Abstract
In this paper a characterization is obtained of those bounded operators on a Hilbert space at which the spectrum is continuous, where the spectrum is considered as a function whose domain is the set of all operators with the norm topology and whose range is the set of compact subsets of the plane with the Hausdorff metric. Similar characterizations of the points of continuity of the Weyl spectrum, the spectral radius, and the essential spectral radius are also obtained.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Apostol, C. Foias, and D. Voiculescu, "Some results on nonquasitriangular operators", II,Rev. Roum. Math. Pures et Appl., 18 (1973), 159–181.
C. Apostol, C. Foias, and D. Voiculescu, "Some results on nonquasitriangular operators", III,Rev. Roum. Math. Pures et Appl., 18, (1973), 309–324.
C. Apostol, C. Foias, and D. Voiculescu, "Some results on nonquasitriangular operators", IV,Rev. Roum. Math. Pures et Appl., 18 (1973), 487–514.
C. Apostol and B. Morrel, "On uniform approximation of operators by simple models, "Indiana University Math. J., 26 (1977), 427–442.
N. J. Bezak and M. Eisen, "Continuity properties of operator spectra",Canad. J. Math., 29 (1977), 429–437.
L. G. Brown, R. G. Douglas, and P. A. Fillmore,Unitary equivalence modulo the compact operators and extensions of C * -algebras, Proc. of a Conference on Operator Theory, Halifax, Springer-Verlag Lecture Notes in Mathematics, vol. 345 (1973).
R. G. Douglas and C. Pearcy, "A note on quasitriangular operators",Duke Math. J., 37 (1970), 177–188.
R. G. Douglas and C. Pearcy,Invariant subspaces of nonquasitriangular operators, Proc. of a conference on operator theory, Springer-Verlag Lecture Notes in Mathematics, vol. 345, pp. 13–57.
J. Dugundji,Topology, Allyn and Bacon, Inc., Boston (1966).
N. Dunford and J. Schwartz,Linear operators, Part 1, Interscience, New York (1958).
P. A. Fillmore, J. G. Stampfli and J. P. Williams, "On the essential numerical range, the essential spectrum, and a problem of Halmos",Acta Sci. Math. (Szeged), 33 (1972), 179–192.
L. Gillman and M. Jerison,Rings of Continuous Functions, D. Van Nostrand Co., Inc., Princeton (1960).
P. R. Halmos,A Hilbert Space Problem Book, D. Van Nostrand Co., Inc., Princeton (1967).
P. R. Halmos and G. Lumer, "Square roots of operators, II",Proc. Amer. Math. Soc., 5 (1954), 589–595.
D. A. Herrero, "On multicyclic operators",Integral Eq. and Operator Theory, 1 (1978), 57–102.
T. Kato,Perturbation Theory for Linear Operators, Springer-Verlag, New York (1966).
J. S. Lancaster, "Lifting from the Calkin algebra", Indiana University Ph.D. Dissertation, 1972.
J. D. Newburgh, "The variation of spectra",Duke Math. J., 18 (1951), 165–176.
M. Schechter, "Invariance of the essential spectrum",Bull. Amer. Math. Soc., 71 (1965), 365–367.
J. G. Stampfli, "Compact perturbations, normal eigenvalues, and a problem of Salinas",J. London Math. Soc., 9 (1974), 165–175.
Author information
Authors and Affiliations
Additional information
The first author was supported by National Science Foundation Grant MCS 77-28396.
Rights and permissions
About this article
Cite this article
Conway, J.B., Morrel, B.B. Operators that are points of spectral continuity. Integr equ oper theory 2, 174–198 (1979). https://doi.org/10.1007/BF01682733
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01682733