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The Weighted g-drazin inverse for operators

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

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: [email protected], [email protected]
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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 81 , Issue 3 , December 2006 , pp. 405 - 423
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Barnes, B. A., ‘Common operator properties of the linear operators RS and SR’, Proc. Amer. Math. Soc. 126 (1998), 1055–1061.CrossRefGoogle Scholar
[2]Bonsall, F. F. and Duncan, J., Complete normed algebras (Springer, Berlin, 1973).CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
[8]Koliha, J. J., ‘Isolated spectral points’, Proc. Amer. Math. Soc. 124 (1996), 3417–3424.CrossRefGoogle 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.CrossRefGoogle Scholar
[11]Mbekhta, M., ‘Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux’, Glasgow Math. J. 29 (1987), 159–175.CrossRefGoogle 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. and Wei, Y., ‘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.CrossRefGoogle 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