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.
@article{bwmeta1.element.bwnjournal-article-smv101i3p215bwm,
author = {Bruce Barnes},
title = {Closed operators affiliated with a Banach algebra of operators},
journal = {Studia Mathematica},
volume = {103},
year = {1992},
pages = {215-240},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv101i3p215bwm}
}
Barnes, Bruce. Closed operators affiliated with a Banach algebra of operators. Studia Mathematica, Tome 103 (1992) pp. 215-240. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv101i3p215bwm/
[00000] [1] W. Arendt and A. Sourour, Perturbation of regular operators and the order essential spectrum, Nederl. Akad. Wentensch. Indag. Math. 48 (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, G. Murphy, R. Smyth, and T. T. West, Riesz and Fredholm Theory in Banach Algebras, Res. Notes in Math. 67, Pitman, Boston 1982.
[00006] [7] F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin 1973.
[00007] [8] N. Dunford and J. Schwartz, Linear Operators, Part I, Interscience, New York 1964.
[00008] [9] S. Goldberg, Unbounded Linear Operators, McGraw-Hill, New York 1966.
[00009] [10] E. Hille and R. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence 1957.
[00010] [11] K. Jörgens, Linear Integral Operators, Pitman, Boston 1982.
[00011] [12] R. Kress, Linear Integral Equations, Springer, Berlin 1989.
[00012] [13] G. Lumer and R. Phillips, Dissipative operators in a Banach space, Pacific J. Math. 11 (1961), 679-698.
[00013] [14] R. Nagel et al., One-parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer, Berlin 1986.
[00014] [15] M. Schechter, Principles of Functional Analysis, Academic Press, New York 1971.