PERTURBATION-THEORY RELATIVE TO A BANACH ALGEBRA OF OPERATORS

Let B be a Banach algebra of bounded linear operators on a Banach space X. Let S be a closed linear operator in X, and let R be a linear operator in X. In this paper the spectral and Fredholm theory relative to B of the perturbed operator S + R is developed. In particular, the situation where R is S-inessential relative to B is studied. Several examples are given to illustrate the usefulness of these concepts.