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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


The Banach algebra generated by a contraction

Author: H. S. Mustafayev
Journal: Proc. Amer. Math. Soc. 134 (2006), 2677-2683
MSC (2000): Primary 47Axx
Published electronically: March 23, 2006
MathSciNet review: 2213747
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Abstract: Let $ T$ be a contraction on a Banach space and $ A_{T}$ the Banach algebra generated by $ T$. Let $ \sigma _{u}(T)$ be the unitary spectrum (i.e., the intersection of $ \sigma (T)$ with the unit circle) of $ T$. We prove the following theorem of Katznelson-Tzafriri type: If $ \sigma _{u}(T)$ is at most countable, then the Gelfand transform of $ R\in A_{T}$ vanishes on $ \sigma _{u}(T)$ if and only if $ \lim_{n\rightarrow \infty }\left\Vert T^{n}R\right\Vert =0.$

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Additional Information

H. S. Mustafayev
Affiliation: Department of Mathematics, Faculty of Arts and Sciences, Yüzüncü Yil University, 65080, Van, Turkey

PII: S 0002-9939(06)08302-X
Keywords: Contraction, Banach algebra, spectrum, semisimplicity
Received by editor(s): February 25, 2005
Received by editor(s) in revised form: April 5, 2005
Published electronically: March 23, 2006
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.