Closed Fredholm and semi-Fredholm operators, essential spectra and perturbations

Abstract

In these notes we provide rather extensive characterizations of closed densely defined Fredholm and semi-Fredholm operators on a Banach space, and their perturbation classes. We make use of the notions of measure of noncompactness, special norm equalities, and certain “pseudo” Banach algebra concepts as they pertain to closed operators. Classes of perturbations of closed Fredholm and semi-Fredholm operators are effectively identified, respectively, with classes of perturbations of the Wolf, Schechter, and Gustafson-Weidman essential spectra for closed operators. A by-product of this identification is a generalization of the celebrated Weyl theorem which characterizes the essential spectrum of a compact self-adjoint operator on a Hilbert space. We obtain spectral mapping theorems for some particular Wolf and Schechter essential spectra.