Skip to main content
SpringerLink
Log in

Weyl Type Theorems for Left and Right Polaroid Operators

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Aiena Fredholm and local spectral theory, with application to multipliers. Kluwer Acad. Publishers (2004).

  2. Aiena P.: Classes of Operators Satisfying a-Weyl’s theorem. Studia Math. 169, 105–122 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. P. Aiena Quasi Fredholm operators and localized SVEP, Acta Sci. Math. (Szeged), 73 (2007), 251-263.

    Google Scholar 

  4. Aiena P., Biondi M.T., Villafãne F.: Property (w) and perturbations III. J. Math. Anal. Appl. 353, 205–214 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Aiena P., Biondi M.T., Carpintero C.: On Drazin invertibility. Proc. Amer. Math. Soc. 136, 2839–2848 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Aiena P., Carpintero C., Rosas E.: Some characterization of operators satisfying a-Browder theorem. J. Math. Anal. Appl. 311, 530–544 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Aiena P., Guillen J.R.: Weyl’s theorem for perturbations of paranormal operators. Proc. Amer. Math. Soc. 35, 2433–2442 (2007)

    MathSciNet  Google Scholar 

  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. Aiena P., Miller T.L.: On generalized a-Browder’s theorem. Studia Math. 180((3), 285300 (2007)

    MathSciNet  Google Scholar 

  10. Aiena P., Peña P.: A variation on Weyl’s theorem. J. Math. Anal. Appl. 324, 566–579 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Aiena P., Sanabria J.E.: On left and right poles of the resolvent. Acta Sci. Math. 74, 669–687 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Aiena P., Villafãne F.: Weyl’s theorem for some classes of operators. Int. Equa. Oper. Theory 53, 453–466 (2005)

    Article  MATH  Google Scholar 

  13. An J., Han Y.M.: Weyl’s theorem for algebraically Quasi-class A operators. Int. Equa. Oper. Theory 62, 1–10 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Amouch M., Zguitti H.: On the equivalence of Browder’s and generalized Browder’s theorem. Glasgow Math. Jour. 48, 179–185 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Berkani M., Amouch M.: On the property (gw). Mediterr. J. Math. 5(3), 371–378 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Berkani M.: On a class of quasi-Fredholm operators. Int. Equa. Oper. Theory 34(1), 244–249 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Berkani M., Sarih M.: On semi B-Fredholm operators. Glasgow Math. J. 43, 457–465 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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.

  19. M. Berkani, J. J. Koliha Weyl type theorems for bounded linear operators, Acta Sci. Math. (Szeged) 69 (2003), no. 1-2, 359–376.

  20. Cao X., Guo M., Meng B.: Weyl type theorems for p-hyponormal and Mhyponormal operators. Studia Math. 163(2), 177–187 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Cao X.: Weyl’s type theorem for analytically hyponormal operators. Linear Alg. and Appl. 405, 229–238 (2005)

    Article  MATH  Google Scholar 

  22. Cao X.: Topological uniform descent and Weyl’s type theorem. Linear Alg. and Appl. 420, 175–182 (2007)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Drazin M.P.: Pseudoinverse in associative rings and semigroups. Amer. Math. Monthly 65, 506–514 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  27. Grabiner S.: Uniform ascent and descent of bounded operators. J. Math. Soc. Japan 34, 317–337 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  28. Heuser H.: Functional Analysis. Marcel Dekker, New York (1982)

    MATH  Google Scholar 

  29. Koliha J.J.: Isolated spectral points Proc. Amer. Math. Soc. 124, 3417–3424 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  30. J.P. Labrousse Les opérateurs quasi-Fredholm., Rend. Circ. Mat. Palermo, XXIX 2, (1980)

  31. Lay D.C.: Spectral analysis using ascent, descent, nullity and defect. Math. Ann. 184, 197–214 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  32. Laursen K.B., Neumann M.M.: Introduction to local spectral theory. Clarendon Press, Oxford (2000)

    MATH  Google Scholar 

  33. Mbekhta M., Müller V.: On the axiomatic theory of the spectrum II. Studia Math. 119, 129–147 (1996)

    MATH  MathSciNet  Google Scholar 

  34. Oudghiri M.: Weyl’s and Browder’s theorem for operators satisfying the SVEP. Studia Math. 163(1), 85–101 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  35. Zguitti H.: A note on generalized Weyl’s theorem. J. Math. Anal. Appl. 316(1), 373–381 (2006)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Metodi e Modelli Matematici, Facoltà di Ingegneria, Università di Palermo, 90128, Palermo, Italy

    Pietro Aiena

  2. Departamento de Matemáticas, Facultád de Ciencias UCLA, Barquisimeto, Venezuela

    Elvis Aponte

  3. Departamento de Matemáticas, Facultád de Ciencias, Universidad del Zulia, Maracaibo, Venezuela

    Edixon Balzan

Authors
  1. Pietro Aiena
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elvis Aponte
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edixon Balzan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pietro Aiena.

Additional information

The first author was supported by ex-60 2008, Fondi Universitá di Palermo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aiena, P., Aponte, E. & Balzan, E. Weyl Type Theorems for Left and Right Polaroid Operators. Integr. Equ. Oper. Theory 66, 1–20 (2010). https://doi.org/10.1007/s00020-009-1738-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-009-1738-2

Mathematics Subject Classification (2010)

Keywords

Search

Navigation