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THE DRAZIN INVERSES OF PRODUCTS AND DIFFERENCES OF PROJECTIONS IN A C*-ALGEBRA

Part of: Basic linear algebra Numerical linear algebra

Published online by Cambridge University Press:  01 April 2009

YUAN LI
YUAN LI*
Affiliation:
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, People’s Republic of China (email: [email protected])
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Abstract

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For two given projections p and q in a C*-algebra, we investigate how to express the Drazin inverses of the product pq and the difference p−q, and give applications. As a special case, we obtain the results of [C. Y. Deng, ‘The Drazin inverses of products and differences of orthogonal projections’, J. Math. Anal. Appl.335 (2007) 64–71], with considerably simpler proofs.

Keywords

Drazin inverseprojection

MSC classification

Secondary: 15A09: Matrix inversion, generalized inverses 65F20: Overdetermined systems, pseudoinverses
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 2 , April 2009 , pp. 189 - 198
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Castro, G. and Koliha, J. J., ‘New additive results for the g-Drazin inverse’, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 1085–1097.Google Scholar
[2]Cheng, S. and Tian, Y., ‘Moore–Penrose inverses of products and differences of orthogonal projectors’, Acta Sci. Math. (Szeged) 69 (2003), 533–542.Google Scholar
[3]Cline, R. E., ‘An application of representation of a matrix’, MRC Technical Report 592 (1965).Google Scholar
[4]Cvetković-Ilić, D. S., Djordjević, D. S. and Wei, Y., ‘Additive results for the generalized Drazin inverse in a Banach algebra’, Linear Algebra Appl. 418 (2006), 53–61.CrossRefGoogle Scholar
[5]Dajic, A. and Koliha, J. J., ‘The weighted g-Drazin inverse for operators’, J. Aust. Math. Soc. 82 (2007), 163–181.CrossRefGoogle Scholar
[6]Deng, C. Y., ‘The Drazin inverses of products and differences of orthogonal projections’, J. Math. Anal. Appl. 335 (2007), 64–71.CrossRefGoogle Scholar
[7]Djordjević, D. S. and Wei, Y., ‘Additive results for the generalized Drazin inverse’, J. Aust. Math. Soc. 73 (2002), 115–125.CrossRefGoogle Scholar
[8]Drazin, M. P., ‘Pseudoinverse in associative rings and semigroups’, Amer. Math. Monthly 65 (1958), 505–514.CrossRefGoogle Scholar
[9]Harte, R. E. and Mbekhta, M., ‘On generalized inverses in C *-algebras’, Studia Math. 103 (1992), 71–77.CrossRefGoogle Scholar
[10]Harte, R. E. and Mbekhta, M., ‘Generalized inverses in C *-algebras II’, Studia Math. 106 (1993), 129–138.CrossRefGoogle Scholar
[11]Hartwig, R. E., Wang, G. R. and Wei, Y., ‘Some additive results on Drazin inverse’, Linear Algebra Appl. 322 (2001), 207–217.CrossRefGoogle Scholar
[12]Koliha, J. J., ‘A generalized Drazin inverse’, Glasgow Math. J. 38 (1996), 367–381.CrossRefGoogle Scholar
[13]Koliha, J. J., ‘The Drazin and Moore–Penrose inverse in C *-algebras’, Math. Proc. R. Ir. Acad. 99A (1999), 17–27.Google Scholar
[14]Koliha, J. J., ‘Elements of C *-algebras commuting with their Moore–Penrose inverse’, Studia Math. 139 (2000), 81–90.CrossRefGoogle Scholar
[15]Koliha, J. J., ‘Continuity and differentiability of the Moore–Penrose inverse in C *-algebras’, Math. Scand. 88 (2001), 154–160.CrossRefGoogle Scholar
[16]Koliha, J. J. and Rakočević, V., ‘Fredholm properties of the difference of orthogonal projection in a Hilbert space’, Integral Equations Operator Theory 52 (2005), 125–134.CrossRefGoogle Scholar
[17]Murphy, G. J., C *-Algebras and Operator Theory (Academic Press, New York, 1990).Google Scholar
[18]Wedin, P. A., ‘Perturbation theory for pseudo-inverses’, BIT 13 (1973), 217–232.CrossRefGoogle Scholar