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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.

Weakly demicompact linear operators and axiomatic measures of weak noncompactness