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.
@article{bwmeta1.element.bwnjournal-article-smv106i2p153bwm,
author = {Bruce Barnes},
title = {Perturbation theory relative to a Banach algebra of operators},
journal = {Studia Mathematica},
volume = {104},
year = {1993},
pages = {153-174},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv106i2p153bwm}
}
Barnes, Bruce. Perturbation theory relative to a Banach algebra of operators. Studia Mathematica, Tome 104 (1993) pp. 153-174. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv106i2p153bwm/
[00000] [1] W. Arendt and A. Sourour, Perturbation of regular operators and the order essential spectrum, Nederl. Akad. Wetensch. Proc. 89 (1986), 109-122.
[00001] [2] B. Barnes, Fredholm theory in a Banach algebra of operators, Proc. Roy. Irish Acad. 87A (1987), 1-11.
[00002] [3] B. Barnes, The spectral and Fredholm theory of extensions of bounded linear operators, Proc. Amer. Math. Soc. 105 (1989), 941-949.
[00003] [4] B. Barnes, Interpolation of spectrum of bounded operators on Lebesgue spaces, Rocky Mountain J. Math. 20 (1990), 359-378.
[00004] [5] B. Barnes, Essential spectra in a Banach algebra applied to linear operators, Proc. Roy. Irish Acad. 90A (1990), 73-82.
[00005] [6] B. Barnes, Closed operators affiliated with a Banach algebra of operators, Studia Math. 101 (1992), 215-240.
[00006] [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.
[00007] [8] F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin 1973.
[00008] [9] H. G. Dales, On norms on algebras, in: Proc. Centre Math. Anal. Austral. Nat. Univ.21, 1989, 61-95.
[00009] [10] N. Dunford and J. Schwartz, Linear Operators, Part I, Interscience, New York 1964.
[00010] [11] S. Goldberg, Unbounded Linear Operators, McGraw-Hill, New York 1966.
[00011] [12] K. Jörgens, Linear Integral Operators, Pitman, Boston 1982.
[00012] [13] T. Kato, Perturbation Theory for Linear Operators, Springer, New York 1966.
[00013] [14] D. Kleinecke, Almost-finite, compact, and inessential operators, Proc. Amer. Math. Soc. 14 (1963), 863-868.
[00014] [15] R. Kress, Linear Integral Equations, Springer, Berlin 1989.
[00015] [16] W. Pfaffenberger, On the ideals of strictly singular and inessential operators, Proc. Amer. Math. Soc. 25 (1970), 603-607.
[00016] [17] M. Schechter, Principles of Functional Analysis, Academic Press, New York 1971.