Compactness of composition operators on the Bergman space of the bidisc

Abstract

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