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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