Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet Space - A. Abdollahi; S. Mehrangiz (Abstract and Applied Analysis)
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 𝐶∗𝜑𝐶𝜑,𝐶𝜑𝐶∗𝜑 and self-commutators of 𝐶𝜑, which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.