Self-commutators of automorphic composition operators on the Dirichlet space

Abdollahi, A.

doi:10.1090/S0002-9939-08-09299-X

Self-commutators of automorphic composition operators on the Dirichlet space
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Proc. Amer. Math. Soc. 136 (2008), 3185-3193 Request permission

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.

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