Abstract
In this paper, we introduce the concept of generalized weakly S-demicompact operators with respect to a weakly closed linear operator S. We study the general setting of Fredholm theory. We give some perturbation results on the behaviour of relative essential spectra of the sum of two bounded linear operators by means of the relative essential spectra of each one. A characterization of generalized weakly S-demicompact operators by means of measures of weak noncompactness is given. Moreover, we impose some sufficient conditions on the inputs of a closable block operator matrix to ensure the generalized weak S-demicompactness of its closure.
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