Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet Space

Abstract EN

Let φ(z)=zn,z∈U, for some positive integer n and Cφ the composition operator on the Dirichlet space D induced by φ.

In this paper, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators Cφ∗Cφ, CφCφ∗ and self-commutators of Cφ, which expose that the spectrum and point spectrum coincide.

We also find the eigenfunctions of the operators.