Fredholm perturbation theory and some essential spectra in Banach algebra with respect to subalgebra

The aim of this paper is to enlarge some known results from Fredholm and perturbation theory in the Banach algebra of bounded operators on a Banach space to the Fredholm theory in Banach algebra with respect to a subalgebra. The obtained results allow us to characterize the stability of some essential spectra with respect to a subalgebra.