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Journal of Operator Theory

Volume 46, Issue 1, Summer 2001  pp. 175-181.

Schatten class composition operators on weighted Bergman spaces of the disk

Authors Kehe Zhu
Author institution: Department of Mathematics, State University of New York, Albany, NY 12222, USA

Summary:  If $\varphi$ is an analytic self-map of the open unit disk $\D$ with bounded valence and $2\le p<+\infty$, we show that the composition operator $C_\varphi$, acting on the weighted Bergman $L^2$ space of $\D$ with radial weight $(1-|z|^2)^\alpha$ ($\alpha>-1$), belongs to the Schatten class $S_p$ if and only if $$\int\limits_{\D}\left(\frac{1-|z|^2}{1-|\varphi(z)|^2} \right)^{p(\alpha+2)/2}\,{\rm d}\lambda(z)<\infty,$$ where ${\rm d}\lambda$ is the M\"obius invariant measure of $\D$.

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