Compactness of Hankel and Toeplitz operators on convex Reinhardt domains in $\mathbb{C}^2$

TL;DR

This paper characterizes the compactness of Hankel and Toeplitz operators on Bergman spaces over convex Reinhardt domains in a72, showing it is equivalent to the vanishing of their Berezin transforms on the boundary.

Contribution

It establishes a precise boundary condition criterion for the compactness of these operators on convex Reinhardt domains in a72, extending previous understanding.

Findings

Toeplitz operators are compact iff their Berezin transforms vanish on the boundary.

Hermitian squares of Hankel operators are compact iff their Berezin transforms vanish on the boundary.

The results apply to symbols continuous on the domain closure.

Abstract

We study compactness of Hankel and Toeplitz operators on Bergman spaces of convex Reinhardt domains in $\mathbb{C}^2$ and we restrict the symbols to the class of functions that are continuous on the closure of the domain. We prove that Toeplitz operators as well as the Hermitian squares of Hankel operators are compact if and only if the Berezin transforms of the operators vanish on the boundary of the domain.