Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Note on the Generalized Invertibility of a-xy*

  • DU, FAPENG (Department of Mathematics, Southeast University, Department of Mathematics, School of mathematical & Physical Sciences, Institute of Technology) ;
  • XUE, YIFENG (Department of Mathematics, East China Normal University)
  • Received : 2013.07.25
  • Accepted : 2014.01.29
  • Published : 2015.06.23
Download PDF
⟨ Previous Next ⟩

Abstract

Let $\mathcal{A}$ be a unital $C^*$-algebra, a, x and y are elements in $\mathcal{A}$. In this paper, we present the expression of the Moore-Penrose inverse and the group inverse of a-$xy^*$ under the conditions $x=aa^+x,y^*=y^*a^+a$, respectively.

Keywords

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. https://doi.org/10.1016/S0024-3795(98)10066-6
  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. https://doi.org/10.1002/nla.646
  4. C. Deng, A generalization of the Sherman-Morrison-Woodbury formula, Appl. Math. Lett., 24(2011), 1561-1564. https://doi.org/10.1016/j.aml.2011.03.046
  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. https://doi.org/10.1137/1023004
  9. W. W. Hager, Updating the inverse of a matrix, Siam Review, 31(1989), 221-239. https://doi.org/10.1137/1031049
  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. https://doi.org/10.1016/j.laa.2004.10.028
  13. Y. Xue, Stable Perturbations of Operators and Related Topics, World Scientific, (2012).