Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Essential spectrum for a Hilbert space operator


Author: Richard Bouldin
Journal: Trans. Amer. Math. Soc. 163 (1972), 437-445
MSC: Primary 47.30
DOI: https://doi.org/10.1090/S0002-9947-1972-0284837-5
Erratum: Trans. Amer. Math. Soc. 199 (1974), 429.
MathSciNet review: 0284837
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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 40 #3335. MR 0250094 (40:3335)
  • [2] L. A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285-288. MR 34 #1846. MR 0201969 (34:1846)
  • [3] I. C. Gohberg and M. G. Kreĭn, The basic propositions on defect numbers, root numbers and indices of linear operators, Uspehi Mat. Nauk 12 (1957), no. 2 (74), 43-118; English transl., Amer. Math. Soc. Transl. (2) 13 (1960), 185-264. MR 20 #3459; MR 22 #3984. MR 0113146 (22:3984)
  • [4] P. R. Halmos, A Hilbert space problem book, Van Nostrand, Princeton, N. J., 1967. MR 34 #8178. MR 0208368 (34:8178)
  • [5] V. Isträtescu, Weyl's theorem for a class of operators, Rev. Roumaine Math. Pures Appl. 13 (1968), 1103-1105. MR 38 #6381. MR 0238105 (38:6381)
  • [6] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der math. Wissenschaften, Band 132, Springer-Verlag, New York, 1966. MR 34 #3324. MR 0203473 (34:3324)
  • [7] -, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261-322. MR 21 #6541. MR 0107819 (21:6541)
  • [8] P. D. Lax, Symmetrizable linear transformations, Comm. Pure Appl. Math. 7 (1954), 633-647. MR 16, 832. MR 0068116 (16:832d)
  • [9] G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29-43. MR 24 #A2860. MR 0133024 (24:A2860)
  • [10] Joseph Nieto, On the essential spectrum of symmetrizable operators, Math. Ann. 178 (1968), 145-153. MR 38 #1544. MR 0233221 (38:1544)
  • [11] George Orland, On a class of operators, Proc. Amer. Math. Soc. 15 (1964), 75-79. MR 28 #480. MR 0157244 (28:480)
  • [12] Martin Schechter, On the essential spectrum of an arbitrary operator. I, J. Math. Anal. Appl. 13 (1966), 205-215. MR 32 #6230. MR 0188798 (32:6230)
  • [13] -, Invariance of the essential spectrum, Bull. Amer. Math. Soc. 71 (1965), 365-367. MR 30 #5167. MR 0174979 (30:5167)
  • [14] K. Yosida, Functional analysis, Die Grundlehren der math. Wissenschaften, Band 123, Academic Press, New York; Springer-Verlag, Berlin, 1965. MR 31 #5054. MR 0180824 (31:5054)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47.30

Retrieve articles in all journals with MSC: 47.30


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0284837-5
Keywords: Essential spectrum, operator on a Hilbert space, eigenvalue, algebraic multiplicity, geometric multiplicity, Fredholm operator, index, closed range
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society