On the Berezin range of Toeplitz and weighted composition operators on weighted Bergman spaces

TL;DR

This paper characterizes the Berezin range of Toeplitz and weighted composition operators on weighted Bergman spaces, introduces a new class of such operators, and explores the convexity and geometric properties of their Berezin ranges.

Contribution

It provides a complete characterization of the Berezin range for harmonic-symbol Toeplitz operators and introduces a new class of weighted composition operators with their Berezin properties.

Findings

Berezin range of harmonic Toeplitz operators is fully characterized

A new class of weighted composition operators is introduced and analyzed

The convexity of the Berezin range is studied, showing the origin's position in its closure

Abstract

In this article, we completely characterize the Berezin range of Toeplitz operators with harmonic symbols acting on weighted Bergman spaces, illustrating the necessity of the harmonicity condition through examples. We then introduce a new class of weighted composition operators on these spaces, investigating their fundamental properties and determining their Berezin range and Berezin number. Finally, we study the convexity of the Berezin range of composition operators on weighted Bergman spaces and show that the origin lies in its closure of Berezin range but not in the range itself.