Necessary condition for compactness of a difference of composition operators on the Dirichlet space

TL;DR

This paper establishes a necessary condition for the compactness of the difference of two composition operators on the Dirichlet space and characterizes when their commutator is compact, advancing understanding of operator behavior in complex analysis.

Contribution

It provides a necessary condition for the compactness of differences of composition operators on the Dirichlet space and characterizes automorphisms with compact commutators.

Findings

Necessary condition for compactness of operator differences.

Characterization of automorphisms with compact commutators.

Insights into the structure of composition operators on the Dirichlet space.

Abstract

Let $\varphi$ be a self-map of the unit disk and let $C_\varphi$ denote the composition operator acting on the standard Dirichlet space $\mathcal{D}$. A necessary condition for compactness of a difference of two bounded composition operators acting on $\mathcal{D}$, is given. As an application, a characterization of disk automorphisms $\varphi$ and $\psi$ for which the commutator $[C^*_\psi,C_\varphi]$ is compact, is given.