Compactness of composition operators on the Bergman space of the bidisc

Abstract

Let $\phi$ be a holomorphic self map of the bidisc that is Lipschitz on the closure. We show that the composition operator $C_{\phi }$ is compact on the Bergman space if and only if $\phi (\overline{{\mathbb{D}}^2})\cap {\mathbb{T}}^2=\varnothing$ and $\phi (\overline{{\mathbb{D}}^2}\setminus {\mathbb{T}}^2)\cap b{\mathbb{D}}^2=\varnothing$.

13 pages; minor correction in comments; to appear in Integral Equations Operator Theory

References in corpus (1)

• Compactness of Toeplitz operators with continuous symbols on pseudoconvex domains in ${\mathbb{C}}^n$