Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B33, 47A10, 47E20, 47B47

Retrieve articles in all journals with MSC (2000): 47B33, 47A10, 47E20, 47B47


Additional Information

A. Abdollahi
Affiliation: Department of Mathematics, Shiraz University, Shiraz 71454, Iran
Email: abdolahi@shirazu.ac.ir

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09299-X
PII: S 0002-9939(08)09299-X
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.