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Schatten class composition operators on the Hardy space

Part of: Holomorphic functions of several complex variables Special classes of linear operators Spaces and algebras of analytic functions

Published online by Cambridge University Press:  24 July 2023

Wenwan Yang  and
Cheng Yuan
Wenwan Yang
Affiliation:
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, Guangdong 510520, China (wenwan_yang@163.com, yuancheng1984@163.com)
Cheng Yuan
Affiliation:
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, Guangdong 510520, China (wenwan_yang@163.com, yuancheng1984@163.com)
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Abstract

Suppose $2< p<\infty$ and $\varphi$ is a holomorphic self-map of the open unit disk $\mathbb {D}$. We show the following assertions:

  1. (1) If $\varphi$ has bounded valence and0.1

    \begin{equation} \int_{\mathbb{D}} \left(\frac{1-|z|^2}{1-|\varphi(z)|^2}\right)^{p/2}\frac{\mathrm{d} A(z)}{(1-|z|^2)^2}<\infty, \end{equation}
    then $C_{\varphi }$ is in the Schatten $p$-class of the Hardy space $H^2$.

  2. (2) There exists a holomorphic self-map $\varphi$ (which is, of course, not of bounded valence) such that the inequality (0.1) holds and $C_{\varphi }: H^2\to H^2$ does not belong to the Schatten $p$-class.

Keywords

composition operatorsHardy spacesSchatten p-class

MSC classification

Primary: 30H10: Hardy spaces
Secondary: 32A35: $H^p$-spaces, Nevanlinna spaces 47B10: Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) 47B33: Composition operators
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 14
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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