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Proceedings of the American Mathematical Society

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Operators with bounded conjugation orbits


Authors: D. Drissi and M. Mbekhta
Journal: Proc. Amer. Math. Soc. 128 (2000), 2687-2691
MSC (1991): Primary 47B10, 47B15
DOI: https://doi.org/10.1090/S0002-9939-00-05338-7
Published electronically: April 27, 2000
MathSciNet review: 1664345
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Abstract:

For a bounded invertible operator $ A $ on a complex Banach space $ X,$ let $ B_A $ be the set of operators $ T $ in $ \mathcal{L} (X) $ for which $ \sup_{n \geq 0} \Vert A^n T A^{-n}\Vert < \infty.$ Suppose that $ Sp(A) = \{1\} $ and $ T $ is in $ B_A \cap B_{A^{-1}}. $ A bound is given on $ \Vert ATA^{-1} - T\Vert $ in terms of the spectral radius of the commutator. Replacing the condition $ T $ in $ B_{A^{-1}} $ by the weaker condition $\Vert A^{-n} T A^n\Vert = o(e^{\epsilon\sqrt{n}}), $ as $ n \to \infty$ for every $\epsilon>0$, an extension of the Deddens-Stampfli-Williams results on the commutant of $ A $ is given.


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Additional Information

D. Drissi
Affiliation: Department of Mathematics and Computer Science, Faculty of Science, Kuwait University, P. O. Box 5969, Safat 13060, Kuwait
Email: drissi@math-1.sci.kuniv.edu.kw

M. Mbekhta
Affiliation: URA 751 au CNRS & UFR de Mathematiques, Université de Lille I, F-59655, Villeneuve d’asq, France; Université de Galatasaray, Ciragan Cad no. 102, Ortakoy 80840, Istanbul, Turquie
Email: Mostafa.Mbekhta@univ-lille1.fr

DOI: https://doi.org/10.1090/S0002-9939-00-05338-7
Keywords: Bounded conjugation orbit, spectrum, spectral radius
Received by editor(s): June 23, 1998
Received by editor(s) in revised form: October 27, 1998
Published electronically: April 27, 2000
Additional Notes: The first author acknowledges support from Kuwait University
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society