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

TL;DR

This paper characterizes compact operators on weighted Bergman spaces over the unit ball, showing they are precisely those in the Toeplitz algebra with boundary-vanishing Berezin transform.

Contribution

It provides a complete characterization of compact operators on $A^p_eta(all_n)$ in terms of Toeplitz algebra membership and Berezin transform behavior.

Findings

Operators are compact iff they are in the Toeplitz algebra and their Berezin transform vanishes at the boundary.

The main theorem links compactness to boundary behavior of the Berezin transform.

The results extend understanding of operator theory on weighted Bergman spaces.

Abstract

In this paper we characterize the compact operators on $A^p_\alpha(\mathbb{B}_n)$ when $1<p<\infty$ and $\alpha>-1$. 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.