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

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.