The Essential Norm of Operators on the Bergman Space of Vector--Valued Functions on the Unit Ball

Abstract

Let $A_{\alpha }^p({\mathbb{B}}^n;{\mathbb{C}}^d)$ be the weighted Bergman space on the unit ball ${\mathbb{B}}^n$ of ${\mathbb{C}}^n$ of functions taking values in ${\mathbb{C}}^d$. For $1<p<\infty$ let ${\mathcal{T}}_{p,\alpha }$ be the algebra generated by finite sums of finite products of Toeplitz operators with bounded matrix--valued symbols (this is called the Toeplitz algebra in the case $d=1$). We show that every $S\in {\mathcal{T}}_{p,\alpha }$ can be approximated by localized operators. This will be used to obtain several equivalent expressions for the essential norm of operators in ${\mathcal{T}}_{p,\alpha }$. We then use this to characterize compact operators in $A_{\alpha }^p({\mathbb{B}}^n;{\mathbb{C}}^d)$. The main result generalizes previous results and states that an operator in $A_{\alpha }^p({\mathbb{B}}^n;{\mathbb{C}}^d)$ is compact if only if it is in ${\mathcal{T}}_{p,\alpha }$ and its Berezin transform vanishes on the boundary.

arXiv commentv2: 33 pages. In order to be self-contained, this paper draws heavily from arXiv:1204.5548