Stability of relative essential spectra of the transport operator in L1-space by means of measures of noncompactness and relative weak demicompactness

The paper is devoted to some new sufficient conditions to ensure the upper-Fredholmness and Fredholmness of an unbounded densely defined linear operator T acting on a Banach space. Some characterizations of the relative essential spectra of T are also given. These characterizations are developed by introducing new classes of perturbations containing weakly compact operators and involving the so-called relatively weakly demicompact operators. These classes are used to discuss their incidence on the behavior of the relative essential spectra of T . The main results are formulated in terms of the De Blasi measure of weak noncompactness. The encountered technical problems are solved with an assumption of convergence and the Dunford-Pettis property. Further, these theoretical results are applied for describing the S 0 -essential spectrum of the transport operator in L 1 -space, where S 0 is a given multiplicative operator.