Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-23T10:48:01.277Z Has data issue: false hasContentIssue false
  • English
  • Français

Additive results for the generalized Drazin inverse

Part of: General theory of linear operators Basic linear algebra

Published online by Cambridge University Press:  09 April 2009

Dragan S. Djordjević  and
Yimin Wei
Dragan S. Djordjević
Affiliation:
Department of Mathematics, Faculty of Sciences, University of Niš, P.O. Box 224, 18000 Niš, Yugoslavia e-mail: dragan@pmf.pmf.ni.ac.yu
Yimin Wei
Affiliation:
Department of Mathematics, and Laboratory of Mathematics, for Nonlinear Sciences, Fudan University, Shanghai 200433, P.R. of China e-mail: ymwei@fudan.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Additive perturbation results for the generalized Drazin inverse of Banach space operators are presented. Precisely, if Ad denotes the generalized Drazin inverse of a bounded linear operator A on an arbitrary complex Banach space, then in some special cases (A + B)d is computed in terms of Ad and Bd. Thus, recent results of Hartwig, Wang and Wei (Linear Algebra Appl. 322 (2001), 207–217) are extended to infinite dimensional settings with simplified proofs.

Keywords

generalized Drazin inverseperturbationsadditive results

MSC classification

Secondary: 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A55: Perturbation theory 15A09: Matrix inversion, generalized inverses
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 73 , Issue 1 , August 2002 , pp. 115 - 126
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Ben-Israel, A. and Greville, T. N. E., Generalized inverses: theory and applications (Wiley-Interscience, New York, 1974).Google Scholar
[2]Campbell, S. L. and Meyer, C. D., Generalized inverses of linear transformations (Pitman, New York, 1979).Google Scholar
[3]Caradus, S. R., Generalized inverses and operator theory, Queen's Papers in Pure and Appl. Math. 50 (Queen's University, Kingston, Ontario, 1978).Google Scholar
[4]Rakočević, V., ‘Continuity of the Drazin inverse’, J. Operator Theory 41 (1999), 55–68.Google Scholar
[5]Rakočević, V. and Wei, Y., ‘The perturbation theory for the Drazin inverse and its applications II’, J. Austral. Math. Soc. 70 (2001), 189–197.Google Scholar
[6]Djordjević, D. S. and Stanimirović, P. S., ‘On the generalized Drazin inverse and generalized resolvent’, Czechoslovak Math. J. 51 (126) (2001), 617–634.Google Scholar
[7]Drazin, M. P., ‘Pseudoinverses in associative rings and semigroups’, Amer. Math. Monthly 65 (1958), 506–514.Google Scholar
[8]González, N. Castro and Koliha, J. J., ‘Perturbation of the Drazin inverse for closed linear operators’, Integral Equations Operator Theory 36 (2000), 92–106.Google Scholar
[9]Harte, R. E., ‘Spectral projections’, Irish Math. Soc. Newsletter 11 (1984), 10–15.Google Scholar
[10]Harte, R. E., Invertibility and singularity for bounded linear operators (Marcel Dekker, New York, 1988).Google Scholar
[11]Harte, R. E., ‘On quasinilpotents in rings’, Panamer. Math. J. 1 (1991), 10–16.Google Scholar
[12]Hartwig, R. E. and Shoaf, J. M., ‘Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices’, J. Austral. Math. Soc. Ser. A 24 (1977), 10–34.Google Scholar
[13]Hartwig, R. E., Wang, G. and Wei, Y., ‘Some additive results on Drazin inverse’, Linear Algebra Appl. 322 (2001), 207–217.Google Scholar
[14]Koliha, J. J., ‘A generalized Drazin inverse’, Glasgow Math. J. 38 (1996), 367–381.Google Scholar
[15]Koliha, J. J. and Rakočević, V., ‘Continuity of the Drazin inverse II’, Studia Math. 131 (1998), 167–177.Google Scholar
[16]Meyer, C. D. Jr, ‘The condition number of a finite Markov chains and perturbation bounds for the limiting probabilities’, SIAM J. Algebraic Discrete Methods 1 (1980), 273–283.CrossRefGoogle Scholar
[17]Meyer, C. D. Jr, and Rose, N. J., ‘The index and the Drazin inverse of block triangular matrices’, SIAM J. Appl. Math. 33 (1977), 1–7.CrossRefGoogle Scholar
[18]Meyer, C. D. Jr, and Shoaf, J. M., ‘Updating finite Markov chains by using techniques of group inversion’, J. Statist. Comput. Simulation 11 (1980), 163–181.CrossRefGoogle Scholar
[19]Wei, Y., ‘On the perturbation of the group inverse and the oblique projection’, Appl. Math. Comput. 98 (1999), 29–42.Google Scholar
[20]Wei, Y. and Wang, G., ‘The perturbation theory for the Drazin inverse and its applications’, Linear Algebra Appl. 258 (1997), 179–186.Google Scholar