The Essential Norm of Operators on $A^p(\mathbb{D}^n)$
  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 Toeplitz algebra $\mathcal{T}_{p}$ and its Berezin transform vanishes on the boundary. 
Mishko Mitkovski, Brett D. Wick
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