Approximation numbers of composition operators  on $H^p$ spaces of Dirichlet series

Bayart, Frédéric; Queffélec, Hervé; Seip, Kristian

doi:10.5802/aif.3019

Approximation numbers of composition operators on

H^{p}

spaces of Dirichlet series

Bayart, Frédéric ^1,²; Queffélec, Hervé ³; Seip, Kristian ⁴

¹ CNRS, UMR 6620 Laboratoire de Mathématiques 63177 Aubière (France)
² Clermont Université, Université Blaise Pascal, Laboratoire de Mathématiques, BP 10448 63000 Clermont-Ferrand (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)

Annales de l'Institut Fourier, Volume 66 (2016) no. 2, pp. 551-588.

Abstract
Résumé

By a theorem of the first named author, $ϕ$ generates a bounded composition operator on the Hardy space $ℋ^{p}$ of Dirichlet series $(1 \leq p < \infty)$ 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 $Re s > 1 / 2$ 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 $n$ th approximation numbers of bounded composition operators on $ℋ^{p}$ 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 $ℋ^{2}$ , developed in an earlier paper, using estimates of solutions of the $\bar{\partial}$ equation. A transference principle from $H^{p}$ of the unit disc is discussed, leading to explicit examples of compact composition operators on $ℋ^{1}$ 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 $ℋ^{p}$ to be compact.

Un théorème du premier auteur affirme que $ϕ$ définit un opérateur de composition borné sur l’espace de Hardy $ℋ^{p}$ des séries de Dirichlet ( $1 \leq p < \infty$ ) 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 $Re s > 1 / 2$ 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 $ℋ^{2}$ utilisant des estimations des solutions d’une équation $\bar{\partial}$ . 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 $ℋ^{p}$ .

Received: 2014-07-16
Revised: 2015-03-30
Accepted: 2015-09-10
Published online: 2016-02-17

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

Author's affiliations:

Bayart, Frédéric ^{1, 2}; Queffélec, Hervé ³; Seip, Kristian ⁴

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     title = {Approximation numbers of composition operators  on $H^p$ spaces of {Dirichlet} series},
     journal = {Annales de l'Institut Fourier},
     pages = {551--588},
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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, Volume 66 (2016) no. 2, pp. 551-588. doi : 10.5802/aif.3019. https://aif.centre-mersenne.org/articles/10.5802/aif.3019/

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