Berezin regularity of domains in C^n and the essential norms of Toeplitz operators

TL;DR

This paper investigates the Berezin regularity of certain pseudoconvex domains in complex spaces and explores how boundary geometry influences this regularity, also relating Toeplitz operator norms to Berezin transforms.

Contribution

It extends the concept of BC-regularity from the unit disc to higher-dimensional pseudoconvex domains and links boundary geometry with operator properties.

Findings

Boundary geometry significantly affects BC-regularity.

Established a relationship between Toeplitz operator norms and Berezin transforms.

Identified conditions under which pseudoconvex domains are BC-regular.

Abstract

For the open unit disc $\mathbb{D}$ in the complex plane, it is well known that if $\phi \in C(\overline{\mathbb{D}})$ then its Berezin transform $\widetilde{\phi}$ also belongs to $C(\overline{\mathbb{D}})$. We say that $\mathbb{D}$ is BC-regular. In this paper we study BC-regularity of some pseudoconvex domains in $\mathbb{C}^n$ and show that the boundary geometry plays an important role. We also establish a relationship between the essential norm of an operator in a natural Toeplitz subalgebra and its Berezin transform.