In this paper, we study the relationship between the class of weakly demicompact linear operators, introduced in [KRICHEN, B.—O’REGAN, D.: On the class of relatively weakly demicompact nonlinear operators, Fixed Point Theory 19 (2018), 625–630], and measures of weak noncompactness of linear operators with respect to an axiomatic one. Moreover, some Fredholm and perturbation results involving the class of weakly demicompact linear operators are investigated. Our results are then used to investigate the relationship between the relative essential spectrum of the sum of two linear operators and the relative essential spectrum of each of these operators.

The aim of the present paper is to give some spectral results about generalized Fredholm operators and the so called S-generalized Fredholm operators, where S is a given bounded linear operator acting on a Banach space X. When X possesses some properties, we provide then some sufficient conditions for which a bounded linear operator will be a generalized Fredholm. The obtained results are applied to characterize the so called generalized S-essential spectrum, in particular the generalized Jeribi S-essential spectrum [17]. These results are formulated by means of measure of weak noncompactness.