Numerical range and Berezin range of weighted composition operators on weighted Dirichlet spaces

TL;DR

This paper studies the numerical and Berezin ranges of weighted composition operators on weighted Dirichlet spaces, providing geometric characterizations and conditions for convexity, with new insights into the structure of these operator ranges.

Contribution

It introduces conditions for the numerical range to contain the origin, analyzes the geometric shape of the numerical range, and characterizes the convexity of the Berezin range for these operators.

Findings

Numerical range contains the origin under specific conditions.

Numerical range can be circular or elliptical with computed radius.

Convexity of the Berezin range is characterized for weighted composition operators.

Abstract

We investigate the numerical ranges of weighted composition operators on weighted Dirichlet spaces, focusing on the properties of the inducing functions. We identify conditions on these functions under which the origin lies in the interior of the numerical range. The geometric structure of the numerical range is also analyzed, determining when it contains a circular or elliptical disc and computing the corresponding radius. Next, we introduce a class of Weyl-type weighted composition operators and obtain their Berezin range and Berezin number. Finally, we characterize the convexity of the Berezin range for weighted composition operators on these spaces.