## 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.

### MSC:

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

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