Additive results for the generalized Drazin inverse in a Banach algebra. (English) Zbl 1104.47040

Cvetković-Ilić, Dragana S.; Djordjević, Dragan S.; Wei, Yimin

Linear Algebra Appl. 418, No. 1, 53-61 (2006).

Summary: There are investigated additive properties of the generalized Drazin inverse in a Banach algebra. The authors give some new conditions under which the generalized Drazin inverse of the sum \(a + b\) could be explicitly expressed in terms of \(a\), \(b\) and their generalized Drazin inverse \(a^{d}\), \(b^{d}\). Also, some recent results of N. Castro González and J. J. Koliha [Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 6, 1085–1097 (2004; Zbl 1088.15006)] are extended.

Reviewer: D. Przeworska-Rolewicz (Warszawa)

MSC:

47C05	Linear operators in algebras
15A09	Theory of matrix inversion and generalized inverses
46H30	Functional calculus in topological algebras
46H05	General theory of topological algebras

Keywords:

Banach algebra; Drazin inverse; generalized Drazin inverse

Citations:

Zbl 1088.15006

References:

[1]	Drazin, M.P., Pseudoinverse in associative rings and semigroups, Am. math. month., 65, 506-514, (1958) · Zbl 0083.02901
[2]	Koliha, J.J., A generalized Drazin inverse, Glasgow math. J., 38, 367-381, (1996) · Zbl 0897.47002
[3]	Harte, R.E., Spectral projections, Irish math. soc. newslett., 11, 10-15, (1984) · Zbl 0556.47001
[4]	Harte, R.E., Invertibility and singularity for bounded linear operators, (1988), Marcel Dekker New York · Zbl 0636.47001
[5]	Harte, R.E., On quasinilpotents in rings, Panam. math. J., 1, 10-16, (1991) · Zbl 0761.16009
[6]	Castro González, N.; Koliha, J.J.; Rakocevic, V., Continuity and general perturbation of the Drazin inverse for closed linear operators, Abstract appl. anal., 7, 335-347, (2002) · Zbl 1009.47002
[7]	Koliha, J.J.; Rakocevic, V., Holomorphic and meromorphic properties of the g-Drazin inverse, Demonstratio Mathematica, 38, 657-666, (2005) · Zbl 1092.47001
[8]	Koliha, J.J.; Rakocevic, V., Differentiability of the g-Drazin inverse, Stud. math., 168, 193-201, (2005) · Zbl 1071.47019
[9]	Djordjević, D.S.; Wei, Y., Additive results for the generalized Drazin inverse, J. austral. math. soc., 73, 115-125, (2002) · Zbl 1020.47001
[10]	Castro González, N.; Koliha, J.J., New additive results for the g-Drazin inverse, Proc. roy. soc. Edinburgh sect. A, 134, 1085-1097, (2004) · Zbl 1088.15006
[11]	Hartwig, R.E.; Wang, G.; Wei, Y., Some additive results on Drazin inverse, Linear algebra appl., 322, 207-217, (2001) · Zbl 0967.15003
[12]	Meyer, C.D.; Rose, N.J., The index and the Drazin inverse of block triangular matrices, SIAM J. appl. math., 33, 1, 1-7, (1977) · Zbl 0355.15009
[13]	Djordjević, D.S.; Stanimirović, P.S., On the generalized Drazin inverse and generalized resolvent, Czechoslovak math. J., 51, 126, 617-634, (2001) · Zbl 1079.47501