On the poles of the resolvent in Calkin algebra

Bel Hadj Fredj, O.

doi:10.1090/S0002-9939-07-08733-3

On the poles of the resolvent in Calkin algebra
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by O. Bel Hadj Fredj PDF

Proc. Amer. Math. Soc. 135 (2007), 2229-2234 Request permission

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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