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

 

Linear maps preserving the set of Fredholm operators


Author: Mostafa Mbekhta
Journal: Proc. Amer. Math. Soc. 135 (2007), 3613-3619
MSC (2000): Primary 47B48, 47A10, 46H05
Published electronically: June 29, 2007
MathSciNet review: 2336577
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Abstract: Let $ H$ be an infinite-dimensional separable complex Hilbert space and $ \mathcal{B}(H)$ the algebra of all bounded linear operators on $ H$. In this paper we characterize surjective linear maps $ \phi : F\mathcal{B}(H)\to \mathcal{B}(H)$ preserving the set of Fredholm operators in both directions. As an application we prove that $ \phi$ preserves the essential spectrum if and only if the ideal of all compact operators is invariant under $ \phi$ and the induced linear map $ \varphi$ on the Calkin algebra is either an automorphism, or an anti-automorphism. Moreover, we have, either $ ind(\phi(T)) = ind(T)$ or $ ind(\phi(T)) = - ind(T)$ for every Fredholm operator $ T$.


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Additional Information

Mostafa Mbekhta
Affiliation: Université de Lille I, UFR de Mathématiques, 59655 Villeneuve d’Ascq Cedex, France
Email: mostafa.mbekhta@math.univ-lille1.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08874-0
PII: S 0002-9939(07)08874-0
Keywords: Fredholm operators, Calkin algebra, linear preservers
Received by editor(s): April 4, 2006
Received by editor(s) in revised form: August 18, 2006
Published electronically: June 29, 2007
Additional Notes: The work of the author is partially supported by “Action integrée Franco-Marocaine, Programme Volubilis, $N^{o}$ MA/03/64” and by I+D MEC project MTM 2004-03882.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society