J. Aust. Math. Soc. 83 (2007), no. 2, pp. 271–284.
Stable perturbation in Banach algebras
Yifeng Xue
Department of Mathematics
East China Normal University
Shanghai 200062
P.R. China
xyf63071@public9.sta.net.cn
yfxue@math.edu.cn
Received 19 March 2005; revised 22 September 2006
Communicated by A. Pryde

Abstract

Let \mathcal {A} be a unital Banach algebra. Assume that a has a generalized inverse a^+. Then \bar a=a+\delta a\in \mathcal {A} is said to be a stable perturbation of a if \bar a\mathcal {A}\cap (1-aa^+)\mathcal {A}=\{0\}. In this paper we give various conditions for stable perturbation of a generalized invertible element and show that the equation \bar a\mathcal {A}\cap (1-aa^+)\mathcal {A}=\{0\} is closely related to the gap function \mathop {\hat \delta }(\bar a\mathcal {A},a\mathcal {A}). These results will be applied to error estimates for perturbations of the Moore–Penrose inverse in C^*-algebras and the Drazin inverse in Banach algebras.

Download the article in PDF format (size 160 Kb)

2000 Mathematics Subject Classification: primary 46H99, 46N40; secondary 65J05
(Metadata: XML, RSS, BibTeX)

References

  1. B. A. Barnes, ‘The Fredholm elements of a ring’, Canad. J. Math. 21 (1969), 84–95. MR237542
  2. N. Castro-González and J. J. Koliha, ‘Perturbation of Drazin inverse for closed linear operators’, Integral Equations Operator Theory 36 (2000), 92–106. MR1736919
  3. N. Castro-González, J. J. Koliha and V. Rakočević, ‘Continuity and general perturbation of Drazin inverse for closed linear operators’, Abstr. Appl. Anal. 7 (2002), 355–347. MR1920147
  4. N. Castro-González, J. J. Koliha and Y. Wei, ‘Error bounds for perturbation of the Drazin inverse of closed operators with equal spectral idempotents’, Appl. Anal. 81 (2002), 915–928. MR1929553
  5. G. Chen, M. Wei and Y. Xue, ‘Perturbation analysis of the least squares solution in Hilbert spaces’, Linear Algebra Appl. 244 (1996), 69–80. MR1403276
  6. G. Chen and Y. Xue, ‘Perturbation analysis for the operator equation Tx=b in Banach spaces’, J. Math. Anal. Appl. 212 (1997), 107–125. MR1460188
  7. G. Chen and Y. Xue, ‘The expression of the generalized inverse of the perturbed operator under type I perturbation in Hilbert spaces’, Linear Algebra Appl. 285 (1998), 1–6. MR1653550
  8. J. Ding and L. J. Huang, ‘Perturbation of generalized inverses of linear operators in Hilbert spaces’, J. Math. Anal. Appl. 198 (1996), 506–515. MR1376277
  9. M. P. Drazin, ‘Pseudo-inverses in associative rings and semigroups’, Amer. Math. Monthly 65 (1958), 506–514. MR98762
  10. V. Rakočević, ‘Continuity of the Drazin inverse’, J. Operator Theory 41 (1999), 55–68. MR1675243
  11. R. Harte and M. Mbekhta, ‘On generalized inverse in C^{*}-algebras’, Studia Math. 103 (1992), 71–77. MR1184103
  12. R. Harte and M. Mbekhta, ‘On generalized inverse in C^{*}-algebras (II)’, Studia Math. 106 (1993), 129–138. MR1240309
  13. J. J. Koliha, ‘Error bounds for a general perturbation of Drazin inverse’, Appl. Math. Comput. 126 (2002), 181–185. MR1879156
  14. J. J. Koliha and V. Rakočević, ‘Continuity of the Drazin inverse II’, Studia Math. 131 (1998), 167–177. MR1636348
  15. G. K. Pedersen, C^{*}-algebras and Their Automorphism Groups (Academic Press, Boston, 1979). MR548006
  16. V. Rakočević, ‘On the continuity of the Moore–Penrose inverse in C^{*}-algebras’, Math. Montisnigri 2 (1993), 89–92. MR1284899
  17. V. Rakočević and Y. Wei, ‘The perturbation theory for the Drazin inverse and its applications (II)’, J. Aust. Math. Soc. 70 (2001), 189–197. MR1815279
  18. C. Rickart, General Theory of Banach Algebras (Van Nostrand Reinhold, New York, 1960). MR115101
  19. S. Roch and B. Silbermann, ‘Continuity of generalized inverses in Banach algebras’, Studia Math. 136 (1999), 197–227. MR1724245
  20. Y. Xue and G. Chen, ‘Some equivalent conditions of stable perturbation of operators in Hilbert spaces’, Applied Math. Comput. 147 (2004), 765–772. MR2011086