Compactness of composition operators on the Bergman space of the bidisc

TL;DR

This paper characterizes when composition operators induced by Lipschitz holomorphic self-maps of the bidisc are compact on the Bergman space, based on boundary behavior conditions.

Contribution

It provides a precise boundary condition characterization for the compactness of composition operators on the Bergman space of the bidisc.

Findings

Compactness of $C_{}$ is equivalent to specific boundary intersection conditions.

Boundary behavior determines operator compactness.

Results extend understanding of composition operators on product domains.

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 $\varphi(\overline{\mathbb{D}^2}\setminus \mathbb{T}^2)\cap b\mathbb{D}^2=\emptyset$.