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Property (R) under Perturbations

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Abstract

Property (R) holds for a bounded linear operator \({T \in L(X)}\), defined on a complex infinite dimensional Banach space X, if the isolated points of the spectrum of T which are eigenvalues of finite multiplicity are exactly those points λ of the approximate point spectrum for which λI − T is upper semi-Browder. In this paper we consider the permanence of this property under quasi nilpotent, Riesz, or algebraic perturbations commuting with T.

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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  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Aiena P.: Property (w) and perturbations II, J. Math. Anal. and Appl. 342, 830–837 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aiena P., Aponte E., Bazan E.: Weyl type theorems for left and right polaroid operators. Integral Equations Oper. Theory 66, 1–20 (2010)

    Article  MATH  Google Scholar 

  6. P. Aiena and E. Aponte, Polaroid type operators under perturbations, (2010), preprint.

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Aiena, M. Chō and M. González, Polaroid type operator under quasiaffinities, J. Math. Anal. Appl. 371 (2010), no.2, 485–495.

    Google Scholar 

  11. Aiena P., Guillen J.R.: Weyl’s theorem for perturbations of paranormal operators. Proc. Amer. Math. Soc 35, 2433–2451 (2007)

    MathSciNet  Google Scholar 

  12. P. Aiena, J. Guillen and P. Peña, Property (R) for bounded linear operators, (2010), To appear in Mediterr. J. Math.

  13. Aiena P., Guillen J., Peña P.: Property (w) for perturbations of polaroid operators. Linear Algebra and its Appl 428, 1791–1802 (2008)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Aiena P., Sanabria J.E.: On left and right Drazin invertibility. Acta Sci. Math. (Szeged) 74, 669–687 (2008)

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Coburn L.A.: Weyl’s theorem for nonnormal operators. Michigan Math. J 20, 529–544 (1970)

    Google Scholar 

  19. Curto R.E., Han Y.M.: Weyl’s theorem, a-Weyl’s theorem, and local spectral theory. J. London Math. Soc 2(67), 499–509 (2003)

    Article  MathSciNet  Google Scholar 

  20. D. S. Djordjević, Operators obeying a-Weyl’s theorem, Publicationes Math. Debrecen 55, 3-4, no. 3 (1999), 283–298.

  21. Duggal B.P.: Hereditarily polaroid operators, SVEP and Weyl’s theorem. J. Math. Anal. Appl 340, 366–373 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Finch J.K.: The single valued extension property on a Banach space, Pacific J. Math. 58, 61–69 (1975)

    MathSciNet  MATH  Google Scholar 

  23. Harte R., Young Lee Woo: Another note on Weyl’s theorem. Trans. Amer. Math. Soc 349, 2115–2124 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Oudghiri M.: Weyl’s and perturbations. Integral Equa. Oper. Theory 53(4), 535–545 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Oudghiri M.: a-Weyl’s and perturbations. Studia Math 173(2), 193–201 (2004)

    Article  MathSciNet  Google Scholar 

  29. Rakočević V.: On a class of operators. Mat. Vesnik 37, 423–426 (1985)

    MathSciNet  MATH  Google Scholar 

  30. Rakočević V.: On a class of operators II. Radovi Matematicki 2, 75–80 (1986)

    MathSciNet  MATH  Google Scholar 

  31. V. Rakočević, Operators obeying a-Weyl’s theorem, Rev. Roumaine Math. Pures Appl. 34 (1989), no. 10, 915–919.

  32. Rakočević V.: Semi-Browder’s operators and perturbations. Studia Math 122, 131–137 (1997)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Metodi e Modelli Matematici, Facoltà di Ingegneria, Università di Palermo, Via delle Scienze, Ed. 8, 90128, Palermo, Italy

    Pietro Aiena

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

    Elvis Aponte

  3. Departamento de Matemáticas, Facultad de Ciencias, ULA, Merida, Venezuela

    Jesús R. Guillén

  4. Departamento de Física y Matemáticas, NURR, ULA, Trujillo, Venezuela

    Pedro Peña

Authors
  1. Pietro Aiena
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  2. Elvis Aponte
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  3. Jesús R. Guillén
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  4. Pedro Peña
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Corresponding author

Correspondence to Pietro Aiena.

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Aiena, P., Aponte, E., Guillén, J.R. et al. Property (R) under Perturbations. Mediterr. J. Math. 10, 367–382 (2013). https://doi.org/10.1007/s00009-012-0174-8

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  • DOI: https://doi.org/10.1007/s00009-012-0174-8

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