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Elements of rings with equal spectral idempotents

Part of: Topological algebras, normed rings and algebras, Banach algebras Selfadjoint operator algebras Basic linear algebra

Published online by Cambridge University Press:  09 April 2009

J. J. Koliha  and
Pedro Patricio
J. J. Koliha
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia e-mail: koliha@unimelb.edu.au
Pedro Patricio
Affiliation:
CMAT—Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal e-mail: pedro@math.uminho.pt
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Abstract

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In this paper we define and study a generalized Drazin inverse xD for ring elements x, and give a characterization of elements a, b for which aaD = bbD. We apply our results to the study of EP elements in a ring with involution.

Keywords

associative ringinvolutionidempotentDrazin inverseMoore–Penrose inverseEP element

MSC classification

Secondary: 15A09: Matrix inversion, generalized inverses 46H05: General theory of topological algebras 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 1 , February 2002 , pp. 137 - 152
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Campbell, S. L. and Meyer, C. D., Generalized inverses of linear transformations (Dover, New York, 1991).Google Scholar
[2]González, N. Catro, Koliha, J. J. and Wei, Y., ‘Perturbation of the Drazin inverse for matrices with equal eigenprojection at zero’, Linear Algebra Appl. 312 (2000), 181–189.CrossRefGoogle Scholar
[3]Drazin, M. P., ‘Pseudo-inverses in associative rings and semigroups’, Amer. Math. Monthly 65 (1958), 506–514.CrossRefGoogle Scholar
[4]Harte, R. E., Invertibility and singularity for bounded linear operators (Marcel Dekker, New York, 1988).Google Scholar
[5]Harte, R. E., ‘On quasinilpotents in rings’, Panam. Math. J. 1 (1991), 10–16.Google Scholar
[6]Hartwig, R. E., ‘Block generalized inverses’, Arch. Rational Mech. Anal. 61 (1976), 197–251.CrossRefGoogle Scholar
[7]Hartwig, R. E., ‘Spectral inverse and the row-space equations’, Linear Algebra Appl. 20 (1978), 157–68.CrossRefGoogle Scholar
[8]Koliha, J. J., ‘A generalizedd Drazin inverse’, Glasgow Math. J. 38 (1996), 367–381.CrossRefGoogle Scholar
[9]Koliha, J. J., ‘The Drazin and Moore–Penrose inverse in C*-algebra’, Math. Proc. Roy. Irish Acad. 99A (1999), 17–27.Google Scholar
[10]Koliha, J. J., ‘Elements of C*-algebras commuting with their Moore–Penrose inverse’, Studia Math. 139 (2000), 81–90.CrossRefGoogle Scholar
[11]Penrose, R., ‘A generalized inverse for matrices’, Proc. Cambridge Philos. Soc. 51 (1955), 406–413.CrossRefGoogle Scholar
[12]Puystjens, R. and Robinson, D. W., ‘The Moore–Penrose inverse of a morphism with factorization’, Linear Algebra Appl. 40 (1981), 129–141.CrossRefGoogle Scholar
[13]Roch, S. and Silberman, B., ‘Continutiy of generalized inverse in Banach algebras’, Studia Math. 136 (1999), 197–227.Google Scholar