Essential spectrum for a Hilbert space operator

Bouldin, Richard

doi:10.1090/S0002-9947-1972-0284837-5

Essential spectrum for a Hilbert space operator
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Trans. Amer. Math. Soc. 163 (1972), 437-445 Request permission

Erratum: Trans. Amer. Math. Soc. 199 (1974), 429.

Abstract:

Various notions of essential spectrum have been defined for densely defined closed operators on a Banach space. This paper shows that the theory for those notions of essential spectrum simplifies if the underlying space is a Hilbert space and the operator is reduced by its finite-dimensional eigenspaces. In that situation this paper classifies each essential spectrum in terms of the usual language for the spectrum of a Hilbert space operator. As an application this paper deduces the main results of several recent papers dealing with generalizations of the Weyl theorem.

References

S. K. Berberian, An extension of Weyl’s theorem to a class of not necessarily normal operators, Michigan Math. J. 16 (1969), 273–279. MR 250094
L. A. Coburn, Weyl’s theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285–288. MR 201969
I. C. Gohberg and M. G. Kreĭn, The basic propositions on defect numbers, root numbers and indices of linear operators, Amer. Math. Soc. Transl. (2) 13 (1960), 185–264. MR 0113146
Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
Vasile Istrăţescu, Weyl’s theorem for a class of operators, Rev. Roumaine Math. Pures Appl. 13 (1968), 1103–1105. MR 238105
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Tosio Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322. MR 107819, DOI 10.1007/BF02790238
Peter D. Lax, Symmetrizable linear transformations, Comm. Pure Appl. Math. 7 (1954), 633–647. MR 68116, DOI 10.1002/cpa.3160070403
G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29–43. MR 133024, DOI 10.1090/S0002-9947-1961-0133024-2
Joseph I. Nieto, On the essential spectrum of symmetrizable operators, Math. Ann. 178 (1968), 145–153. MR 233221, DOI 10.1007/BF01350656
George H. Orland, On a class of operators, Proc. Amer. Math. Soc. 15 (1964), 75–79. MR 157244, DOI 10.1090/S0002-9939-1964-0157244-4
Martin Schechter, On the essential spectrum of an arbitrary operator. I, J. Math. Anal. Appl. 13 (1966), 205–215. MR 188798, DOI 10.1016/0022-247X(66)90085-0
Martin Schechter, Invariance of the essential spectrum, Bull. Amer. Math. Soc. 71 (1965), 365–367. MR 174979, DOI 10.1090/S0002-9904-1965-11296-4
Kôsaku Yosida, Functional analysis, Die Grundlehren der mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824