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New Classes of Operators Admitting a Perturbative Characterization

  • Manuel González
  • Antonio Martínez-AbejónEmail author
Article
  • 24 Downloads

Abstract

We study the semigroups \(({\mathcal {C}}_p)_+\), \(({\mathcal {C}}_p^{\text {dual}})_-\), \({\mathcal {G}}_-\) and \({\mathcal {H}}_-\) associated with the operator ideals \({\mathcal {C}}_p\) of p-converging operators (\(1<p<\infty \)), \({\mathcal {G}}\) of Grothendieck operators, and \({\mathcal {H}}\) of \(c_0\)-cosingular operators. We show that these semigroups admit a perturbative characterization and satisfy the 3-operator property.

Keywords

Three-operator property perturbative characterization semigroup operator ideal 

Mathematics Subject Classification

Primary 46B03 Secondary 46B10 

Notes

References

  1. 1.
    Aiena, P., González, M., Martínez-Abejón, A.: Operator semigroups in Banach space theory. Boll. Un. Mat. Ital. 8(4-B), 157–205 (2001)Google Scholar
  2. 2.
    Albiac, F., Kalton, N.J.: Topics in Banach Space Theory. Graduate Texts in Mathematics, vol. 233. Springer, Berlin (2006)zbMATHGoogle Scholar
  3. 3.
    Castillo, J.M.F.: \(p\)-Converging and weakly-p-compact operators in \(L_p\)-spaces. Extracta Mathematicae, vol. especial. II Congress on Functional Analysis in Jarandilla, pp. 46–54 (1990)Google Scholar
  4. 4.
    Castillo, J.M.F.: On Banach spaces \(X\) such that \(L(X, L_p)=K(X, L_p)\). Extr. Math. 10, 27–36 (1995)zbMATHGoogle Scholar
  5. 5.
    Castillo, J.M.F., Chō, M., González, M.: Three-operator problems in Banach spaces. Extr. Math. 33, 149–165 (2018)Google Scholar
  6. 6.
    Castillo, J.M.F., González, M.: Three-Space Problems in Banach Space Theory. Lecture Notes in Mathematics, vol. 1667. Springer, Berlin (1997)CrossRefGoogle Scholar
  7. 7.
    Castillo, J.M.F., González, M., Martínez-Abejón, A.: Classes of operators preserved by extensions or liftings. J. Math. Anal. Appl. 462, 471–482 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Castillo, J.M.F., López Molina, J.A.: Operators defined on projective and natural tensor products. Mich. J. Math. 40, 411–415 (1993)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Castillo, J.M.F., Sánchez, F.: Dunford-Pettis-like properties of continuous vector function spaces. Rev. Mat. Univ. Complut. Madrid 6, 43–59 (1993)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen, D., Chávez-Domínguez, A., Li, L.: \(p\)-converging operators and Dunford–Pettis property of order \(p\). J. Math. Anal. Appl. 461, 1053–1066 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ghenciu, I.: The \(p\)-Gelfand–Phillips property in spaces of operators and Dunford–Pettis like sets. Acta Math. Hung. 155, 439–457 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    González, M., Martínez-Abejón, A.: Lifting unconditionally converging series and semigroups of operators. Bull. Aust. Math. Soc. 57, 135–145 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    González, M., Martínez-Abejón, A.: Tauberian Operators. Operator Theory: Advances and Applications, vol. 194. Birkhäuser, Basel (2010)CrossRefGoogle Scholar
  14. 14.
    González, M., Onieva, V.M.: Lifting results for sequences in Banach spaces. Math. Proc. Camb. Philos. Soc. 105, 117–121 (1989)MathSciNetCrossRefGoogle Scholar
  15. 15.
    González, M., Onieva, V.M.: Characterizations of Tauberian operators and other semigroups of operators. Proc. Am. Math. Soc. 108, 399–405 (1990)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Howard, J.: \({\cal{F}}\)-singular and \({\cal{G}}\)-cosingular operators. Colloq. Math. 22, 85–89 (1970)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kalton, N., Wilansky, A.: Tauberian operators on Banach spaces. Proc. Am. Math. Soc. 57, 251–255 (1976)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pietsch, A.: Operator Ideals. North-Holland, Amsterdam (1980)zbMATHGoogle Scholar
  19. 19.
    Räbiger, F.: Beiträge zur Strukturtheorie der Grothendieck-Räume. Sitzungsberichte der Heidelberger Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse. Ph.D. Dissertation. Springer, Berlin (1985)CrossRefGoogle Scholar
  20. 20.
    Tacon, D.G.: Generalized semi-Fredholm transformations. J. Aust. Math. Soc. 34, 60–70 (1983)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zeekoei, E.D., Fourie, J.H.: On \(p\)-convergent operators on Banach lattices. Acta Math. Sin. 34, 873–890 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zeng, Q., Zhong, H.: Three-space theorem for semi-Fredholmness. Arch. Math. (Basel) 100, 55–61 (2013)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Universidad de CantabriaSantanderSpain
  2. 2.Universidad de OviedoOviedoSpain

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