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

Abstract

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

v1: 32 pages; v2: 32 pages, typos corrected; v3: 32 pages, typos corrected, presentation improved based on referee comments