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Elements with generalized bounded conjugation orbits

Authors: Driss Drissi and Mostafa Mbekhta
Journal: Proc. Amer. Math. Soc. 129 (2001), 2011-2016
MSC (2000): Primary 47B10, 47B15
Published electronically: January 17, 2001
MathSciNet review: 1825911
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Abstract: For a pair of linear bounded operators $ T $ and $ A $ on a complex Banach space $ X$, if $ T $ commutes with $ A, $ then the orbits $ \{A^n TA^{-n}\} $ of $ T $ under $ A $ are uniformly bounded. The study of the converse implication was started in the 1970s by J. A. Deddens. In this paper, we present a new approach to this type of question using two localization theorems; one is an operator version of a theorem of tauberian type given by Katznelson-Tzafriri and the second one is on power-bounded operators by Gelfand-Hille. This improves former results of Deddens-Stampfli-Williams.

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Additional Information

Driss Drissi
Affiliation: Department of Mathematics and Computer Science, Faculty of Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

Mostafa Mbekhta
Affiliation: UMR-CNRS 8524 & UFR de Mathematiques, Université de Lille I, F-59655, Villeneuve d’asq, France

Keywords: Bounded conjugation orbit, spectrum, spectral radius
Received by editor(s): November 1, 1999
Published electronically: January 17, 2001
Additional Notes: Research of the first author partially supported by grants from Kuwait University.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society