Journal of Operator Theory

Volume 45, Issue 2, Spring 2001 pp. 335-355.

Compact composition operators on weighted Bergman spaces of the unit Ball

Authors: Dana D. Clahane
Author institution: Department of Mathematics, University of California, Irvine, California 92717, USA

Summary: For

$p>0$ and

$\alpha\geq 0$ , let

$A^p_\alpha(B_n)$ be the weighted Bergman spac e of the unit ball

$B_n$ in

$\CC^n$ , and denote the Hardy space by

$H^p(B_n)$ . Suppose that

$\phi:B_n\righta rrow B_n$ is holomorphic. We show that if the composition operator

$C_\phi$ de fined by

$C_\phi(f)=f\circ\phi$ is bounded on

$A^p_\alpha(B_n)$ and satisfies

$\lim_{|z| \rightarrow 1^-} \Big(\frac{1-|z|^2}{1-|\phi(z)|^2}\Big )^{\alpha+2}\| \phi'(z)\|^2=0,$ then

$C_\phi$ is compact on

$A^p_\beta(B_n)$ for all

$\beta\geq \alpha$ . Along the way we prove some comparison results on boundedne ss and compactness of composition operators on

$H^p(B_n)$ and

$A^p_\alpha(B_n)$ , as well as a Carleson measure-type theorem involving these spaces and more gene ral weighted holomorphic Sobolev spaces.

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