Compact differences of composition operators on weighted Dirichlet spaces

TL;DR

This paper characterizes when the difference of two composition operators is compact on weighted Dirichlet spaces, extending results via complex interpolation and applying to operators induced by linear fractional maps.

Contribution

It provides a comprehensive analysis of compact differences of composition operators on weighted Dirichlet spaces, including new results for specific spaces and generalizations through interpolation.

Findings

Characterization of compact differences on Dirichlet space and S^2

Extension of results to general weighted Dirichlet spaces via interpolation

Application to composition operators induced by linear fractional maps

Abstract

Here we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces $\mathcal{D}_\alpha$. Specifically we study differences of composition operators on the Dirichlet space $\mathcal{D}$ and $S^2$, the space of analytic functions whose first derivative is in $H^2$, and then use Calder\'{o}n's complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk.