Self-commutators of composition operators with monomial symbols on the Dirichlet space (Q642720)

Summary: Let \(\varphi(z) = z^n\), \(z \in \mathbb U\), for some positive integer \(n\), and \(C_{\varphi}\) denote the composition operator on the Dirichlet space \(\mathcal D\) induced by \(\varphi\). In this paper, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators \(C^*_\varphi C_\varphi\), \(C_\varphi C^*_\varphi\) and the self-commutators of \(C_\varphi\), which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.