Necessary condition for compactness of a difference 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.