In this article, we consider Drazin invertible operators for study of the relationship between their B-Fredholm spectra and transfer some spectral properties defined through class operators. Among other results, investigate generalized a-Weyl’s theorem from T to inverse S, if it exists.

Our motivation is to derive the Drazin inverse matrix modification formulae utilizing inverses of adequate Peirce corners under some special cases, and a with an additive perturbation. As applications, several new results for expressions modified matrices A ?? CB CDdB are obtained, well known in literature, as Sherman-Morrison-Woodbury formula Jacobson?s Lemma, generalized.

We present new conditions under which Cline?s formula and Jacobson?s lemma for g-Drazin inverse hold. Let A be a Banach algebra, let a,b ? satisfying akbkak = ak+1 some k N. prove that has if only bkak inverse. In this case, (bkak)d bk(ad)2ak ad ak[(bkak)d]k+1. Further, we study in algebra the preceding condition. The common spectral property of bounded linear operators on space is thereby obta...