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

 

Essentially spectrally bounded linear maps


Authors: M. Bendaoud and A. Bourhim
Journal: Proc. Amer. Math. Soc. 137 (2009), 3329-3334
MSC (2000): Primary 47B49; Secondary 47A10, 47D25
Published electronically: June 5, 2009
MathSciNet review: 2515402
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Abstract: Let $ {\mathcal L}({\mathcal{H}})$ be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space $ {\mathcal{H}}$. We characterize essentially spectrally bounded linear maps from $ {\mathcal L}( {\mathcal{H}})$ onto $ {\mathcal L}({\mathcal{H}})$ itself. As a consequence, we characterize linear maps from $ {\mathcal L}( {\mathcal{H}})$ onto $ {\mathcal L}({\mathcal{H}})$ itself that compress different essential spectral sets such as the the essential spectrum, the (left, right) essential spectrum, and the semi-Fredholm spectrum.


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Additional Information

M. Bendaoud
Affiliation: Département de Mathématiques, Université Moulay Ismail, Meknès, Morocco
Email: bendaoudmohamed@gmail.com

A. Bourhim
Affiliation: Département de Mathématiques et de Statistique, Université Laval, Québec G1K 7P4, Canada
Address at time of publication: Department of Mathematics, Syracuse University, 215 Carnegie Building, Syracuse, New York 13244
Email: bourhim@mat.ulaval.ca, abourhim@syr.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09815-3
PII: S 0002-9939(09)09815-3
Keywords: Linear preserver problem, spectrally bounded linear map, essential spectral radius, Fredholm and semi-Fredholm operators
Received by editor(s): June 23, 2008
Published electronically: June 5, 2009
Additional Notes: The second author was supported by an adjunct professorship at Laval University
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society