Perturbation theory relative to a Banach algebra of operators

Bruce Barnes

Studia Mathematica (1993)

  • Volume: 106, Issue: 2, page 153-174
  • ISSN: 0039-3223

Abstract

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Let ℬ be a Banach algebra of bounded linear operators on a Banach space X. Let S be a closed linear operator in X, and let R be a linear operator in X. In this paper the spectral and Fredholm theory relative to ℬ of the perturbed operator S + R is developed. In particular, the situation where R is S-inessential relative to ℬ is studied. Several examples are given to illustrate the usefulness of these concepts.

How to cite

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Barnes, Bruce. "Perturbation theory relative to a Banach algebra of operators." Studia Mathematica 106.2 (1993): 153-174. <http://eudml.org/doc/216010>.

@article{Barnes1993,
abstract = {Let ℬ be a Banach algebra of bounded linear operators on a Banach space X. Let S be a closed linear operator in X, and let R be a linear operator in X. In this paper the spectral and Fredholm theory relative to ℬ of the perturbed operator S + R is developed. In particular, the situation where R is S-inessential relative to ℬ is studied. Several examples are given to illustrate the usefulness of these concepts.},
author = {Barnes, Bruce},
journal = {Studia Mathematica},
keywords = {Banach algebra of operators; Fredholm operator; perturbation theory; essential spectrum; Fredholm theory; perturbed operator},
language = {eng},
number = {2},
pages = {153-174},
title = {Perturbation theory relative to a Banach algebra of operators},
url = {http://eudml.org/doc/216010},
volume = {106},
year = {1993},
}

TY - JOUR
AU - Barnes, Bruce
TI - Perturbation theory relative to a Banach algebra of operators
JO - Studia Mathematica
PY - 1993
VL - 106
IS - 2
SP - 153
EP - 174
AB - Let ℬ be a Banach algebra of bounded linear operators on a Banach space X. Let S be a closed linear operator in X, and let R be a linear operator in X. In this paper the spectral and Fredholm theory relative to ℬ of the perturbed operator S + R is developed. In particular, the situation where R is S-inessential relative to ℬ is studied. Several examples are given to illustrate the usefulness of these concepts.
LA - eng
KW - Banach algebra of operators; Fredholm operator; perturbation theory; essential spectrum; Fredholm theory; perturbed operator
UR - http://eudml.org/doc/216010
ER -

References

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  1. [1] W. Arendt and A. Sourour, Perturbation of regular operators and the order essential spectrum, Nederl. Akad. Wetensch. Proc. 89 (1986), 109-122. 
  2. [2] B. Barnes, Fredholm theory in a Banach algebra of operators, Proc. Roy. Irish Acad. 87A (1987), 1-11. 
  3. [3] B. Barnes, The spectral and Fredholm theory of extensions of bounded linear operators, Proc. Amer. Math. Soc. 105 (1989), 941-949. 
  4. [4] B. Barnes, Interpolation of spectrum of bounded operators on Lebesgue spaces, Rocky Mountain J. Math. 20 (1990), 359-378. 
  5. [5] B. Barnes, Essential spectra in a Banach algebra applied to linear operators, Proc. Roy. Irish Acad. 90A (1990), 73-82. 
  6. [6] B. Barnes, Closed operators affiliated with a Banach algebra of operators, Studia Math. 101 (1992), 215-240. 
  7. [7] B. Barnes, G. Murphy, R. Smyth, and T. T. West, Riesz and Fredholm Theory in Banach Algebras, Pitman Res. Notes in Math. 67, Pitman, Boston 1982. 
  8. [8] F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin 1973. 
  9. [9] H. G. Dales, On norms on algebras, in: Proc. Centre Math. Anal. Austral. Nat. Univ.21, 1989, 61-95. 
  10. [10] N. Dunford and J. Schwartz, Linear Operators, Part I, Interscience, New York 1964. 
  11. [11] S. Goldberg, Unbounded Linear Operators, McGraw-Hill, New York 1966. 
  12. [12] K. Jörgens, Linear Integral Operators, Pitman, Boston 1982. 
  13. [13] T. Kato, Perturbation Theory for Linear Operators, Springer, New York 1966. 
  14. [14] D. Kleinecke, Almost-finite, compact, and inessential operators, Proc. Amer. Math. Soc. 14 (1963), 863-868. 
  15. [15] R. Kress, Linear Integral Equations, Springer, Berlin 1989. 
  16. [16] W. Pfaffenberger, On the ideals of strictly singular and inessential operators, Proc. Amer. Math. Soc. 25 (1970), 603-607. 
  17. [17] M. Schechter, Principles of Functional Analysis, Academic Press, New York 1971. 

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