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Zbl 1079.47501
Djordjević, Dragan S.; Stanimirović, Predrag S.
On the generalized Drazin inverse and generalized resolvent. (English)
[J] Czech. Math. J. 51, No. 3, 617-634 (2001). ISSN 0011-4642; ISSN 1572-9141/e

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 2000:: *47A05 General theory of linear operators
47A10 Spectrum and resolvent of linear operators
46L05 General theory of C*-algebras

Keywords: Drazin inverse; generalized resolvent; limit processes; outer inverses; operator matrices

Cited in: Zbl 1239.47002 Zbl 1148.15001

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