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 $\mathbb D$ 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 $\mathbb D$ with radial weight $(1-|z|^2)^\alpha$ $($$\alpha>-1$$)$, belongs to the Schatten class $S_p$ if and only if $$\int\limits_{\mathbb D}\left(\frac{1-|z|^2}{1-|\varphi(z)|^2} \right)^{p(\alpha+2)/2}\,{\rm d}\lambda(z)<\infty,$$ where ${\rm d}\lambda$ is the Möbius invariant measure of $\mathbb D$.

Keywords: Composition operator, Bergman space, Schatten class, bounded valence

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