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

TL;DR

This paper characterizes compact operators on the Bergman space of the polydisc, showing they are precisely those in the Toeplitz algebra with boundary-vanishing Berezin transform.

Contribution

It provides a complete characterization of compact operators on $A^p( ext{D}^n)$ in terms of Toeplitz algebra membership and Berezin transform behavior.

Findings

Operators are compact iff they are in the Toeplitz algebra with boundary-vanishing Berezin transform.

The main theorem links compactness to algebraic and boundary properties of operators.

Provides a criterion for compactness on multivariable Bergman spaces.

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.