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

Abstract

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.

v1: 45 pages. v2: 45 pages, typos corrected and presentation improved. v3: 45 pages, typos corrected based on referee comments. This paper draws heavily from arXiv:1204.5548 in order to be self-contained

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