Compact Operators on Vector-Valued Bergman Space via the Berezin Transform

TL;DR

This paper characterizes the compactness of certain Toeplitz operator combinations on vector-valued weighted Bergman spaces using the Berezin transform, establishing a boundary vanishing criterion.

Contribution

It provides a new characterization of compact Toeplitz operator sums on vector-valued Bergman spaces via the Berezin transform.

Findings

Compactness is equivalent to Berezin transform vanishing on the boundary.

Finite sums of finite products of Toeplitz operators are compact iff their Berezin transform vanishes at the boundary.

The result extends scalar-valued Bergman space theory to vector-valued settings.

Abstract

In this paper, we characterise compactness of finite sums of finite products of Toeplitz operators acting on the $\mathbb{C}^{d}$-valued weighted Bergman Space, denoted $A_{\alpha}^{p}(\mathbb{B}_{n},\mathbb{C}^{d})$. The main result shows that a finite sum of finite product of Toeplitz operators acting on $A_{\alpha}^{p}(\mathbb{B}_n,\mathbb{C}_d)$ is compact if and only if its Berezin transform vanishes on the boundary of the ball.