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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.$

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

  • 1. A. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North-Holland, Amsterdam, 1988. MR 0967989 (90d:47001)
  • 2. J. Esterle, E. Strouse and F. Zouakia, Theorems of Katznelson-Tzafriri type for contractions, J. Funct. Anal., 94(1990), 273-287. MR 1081645 (92c:47016)
  • 3. G. M. Feldman, The semisimplicity of an algebra generated by isometric operators, Funksional Anal. Prilozhen., 8(1974), 93-94 (Russian). MR 0361800 (50:14245)
  • 4. Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal., 68(1986), 313-328. MR 0859138 (88e:47006)
  • 5. R. Larsen, Banach Algebras, Marcel Dekker, New York, 1973. MR 0487369 (58:7010)
  • 6. L. H. Loomis, The spectral characterization of a class of almost periodic functions, Ann. Math., 72(1960), 362-368. MR 0120502 (22:11255)
  • 7. Vu Quoc Phong, Theorems of Katznelson-Tzafriri type for temigroups of operators, J. Funct. Anal., 103(1992), 74-84. MR 1144683 (93e:47050)

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Additional Information

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

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.

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