Compact Operators on Vector-Valued Bergman Space via the Berezin   Transform
  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. 
Robert S. Rahm
math.CA