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 Cφ denote the composition operator acting on the standard Dirichlet space D. A necessary condition for compactness of a difference of two bounded composition operators acting on D is given. As an application, a characterization of disk automorphisms φ and ψ, for which the commutator [Cψ⁎,Cφ] is compact, is given.