Skip to main content
Journal of Inequalities and Applications
Download PDF

Spectrum of Class Operators

Journal of Inequalities and Applications volume 2007, Article number: 027195 (2007) Cite this article

Abstract

This paper discusses some spectral properties of class operators for,,, and. It is shown that if is a class operator, then the Riesz idempotent of with respect to each nonzero isolated point spectrum is selfadjoint and. Afterwards, we prove that every class operator has SVEP and property, and Weyl's theorem holds for when.

[123456789101112131415161718192021222324252627282930313233]

References

  1. Martin M, Putinar M: Lectures on Hyponormal Operators, Operator Theory: Advances and Applications. Volume 39. Birkhäuser, Basel, Switzerland; 1989:304.

    Book  MATH  Google Scholar 

  2. Xia D: Spectral Theory of Hyponormal Operators, Operator Theory: Advances and Applications. Volume 10. Birkhäuser, Basel, Switzerland; 1983:xiv+241.

    Book  Google Scholar 

  3. Tanahashi K: On log-hyponormal operators. Integral Equations and Operator Theory 1999,34(3):364–372. 10.1007/BF01300584

    Article  MathSciNet  MATH  Google Scholar 

  4. Aluthge A, Wang D: -hyponormal operators. Integral Equations and Operator Theory 2000,36(1):1–10. 10.1007/BF01236285

    Article  MathSciNet  MATH  Google Scholar 

  5. Fujii M, Himeji C, Matsumoto A: Theorems of Ando and Saito for-hyponormal operators. Mathematica Japonica 1994,39(3):595–598.

    MathSciNet  MATH  Google Scholar 

  6. Aluthge A: On-hyponormal operators for. Integral Equations and Operator Theory 1990,13(3):307–315. 10.1007/BF01199886

    Article  MathSciNet  MATH  Google Scholar 

  7. Ito M: Some classes of operators associated with generalized Aluthge transformation. SUT Journal of Mathematics 1999,35(1):149–165.

    MathSciNet  MATH  Google Scholar 

  8. Yang C, Yuan J: Spectrum of classoperators for and. Acta Scientiarum Mathematicarum 2005,71(3–4):767–779.

    MathSciNet  MATH  Google Scholar 

  9. Yang C, Yuan J: On class operators. Acta Mathematica Scientia. Series A. (Chinese Edition) 2007,27(5):769–780.

    MathSciNet  MATH  Google Scholar 

  10. Ito M, Yamazaki T: Relations between two inequalities andand their applications. Integral Equations and Operator Theory 2002,44(4):442–450. 10.1007/BF01193670

    Article  MathSciNet  MATH  Google Scholar 

  11. Fujii M, Jung D, Lee SH, Lee MY, Nakamoto R: Some classes of operators related to paranormal and log-hyponormal operators. Mathematica Japonica 2000,51(3):395–402.

    MathSciNet  MATH  Google Scholar 

  12. Furuta T, Ito M, Yamazaki T: A subclass of paranormal operators including class of log-hyponormal and several related classes. Scientiae Mathematicae 1998,1(3):389–403.

    MathSciNet  MATH  Google Scholar 

  13. Fujii M, Nakamoto R: Some classes of operators derived from Furuta inequality. Scientiae Mathematicae 2000,3(1):87–94.

    MathSciNet  MATH  Google Scholar 

  14. Aluthge A, Wang D: The joint approximate point spectrum of an operator. Hokkaido Mathematical Journal 2002,31(1):187–197.

    Article  MathSciNet  MATH  Google Scholar 

  15. Chō M, Tanahashi K: Isolated point of spectrum of-hyponormal, log-hyponormal operators. Integral Equations and Operator Theory 2002,43(4):379–384. 10.1007/BF01212700

    Article  MathSciNet  MATH  Google Scholar 

  16. Chō M, Yamazaki T: An operator transform from class to the class of hyponormal operators and its application. Integral Equations and Operator Theory 2005,53(4):497–508. 10.1007/s00020-004-1332-6

    Article  MathSciNet  MATH  Google Scholar 

  17. Han YM, Lee JI, Wang D: Riesz idempotent and Weyl's theorem for -hyponormal operators. Integral Equations and Operator Theory 2005,53(1):51–60. 10.1007/s00020-003-1313-1

    Article  MathSciNet  MATH  Google Scholar 

  18. Kimura F: Analysis of non-normal operators via Aluthge transformation. Integral Equations and Operator Theory 2004,50(3):375–384. 10.1007/s00020-003-1231-2

    Article  MathSciNet  MATH  Google Scholar 

  19. Tanahashi K, Uchiyama A: Isolated point of spectrum of -quasihyponormal operators. Linear Algebra and Its Applications 2002,341(1–3):345–350.

    Article  MathSciNet  MATH  Google Scholar 

  20. Uchiyama A: On the isolated points of the spectrum of paranormal operators. Integral Equations and Operator Theory 2006,55(1):145–151. 10.1007/s00020-005-1386-0

    Article  MathSciNet  MATH  Google Scholar 

  21. Uchiyama A, Tanahashi K: On the Riesz idempotent of class operators. Mathematical Inequalities & Applications 2002,5(2):291–298.

    Article  MathSciNet  MATH  Google Scholar 

  22. Uchiyama A, Tanahashi K, Lee JI: Spectrum of class operators. Acta Scientiarum Mathematicarum 2004,70(1–2):279–287.

    MathSciNet  MATH  Google Scholar 

  23. Tanahashi K: Putnam's inequality for log-hyponormal operators. Integral Equations and Operator Theory 2004,48(1):103–114. 10.1007/s00020-999-1172-5

    Article  MathSciNet  MATH  Google Scholar 

  24. Stampfli JG: Hyponormal operators. Pacific Journal of Mathematics 1962,12(4):1453–1458.

    Article  MathSciNet  MATH  Google Scholar 

  25. Huruya T: A note on -hyponormal operators. Proceedings of the American Mathematical Society 1997,125(12):3617–3624. 10.1090/S0002-9939-97-04004-5

    Article  MathSciNet  MATH  Google Scholar 

  26. Chō M, Itoh M: Putnam's inequality for -hyponormal operators. Proceedings of the American Mathematical Society 1995,123(8):2435–2440.

    MathSciNet  MATH  Google Scholar 

  27. Riesz F, Sz.-Nagy B: Functional Analysis. Frederick Ungar, New York, NY, USA; 1955:xii+468.

    MATH  Google Scholar 

  28. Conway JB: A Course in Functional Analysis, Graduate Texts in Mathematics. Volume 96. 2nd edition. Springer, New York, NY, USA; 1990:xvi+399.

    MATH  Google Scholar 

  29. Finch JK: The single valued extension property on a Banach space. Pacific Journal of Mathematics 1975,58(1):61–69.

    Article  MathSciNet  MATH  Google Scholar 

  30. Bishop E: A duality theorem for an arbitrary operator. Pacific Journal of Mathematics 1959,9(2):379–397.

    Article  MathSciNet  MATH  Google Scholar 

  31. Coburn LA: Weyl's theorem for nonnormal operators. Michigan Mathematical Journal 1966,13(3):285–288.

    Article  MathSciNet  MATH  Google Scholar 

  32. Kim IH: On-quasihyponormal operators. Mathematical Inequalities & Applications 2004,7(4):629–638.

    Article  MathSciNet  MATH  Google Scholar 

  33. Lee WY, Lee SH: A spectral mapping theorem for the Weyl spectrum. Glasgow Mathematical Journal 1996,38(1):61–64. 10.1017/S0017089500031268

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LMIB and Department of Mathematics, Beihang University, Beijing, 100083, China

    Jiangtao Yuan & Zongsheng Gao

  2. Dedicated to Professor Daoxing Xia on his 77th birthday with respect and affection, China

    Zongsheng Gao

Authors
  1. Jiangtao Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zongsheng Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiangtao Yuan.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Yuan, J., Gao, Z. Spectrum of Class Operators. J Inequal Appl 2007, 027195 (2007). https://doi.org/10.1155/2007/27195

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/27195

Keywords

  • Spectral Property
  • Class Operator
  • Point Spectrum