Self-commutators of automorphic composition operators on the Dirichlet space

Abstract

Let φ \varphi be a conformal automorphism on the unit disk U \mathbb {U} and C φ : D ⟶ D C_\varphi :\mathcal {D} \longrightarrow \mathcal {D} be the composition operator on the Dirichlet space D \mathcal {D} induced by φ \varphi . In this article we completely determine the point spectrum, spectrum, essential spectrum and essential norm of the operators C φ ∗ C φ , C φ C φ ∗ C^*_\varphi C_\varphi , C_\varphi C^*_\varphi and self-commutators of C φ C_\varphi , which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.