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

Let $\phi$ be a self-map of the unit disk and let $C_{\phi }$ 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 $\phi$ and $\psi$ for which the commutator $[C_{\psi }^{\ast },C_{\phi }]$ is compact, is given.