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In this paper we characterize the compact operators on the weighted Bergman spaces \({A^p_\alpha(\mathbb{B}_n)}\) when 1 < p < ∞ and α > −1. The main result shows that an operator on \({A^p_\alpha(\mathbb{B}_n)}\) is compact if and only if it belongs to the Toeplitz algebra and its Berezin transform vanishes on the boundary of the ball.

The Essential Norm of Operators on $${A^p_\alpha(\mathbb{B}_n)}$$