Advertisement

Positivity

, Volume 22, Issue 1, pp 59–62 | Cite as

On the “function” and “lattice” definitions of a narrow operator

Article

Abstract

We prove that the function and lattice definitions of a narrow operator defined on a Köthe Banach space E on a finite atomless measure space \((\Omega , \Sigma , \mu )\) are equivalent if and only if the set of all simple functions is dense in E. This answers Problem 10.3 from Popov and Randrianantoanina (Narrow operators on function spaces and vector lattices, De Gruyter studies in mathematics 45, De Gruyter, Berlin, 2013).

Keywords

Köthe Banach space Narrow operator Vector lattice 

Mathematics Subject Classification

Primary 46B20 Secondary 46B03 46B10 

References

  1. 1.
    Albiac, F., Kalton, N.: Topics in Banach Space Theory, Graduate Texts in Mathematics, 233. Springer, New York (2006)Google Scholar
  2. 2.
    Krasikova, I.V., Martín, M., Merí, J., Mykhaylyuk, V., Popov, M.: On order structure and operators in \(L_\infty (\mu )\). Cent. Eur. J. Math. 7(4), 683–693 (2009)MathSciNetMATHGoogle Scholar
  3. 3.
    Maslyuchenko, O.V., Mykhaylyuk, V.V., Popov, M.M.: A lattice approach to narrow operators. Positivity 13, 459–495 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Mykhaylyuk, V., Pliev, M., Popov, M., Sobchuk, O.: Dividing measures and narrow operators. Stud. Math. 231, 97–116 (2015)MathSciNetMATHGoogle Scholar
  5. 5.
    Plichko, A.M., Popov, M.M.: Symmetric function spaces on atomless probability spaces. Diss. Math. (Rozprawy Mat.) 306, 1–85 (1990)MathSciNetMATHGoogle Scholar
  6. 6.
    Pliev, M.A., Popov, M.M.: Narrow orthogonally additive operators. Positivity 18(4), 641–667 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Popov, M.M.: Narrow operators (a survey). Banach Center Publ. 92, 299–326 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Popov, M., Randrianantoanina, B.: Narrow Operators on Function Spaces and Vector Lattices, De Gruyter Studies in Mathematics 45. De Gruyter, Berlin (2013)Google Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Institute of MathematicsPomeranian University in SłupskSłupskPoland
  2. 2.Department of Mathematics and InformaticsVasyl Stefanyk Precarpathian National UniversityIvano-FrankivskUkraine
  3. 3.Department of Mathematics and InformaticsChernivtsi National UniversityChernivtsiUkraine

Personalised recommendations