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In this paper, we characterise compactness of finite sums of finite productsof Toeplitz operators acting on the $\mathbb{C}^{d}$-valued weighted BergmanSpace, denoted $A_{\alpha}^{p}(\mathbb{B}_{n},\mathbb{C}^{d})$. The main resultshows 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 itsBerezin transform vanishes on the boundary of the ball.

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