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In this paper, we introduce and investigate a new concept that we call demicompact elements in Banach algebras as a generalization of demicompact linear operators acting on Banach spaces. Our concept is used to construct a new class of Fredholm perturbations with respect to a given Banach subalgebra  B , that contains an inessential idealkB{k_{B}}and the set of left Fredholm perturbations introduced in [A. Ben Ali and N. Moalla, Fredholm perturbation theory and some essential spectra in Banach algebra with respect to subalgebra, Indag. Math. (N. S.) 28 2017, 2, 276–286]. We show that it is a two-sided closed ideal of  B  playing an important role in the characterization of the Weyl spectrum of elements affiliated with  B .

Demicompactness perturbation in Banach algebras and some stability results