On the generalized Drazin inverse and generalized resolvent. (English) Zbl 1079.47501

Summary: 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, \(2\times 2\) operator matrices are considered. As corollaries, we get some well-known results.


47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A10 Spectrum, resolvent
46L05 General theory of \(C^*\)-algebras
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