University of Nis, Faculty of Sciences, Department of Mathematics, Cirila i Metodija 2, 18000 Nis, Yugoslavia, e-mails: dragan@archimed.filfak.ni.ac.yu, dragan@filfak.filfak.ni.ac.yu, pecko@archimed.filfak.ni.ac.yu, pecko@filfak.filfak.ni.ac.yu
Abstract: We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in $C^*$-algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel. Also, $2\times2$ operator matrices are considered. As corollaries, we get some well-known results.
Keywords: Drazin inverse, generalized resolvent, limit processes, outer inverses, operator matrices
Classification (MSC 2000): 47A05, 47A10, 46L05
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