Note on the Generalized Invertibility of a-xy*

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 2,  2015, pp.251-258
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.2.251
Title & Authors
Note on the Generalized Invertibility of a-xy*
DU, FAPENG; XUE, YIFENG;

Abstract
Let $\small{\mathcal{A}}$ be a unital $\small{C^*}$-algebra, a, x and y are elements in $\small{\mathcal{A}}$. In this paper, we present the expression of the Moore-Penrose inverse and the group inverse of a-$\small{xy^*}$ under the conditions $x Keywords Generalized inverse;Moore-Penrose inverse;Group inverse; Language English Cited by References 1. G. Chen, Y. Xue, The expression of generalized inverse of the perturbed operators under type I perturbation in Hilbert spaces, Linear Algebra Appl., 285(1998), 1-6. 2. Y. Chen, X. Hu, Q. Xu, The Moore-Penrose inverse of A -$XY^*$, J. Shanghai Normal Univer., 38(2009), 15-19. 3. C. Deng, On the invertibility of the operator A - XB, Numb. Linear Algebra Appl., 16(2009), 817-831. 4. C. Deng, A generalization of the Sherman-Morrison-Woodbury formula, Appl. Math. Lett., 24(2011), 1561-1564. 5. C. Deng, On Moore-Penrose inverse of a kind of operators, Proceedings of the Ninth International Conference on Matrix Theory and Its Applications in China, (2010), 88-91. 6. C. Deng, Y. Wei, Some New Results of the Sherman-Morrison-Woodbury Formula, Proceeding of The Sixth Iternational Conference of Matrices and Operators, 2(2011), 220-223. 7. F. Du, Y. Xue, The expression of the Moore-Penrose inverse of A -$XY^*$, J. East China Normal Univ. (Nat. Sci.), 5(2010), 33-37. 8. H. V. Hsnderson, Searl S. R., On deriving the inverse of a sum of matrices, Siam Review, 23(1)(1981), 53-60. 9. W. W. Hager, Updating the inverse of a matrix, Siam Review, 31(1989), 221-239. 10. Shani Jose, K. C. Sivakumar, Moore-Penrose Inverse of Perturbed Operators on Hilbert Spaces, Combinatorial Matrix Theory and Generalized Inverses of Matrices, (2013), 119-131. 11. S. Kurt, A. Riedel, A Shermen-Morrison-Woodbury identity for rank augmenting matrices with application to centering, Siam J. Math. Anal., 12(1)(1991), 80-95. 12. T. Steerneman, F. P. Kleij, Properties of the matrix A -$XY^*\$, Linear Algebra Appl., 410(2005), 70-86.

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