Strong variations of Weyl and Browder type theorems for direct sums and restrictions

  • J. SanabriaEmail author
  • L. Vásquez
  • C. Carpintero
  • E. Rosas
  • O. García


In this paper, we study the stability under direct sums and restrictions of some strong variations of Weyl and Browder type theorems recently introduced in Rashid and Prasad (Asia-Eur J Math 8:14, 2015., Sanabria et al. (Rev Colomb Mat 51(2):153–171, 2017a, Acta Math Univ Comen (NS) 86(2):345–356, 2017b, Open Math 16(1):289–297, 2018).


Semi-Fredholm operator Property \((V_{E})\) Direct sums Restrictions 

Mathematics Subject Classification

Primary 47A10 47A11 Secondary 47A53 47A55 



The authors thank the referee for his valuable comments and suggestions which have greatly contributed to this paper.


  1. 1.
    Aiena, P.: Fredholm and Local Spectral Theory, with Applications to Multipliers. Kluwer Academic Publishers, Dordrecht (2004)zbMATHGoogle Scholar
  2. 2.
    Aiena, P.: Quasi-Fredholm operators and localized SVEP. Acta Sci. Math. (Szeged) 73(1–2), 251–263 (2007)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Aiena, P., Aponte, E., Balzan, E.: Weyl type Theorems for left and right polaroid operators. Integr. Equ. Oper. Theory 66(1), 1–20 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arroud, A., Zariouh, H.: Browder-type theorems for direct sums of operators. Funct. Anal. Approx. Comput. 7(1), 77–84 (2015)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Berkani, M.: On a class of quasi-Fredholm operators. Integr. Equ. Oper. Theory 34(2), 244–249 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berkani, M.: Restriction of an operator to the range of its powers. Studia Math. 140(2), 163–175 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Berkani, M., Kachad, M., Zariouh, H.: Extended Weyl-type theorems for direct sums. Demonstr. Math. 47(2), 411–422 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Berkani, M., Zariouh, H.: Weyl type-theorems for direct sums. Bull. Korean Math. Soc. 49(5), 1027–1040 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Carpintero, C., García, O., Muñoz, D., Rosas, E., Sanabria, J.: Weyl type theorems for restrictions of bounded linear operators. Extracta Math. 28(1), 127–139 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Carpintero, C., García, O., Rosas, E., Sanabria, J.: \(B\)-Browder spectra and localized SVEP. Rend. Circ. Mat. Palermo 57(2), 241–255 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Carpintero, C., Muñoz, D., Rosas, E., Sanabria, J., García, O.: Weyl type theorems and restrictions. Mediterr. J. Math. 11(4), 1215–1228 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Carpintero, C., Rosas, E., Rodríguez, J., Muñoz, D., Alcalá, K.: Spectral properties and restrictions of bounded linear operators. Ann. Funct. Anal. 6(2), 173–183 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chen, L., Su, W.: A note on Weyl-type theorems and restrictions. Ann. Funct. Anal. 8(2), 190–198 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Duggal, B.P., Kubrusly, C.S.: Weyl’s theorem for direct sums. Studia Sci. Math. Hungar. 44(2), 275–290 (2007)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Finch, J.K.: The single valued extension property on a Banach space. Pac. J. Math. 58(1), 61–69 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gupta, A., Kashyap, N.: Generalized \(a\)-Weyl theorem for direct sums. Mat. Vesn. 62(4), 265–270 (2010)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Heuser, H.: Functional Analysis. Marcel Dekker, New York (1982)zbMATHGoogle Scholar
  18. 18.
    Liu, A.: A note on property \((W_{E})\). Acta Math. Hungar. (2017). zbMATHGoogle Scholar
  19. 19.
    Rashid, M.H.M., Prasad, T.: Property \((Sw)\) for bounded linear operators. Asia-Eur. J. Math. 8, 14 (2015). MathSciNetzbMATHGoogle Scholar
  20. 20.
    Sanabria, J., Carpintero, C., Rosas, E., García, O.: On generalized property \((v)\) for bounded linear operators. Studia Math. 212(2), 141–154 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sanabria, J., Carpintero, C., Rosas, E., García, O.: On property \((Saw)\) and others spectral properties type Weyl–Browder theorems. Rev. Colomb. Mat. 51(2), 153–171 (2017a)Google Scholar
  22. 22.
    Sanabria, J., Vásquez, L., Carpintero, C., Rosas, E., García, O.: On strong variations of Weyl type theorems. Acta Math. Univ. Comen. (N.S.) 86(2), 345–356 (2017b)Google Scholar
  23. 23.
    Sanabria, J., Carpintero, C., Rodríguez, J., Rosas, E., García, O.: On new strong versions of Browder type theorems. Open Math. 16(1), 289–297 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zariouh, H.: New version of property \((az)\). Mat. Vesn. 66(3), 317–322 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  • J. Sanabria
    • 1
    • 2
    Email author
  • L. Vásquez
    • 1
  • C. Carpintero
    • 1
    • 3
  • E. Rosas
    • 1
    • 4
  • O. García
    • 1
    • 5
  1. 1.Departamento de MatemáticasUniversidad de OrienteCumanáVenezuela
  2. 2.Facultad de Ciencias BásicasUniversidad del AtlánticoBarranquillaColombia
  3. 3.Vicerrectoría de InvestigaciónUniversidad Autónoma del CaribeBarranquillaColombia
  4. 4.Departamento de Ciencias Naturales y ExactasUniversidad de la CostaBarranquillaColombia
  5. 5.Departamento de Ciencias Básicas, Ingeniería y ArquitecturaCorporación Universitaria del Caribe-CECARSincelejoColombia

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