Integral Equations and Operator Theory

, Volume 66, Issue 1, pp 1–20 | Cite as

Weyl Type Theorems for Left and Right Polaroid Operators

  • Pietro AienaEmail author
  • Elvis Aponte
  • Edixon Balzan


A bounded operator defined on a Banach space is said to be polaroid if every isolated point of the spectrum is a pole of the resolvent. In this paper we consider the two related notions of left and right polaroid, and explore them together with the condition of being a-polaroid. Moreover, the equivalences of Weyl type theorems and generalized Weyl type theorems are investigated for left and a-polaroid operators. As a consequence, we obtain a general framework which allows us to derive in a unified way many recent results, concerning Weyl type theorems (generalized or not) for important classes of operators.

Mathematics Subject Classification (2010)

Primary 47A10 47A11 Secondary 47A53 47A55 


Localized SVEP semi B-Brower operators left and right Drazin invertibility Weyl’s theorem property (w


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. Aiena Fredholm and local spectral theory, with application to multipliers. Kluwer Acad. Publishers (2004).Google Scholar
  2. 2.
    Aiena P.: Classes of Operators Satisfying a-Weyl’s theorem. Studia Math. 169, 105–122 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    P. Aiena Quasi Fredholm operators and localized SVEP, Acta Sci. Math. (Szeged), 73 (2007), 251-263.Google Scholar
  4. 4.
    Aiena P., Biondi M.T., Villafãne F.: Property (w) and perturbations III. J. Math. Anal. Appl. 353, 205–214 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Aiena P., Biondi M.T., Carpintero C.: On Drazin invertibility. Proc. Amer. Math. Soc. 136, 2839–2848 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Aiena P., Carpintero C., Rosas E.: Some characterization of operators satisfying a-Browder theorem. J. Math. Anal. Appl. 311, 530–544 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Aiena P., Guillen J.R.: Weyl’s theorem for perturbations of paranormal operators. Proc. Amer. Math. Soc. 35, 2433–2442 (2007)MathSciNetGoogle Scholar
  8. 8.
    Aiena P., Guillen J., Peña P.: Property (w) for perturbation of polaroid operators. Linear Alg. and Appl. 424, 1791–1802 (2008)Google Scholar
  9. 9.
    Aiena P., Miller T.L.: On generalized a-Browder’s theorem. Studia Math. 180((3), 285300 (2007)MathSciNetGoogle Scholar
  10. 10.
    Aiena P., Peña P.: A variation on Weyl’s theorem. J. Math. Anal. Appl. 324, 566–579 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Aiena P., Sanabria J.E.: On left and right poles of the resolvent. Acta Sci. Math. 74, 669–687 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Aiena P., Villafãne F.: Weyl’s theorem for some classes of operators. Int. Equa. Oper. Theory 53, 453–466 (2005)zbMATHCrossRefGoogle Scholar
  13. 13.
    An J., Han Y.M.: Weyl’s theorem for algebraically Quasi-class A operators. Int. Equa. Oper. Theory 62, 1–10 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Amouch M., Zguitti H.: On the equivalence of Browder’s and generalized Browder’s theorem. Glasgow Math. Jour. 48, 179–185 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Berkani M., Amouch M.: On the property (gw). Mediterr. J. Math. 5(3), 371–378 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Berkani M.: On a class of quasi-Fredholm operators. Int. Equa. Oper. Theory 34(1), 244–249 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Berkani M., Sarih M.: On semi B-Fredholm operators. Glasgow Math. J. 43, 457–465 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    M. Berkani Index of B-Fredholm operators and generalization of a Weyl’s theorem, Proc. Amer. Math. Soc., vol. 130, 6, (2001), 1717-1723.Google Scholar
  19. 19.
    M. Berkani, J. J. Koliha Weyl type theorems for bounded linear operators, Acta Sci. Math. (Szeged) 69 (2003), no. 1-2, 359–376.Google Scholar
  20. 20.
    Cao X., Guo M., Meng B.: Weyl type theorems for p-hyponormal and Mhyponormal operators. Studia Math. 163(2), 177–187 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Cao X.: Weyl’s type theorem for analytically hyponormal operators. Linear Alg. and Appl. 405, 229–238 (2005)zbMATHCrossRefGoogle Scholar
  22. 22.
    Cao X.: Topological uniform descent and Weyl’s type theorem. Linear Alg. and Appl. 420, 175–182 (2007)zbMATHCrossRefGoogle Scholar
  23. 23.
    Curto R.E., Han Y.M.: Generalized Browder’s and Weyl’s theorems for Banach spaces operators. J. Math. Anal. Appl. 2, 1424–1442 (2007)CrossRefMathSciNetGoogle Scholar
  24. 24.
    R. E. Curto, Y. M. Han Weyl’s theorem for algebraically paranormal operators, Integ. Equa. Oper. Theory 50, (2004), No.2, 169-196.Google Scholar
  25. 25.
    Drazin M.P.: Pseudoinverse in associative rings and semigroups. Amer. Math. Monthly 65, 506–514 (1958)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Duggal B.P., Jeon I.H., Kim I.H.: On Weyl’s theorem for quasi-class A operators. J. Korean Math. Soc. 43, 899–909 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Grabiner S.: Uniform ascent and descent of bounded operators. J. Math. Soc. Japan 34, 317–337 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Heuser H.: Functional Analysis. Marcel Dekker, New York (1982)zbMATHGoogle Scholar
  29. 29.
    Koliha J.J.: Isolated spectral points Proc. Amer. Math. Soc. 124, 3417–3424 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    J.P. Labrousse Les opérateurs quasi-Fredholm., Rend. Circ. Mat. Palermo, XXIX 2, (1980)Google Scholar
  31. 31.
    Lay D.C.: Spectral analysis using ascent, descent, nullity and defect. Math. Ann. 184, 197–214 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Laursen K.B., Neumann M.M.: Introduction to local spectral theory. Clarendon Press, Oxford (2000)zbMATHGoogle Scholar
  33. 33.
    Mbekhta M., Müller V.: On the axiomatic theory of the spectrum II. Studia Math. 119, 129–147 (1996)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Oudghiri M.: Weyl’s and Browder’s theorem for operators satisfying the SVEP. Studia Math. 163(1), 85–101 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Zguitti H.: A note on generalized Weyl’s theorem. J. Math. Anal. Appl. 316(1), 373–381 (2006)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  1. 1.Dipartimento di Metodi e Modelli Matematici, Facoltà di IngegneriaUniversità di PalermoPalermoItaly
  2. 2.Departamento de MatemáticasFacultád de Ciencias UCLABarquisimetoVenezuela
  3. 3.Departamento de Matemáticas, Facultád de CienciasUniversidad del ZuliaMaracaiboVenezuela

Personalised recommendations