Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

OPERATORS SIMILAR TO NORMALOID OPERATORS

  • Zhu, Sen (Department of Mathematics Jilin University) ;
  • Li, Chun Guang (Institute of Mathematics Jilin University)
  • Received : 2010.07.07
  • Published : 2011.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, the authors investigate the structure of operators similar to normaloid and transloid operators. In particular, we characterize the interior of the set of operators similar to normaloid (transloid, respectively) operators. This gives a concise spectral condition to determine when an operator is similar to a normaloid or transloid operator. Also it is proved that any Hilbert space operator has a compact perturbation with transloid property. This is used to give a negative answer to a problem posed by W. Y. Lee, concerning Weyl's theorem.

Keywords

References

  1. C. Apostol, L. A. Fialkow, D. A. Herrero, and D. Voiculescu, Approximation of Hilbert space operators. Vol. II, Research Notes in Mathematics, 102. Pitman (Advanced Pub-lishing Program), Boston, MA, 1984.
  2. C. Apostol and B. B. Morrel, On uniform approximation of operators by simple models, Indiana Univ. Math. J. 26 (1977), no. 3, 427-442. https://doi.org/10.1512/iumj.1977.26.26033
  3. J. B. Conway, A Course in Functional Analysis, Second edition.Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1990.
  4. J. B. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs, 36. American Mathematical Society, Providence, RI, 1991.
  5. F. Gilfeather, Norm conditions on resolvents of similarities of Hilbert space operators and applications to direct sums and integrals of operators, Proc. Amer. Math. Soc. 68 (1978), no. 1, 44-48. https://doi.org/10.1090/S0002-9939-1978-0475583-4
  6. P. R. Halmos, A Hilbert Space Problem Book, Second edition.Graduate Texts in Mathe-matics, 19. Encyclopedia of Mathematics and its Applications, 17. Springer-Verlag, New York-Berlin, 1982.
  7. D. A. Herrero, A Rota universal model for operators with multiply connected spectrum, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 1, 15-23.
  8. D. A. Herrero, Approximation of Hilbert Space Operators. Vol. 1, Second edition.Pitman Research Notes in Mathematics Series, 224. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.
  9. C. L. Jiang and Z. Y. Wang, Structure of Hilbert Space Operators, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
  10. W. Y. Lee, Lecture Notes on Operator Theory, Seoul National University, Seoul, 2010.
  11. H. Radjavi and P. Rosenthal, Invariant Subspaces, Second edition.Dover Publications, Inc., Mineola, NY, 2003.
  12. G.-C. Rota, On models for linear operators, Comm. Pure Appl. Math. 13 (1960), 469-472. https://doi.org/10.1002/cpa.3160130309
  13. J. von Neumann, Eine Spektraltheorie fur allgemeine Operatoren eines unitaren Raumes, Math. Nachr. 4 (1951), 258-281.
  14. A. Wintner, Zur theorie der beschrankten bilinearformen, Math. Z. 30 (1929), no. 1, 228-281.