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In this paper we characterize the compact operators on the Bergman space$A^p(\mathbb{D}^n)$. The main result shows that an operator on$A^p(\mathbb{D}^n)$ is compact if and only if it belongs to the Toeplitzalgebra $\mathcal{T}_{p}$ and its Berezin transform vanishes on the boundary.

The Essential Norm of Operators on $A^p(\mathbb{D}^n)$