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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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