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The weighted g-Drazin inverse for operators

Part of: Topological algebras, normed rings and algebras, Banach algebras General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

A. Dajić  and
J. J. Koliha
J. J. Koliha
Affiliation:
Department of Mathematics and Statistics The University of MelbourneVIC 3010Australia e-mail: a.dajic@ms.unimelb.edu.au, j.koliha@ms.unimelb.edu.au
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Abstract

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The paper introduces and studies the weighted g-Drazin inverse for bounded linear operators between Banach spaces, extending the concept of the weighted Drazin inverse of Rakočević and Wei (Linear Algebra Appl. 350 (2002), 25–39) and of Cline and Greville (Linear Algebra Appl. 29 (1980), 53–62). We use the Mbekhta decomposition to study the structure of an operator possessing the weighted g-Drazin inverse, give an operator matrix representation for the inverse, and study its continuity. An open problem of Rakočević and Wei is solved.

Keywords

Banach algebra without unitg-Drazin inversebounded linear operatorweighted g-Drazin inversethe Mbekhta decompositionascent and descent.

MSC classification

Secondary: 47A60: Functional calculus 46H30: Functional calculus in topological algebras 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A10: Spectrum, resolvent
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 2 , April 2007 , pp. 163 - 181
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Barnes, B. A., ‘Common operator properties of the linear operators RS and SR’, Proc. Amer. Math. Soc. 126 (1998), 1055–1061.Google Scholar
[2]Bonsall, F. F. and Duncan, J., Complete normed algebras (Springer, Berlin, 1973).Google Scholar
[3]Buoni, J. J. and Faires, J. D., ‘Ascent, descent, nullity and defect of products of operators’, Indiana Univ. Math. J. 25 (1976), 703–707.Google Scholar
[4]Campbell, S. L. and Meyer, C. D., Generalized inverses of linear transformations (Pitman, London, 1979).Google Scholar
[5]Cline, R. E. and Greville, T. N. E., ‘A Drazin inverse for rectangular matrices’, Linear Algebra Appl. 29 (1980), 53–62.Google Scholar
[6]González, N. Castro, Koliha, J. J. and Wei, Y., ‘On integral representation of the Drazin inverse in Banach algebras’, Proc. Edinburgh Math. Soc. 45 (2002), 327–331.CrossRefGoogle Scholar
[7]Koliha, J. J., ‘A generalized Drazin inverse’, Glasgow Math. J. 38 (1996), 367–381.Google Scholar
[8]Koliha, J. J., ‘Isolated spectral points’, Proc. Amer. Math. Soc. 124 (1996), 3417–3424.Google Scholar
[9]Koliha, J. J. and Rakočević, V., ‘Continuity of the Drazin inverse II’, Studia Math. 131 (1998), 167–177.Google Scholar
[10]Koliha, J. J. and Poon, P. W., ‘Spectral sets II’, Rend. Circ. Mat. Palermo (2) 47 (1998), 293–310.Google Scholar
[11]Mbekhta, M., ‘Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux’, Glasgow Math. J. 29 (1987), 159–175.Google Scholar
[12]Qiao, S. Z., ‘The weighted Drazin inverse of a linear operator on a Banach space and its approximation’, Numer. Math. J. Chinese Univ. 3 (1981), 296–305 (Chinese).Google Scholar
[13]Rakočević, V. and Wei, Y., ‘A weighted Drazin inverse and applications’, Linear Algebra Appl. 350 (2002), 25–39.CrossRefGoogle Scholar
[14]Rakočević, V., ‘The representation and approximation of the W-weighted Drazin inverse of linear operators in Hilbert spaces’, Appl. Math. Comput. 141 (2003), 455–470.Google Scholar
[15]Schmoeger, Ch., ‘On isolated points of the spectrum of a bounded linear operator’, Proc. Amer. Math. Soc. 117 (1993), 715–719.Google Scholar
[16]Wang, G. R., ‘Approximation methods for the W-weighted Drazin inverse of linear operators in Banach spaces’, Numer. Math. J. Chinese Univ. 10 (1988), 74–81 (Chinese).Google Scholar
[17]Wang, G. R., ‘Iterative methods for computing the Drazin inverse and the W-weighted Drazin inverse of linear operators based on functional interpolation’, Numer. Math. J. Chinese Univ. 11 (1989), 269–280.Google Scholar
[18]Wei, Y., ‘Integral representation of the W-weighted Drazin inverse’, Appl. Math. Comput. 144 (2003), 3–10.Google Scholar
[19]Yukhno, L. F., ‘An eigenvalue property of the product of two rectangular matrices’, Comput. Math. Math. Phys. 36 (1996), 555–557.Google Scholar