Continuity of generalized inverses in Banach algebras

Steffen Roch; Bernd Silbermann

Studia Mathematica (1999)

  • Volume: 136, Issue: 3, page 197-227
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Roch, Steffen, and Silbermann, Bernd. "Continuity of generalized inverses in Banach algebras." Studia Mathematica 136.3 (1999): 197-227. <http://eudml.org/doc/216668>.

@article{Roch1999,
abstract = {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.},
author = {Roch, Steffen, Silbermann, Bernd},
journal = {Studia Mathematica},
keywords = {generalized inverses; Drazin inverses; Moore-Penrose inverses; group inverses; symmetric inverses; asymptotic generalized invertibility; problem of continuity of a generalized inverse; regularization; asymptotic splitting of approximation numbers},
language = {eng},
number = {3},
pages = {197-227},
title = {Continuity of generalized inverses in Banach algebras},
url = {http://eudml.org/doc/216668},
volume = {136},
year = {1999},
}

TY - JOUR
AU - Roch, Steffen
AU - Silbermann, Bernd
TI - Continuity of generalized inverses in Banach algebras
JO - Studia Mathematica
PY - 1999
VL - 136
IS - 3
SP - 197
EP - 227
AB - 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.
LA - eng
KW - generalized inverses; Drazin inverses; Moore-Penrose inverses; group inverses; symmetric inverses; asymptotic generalized invertibility; problem of continuity of a generalized inverse; regularization; asymptotic splitting of approximation numbers
UR - http://eudml.org/doc/216668
ER -

References

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  1. [1] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973. Zbl0271.46039
  2. [2] A. Böttcher, On the approximation numbers of large Toeplitz matrices, Doc. Math. 2 (1997), 1-29. Zbl0873.47014
  3. [3] N. Bourbaki, Éléments de Mathématique, Fascicule XXXII, Théories spectrales, Hermann, Paris, 1967. 
  4. [4] I. Gohberg and N. Krupnik, Introduction to the Theory of One-Dimensional Singular Integral Operators, Birkhäuser, Basel, 1992. Zbl0743.45004
  5. [5] R. Harte and M. Mbekhta, On generalized inverses in C*-algebras, Studia Math. 103 (1992), 71-77. Zbl0810.46062
  6. [6] R. Harte and M. Mbekhta, Generalized inverses in C*-algebras, II, ibid. 106 (1993), 129-138. Zbl0810.46063
  7. [7] D. R. Huang, Generalized inverses over Banach algebras, Integral Equations Operator Theory 15 (1992), 454-469. Zbl0761.46038
  8. [8] D. R. Huang, Group inverses and Drazin inverses over Banach algebras, ibid. 17 (1993), 54-67. Zbl0790.15005
  9. [9] J. J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38 (1996), 367-381. Zbl0897.47002
  10. [10] J. J. Koliha and V. Rakočević, Continuity of the Drazin inverse, II, Studia Math. 131 (1998), 167-177. Zbl0926.47001
  11. [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. [12] S. K. Mitra and C. R. Rao, Generalized Inverse of Matrices and its Applications, Wiley, New York, 1971. Zbl0236.15004
  13. [13] R. H. Moore and M. Z. Nashed, Approximation of generalized inverses of linear operators, SIAM J. Appl. Math. 27 (1974), 1-16. Zbl0295.41018
  14. [14] M. Z. Nashed (ed.), Generalized Inverses and Applications, Academic Press, New York, 1976. 
  15. [15] V. Rakočević, Continuity of the Drazin inverse, J. Operator Theory 41 (1999), 55-68. Zbl0990.47002
  16. [16] S. Roch and B. Silbermann, C*-algebra techniques in numerical analysis, ibid. 35 (1996), 241-280. Zbl0865.65035
  17. [17] S. Roch and B. Silbermann, Asymptotic Moore-Penrose invertibility of singular integral operators, Integral Equations Operator Theory 26 (1996), 81-101. Zbl0860.65145
  18. [18] S. Roch and B. Silbermann, Index calculus for approximation methods, and singular value decomposition, J. Math. Anal. Appl. 225 (1998), 401-426. Zbl0923.65026
  19. [18] B. Silbermann, Asymptotic Moore-Penrose inversion of Toeplitz operators, Linear Algebra Appl. 256 (1997), 219-234. Zbl0880.15002

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