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

TL;DR

This paper characterizes the essential norm and compactness of operators on vector-valued weighted Bergman spaces on the unit ball, extending previous results to matrix-valued symbols and Toeplitz algebra.

Contribution

It introduces approximation of operators by localized operators in the Toeplitz algebra and characterizes compact operators via Berezin transform vanishing.

Findings

Operators in the Toeplitz algebra can be approximated by localized operators.

The essential norm of these operators can be expressed in multiple equivalent ways.

Compactness is characterized by the vanishing of the Berezin transform on the boundary.

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.