Pseudo B-Fredholm linear relations and spectral theory

摘要

Abstract The notions of pseudo B-Fredholm and pseudo B-Weyl linear relations in Banach spaces are introduced and studied. We prove that if T is an everywhere defined closed linear relation with a nonempty resolvent set, then T is Fredholm (resp. Weyl) if and only if T is pseudo B-Fredholm (resp. pseudo B-Weyl) and 0 is non-isolated in the Fredholm (resp. Weyl) spectrum of T . We also show that T is pseudo B-Fredholm (resp. pseudo B-Weyl) if and only if T admits a generalized Kato decomposition and 0 is not an accumulation point of the Fredholm (resp. Weyl) spectrum of T . As an application, we calculate the pseudo B-Fredholm, the pseudo B-Weyl and the generalized Kato spectra of the inverse of the backward unilateral shift operator regarded as a linear relation.