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1999 | 136 | 3 | 197-227
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Continuity of generalized inverses in Banach algebras

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The main topic of the paper is the continuity of several kinds of generalized inversion of elements in a Banach algebra with identity. We introduce the notion of asymptotic generalized invertibility and completely characterize sequences of elements with this property. Based on this result, we derive continuity criteria which generalize the well known criteria from operator theory.
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Bibliografia
  • [1] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973.
  • [2] A. Böttcher, On the approximation numbers of large Toeplitz matrices, Doc. Math. 2 (1997), 1-29.
  • [3] N. Bourbaki, Éléments de Mathématique, Fascicule XXXII, Théories spectrales, Hermann, Paris, 1967.
  • [4] I. Gohberg and N. Krupnik, Introduction to the Theory of One-Dimensional Singular Integral Operators, Birkhäuser, Basel, 1992.
  • [5] R. Harte and M. Mbekhta, On generalized inverses in C*-algebras, Studia Math. 103 (1992), 71-77.
  • [6] R. Harte and M. Mbekhta, Generalized inverses in C*-algebras, II, ibid. 106 (1993), 129-138.
  • [7] D. R. Huang, Generalized inverses over Banach algebras, Integral Equations Operator Theory 15 (1992), 454-469.
  • [8] D. R. Huang, Group inverses and Drazin inverses over Banach algebras, ibid. 17 (1993), 54-67.
  • [9] J. J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 (1996), 367-381.
  • [10] J. J. Koliha and V. Rakočević, Continuity of the Drazin inverse, II, Studia Math. 131 (1998), 167-177.
  • [11] S. G. Michlin und S. Prössdorf, Singuläre Integraloperatoren, Akademie-Verlag, Berlin, 1980 (extended English translation: Singular Integral Operators, Akademie-Verlag and Springer, 1986).
  • [12] S. K. Mitra and C. R. Rao, Generalized Inverse of Matrices and its Applications, Wiley, New York, 1971.
  • [13] R. H. Moore and M. Z. Nashed, Approximation of generalized inverses of linear operators, SIAM J. Appl. Math. 27 (1974), 1-16.
  • [14] M. Z. Nashed (ed.), Generalized Inverses and Applications, Academic Press, New York, 1976.
  • [15] V. Rakočević, Continuity of the Drazin inverse, J. Operator Theory 41 (1999), 55-68.
  • [16] S. Roch and B. Silbermann, C*-algebra techniques in numerical analysis, ibid. 35 (1996), 241-280.
  • [17] S. Roch and B. Silbermann, Asymptotic Moore-Penrose invertibility of singular integral operators, Integral Equations Operator Theory 26 (1996), 81-101.
  • [18] S. Roch and B. Silbermann, Index calculus for approximation methods, and singular value decomposition, J. Math. Anal. Appl. 225 (1998), 401-426.
  • [18] B. Silbermann, Asymptotic Moore-Penrose inversion of Toeplitz operators, Linear Algebra Appl. 256 (1997), 219-234.
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bwmeta1.element.bwnjournal-article-smv136i3p197bwm
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