Compactness of composition operators on the Bergman space of bounded pseudoconvex domains in $\mathbb{C}^n$

TL;DR

This paper investigates the conditions under which composition operators are compact on Bergman spaces of bounded pseudoconvex domains in complex n-space, providing a characterization for the polydisk case.

Contribution

It offers a new characterization of the compactness of composition operators with continuous symbols on Bergman spaces of certain pseudoconvex domains.

Findings

Characterization of compactness for composition operators on the polydisk

Identification of boundary conditions affecting compactness

Extension of results to domains with analytic disks in the boundary

Abstract

We study the compactness of composition operators on the Bergman spaces of certain bounded pseudoconvex domains in $\mathbb{C}^n$ with non-trivial analytic disks contained in the boundary. As a consequence we characterize that compactness of the composition operator with a holomorphic, continuous symbol (up to the closure) on the Bergman space of the polydisk.