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Pseudo-B-Fredholm operators, poles of the resolvent and mean convergence in the calkin algebra

Berkani Mohammed (Science faculty of Oujda, University Mohammed I, Laboratory LAGA, Oujda, Morocco)
Živković-Zlatanović Snežana Č. (University of Niš,Faculty of Sciences and Mathematics, Niš, Serbia)

We define here a pseudo B-Fredholm operator as an operator such that 0 is isolated in its essential spectrum, then we prove that an operator T is pseudo-B-Fredholm if and only if T = R + F where R is a Riesz operator and F is a B-Fredholm operator such that the commutator [R,F] is compact. Moreover, we prove that 0 is a pole of the resolvent of an operator T in the Calkin algebra if and only if T = K + F, where K is a power compact operator and F is a B-Fredholm operator, such that the commutator [K,F] is compact. As an application, we characterize the mean convergence in the Calkin algebra.

Keywords: Pseudo-B-Fredholm, essential ascent, essential descent, poles of the resolvent, Calkin algebra, mean convergence