Acta Mathematica Vietnamica

, Volume 42, Issue 4, pp 747–759 | Cite as

Property (a B w) and Weyl Type Theorems

  • Mohammad H. M. RashidEmail author


In this paper, we introduces the property (a B w), a variant of generalized a-Weyl’s theorem for a bounded linear operator T on an infinite-dimensional Banach space \(\mathbb {X}\). We establish several sufficient and necessary conditions for which property (a B w) holds. Also, we prove that if \(T\in \mathbf {L(\mathbb {X})}\) satisfies property (a B w) then T satisfies property (B w). Certain conditions are explored on Hilbert space operators T and S so that TS obeys property (a B w).


Generalized Weyl’s theorem Generalized a-Weyl’s theorem Polaroid operators SVEP Property (aBw

Mathematics Subject Classification (2010)

47A53 47A55 47A10 47A11 47A20 



The author would like to express their sincere appreciation to the referees for their very helpful suggestions and many kind comments.


  1. 1.
    Aiena, P., Monsalve, O.: The single valued extension property and the generalized Kato decomposition property. Acta Sci. Math. (Szeged) 67, 461–477 (2001)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aiena, P., Colasante, M.L., Gonzalez, M.: Operators which have a closed quasi-nilpotent part. Proc. Am. Math. Soc. 130(9), 2701–2710 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aiena, P.: Fredholm and Local Spectral Theory with Applications to Multipliers. Kluwer Acadmic Publishing, Dordrecht (2004)zbMATHGoogle Scholar
  4. 4.
    Aiena, P., Carpintero, C.: Weyl’s theorem, a-Weyl’s theorem and single-valued extension property. Extracta Math. 20, 25–41 (2005)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Aiena, P., Biondi, M.T.: Property (w) and perturbations. J. Math. Anal. Appl. 336, 683–692 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Aiena, P., Biondi, M.T., Villafañe, F.: Property (w) and perturbations III. J. Math. Anal. Appl. 353, 205–214 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Amouch, M.: Generalized a-Weyl’s theorem and the single-valued extension property. Extracta Math. 21(1), 51–65 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Amouch, M., Zguitti, H.: On the equivalence of Browder’s and generalized Browder’s theorem. Glasg. Math. J. 48, 179–185 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berkani, M.: On a class of quasi-Fredholm operators. Integr. Equ. Oper. Theory 34(2), 244–249 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Berkani, M.: Index of B-Fredholm operators and generalization of a Weyl theorem. Proc. Am. Math. Soc. 130, 1717–1723 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Berkani, M.: B-Weyl spectrum and poles of the resolvent. J. Math. Anal. Appl. 272, 596–603 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Berkani, M., Koliha, J.: Weyl type theorems for bounded linear operators. Acta Sci. Math. (Szeged) 69(1-2), 359–376 (2003)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Berkani, M., Arroud, A.: Generalized Weyl’s theorem and hyponormal operators. J. Austral. Math. Soc. 76, 1–12 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Berkani, M.: On the equivalence of Weyl theorem and generalized Weyl theorem. Acta Math. Sinica 272(1), 103–110 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Berkani, M., Zariouh, H.: New extended Weyl type theorems. Math. Bohem. 62(2), 145–154 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Coburn, L.A.: Weyl’s theorem for nonnormal operators. Mich. Math. J. 13, 285–288 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Curto, R.E., Han, Y.M.: Weyl’s theorem for algebraically paranormal operators. Integr. Equ. Oper. Theory 47, 307–314 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Duggal, B.P.: SVEP and generalized Weyls theorem. Mediterr. J. Math. 4, 309–320 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Duggal, B.P.: Perturbations of operators satisfying a local growth condition. Extracta Math. 23(1), 29–42 (2008)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Duggal, B.P., Djordjevic, S.V.: Generalized Weyl’s theorem for a class of operators satisfying a norm condition II. Math. Proc. Royal Irish Acad. 104A, 1–9 (2006)CrossRefzbMATHGoogle Scholar
  21. 21.
    Dunford, N., Schwartz, J.T.: Linear Operators, Parts I and III. Inter-science, New York (1964, 1971)Google Scholar
  22. 22.
    Finch, J.K.: The single valued extension property on a Banach space. Pac. J. Math. 58, 61–69 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gupta, A., Kashyap, N.: Property (B w) and Weyl type theorems. Bullet. Math. Anal. Appl. 3(1), 1–7 (2011)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Heuser, H.: Functional Analysis. Marcel Dekker, New York (1982)zbMATHGoogle Scholar
  25. 25.
    Jafarian, A.A., Radjabalipour, M.: Transitive algebra problem and local resolvent techniques. J. Oper. Theory 1, 273–285 (1979)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Laursen, K.B., Neumann, M.M.: An Introduction to Local Spectral Theory. Clarendon, Oxford (2000)zbMATHGoogle Scholar
  27. 27.
    Lahrouz, M., Zohry, M.: Weyl type theorems and the approximate point spectrum. Irish Math. Soc. Bullet. 55, 41–51 (2005)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mbekhta, M.: Sur la théoric spectrale locale et limite de nilpotents. Proc. Am. Math. Soc. 3, 621–631 (1990)zbMATHGoogle Scholar
  29. 29.
    Rakoc~ević, V.: Operators obeying a-Weyl’s theorem. Rev. Roumaine Math. Pures Appl. 10, 915–919 (1986)MathSciNetGoogle Scholar
  30. 30.
    Stampfli, J.G.: A local spectral theory for operators: Spectral subspaces for hyponormal operators. Trans. Am. Math. Soc. 217, 359–365 (1976)MathSciNetzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of ScienceMu’tah UniversityAl-karakJordan

Personalised recommendations