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On the poles of the resolvent in Calkin algebra


Author: O. Bel Hadj Fredj
Journal: Proc. Amer. Math. Soc. 135 (2007), 2229-2234
MSC (2000): Primary 47L10
DOI: https://doi.org/10.1090/S0002-9939-07-08733-3
Published electronically: March 2, 2007
MathSciNet review: 2299500
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Abstract: In the present note, we study the problem of lifting poles in Calkin algebra on a separable infinite-dimensional complex Hilbert space $ H$. We show by an example that such lifting is not possible in general, and we prove that if zero is a pole of the resolvent of the image of an operator $ T$ in the Calkin algebra, then there exists a compact operator $ K$ for which zero is a pole of $ T+K$ if and only if the index of $ T-\lambda$ is zero on a punctured neighbourhood of zero. Further, a useful characterization of poles in Calkin algebra in terms of essential ascent and descent is provided.


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Additional Information

O. Bel Hadj Fredj
Affiliation: Université Lille 1, UFR de Mathématiques, UMR-CNRS 8524, 59655 Villeneuve d’Ascq, France
Email: Olfa.Bel-Hadj-Fredj@math.univ-lille1.fr

DOI: https://doi.org/10.1090/S0002-9939-07-08733-3
Keywords: Poles, Calkin algebra, essential ascent and essential descent
Received by editor(s): February 6, 2006
Received by editor(s) in revised form: March 28, 2006
Published electronically: March 2, 2007
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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