Geodesic
On the generalized Drazin inverse and generalized resolvent
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 3, pp. 617-634.

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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.
Classification : 46H30, 46L05, 47A05, 47A10
Keywords: Drazin inverse; generalized resolvent; limit processes; outer inverses; operator matrices 
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     author = {Djordjevi\'c, Dragan S. and Stanimirovi\'c, Predrag S.},
     title = {On the generalized {Drazin} inverse and generalized resolvent},
     journal = {Czechoslovak Mathematical Journal},
     pages = {617--634},
     publisher = {mathdoc},
     volume = {51},
     number = {3},
     year = {2001},
     mrnumber = {1851551},
     zbl = {1079.47501},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2001__51_3_a12/}
}
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Djordjević, Dragan S.; Stanimirović, Predrag S. On the generalized Drazin inverse and generalized resolvent. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 3, pp. 617-634. http://geodesic.mathdoc.fr/item/CMJ_2001__51_3_a12/