On the poles of the resolvent in Calkin algebra

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 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.