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

Bayart, Frédéric; Queffélec, Hervé; Seip, Kristian

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

Un théorème du premier auteur affirme que $\varphi$ définit un opérateur de composition borné sur l’espace de Hardy ${\mathscr{H}}^p$ des séries de Dirichlet ($1\le p<\infty$) dès lors que $\varphi \left(s\right)=c_{0}s+\psi \left(s\right)$, où $c_0$ est un entier positif ou nul et $\psi$ 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 ${\mathscr{H}}^2$ utilisant des estimations des solutions d’une équation $\overline{\partial }$. 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 ${\mathscr{H}}^p$.

By a theorem of the first named author, $\varphi$ generates a bounded composition operator on the Hardy space ${\mathscr{H}}^p$of Dirichlet series $(1\le p<\infty )$ only if $\varphi \left(s\right)=c_{0}s+\psi \left(s\right)$, where $c_0$ is a nonnegative integer and $\psi$ a Dirichlet series with the following mapping properties: $\psi$ 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 $n$th approximation numbers of bounded composition operators on ${\mathscr{H}}^p$ 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 ${\mathscr{H}}^2$, developed in an earlier paper, using estimates of solutions of the $\overline{\partial }$ equation. A transference principle from $H^p$ of the unit disc is discussed, leading to explicit examples of compact composition operators on ${\mathscr{H}}^1$ 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 ${\mathscr{H}}^p$ to be compact.

Classification : 47B33, 30B50, 30H10, 47B07
Keywords: Dirichlets series, composition operators, approximation numbes
Mot clés : Séries de Dirichlet, opérateurs de composition, nombres d’approximation

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