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

Let $φ$ be a self-map of the unit disk and let $C_φ$ 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 $φ$ and $ψ$ for which the commutator $[C^*_ψ,C_φ]$ is compact, is given.

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