Approximation numbers of composition operators on H p spaces of Dirichlet series
Annales de l'Institut Fourier, Volume 66 (2016) no. 2, p. 551-588

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.

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 .

Received : 2014-07-16
Revised : 2015-03-30
Accepted : 2015-09-10
Published online : 2016-02-17
DOI : https://doi.org/10.5802/aif.3019
Classification:  47B33,  30B50,  30H10,  47B07
Keywords: Dirichlets series, composition operators, approximation numbes
@article{AIF_2016__66_2_551_0,
     author = {Bayart, Fr\'ed\'eric and Queff\'elec, Herv\'e and Seip, Kristian},
     title = {Approximation numbers of composition operators  on $H^p$ spaces of Dirichlet series},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {2},
     year = {2016},
     pages = {551-588},
     doi = {10.5802/aif.3019},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_2_551_0}
}
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. http://www.numdam.org/item/AIF_2016__66_2_551_0/

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