Weighted composition operators on weighted Dirichlet spaces: boundedness, compactness and spectral properties

TL;DR

This paper characterizes when weighted composition operators are bounded or compact on weighted Dirichlet spaces and analyzes their spectra, extending previous unweighted results and linking operator behavior with function properties.

Contribution

It provides necessary and sufficient conditions for boundedness and compactness of weighted composition operators on weighted Dirichlet spaces, and determines their spectra.

Findings

Established criteria for boundedness and compactness.

Determined the spectrum of specific weighted composition operators.

Extended unweighted operator results to weighted cases.

Abstract

We establish necessary and sufficient conditions for the boundedness and compactness of weighted composition operators acting on weighted Dirichlet spaces and determine the spectrum of a certain class of such operators. Our results extend earlier work on unweighted composition operators and highlight the close interplay between the operator theoretic behavior of weighted composition operators and the function theoretic properties of their inducing functions. Several examples are provided to illustrate the applicability of the obtained results.