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

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.

arXiv comment16 pages. In order to be self contained, this paper draws from arXiv:1407.4786