Approximation numbers of composition operators on H p spaces of Dirichlet series
[Nombres d’approximation des opérateurs de composition sur les espaces H p des séries de Dirichlet]
Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 551-588.

Un théorème du premier auteur affirme que ϕ définit un opérateur de composition borné sur l’espace de Hardy p des séries de Dirichlet (1p<) dès lors que ϕ(s)=c 0 s+ψ(s), où c 0 est un entier positif ou nul et ψ est une série de Dirichlet qui envoie le demi-plan droit sur le demi-plan Res>1/2 lorsque c 0 =0 et soit est identiquement nulle, soit envoie le demi-plan droit dans lui-même si c 0 >0. Nous prouvons que le n-ième nombre d’approximation de ces opérateurs de composition est minoré, à une constante multiplicative près, par r n , 0<r<1 si c 0 =0 et par n -A , A>0, si c 0 >0. Ces minorations sont optimales et reposent sur une combinaison d’outils venant à la fois de la théorie des espaces de Banach (type et cotype, inégalités de Weyl, bases de Schauder) et sur une méthode d’interpolation pour 2 utilisant des estimations des solutions d’une équation ¯. Un principe de transfert avec les espaces H p du disque est discuté, conduisant à des exemples explicites d’opérateurs de composition ayant des nombres d’approximation avec divers types de décroissance sous-exponentielle. Enfin, une nouvelle formule de Littlewood-Paley est établie, conduisant à une condition suffisante de compacité pour un opérateur de composition sur p .

By a theorem of the first named author, ϕ generates a bounded composition operator on the Hardy space p of Dirichlet series (1p<) only if ϕ(s)=c 0 s+ψ(s), where c 0 is a nonnegative integer and ψ a Dirichlet series with the following mapping properties: ψ maps the right half-plane into the half-plane Res>1/2 if c 0 =0 and is either identically zero or maps the right half-plane into itself if c 0 is positive. It is shown that the nth approximation numbers of bounded composition operators on p are bounded below by a constant times r n for some 0<r<1 when c 0 =0 and bounded below by a constant times n -A for some A>0 when c 0 is positive. Both results are best possible. Estimates rely on a combination of soft tools from Banach space theory (s-numbers, type and cotype of Banach spaces, Weyl inequalities, and Schauder bases) and a certain interpolation method for 2 , developed in an earlier paper, using estimates of solutions of the ¯ equation. A transference principle from H p of the unit disc is discussed, leading to explicit examples of compact composition operators on 1 with approximation numbers decaying at a variety of sub-exponential rates. Finally, a new Littlewood–Paley formula is established, yielding a sufficient condition for a composition operator on p to be compact.

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DOI : 10.5802/aif.3019
Classification : 47B33, 30B50, 30H10, 47B07
Keywords: Dirichlets series, composition operators, approximation numbes
Mot clés : Séries de Dirichlet, opérateurs de composition, nombres d’approximation
Bayart, Frédéric 1, 2 ; Queffélec, Hervé 3 ; Seip, Kristian 4

1 Clermont Université, Université Blaise Pascal, Laboratoire de Mathématiques, BP 10448 63000 Clermont-Ferrand (France)
2 CNRS, UMR 6620 Laboratoire de Mathématiques 63177 Aubière (France)
3 Université Lille Nord de France, USTL Laboratoire Paul Painlevé UMR. CNRS 8524, 59 655 Villeneuve d’Ascq Cedex (France)
4 Department of Mathematical Sciences Norwegian University of Science and Technology 7491 Trondheim (Norway)
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     title = {Approximation numbers of composition operators  on $H^p$ spaces of {Dirichlet} series},
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Bayart, Frédéric; Queffélec, Hervé; Seip, Kristian. Approximation numbers of composition operators  on $H^p$ spaces of Dirichlet series. Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 551-588. doi : 10.5802/aif.3019. http://archive.numdam.org/articles/10.5802/aif.3019/

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