − π^{2}) if q > \sqrt{1 + π^{2}/2} (see [63, Corollary 9.4] or the updated [64, Corollary 10.2.1].)

Our computations show that the last estimate of the Sidon constant has the right order in q^{−1} for geometric progressions.

Proposition 11.3. Let C be the Sidon constant of the geometric progression {1, q, q^{2}, …}, where q ⩾ 3 is an integer. Then

1 + π^{2}/8(q + 1)^{2} ⩽ sec(π/2(q + 1)) ⩽ C ⩽ 1 + π^{2}/ (2q^{2} − 2 − π^{2}) .

One initial motivation for this work was to decide whether there are sets {λ_{j} }_{j⩾1} with |λ_{j+1} | ⩾ q|λ_{j} | whose Sidon constant is arbitrarily close to 1 and to find evidence among sets with three elements. That there are such sets, arbitrarily large albeit finite, may in fact be proved by the method of Riesz products in [47, Appendix V, §1.II]; see also [64, Proposition 13.1.3]. The case of infinite sets remains open.

A second motivation was to show that the real and complex unconditional constants of the basis (e_{λ_{1}} , e_{λ_{2}} , e_{λ_{3}} ) of C_{Λ} are different; we prove however that they coincide, and it remains an open question whether they may be different for larger sets. The real unconditional constant of (e_{λ_{1}} , e_{λ_{2}} , e_{λ_{3}} ) is the maximum of the norm of the eight unimodular relative Fourier multipliers (t_{1}, t_{2}, t_{3}) such that t_{k} = 0 modulo π. Let i, j, k be a permutation of 1, 2, 3 such that the power of 2 in λ_{i} − λ_{k} and in λ_{j} − λ_{k} are equal. Lemma 2.1 shows that the four relative multipliers satisfying t_{i} = t_{j} modulo 2π are isometries and that the norm of any of the four others, satisfying t_{i} ≠ t_{j} modulo 2π, gives the real unconditional constant. In general, the complex unconditional constant is bounded by π/2 times the real unconditional constant, as proved in [93]; in our case, they are equal.

Corollary 11.4. The complex unconditional constant of the basis (e_{λ_{1}} , e_{λ_{2}} , e_{λ_{3}} ) of C_{Λ} is equal to its real unconditional constant.

Acknowledgement. I would like to thank Henri Comman, Hervé Queffélec and Harold S. Shapiro for their encouragement at crucial moments of this research. The Hungarian-French Scientific and Technological Governmental Cooperation, Project #F-10/04, eased scientific contacts that were helpful to this work.

Keywords. Trigonometric trinomial, maximum modulus, exposed point, extreme point, Mandel′shtam problem, extremal problem, relative Fourier multiplier, Sidon constant, unconditional constant.

2000 Mathematics Subject Classification. Primary 42A05; Secondary 26D05, 30C10, 42A45, 42A55, 46B15, 46B20.

On the Sidon constant of {0, 1, 2, 3}

Abstract

We study an elementary extremal problem on trigonometric polynomials of degree 3. We discover a distinguished torus of extremal functions.

1 Introduction

Let Λ = {λ_{0}, λ_{1}, …, λ_{n−1} } be a set of n frequencies. We study the following extremal problem:

(†)

Given n positive intensities ϱ_{0}, ϱ_{1}, …, ϱ_{n−1} , to find n phases ϑ_{0}, ϑ_{1}, …, ϑ_{n−1} such that the

maximum max_{t} | ∑ ϱ_{j} e^{iϑ_{j}} e^{iλ_{j} t} | is minimal.

This should help us to study the following extremal problem:

(‡)

To find n complex coefficients c_{0}, c_{1}, …, c_{n−1} with given moduli sum |c_{0}|+|c_{1}|+⋯ +|c_{n−1} | = 1

such that the maximum max_{t} | ∑ c_{j} e^{iλ_{j} t} | is minimal.

Note that this maximum’s inverse is the Sidon constant S(Λ). D. J. Newman (see [94, Chapter 3]) obtained the following upper bound for S({0, 1, …, n}): by Parseval’s theorem on ℓ^{2}({1, e^{2iπ/n} , …, e^{2i(n−1)π/n} }), putting ∑ c_{j} e^{ijt} = f(t),

max_{t}|f(t)|^{2} = max_{t}|f(t)|^{2} ∨ |f(t + 2π/n)|^{2} ∨ ⋯∨ |f(t + 2(n − 1)π/n)|^{2}

⩾ max_{t}(|f(t)|^{2} + |f(t + 2π/n)|^{2} + ⋯+ |f(t + 2(n − 1)π/n)|^{2})/n

= max_{t}|c_{0} + c_{n} e^{int} |^{2} + |c_{1}|^{2} + ⋯+ |c_{n−1} |^{2}

= (|c_{0}| + |c_{n}|)^{2} + |c_{1}|^{2} + |c_{2}|^{2} + ⋯+ |c_{n−1} |^{2}  (I.1)

⩾ (|c_{0}| + |c_{1}| + ⋯+ |c_{n}|)^{2}/n,

and H. S. Shapiro showed (ibid.) that equality can hold exactly if n ∈ {1, 2, 4}. If n = 2, equality holds exactly for multiples and translates of f(t) = 1 + 2ie^{it} + e^{i2t}. If n = 3, the functions

\frac{i2\sqrt{2}cosτ − 1 − 3 sinτ}{15} + \frac{3 + sinτ}{10} e^{it} + \frac{3 − sinτ}{10} e^{i2t} + \frac{i2\sqrt{2}cosτ − 1 + 3 sinτ}{15} e^{i3t}

have their modulus bounded by 3/5 for each τ, so that 5/3 ⩽ S(\{0, 1, 2, 3\}) < \sqrt{3}.

The motivation is that we wish to know whether the real and complex unconditionality constants are distinct for basic sequences of characters e^{int}.

Notation. 𝕋 = { z ∈ ℂ : |z| = 1 } and e_{λ}(z) = z^{λ} for z ∈ 𝕋 and λ ∈ ℤ.

118  I. ON THE SIDON CONSTANT OF {0, 1, 2, 3}

2 Necessary conditions for solutions to Extremal problem (†)

Let us the state the theorem of Chebotarev (see [26, Th. VI.3.2, (3.6)]) in our context. Theorem 2.1. Let Φ: ℝ^{k} × ℝ → ℝ be a continuous function that is C^{1} in the first variable and periodic in the second variable. If Φ^{∗}: x ↦ max_{y} Φ(x, y) achieves its minimum at x^{∗}, then there are r points y_{1}, …, y_{r} such that 1 ⩽ r ⩽ k + 1 and Φ(x^{∗}, y_{1}) = ⋯= Φ(x^{∗}, y_{r}) = Φ^{∗}(x^{∗}), and

∃ α_{1}, …, α_{r} ⩾ 0 ∑ α_{i} = 1 ∑ α_{i} \frac{∂Φ}{∂x} (x^{∗}, y_{i}) = 0.

Let us apply this theorem to our problem. For n pairwise distinct integers λ_{0}, λ_{1}, …, λ_{n−1} , n positive real numbers ϱ_{0}, ϱ_{1}, …, ϱ_{n−1} and arguments t and ϑ = (ϑ_{0}, ϑ_{1}, …, ϑ_{n−1} ) let

f(t, ϱ, ϑ) = \frac{∑ ϱ_{j} e^{i(λ_{j} t+ϑ_{j} )}}{∑ ϱ_{j}} = R(t, ϱ, ϑ)e^{iΘ(t,ϱ,ϑ)} with R(t, ϱ, ϑ) ⩾ 0 and Φ = R^{2}.

Define

Φ^{∗}(ϱ, ϑ) = max_{t}Φ(t, ϱ, ϑ), Φ_{∗}^{∗} = minΦ^{∗}

and, for fixed ϱ_{j} > 0,

Φ_{ϱ}(t, ϑ) = Φ(t, ϱ, ϑ), Θ_{ϱ}(t, ϑ) = Θ(t, ϱ, ϑ), Φ_{ϱ}^{∗}(ϑ) = Φ^{∗}(ϱ, ϑ), (Φ_{ϱ}^{∗})_{∗} = minΦ_{ϱ}^{∗}.

Note that Φ^{∗} is continuous (see [81, Chapter 5.4]). Then

Φ = \frac{∑ ∑ ρ_{j} ρ_{k} cos((λ_{k} − λ_{j})t + ϑ_{k} − ϑ_{j})}{(∑ ϱ_{j})^{2}}
\frac{∂Φ}{∂t} = \frac{∑ ∑ (λ_{j} − λ_{k})ϱ_{j} ϱ_{k} sin((λ_{k} − λ_{j})t + ϑ_{k} − ϑ_{j})}{(∑ ϱ_{j})^{2}}  = 2ℜ( f̄\frac{∂f}{∂t} )
\frac{∂Φ}{∂ϑ_{k}} = −2ϱ_{k} \frac{∑ ϱ_{j} sin((λ_{k} − λ_{j})t + ϑ_{k} − ϑ_{j})}{(∑ ϱ_{j})^{2}} = \frac{−2ϱ_{k}}{∑ ϱ_{j}} ℑ(f̄e^{i(λ_{k} t+ϑ_{k})} )  (I.2)

Note that our problem depends in fact on n − 2 phases only; we have

∑ \frac{∂Φ}{∂ϑ_{k}} = 0 , ∑ λ_{k} \frac{∂Φ}{∂ϑ_{k}} = \frac{∂Φ}{∂t} .  (I.3)

We get thus Lemma 2.2. Let Φ_{ϱ}^{∗} achieve its minimum at ϑ and M = { t : Φ_{ϱ}(t, ϑ) = (Φ_{ϱ}^{∗})_{∗} }.

(a) If M is a singleton {t}, then (t, ϑ) is a critical point of Φ_{ϱ}. Then all λ_{k}t + ϑ_{k} are congruent to Θ_{ϱ}(t, ϑ) modulo π.

(b) If M is a pair {t, u} and (t, ϑ), (u, ϑ) are not critical points of Φ_{ϱ}, then

∃ μ, ν > 0 ∀ k μ\frac{∂Φ}{∂ϑ_{k}} (t, ϑ) + ν \frac{∂Φ}{∂ϑ_{k}} (u, ϑ) = 0.

Then μ sin(Θ_{ϱ}(t, ϑ) − λ_{k}t − ϑ_{k}) + ν sin(Θ_{ϱ}(u, ϑ) − λ_{k}u − ϑ_{k}) = 0 for all k. Furthermore one may compute

\frac{∂Φ}{∂ϱ_{k}} = 2\frac{∑ ϱ_{j} cos((λ_{k} − λ_{j})t + ϑ_{k} − ϑ_{j})}{(∑ ϱ_{j})^{2}} − \frac{2Φ}{∑ ϱ_{j}} = \frac{2}{∑ ϱ_{j}} (ℜ(f̄e^{i(λ_{k} t+ϑ_{k})} ) − Φ).

Note that our problem depends on n − 1 intensities only; we have

∑ ϱ_{k} \frac{∂Φ}{∂ϱ_{k}} = 0.

Thus

\frac{∑ ϱ_{j}}{2} \frac{∂Φ}{∂ϱ_{k}} − i \frac{∑ ϱ_{j}}{2ϱ_{k}} \frac{∂Φ}{∂ϑ_{k}} = R(e^{i(λ_{k} t+ϑ_{k} −Θ)} − R).

3. NECESSARY CONDITIONS FOR SOLUTIONS TO EXTREMAL PROBLEM (‡)  119

Lemma 2.3. Let Φ^{∗} achieve its minimum at (ϱ, ϑ) and M = { t : Φ(t, ϱ, ϑ) = Φ_{∗}^{∗} }.

(a) If M is a singleton {t}, then (t, ϱ, ϑ) is a critical point of Φ. Then Φ_{∗}^{∗} = 1.

(b) If M is a pair {t, u} and Φ_{∗}^{∗} ≠ 1, then

∃ μ, ν > 0 ∀ k μ( \frac{∂Φ}{∂ϑ_{k}} , \frac{∂Φ}{∂ϱ_{k}} ) (t, ϱ, ϑ) + ν( \frac{∂Φ}{∂ϑ_{k}} , \frac{∂Φ}{∂ϱ_{k}} ) (u, ϱ, ϑ) = 0.  (I.4)

If in turn Φ_{∗}^{∗} ≠ 1 and (I.4) holds, then there is an α and there are signs ε_{j} ∈ {−1, 1} not all equal such that, with d_{ijk} = gcd(λ_{k} − λ_{i}, λ_{j} − λ_{i}) and δ_{ijk} ∈ {0, 1}

R(t, ϱ, ϑ) = R(u, ϱ, ϑ) = cosα
∑ ε_{i}ϱ_{i} = ∑ ε_{i}λ_{i}ϱ_{i} = 0
∀ i, j, k ((λ_{j} − λ_{k})ε_{i} + (λ_{k} − λ_{i})ε_{j} + (λ_{i} − λ_{j})ε_{k})α ≡
(λ_{j} − λ_{k})ϑ_{i} + (λ_{k} − λ_{i})ϑ_{j} + (λ_{i} − λ_{j})ϑ_{k} ≡ δ_{ijk} d_{ijk} π mod 2d_{ijk} π
∀ j Θ(t, ϱ, ϑ) − λ_{j} t − ϑ_{j} = ε_{j} α = −(Θ(u, ϱ, ϑ) − λ_{j} u − ϑ_{j}).

Proof. Let R = R(t, ϱ, ϑ) = R(u, ϱ, ϑ), Θ_{t} = Θ(t, ϱ, ϑ), Θ_{u} = Θ(u, ϱ, ϑ). Then

∀ k μ(e^{i(λ_{k} t+ϑ_{k} −Θ_{t})} − R) + ν(e^{i(λ_{k} u+ϑ_{k} −Θ_{u} )} − R) = 0.

Let us suppose μ + ν = 1. By taking moduli, we get then

∀ k cos(λ_{k}t + ϑ_{k} − Θ_{t}) = \frac{R^{2} + μ^{2} − ν^{2}}{2μR} , cos(λ_{k}u + ϑ_{k} − Θ_{u}) = \frac{R^{2} + ν^{2} − μ^{2}}{2νR} ;

Returning to the definition of f, we have then

∀ k cos(λ_{k}t + ϑ_{k} − Θ_{t}) = R = cos(λ_{k}u + ϑ_{k} − Θ_{u}),

so that furthermore μ = ν. Thus we may choose signs ε_{k} ∈ {−1, 1} such that

∀ j, k ε_{j} sin(λ_{j} t + ϑ_{j} − Θ_{t}) = ε_{k} sin(λ_{k}t + ϑ_{k} − Θ_{t}) = −ε_{k} sin(λ_{k}u + ϑ_{k} − Θ_{u}).

These sines do not vanish, because otherwise R = 1. The expressions of f and ∂Φ/∂t yield therefore

∑ ε_{j} ϱ_{j} = 0, ∑ ε_{j} λ_{j} ϱ_{j} = 0.

We can therefore choose α, α′ and signs ε_{j} ∈ {−1, 1} such that

∀ j Θ_{t} − λ_{j} t − ϑ_{j} = ε_{j} α , Θ_{u} − λ_{j} u − ϑ_{j} = ε_{j} α′.

(λ_{k} − λ_{j})t + ϑ_{k} − ϑ_{j} = \{ \begin{matrix}0 & if ε_{j} = ε_{k} \\ 2ε_{j} α & otherwise. \end{matrix}

and similarly for u. Finally

\frac{λ_{j} − λ_{k}}{d_{ijk}} ϑ_{i} + \frac{λ_{k} − λ_{i}}{d_{ijk}} ϑ_{j} + \frac{λ_{i} − λ_{j}}{d_{ijk}} ϑ_{k} = ±( \frac{λ_{j} − λ_{k}}{d_{ijk}} ε_{i} + \frac{λ_{k} − λ_{i}}{d_{ijk}} ε_{j} + \frac{λ_{i} − λ_{j}}{d_{ijk}} ε_{k}) α.

3 Necessary conditions for solutions to Extremal problem (‡)

Let (ϱ, ϑ) solve the extremal problem. If the point t such that Φ^{∗}(ϱ, ϑ) = Φ(t, ϱ, ϑ) were unique, then Φ^{∗}(ϱ, ϑ) = 1. If there are exactly two points t, u such that Φ^{∗}(ϱ, ϑ) = Φ(t, ϱ, ϑ) = Φ(u, ϱ, ϑ), then either Φ^{∗}(ϱ, ϑ) = 1 or there is an α and there are signs ε_{0}, ε_{1}, ε_{2} ∈ {−1, 1} such that, with d = gcd(λ_{1} − λ_{0}, λ_{2} − λ_{0}),

R(t, ϱ, ϑ) = R(u, ϱ, ϑ) = cosα
ε_{0}ϱ_{0} + ε_{1}ϱ_{1} + ε_{2}ϱ_{2} = ε_{0}λ_{0}ϱ_{0} + ε_{1}λ_{1}ϱ_{1} + ε_{2}λ_{2}ϱ_{2} = 0
((λ_{1} − λ_{2})ε_{0} + (λ_{2} − λ_{0})ε_{1} + (λ_{0} − λ_{1})ε_{2})α ≡
(λ_{1} − λ_{2})ϑ_{0} + (λ_{2} − λ_{0})ϑ_{1} + (λ_{0} − λ_{1})ϑ_{2} ≡ dπ mod 2dπ

We need not consider the case (λ_{1} − λ_{2})ϑ_{0} + (λ_{2} − λ_{0})ϑ_{1} + (λ_{0} − λ_{1})ϑ_{2} ≡ 0 mod 2dπ, for which the system

λ_{0}t + ϑ_{0} = λ_{1}t + ϑ_{1} = λ_{2}t + ϑ_{2}

has a solution, so that Φ^{∗}(ϱ, ϑ) = 1. Then

ε_{0} = sgn(λ_{1} − λ_{2})ε, ε_{1} = sgn(λ_{2} − λ_{0})ε, ε_{2} = sgn(λ_{0} − λ_{1})ε for some ε ∈ {−1, 1},

so that

ϱ_{0} = |λ_{1} − λ_{2}|σ, ϱ_{1} = |λ_{2} − λ_{0}|σ, ϱ_{2} = |λ_{0} − λ_{1}|σ for some σ > 0.

Choose {i, j, k} = {0, 1, 2} such that ε_{i} ≠ ε_{j} = ε_{k}. Then 2|λ_{j} − λ_{k}|α ≡ dπ mod 2dπ, so that R ⩽ cos(dπ/2|λ_{j} − λ_{k}|). Further ϱ_{k} + ϱ_{j} = ϱ_{i}, and the system

λ_{j} t + ϑ_{j} = λ_{k}t + ϑ_{k} = λ_{i}t + ϑ_{i} − dπ/(λ_{j} − λ_{k})

has a solution t, for which

R = \frac{|ϱ_{k} + ϱ_{j} + ϱ_{i}e^{idπ/(λ_{j} −λ_{k})} |}{2ϱ_{i}} = cos \frac{dπ}{2|λ_{j} − λ_{k}|} .

This solves the problem for Λ = {0, 1, 2}, as t ↦ Φ(t, ϱ, ϑ) has at most two maxima.

4 The case {0, 1, 2, 3}: a distinguished family of polynomials

Let Λ = {0, 1, 2, 3}. Let (ϱ, ϑ) solve the extremal problem. If the point t such that Φ^{∗}(ϱ, ϑ) = Φ(t, ϱ, ϑ) were unique, then Φ_{∗}^{∗} = 1. If there were exactly two points t, u such that Φ^{∗}(ϱ, ϑ) = Φ(t, ϱ, ϑ) = Φ(u, ϱ, ϑ), then Φ_{∗}^{∗} = \sqrt{2}. There are therefore exactly three points t, u, t′′ such that Φ^{∗}(ϱ, ϑ) = Φ(t, ϱ, ϑ) = Φ(u, ϱ, ϑ) = Φ(t′′, ϱ, ϑ).

Let f(t, τ) be given by

\frac{i2\sqrt{2}cosτ − 1 − 3 sinτ}{15} + \frac{3 + sinτ}{10} e^{it} + \frac{3 − sinτ}{10} e^{i2t} + \frac{i2\sqrt{2}cosτ − 1 + 3 sinτ}{15} e^{i3t}.

One computes that the moduli sum of the coefficients is 1, independently of τ. Note that f(t, −τ) = e^{i3t}f(−t, τ) and f(t, τ + π) = e^{i3t} \overline{f(t, τ)}, so that we shall restrict the parameter τ to [0, π/2]. Let Φ(t, τ) = |f(t, τ)|^{2}. We get

Φ(t, τ) = \frac{2\sqrt{2}sin2τ}{75}(sint − sin2t+ 2 sin3t) + \frac{247 − 13 cos2τ}{900}
+ (1 + cos2τ)( \frac{cost}{20} − \frac{cos2t}{25}) +\frac{1 + 17 cos2τ}{225} cos3t.

Let us put

M = ( \begin{matrix}\frac{2 sin2t}{25} − \frac{sint}{20} − \frac{17 sin3t}{75} & \frac{2\sqrt{2}}{75}(cost − 2 cos2t+ 6 cos3t) \\ \frac{2\sqrt{2}}{75}(sint − sin2t+ 2 sin3t) & \frac{13}{900} − \frac{cost}{20} + \frac{cos2t}{25} − \frac{17 cos3t}{225} \end{matrix} )

The critical points (t, τ) of Φ satisfy

M ( \begin{matrix}cos & 2τ \\ sin & 2τ \end{matrix}) = ( \begin{matrix}\frac{sint}{20} − \frac{2 sin2t}{25} + \frac{sin3t}{75} \\ 0 \end{matrix} )

We have

detM = \frac{1}{6750} sint( cost − \frac{1}{4}) (4 cost − 11)(16 cos^{3} t − 72 cos^{2} t + 33 cost − 41),

5. THE REAL UNCONDITIONAL CONSTANT OF {0, 1, 2, 3}  121

which vanishes exactly if cost ∈ {−1, 1/4, 1}. Otherwise we get

\{ \begin{matrix}cos2τ & = −\frac{272 cos^{3} t − 72 cos^{2} t − 159 cost + 23}{16 cos^{3} t − 72 cos^{2} t + 33 cost − 41} = C(t) \\ sin2τ & = −\frac{24\sqrt{2}sint(4 cost + 1)(2 cost − 1)}{16 cos^{3} t − 72 cos^{2} t + 33 cost − 41} = S(t) \end{matrix}  (I.5)

Note that this solution is consistent, as C^{2} +S^{2} = 1. For such τ, Φ(t, τ) = 9/25. Checking the special cases cost ∈ {−1, 1/4, 1} yields that all local maxima are given by the above formulas, that Φ attains its global minimum, 0, exactly for τ = 0 and t = π, and has exactly one other local minimum, of value 49/225, for τ = π/2 and t = 0. There is exactly one other critical point, of value 5/18, that is a saddle point, given by τ = arccos(17/37)/2, t = arccos1/4.

As C(0) = 1, C(±π/3) = −1, C(± arccos(−1/4)) = 1, C(π) = −1, the intermediate values theorem shows that for a given τ, there are exactly three solutions t to system (I.5), for which Φ(t, τ) achieves then its global maximum, 9/25.

These formulas yield in turn that for a sign ε ∈ {−1, +1}

\{ \begin{matrix}cosτ & = \frac{8ε(2 cost − 1) cost/2}{\sqrt{−16 cos^{3} t + 72 cos^{2} t − 33 cost + 41}} = C_{ε}(t) \\ sinτ & = \frac{3\sqrt{2}ε(4 cost + 1) sint/2}{\sqrt{−16 cos^{3} t + 72 cos^{2} t − 33 cost + 41}} = S_{ε}(t) \end{matrix}  (I.6)

5 The real unconditional constant of {0, 1, 2, 3}

If L is a subspace of the space of complex continuous functions on a compact space T with n dimensions, then every functional l on L extends isometrically to a linear combination of at most 2n Dirac measures: there are m ⩽ 2n points t_{k} ∈ T and coefficients b_{k} ∈ ℂ such that for every f ∈ L one has l(f) = ∑ b_{k}f(t_{k}) and ∥l∥ = ∑ |b_{k}| (see [13, Exercice 6.8].) This implies in particular that there is a function f ∈ L whose maximum modulus points contain the t_{k}.

Let us now specialise to the case L = C_{Λ} (𝕋) with Λ a finite set. Note that a function in L has at most maxΛ − minΛ maximum modulus points; a trigonometric trinomial has at most 2 maximum modulus points up to periodicity.

Let us make the ad hoc hypothesis that the t_{k} are the nth roots of unity, whose set forms the group 𝕌_{n}: this obliges us to restrict our study to those functionals l such that l(e_{j}) = l(e_{j′}) if j ≡ j′ mod n. Then the condition l(f) = ∑ b_{k}f(t_{k}) reads

l(e_{j}) = ∑_{k=0}^{n−1} b_{k}e^{i2jkπ/n} for j ∈ Λ,

which may be interpreted as telling that the l(e_{j}) are the Fourier coefficients of the measure μ on 𝕌_{n} given by

μ = ∑_{k=0}^{n−1} b_{k}δ_{e^{i2kπ/n}}

(where the Dirac measures act on 𝕌_{n}). A solution to these equations is given by

b_{k} = \frac{1}{n} ∑_{j=0}^{n−1} e^{−i2jkπ/n} \{ \begin{matrix}l(e_{j′}) & if there is j′ ≡ j mod n in Λ \\ 0 & otherwise. \end{matrix}

The norm of μ is bounded by

∑_{k=0}^{n−1} |b_{k}|

and is attained at u ∈ C(𝕌_{n}) if and only if u(e^{i2kπ/n} )b_{k} = |b_{k}| for every k, up to a nonzero complex number. This yields an upper bound for the norm of l that becomes an equality if there is an f ∈ C(𝕋) of norm 1 such that f(e^{i2kπ/n} ) = u(e^{i2kπ/n} ).

Here are two applications.

Proposition 5.1. Let Λ be a finite subset of ℤ. The Sidon constant of Λ is at most (# Λ − 1)^{1/2} .

Proof. One may suppose that minΛ = 0 and choose n = maxΛ. Let l be a linear functional l with coefficients l(e_{j}) of modulus 1: one may suppose that l(e_{0}) = l(e_{n}). Then

∥l∥ ⩽ \frac{1}{n} ∑_{k=0}^{n−1} | ∑_{j∈Λ\\{n\}} l(e_{j})e^{−i2jkπ/n} |

⩽ ( \frac{1}{n} ∑_{k=0}^{n−1} | ∑_{j∈Λ\\{n\}} l(e_{j})e^{−i2jkπ/n} |^{2}) ^{1/2}  (I.7)

= ( ∑_{j∈Λ\{n}} |l(e_{j})|^{2}) ^{1/2} = (# Λ − 1)^{1/2} .

Remark 5.2. Hervé Queffélec showed me a more elementary proof of the same fact, that follows the line of D. J. Newman’s computation in the introduction: if f ∈ C_{Λ} (𝕋), then there are only # Λ − 1 squares, and not n, in Inequality (I.1)!

Remark 5.3. If Λ = {0, 1, …, n}, then Inequality (I.7) is an equality if and only if (l(e_{j}))_{j=0}^{n−1} is a biunimodular sequence, that is a unimodular function on 𝕌_{n} whose Fourier transform is also unimodular. In other words, the matrix H = (l(e_{j−k} ))_{0⩽j,k⩽n−1} is a circulant complex Hadamard matrix, where the indices j − k are computed modulo n: it satisfies H^{∗}H = n Id. Such matrices always exist: see [10].

Proposition 5.4. Let Λ = {0, 1, 2, 3}. The real unconditional constant of C_{Λ} (𝕋) is 5/3.

Proof. The polynomials in the previous section show that the real unconditional constant of C_{Λ} (𝕋) is at least 5/3. This constant is the maximum of the norm of linear functionals l with l(e_{j}) ∈ {−1, 1}. As l has the same norm as l̃: f ↦ l(f(· + π)), for which l̃(e_{j}) = (−1)^{j} l(e_{j}), one may suppose that l(e_{0}) = l(e_{3}). Let us now try to lift l to a sum of Dirac measures on the third roots of unity. Such a lifting is either the Dirac measure at 0 or

±(l(e_{j}))_{0⩽j⩽2} ∈ {(−1, 1, 1), (1, −1, 1), (1, 1, −1)}

and these six cases yield the same norm

\frac{1}{3}(|−1 + 1 + 1| + |−1 + e^{i2π/3} + e^{i4π/3} | + |−1 + e^{i4π/3} + e^{i2π/3} |) = 5/3.

6 Trigonometric polynomials of degree 3 with real coefficients

Suppose c_{0}, c_{1}, c_{2}, c_{3} are real and c_{0}c_{3} ≠ 0. Let ε_{i} = sgnc_{i} be their sign. Then

∥c_{0} + c_{1}e^{it} + c_{2}e^{2it} + c_{3}e^{3it}∥_{∞} = |c_{0} + c_{2}| + |c_{1} + c_{3}|

if |||||||||| ε_{0} = ε_{2} and ε_{1} = ε_{3}

or \{ \begin{matrix}ε_{0}ε_{3}(c_{0}c_{1} + c_{1}c_{2} + c_{2}c_{3} + 9c_{0}c_{3}) + 4(c_{0}c_{2} + c_{1}c_{3}) ⩾ 0 \\ 6|c_{0}c_{3}| ⩽ |c_{0}c_{2} + c_{1}c_{3}| \end{matrix}
or ε_{0}ε_{3}(c_{0}c_{1} + c_{1}c_{2} + c_{2}c_{3} + 9c_{0}c_{3}) ⩽ 4(c_{0}c_{2} + c_{1}c_{3})
or ε_{0}ε_{1}ε_{2}ε_{3} = −1 and (|c_{0}c_{2}| + |c_{1}c_{3}|)^{2} ⩽ 4(|c_{0}| + |c_{3}|)c_{0}c_{3}(ε_{0}c_{1} + ε_{3}c_{2}).

Otherwise ∥c_{0} + c_{1}e^{it} + c_{2}e^{2it} + c_{3}e^{3it}∥_{∞}^{2} is equal to

2c_{0}c_{3}( ( \frac{c_{1}}{3c_{0}} + \frac{c_{2}}{3c_{3}} )^{3} + ( ( \frac{c_{1}}{3c_{0}} − \frac{c_{2}}{3c_{3}} )^{2} + ( 1 − \frac{c_{2}}{3c_{0}} ) ( 1 − \frac{c_{1}}{3c_{3}} ) ) ^{3/2} )

+ ( 1 − \frac{c_{1}c_{2}}{3c_{0}c_{3}}) (c_{0}^{2} + c_{3}^{2}) − ( 1 + \frac{c_{1}c_{2}}{3c_{0}c_{3}}) (c_{0}c_{2} + c_{1}c_{3}) + \frac{2}{3}(c_{1}^{2} + c_{2}^{2}).

Suppose c_{0}, c_{1}, c_{2} are real and c_{0}c_{2} ≠ 0. Let ε_{i} = sgnc_{i} be their sign. Then

∥c_{0} + c_{1}e^{it} + c_{2}e^{2it}∥_{∞} = |c_{0} + c_{2}| + |c_{1}| if ε_{0} = ε_{2} or |c_{1}| |c_{0}^{−1} + c_{2}^{−1} | ⩾ 4;

otherwise

∥c_{0} + c_{1}e^{it} + c_{2}e^{2it}∥_{∞}^{2} = (c_{0} − c_{2})^{2}(1 − c_{1}^{2}/4c_{0}c_{2}).

Bibliographie

[1] A. B. Aleksandrov et V. V. Peller – « Hankel and Toeplitz-Schur multipliers », Math. Ann. 324 (2002), no. 2, p. 277–327. (p. 86).

[2] Richard M. Aron et Maciej Klimek – « Supremum norms for quadratic polynomials », Arch. Math. (Basel) 76 (2001), no. 1, p. 73–80. (p. 108).

[3] F. V. Atkinson, G. A. Watterson et P. A. P. Moran – « A matrix inequality », Quart. J. Math. Oxford Ser. (2) 11 (1960), p. 137–140. (pp. 55, 56, 67 et 68).

[4] William D. Banks et Asma Harcharras – « On the norm of an idempotent Schur multiplier on the Schatten class », Proc. Amer. Math. Soc. 132 (2004), no. 7, p. 2121–2125. (p. 39).

[5] William D. Banks, Asma Harcharras, Ilijas Farah et Dominique Lecomte – « The R-C problem », manuscrit, 2003. (p. 48).

[6] Erik Bédos – « On Følner nets, Szegő’s theorem and other eigenvalue distribution theorems », Exposition. Math. 15 (1997), no. 3, p. 193–228, Erratum p. 384. (pp. 75, 78, 79 et 80).

[7] Paul R. Beesack – « Inequalities involving iterated kernels and convolutions », Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 274–301 (1969), p. 11–16. (p. 56).

[8] E. Beller et D. J. Newman – « An l_{1} extremal problem for polynomials », Proc. Amer. Math. Soc. 29 (1971), p. 474–481. (p. 115).

[9] T. Beth, D. Jungnickel et H. Lenz – Design theory, seconde éd., Cambridge University Press, 1999. (pp. 12, 53, 54, 65 et 66).

[10] Göran Björck et Bahman Saffari – « New classes of finite unimodular sequences with unimodular Fourier transforms. Circulant Hadamard matrices with complex entries », C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), p. 319–324. (p. 122).

[11] Béla Bollobás – Extremal graph theory, Academic Press, 1978. (pp. 28, 30 et 65).

[12] J. A. Bondy et M. Simonovits – « Cycles of even length in graphs », J. Combin. Theory Ser. B 16 (1974), p. 97–105. (pp. 10 et 27).

[13] Nicolas Bourbaki – Espaces vectoriels topologiques. Chapitres 1 à 5, nouvelle éd., Masson, Paris, 1981, Éléments de mathématique. (p. 121).

[14] J. Bourgain – « Bounded orthogonal systems and the Λ(p)-set problem », Acta Math. 162 (1989), no. 3-4, p. 227–245. (p. 26).

[15] Jean Bourgain – « Λ_{p}-sets in analysis : results, problems and related aspects », Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, p. 195–232. (p. 84).

[16] Marek Bożejko – « A new group algebra and lacunary sets in discrete noncommutative groups », Studia Math. 70 (1981), no. 2, p. 165–175. (p. 84).

[17] Marek Bożejko et Gero Fendler – « Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group », Boll. Un. Mat. Ital. A (6) 3 (1984), no. 2, p. 297–302. (p. 80).

[18] Nathanial P. Brown et Narutaka Ozawa – C^{∗}-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. (pp. 80 et 81).

[19] D. de Caen et L. A. Székely – « The maximum size of 4- and 6-cycle free bipartite graphs on m, n vertices », Sets, graphs and numbers (Budapest, 1991), North-Holland, 1992, p. 135–142. (p. 70).

[20] — , « On dense bipartite graphs of girth eight and upper bounds for certain configurations in planar point-line systems », J. Combin. Theory Ser. A 77 (1997), no. 2, p. 268–278. (pp. 64, 69 et 70).

[21] Donald I. Cartwright, Robert B. Howlett et John R. McMullen – « Extreme values for the Sidon constant », Proc. Amer. Math. Soc. 81 (1981), no. 4, p. 531–537. (pp. 46, 97 et 115).

[22] P. G. Casazza et N. J. Kalton – « Notes on approximation properties in separable Banach spaces », Geometry of Banach spaces (Strobl, 1989) (P. F. X. Müller et W. Schachermayer, éds.), London Math. Soc. Lect. Notes 158, Cambridge Univ. Press, 1991, p. 49–63. (pp. 14 et 48).

[23] Nikolaĭ Grigor′evich Chebotarëv – « On a general criterion of the minimax », C. R. (Doklady) Acad. Sci. URSS (N.S.) 39 (1943), p. 339–341. (p. 112).

[24] — , « Kriterii minimaksa i ego prilozheniya », Sobranie sochineniĭ. Vol. 2, Izdatel′stvo Akademii Nauk SSSR, 1949, p. 396–409. (pp. 16 et 100).

[25] Jean De Cannière et Uffe Haagerup – « Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups », Amer. J. Math. 107 (1985), no. 2, p. 455–500. (pp. 78, 81 et 91).

[26] Vladimir F. Dem′yanov et Vassili N. Malozëmov – Introduction to minimax, Halsted Press, 1974. (pp. 16, 100, 112 et 118).

[27] Konstantin M. Dyakonov – « Extreme points in spaces of polynomials », Math. Res. Lett. 10 (2003), no. 5-6, p. 717–728. (p. 112).

[28] Paul Erdős – « Some remarks on polynomials », Bull. Amer. Math. Soc. 53 (1947), p. 1169–1176. (p. 115).

[29] — , « Extremal problems in graph theory », Theory of Graphs and its Applications (Smolenice, 1963), Publ. House Czechoslovak Acad. Sci., 1964, p. 29–36. (pp. 21 et 27).

[30] T. W. Gamelin – Uniform algebras and Jensen measures, Cambridge University Press, 1978. (p. 87).

[31] G. Godefroy et N. J. Kalton – « Approximating sequences and bidual projections », Quart. J. Math. Oxford (2) 48 (1997), p. 179–202. (p. 48).

[32] G. Godefroy, N. J. Kalton et P. D. Saphar – « Unconditional ideals in Banach spaces », Studia Math. 104 (1993), p. 13–59. (pp. 14 et 48).

[33] Israel Gohberg et Naum Krupnik – One-dimensional linear singular integral equations. I, II, Birkhäuser Verlag, 1992. (p. 87).

[34] I. Ts. Gokhberg et N. Ya. Krupnik – « Norm of the Hilbert transformation in the L_{p} space », Funct. Anal. Appl. 2 (1968), no. 2, p. 180–181. (p. 87).

[35] E. Györi – « C_{6}-free bipartite graphs and product representation of squares », Discrete Math. 165/166 (1997), p. 371–375. (p. 70).

[36] Uffe Haagerup et Jon Kraus – « Approximation properties for group C^{∗}-algebras and group von Neumann algebras », Trans. Amer. Math. Soc. 344 (1994), no. 2, p. 667–699. (p. 89).

[37] Asma Harcharras – « Fourier analysis, Schur multipliers on S^{p} and non-commutative Λ(p)-sets », Studia Math. 137 (1999), no. 3, p. 203–260. (pp. 10, 21, 23, 24, 25, 26, 27, 39, 52, 53, 84 et 85).

[38] Asma Harcharras, Stefan Neuwirth et Krzysztof Oleszkiewicz – « Lacunary matrices », Indiana Univ. Math. J. 50 (2001), p. 1675–1689. (pp. 39 et 48).

[39] G. H. Hardy, J. E. Littlewood et G. Pólya – Inequalities, seconde éd., Cambridge Univ. Press, 1952. (p. 67).

[40] S. Hartman – « Some problems and remarks on relative multipliers », Colloq. Math. 54 (1987), p. 103–111, Erratum dans Colloq. Math. 57 (1989), 189. (p. 102).

[41] Milan Hladnik et John Holbrook – « Optimal measures for Toeplitz matrices », Linear Algebra Appl. 402 (2005), p. 319–334. (p. 114).

[42] B. Hollenbeck, N. J. Kalton et I. E. Verbitsky – « Best constants for some operators associated with the Fourier and Hilbert transforms », Studia Math. 157 (2003), no. 3, p. 237–278. (p. 87).

[43] Brian Hollenbeck et Igor E. Verbitsky – « Best constants for the Riesz projection », J. Funct. Anal. 175 (2000), no. 2, p. 370–392. (p. 88).

[44] Shlomo Hoory – « The size of bipartite graphs with a given girth », J. Combin. Theory Ser. B 86 (2002), no. 2, p. 215–220. (pp. 12 et 53).

[45] Marius Junge et Zhong-Jin Ruan – « Approximation properties for noncommutative L_{p}-spaces associated with discrete groups », Duke Math. J. 117 (2003), no. 2, p. 313–341. (pp. 77 et 89).

[46] Jean-Pierre Kahane – « Ensembles quasi-indépendants et ensembles de Sidon, théorie de Bourgain (d’après Myriam Déchamps, Li et Queffélec) », http://arxiv.org/0709.4386, 2007. (p. 115).

[47] Jean-Pierre Kahane et Raphaël Salem – Ensembles parfaits et séries trigonométriques, Hermann, 1963. (pp. 94 et 116).

[48] Alexander Koldobsky et Hermann König – « Aspects of the isometric theory of Banach spaces », Handbook of the geometry of Banach spaces, Vol. 1 (W. B. Johnson et J. Lindenstrauss, éds.), North-Holland, 2001, p. 899–939. (p. 45).

[49] Felix Lazebnik, Vasiliy A. Ustimenko et Andrew J. Woldar – « Polarities and 2k-cycle-free graphs », Discrete Math. 197/198 (1999), p. 503–513. (p. 28).

[50] Daniel Li – « Complex unconditional metric approximation property for C_{Λ} (𝕋) spaces », Studia Math. 121 (1996), no. 3, p. 231–247. (pp. 45, 91 et 115).

[51] Daniel Li et Hervé Queffélec – Introduction à l’étude des espaces de Banach, Cours Spécialisés [Specialized Courses], vol. 12, Société Mathématique de France, Paris, 2004, Analyse et probabilités. [Analysis and probability theory]. (p. 91).

[52] John Edensor Littlewood – « A theorem on power series », Proc. London Math. Soc. (2) 23 (1925), p. 94–103 (English). (pp. 16 et 100).

[53] Leo Livshits – « A note on 0-1 Schur multipliers », Linear Algebra Appl. 222 (1995), p. 15–22. (p. 39).

[54] Jorge M. López et Kenneth A. Ross – Sidon sets, Lecture Notes Pure Appl. Math. 13, Marcel Dekker Inc., New York, 1975. (pp. 85, 86, 91 et 115).

[55] Adolf Mader et Otto Mutzbauer – « Double orderings of (0,1)-matrices », Ars Combin. 61 (2001), p. 81–95. (p. 12).

[56] H. van Maldeghem – Generalized polygons, Birkhäuser Verlag, 1998. (pp. 12, 53, 54, 66, 67 et 69).

[57] S. P. H. Mandel et I. M. Hughes – « Change in mean viability at a multi-allelic locus in a population under random mating », Nature 182 (1958), p. 63–64 (English). (p. 55).

[58] Michael Marsalli et Graeme West – « Noncommutative H^{p} spaces », J. Operator Theory 40 (1998), no. 2, p. 339–355. (p. 87).

[59] Gustavo A. Muñoz Fernández et Juan B. Seoane Sepúlveda – « Geometry of Banach spaces of trinomials », J. Math. Anal. Appl. 340 (2008), p. 1069–1087. (p. 112).

[60] H. P. Mulholland et C. A. B. Smith – « An inequality arising in genetical theory », Amer. Math. Monthly 66 (1959), p. 673–683. (p. 55).

[61] F. Nazarov, G. Pisier, S. Treil et A. Volberg – « Sharp estimates in vector Carleson imbedding theorem and for vector paraproducts », J. Reine Angew. Math. 542 (2002), p. 147–171. (p. 89).

[62] Stefan Neuwirth – « The Sidon constant of sets with three elements », http://arxiv.org/abs/ math/0102145, 2001. (p. 103).

[63] — , « Metric unconditionality and Fourier analysis », Studia Math. 131 (1998), p. 19–62. (pp. 42, 47, 49, 51, 97, 98 et 115).

[64] — , « Multiplicateurs et analyse fonctionnelle », Thèse, Université Paris 6, 1999, http://

tel.archives-ouvertes.fr/tel-00010399. (pp. 48, 91, 103, 115 et 116).

[65] — , « The size of bipartite graphs with girth eight », http://arxiv.org/math/0102210, 2001. (p. 53).

[66] — , « Cycles and 1-unconditional matrices », Proc. London Math. Soc. (3) 93 (2006), p. 761– 790. (pp. 80, 81 et 85).

[67] Daniel M. Oberlin – « Translation-invariant operators on L^{p}(G), 0 < p < 1 », Michigan Math. J. 23 (1976), no. 2, p. 119–122. (p. 86).

[68] Timur Oikhberg – « Restricted Schur multipliers and their applications », Proc. Amer. Math. Soc. 138 (2010), no. 5, p. 1739–1750. (p. 92).

[69] Victor Olevskii – « A connection between Fourier and Schur multipliers », Integral Equations Operator Theory 25 (1996), no. 4, p. 496–500. (p. 87).

[70] Władysław Orlicz – Linear functional analysis, Series in Real Analysis, vol. 4, World Scientific Publishing Co. Inc., River Edge, NJ, 1992, Translated from the Chinese edition by Peng Yee Lee, With prefaces by Julian Musielak and Wen Qing Li, With an addendum by Cong Xin Wu. (p. 77).

[71] Vern Paulsen – Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. (p. 88).

[72] Vern I. Paulsen, Stephen C. Power et Roger R. Smith – « Schur products and matrix completions », J. Funct. Anal. 85 (1989), no. 1, p. 151–178. (pp. 35, 40, 78 et 81).

[73] A. Pełczyński et P. Wojtaszczyk – « Banach spaces with finite dimensional expansions of identity and universal bases of finite dimensional subspaces », Studia Math. 40 (1971), p. 91–108. (pp. 14 et 48).

[74] Vladimir V. Peller – « Hankel operators of class 𝔖_{p} and their applications (rational approximation, Gaussian processes, the problem of majorizing operators) », Math. USSR-Sb. 41 (1982), no. 4, p. 443–479. (p. 53).

[75] — , Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. (p. 37).

[76] S. K. Pichorides – « On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov », Studia Math. 44 (1972), p. 165–179. (errata insert). (p. 87).

[77] Gilles Pisier – Non-commutative vector valued L_{p}-spaces and completely p-summing maps, Astérisque, no. 247, Société mathématique de France, 1998. (pp. 34, 37, 39, 52, 77, 78, 80, 82 et 86).

[78] — , « An inequality for p-orthogonal sums in non-commutative L_{p} », Illinois J. Math. 44 (2000), no. 4, p. 901–923. (p. 25).

[79] — , Similarity problems and completely bounded maps, expanded éd., Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 2001, Includes the solution to “The Halmos problem”. (pp. 23, 34, 35, 37, 40, 77 et 81).

[80] Gilles Pisier et Quanhua Xu – « Non-commutative L^{p}-spaces », Handbook of the geometry of Banach spaces, Vol. 2 (W. B. Johnson et J. Lindenstrauss, éds.), North-Holland, 2003, p. 1459– 1517. (pp. 39 et 48).

[81] George Pólya et Gábor Szegő – Problems and theorems in analysis. Vol. I : Series, integral calculus, theory of functions, Springer-Verlag, 1972. (pp. 95, 112 et 118).

[82] Q. I. Rahman et G. Schmeisser – Analytic theory of polynomials, Oxford University Press, 2002. (p. 115).

[83] Narcisse Randrianantoanina – « Hilbert transform associated with finite maximal subdiagonal algebras », J. Austral. Math. Soc. Ser. A 65 (1998), no. 3, p. 388–404. (p. 87).

[84] Szilárd György Révész – « Minimization of maxima of nonnegative and positive definite cosine polynomials with prescribed first coefficients », Acta Sci. Math. (Szeged) 60 (1995), p. 589–608. (pp. 106, 108 et 111).

[85] Éric Ricard – « L’espace H^{1} n’a pas de base complètement inconditionnelle », C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 8, p. 625–628. (p. 91).

[86] — , « Décompositions de H^{1}, multiplicateurs de schur et espaces d’opérateurs », Thèse, Uni-

versité Paris 6, 2001, www.institut.math.jussieu.fr/theses/2001/ricard. (p. 91).

[87] Vojtěch Rödl – « On a packing and covering problem », European J. Combin. 6 (1985), no. 1, p. 69–78. (p. 48).

[88] Walter Rudin – « Trigonometric series with gaps », J. Math. Mech. 9 (1960), p. 203–228. (pp. 9, 21, 27, 84 et 85).

[89] — , Fourier analysis on groups, Interscience, 1962. (p. 39).

[90] Raymond A. Ryan – Introduction to tensor products of Banach spaces, Springer-Verlag, 2002. (p. 35).

[91] Raphaël Salem – « Sur les propriétés extrémales de certains polynomes trigonométriques », C. R. Acad. Sci. Paris 196 (1933), p. 1776–1778 (French). (pp. 16 et 100).

[92] Carsten Schütt – « Unconditionality in tensor products », Israel J. Math. 31 (1978), no. 3-4, p. 209–216. (pp. 36 et 39).

[93] Josef A. Seigner – « Rademacher variables in connection with complex scalars », Acta Math. Univ. Comenian. (N.S.) 66 (1997), p. 329–336. (pp. 39, 94 et 116).

[94] Harold S. Shapiro – Extremal problems for polynomials and power series, Mémoire, Massachusetts Institute of Technology, 1951. (pp. 93, 97, 98, 115 et 117).

[95] — , « On a class of extremal problems for polynomials in the unit circle », Portugal. Math. 20 (1961), p. 67–93. (p. 114).

[96] Barry Simon – Trace ideals and their applications, Cambridge University Press, 1979. (p. 23).

[97] Stefan Straszewicz – « Über exponierte Punkte abgeschlossener Punktmengen », Fund. Math. 24 (1935), p. 139–143. (p. 111).

[98] Nicholas Varopoulos – Some combinatorial problems in harmonic analysis, lecture notes by D. Salinger from a course given by N. Th. Varopoulos at the Summer School in Harmonic Analysis, University of Warwick, 1st July-13th July, 1968, Mathematics Institute, University of Warwick. (pp. 41 et 51).

[99] — , « Tensor algebras and harmonic analysis », Acta Math. 119 (1967), p. 51–112. (pp. 40 et 41).

[100] — , « Tensor algebras over discrete spaces », J. Functional Analysis 3 (1969), p. 321–335. (pp. 23, 39, 40, 41 et 85).

[101] László Zsidó – « On spectral subspaces associated to locally compact abelian groups of operators », Adv. in Math. 36 (1980), no. 3, p. 213–276. (p. 87).