Acta Mathematica Sinica, English Series

, Volume 26, Issue 11, pp 2109–2116 | Cite as

On operators satisfying T*|T 2|TT*|T*|2 T

  • Jun Li ShenEmail author
  • Fei Zuo
  • Chang Sen Yang


Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class denoted by l-*-A, of operators satisfying T*|T 2|TT*|T*|2 T, and we prove the basic properties of these operators. Using these results, we also prove that if T or T* ∈ l-*-A, then w(f(T)) = f(w(T)), σ ea(f(T)) = f(σ ea(T)) for every fH(σ(T)), where H(σ(T)) denotes the set of all analytic functions on an open neighborhood of σ(T).


l-*-A point spectrum joint point spectrum α-Browder’s theorem 

MR(2000) Subject Classification

47B20 47A63 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Chō, M., Tanahashi, K.: Isolated point of spectrum of p-hyponormal, log-hyponormal operators. Integr. Equ. Oper. Theory, 43, 379–384 (2002)zbMATHCrossRefGoogle Scholar
  2. [2]
    Duggal, B. P.: On quasi-similar p-hyponormal operators. Integr. Equ. Oper. Theory, 26, 338–345 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Duggal, B. P.: Tensor products of operators-strong stability and p-hyponormality. Glasg. Math. J., 42, 371–381 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Yang, C., Ding, Y.: Binormal operator and *-Aluthge transformation. Acta Mathematica Sinica, English Series, 24(8), 1369–1378 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Yang, C., Li, H.: A note on p-w-Hyponormal operators. Acta Mathematica Sinica, Chinese Series, 49(1), 19–28 (2006)zbMATHMathSciNetGoogle Scholar
  6. [6]
    Yang, C., Shen, J.: Spectrume of Absolute-*-k-paranormal (0 ≤k≤ 1). Journal of Henan Normal University (Natural Science), 2, 152 (2008)Google Scholar
  7. [7]
    Jeon, I. H., Lee, J. I., Uchiyama, A.: On p-quasihyponormal operators and quasisimilarity. Math. Inequal. Appl., 6, 309–315 (2003)zbMATHMathSciNetGoogle Scholar
  8. [8]
    Tanahashi, K., Uchiyama. A.: Isolated point of spectrum of p-quasihyponormal operator. Linear Algebra Appl., 341, 345–350 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Uchiyama, A.: Inequalities of Putnam and Berger-Shaw for p-quasihyponormal operators. Integr. Equ. Oper. Theory, 34, 91–106 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Harte, R. E.: Invertibility and Singularity for Bounded Linear Operators Dekker, New York, 1988Google Scholar
  11. [11]
    Harte, R. E.: Fredholm, Weyl and Browder theory. Proc. Royal Irish Acad., 85A(2), 151–176 (1985)zbMATHMathSciNetGoogle Scholar
  12. [12]
    Laursen, K. B., Neumann, M. M.: An Introduction to Local Spectral Theory, London Mathematical Society Monographs New Series 20, Clarendon Press, Oxford 2000zbMATHGoogle Scholar
  13. [13]
    Hansen, F.: An equality. Math. Ann., 246, 249–250 (1980)zbMATHCrossRefGoogle Scholar
  14. [14]
    Han, J. K., Lee, H. Y., Lee, W. Y.: Invertible completions of 2 × 2 upper triangular operator matrices. Proc. Amer. Math. Soc., 128, 119–123 (1999)CrossRefMathSciNetGoogle Scholar
  15. [15]
    Han, Y. M., Lee, J. I., Wang, D.: Riesz idempotent and Weyl’s theorem for w-hyponormal operator. Integr. Equ. Oper. Theory, 53, 51–60 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Laursen, K. B.: Operators with finite ascent. Pacific J. Math., 152, 323–336 (1992)zbMATHMathSciNetGoogle Scholar
  17. [17]
    Aiena, P., Monsalve, O.: Operators which do not have the single valued extension property. J. Math. Anal. Appl., 250, 435–448 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Rakočević, V.: Approximate point spectrum and commuting compact perturbations. Glasg. Math. J., 28, 193–198 (1986)zbMATHCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of MathematicsXinxiang UniversityXinxiangP. R. China
  2. 2.College of Mathematics and Information ScienceHe’nan Normal UniversityXinxiangP. R. China

Personalised recommendations