Pseudo-B-Fredholm operators, poles of the resolvent and mean convergence in the Calkin Algebra

Abstract

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.