Examples of nonnormal seminormal operators whose spectra are not spectral sets
Author:
Kevin F. Clancey
Journal:
Proc. Amer. Math. Soc. 24 (1970), 797-800
MSC:
Primary 47.40
DOI:
https://doi.org/10.1090/S0002-9939-1970-0254643-X
MathSciNet review:
0254643
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: An example is given of a nonnormal seminormal operator on a Hilbert space whose spectrum is thin (in the sense of von Neumann) and is therefore not a spectral set. It is shown that every nonnormal subnormal operator is the limit of a sequence of hyponormal and nonsubnormal operators.
- S. K. Berberian, A note on operators whose spectrum is a spectral set, Acta Sci. Math. (Szeged) 27 (1966), 201–203. MR 203458
- Errett Bishop, Spectral theory for operators on a Banach space, Trans. Amer. Math. Soc. 86 (1957), 414–445. MR 100789, DOI https://doi.org/10.1090/S0002-9947-1957-0100789-4
- Tosio Kato, Smooth operators and commutators, Studia Math. 31 (1968), 535–546. MR 234314, DOI https://doi.org/10.4064/sm-31-5-535-546 M. Lavrentieff, Sur les fonctions d’une variable complexe représentables par des séries de polynomes, Actualités Sci. Indust., no. 441, Hermann, Paris, 1936.
- Arnold Lebow, On von Neumann’s theory of spectral sets, J. Math. Anal. Appl. 7 (1963), 64–90. MR 156220, DOI https://doi.org/10.1016/0022-247X%2863%2990078-7
- N. I. Muskhelishvili, Singular integral equations, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 0355494
- Johann von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281 (German). MR 43386, DOI https://doi.org/10.1002/mana.3210040124
- C. R. Putnam, Commutation properties of Hilbert space operators and related topics, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 36, Springer-Verlag New York, Inc., New York, 1967. MR 0217618
- Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528 J. G. Stampfli, On operators related to normal operators, Ph.D. Thesis, University of Michigan, Ann Arbor, Mich., 1959.
- John Wermer, Banach algebras and analytic functions, Advances in Math. 1 (1961), no. fasc. 1, 51–102. MR 143063, DOI https://doi.org/10.1016/0001-8708%2865%2990036-8
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47.40
Retrieve articles in all journals with MSC: 47.40
Additional Information
Keywords:
Seminormal operator,
hyponormal operator,
subnormal operator,
spectral set
Article copyright:
© Copyright 1970
American Mathematical Society