Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

On the poles of the resolvent in Calkin algebra

Author(s): O. Bel Hadj Fredj
Journal: Proc. Amer. Math. Soc. 135 (2007), 2229-2234.
MSC (2000): Primary 47L10
Posted: 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 . 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 in the Calkin algebra, then there exists a compact operator for which zero is a pole of 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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