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Self-commutators of automorphic composition operators on the Dirichlet space

Authors:	A Abdollahi

Institution:	Department of Mathematics, Shiraz University, Shiraz 71454, Iran

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

Keywords:	Dirichlet space composition operator spectrum essential spectrum essential norm self-commutator anti-self-commutator

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