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Schatten class composition operators on the Hardy space
Published online by Cambridge University Press: 24 July 2023
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) 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) 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.
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- Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh