Closed operators affiliated with a Banach algebra of operators

Bruce Barnes

Studia Mathematica (1992)

  • Volume: 101, Issue: 3, page 215-240
  • ISSN: 0039-3223

Abstract

top
Let ℬ be a Banach algebra of bounded linear operators on a Banach space X. If S is a closed operator in X such that (λ - S)^{-1} ∈ ℬ for some number λ, then S is affiliated with ℬ. The object of this paper is to study the spectral theory and Fredholm theory relative to ℬ of an operator which is affiliated with ℬ. Also, applications are given to semigroups of operators which are contained in ℬ.

How to cite

top

Barnes, Bruce. "Closed operators affiliated with a Banach algebra of operators." Studia Mathematica 101.3 (1992): 215-240. <http://eudml.org/doc/215902>.

@article{Barnes1992,
abstract = {Let ℬ be a Banach algebra of bounded linear operators on a Banach space X. If S is a closed operator in X such that (λ - S)^\{-1\} ∈ ℬ for some number λ, then S is affiliated with ℬ. The object of this paper is to study the spectral theory and Fredholm theory relative to ℬ of an operator which is affiliated with ℬ. Also, applications are given to semigroups of operators which are contained in ℬ.},
author = {Barnes, Bruce},
journal = {Studia Mathematica},
keywords = {closed operator; spectrum; Fredholm operator; semigroup of operators; Banach algebra; operators affiliated with a Banach algebra of operators; Banach algebra of bounded linear operators on a Banach space; semigroups of operators},
language = {eng},
number = {3},
pages = {215-240},
title = {Closed operators affiliated with a Banach algebra of operators},
url = {http://eudml.org/doc/215902},
volume = {101},
year = {1992},
}

TY - JOUR
AU - Barnes, Bruce
TI - Closed operators affiliated with a Banach algebra of operators
JO - Studia Mathematica
PY - 1992
VL - 101
IS - 3
SP - 215
EP - 240
AB - Let ℬ be a Banach algebra of bounded linear operators on a Banach space X. If S is a closed operator in X such that (λ - S)^{-1} ∈ ℬ for some number λ, then S is affiliated with ℬ. The object of this paper is to study the spectral theory and Fredholm theory relative to ℬ of an operator which is affiliated with ℬ. Also, applications are given to semigroups of operators which are contained in ℬ.
LA - eng
KW - closed operator; spectrum; Fredholm operator; semigroup of operators; Banach algebra; operators affiliated with a Banach algebra of operators; Banach algebra of bounded linear operators on a Banach space; semigroups of operators
UR - http://eudml.org/doc/215902
ER -

References

top
  1. [1] W. Arendt and A. Sourour, Perturbation of regular operators and the order essential spectrum, Nederl. Akad. Wentensch. Indag. Math. 48 (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, G. Murphy, R. Smyth, and T. T. West, Riesz and Fredholm Theory in Banach Algebras, Res. Notes in Math. 67, Pitman, Boston 1982. 
  7. [7] F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin 1973. 
  8. [8] N. Dunford and J. Schwartz, Linear Operators, Part I, Interscience, New York 1964. 
  9. [9] S. Goldberg, Unbounded Linear Operators, McGraw-Hill, New York 1966. 
  10. [10] E. Hille and R. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence 1957. 
  11. [11] K. Jörgens, Linear Integral Operators, Pitman, Boston 1982. 
  12. [12] R. Kress, Linear Integral Equations, Springer, Berlin 1989. 
  13. [13] G. Lumer and R. Phillips, Dissipative operators in a Banach space, Pacific J. Math. 11 (1961), 679-698. 
  14. [14] R. Nagel et al., One-parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer, Berlin 1986. 
  15. [15] M. Schechter, Principles of Functional Analysis, Academic Press, New York 1971. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.