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Self-commutators of automorphic composition operators on the Dirichlet space

Author: A. Abdollahi
Journal: Proc. Amer. Math. Soc. 136 (2008), 3185-3193
MSC (2000): Primary 47B33; Secondary 47A10, 47E20, 47B47
Published electronically: April 29, 2008
MathSciNet review: 2407083
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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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Additional Information

A. Abdollahi
Affiliation: Department of Mathematics, Shiraz University, Shiraz 71454, Iran

Keywords: Dirichlet space, composition operator, spectrum, essential spectrum, essential norm, self-commutator, anti-self-commutator
Received by editor(s): May 14, 2007
Received by editor(s) in revised form: July 16, 2007
Published electronically: April 29, 2008
Additional Notes: This research was partially supported by a grant from the Shiraz University Research Council
This work was carried out at the Department of Mathematics, University of Auckland, where the author was on the sabbatical leave during the academic year 2006-2007.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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