A note on weighted composition operators on Dirichlet space

TL;DR

This paper investigates the properties of weighted composition operators on Dirichlet space, providing conditions for compactness, characterizing their numerical range, and identifying classes with specific geometric numerical ranges.

Contribution

It offers new sufficient conditions for compactness, characterizes the numerical range, and introduces classes of operators with specific geometric numerical ranges on Dirichlet space.

Findings

Sufficient conditions for compactness of weighted composition operators.

Characterization of the numerical range and criteria for zero inclusion.

Introduction of classes with numerical ranges as circular or elliptical disks.

Abstract

In this paper, we provide some sufficient conditions for the compactness of weighted composition operators on Dirichlet space. Furthermore, we characterize the numerical range of certain classes of weighted composition operators on Dirichlet space and establish the criteria that guarantee the inclusion of zero within the numerical range. Finally, several classes of weighted composition operators are introduced, whose numerical range contains either a circular disk with a given center and radius or an elliptical disk with specified foci and axis lengths.