Self-commutators of automorphic composition operators on the Dirichlet space

Let phi be a conformal automorphism on the unit disk U and C-phi : D -> D be the composition operator on the Dirichlet space D induced by phi. In this article we completely determine the point spectrum, spectrum, essential spectrum and essential norm of the operators C-phi*C-phi, C phi C phi* and self-commutators of C-phi, which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.