, Volume 18, Issue 4, pp 641–667 | Cite as

Narrow orthogonally additive operators

  • Marat Pliev
  • Mikhail PopovEmail author


We generalize the notion of narrow operators to nonlinear maps on vector lattices. The main objects are orthogonally additive operators and, in particular, abstract Uryson operators. Most of the results extend known theorems obtained by O. Maslyuchenko, V. Mykhaylyuk and the second named author published in Positivity 13:459–495 (2009) for linear operators. For instance, we prove that every orthogonally additive laterally-to-norm continuous C-compact operator from an atomless Dedekind complete vector lattice to a Banach space is narrow. Another result asserts that the set \({\mathcal U}_{on}^{lc}(E,F)\) of all order narrow laterally continuous abstract Uryson operators is a band in the vector lattice of all laterally continuous abstract Uryson operators from an atomless vector lattice \(E\) with the principal projection property to a Dedekind complete vector lattice \(F\). The band generated by the disjointness preserving laterally continuous abstract Uryson operators is the orthogonal complement to \({\mathcal U}_n^{lc}(E,F)\).


Narrow operators C-compact operators Orthogonally additive operators Abstract Uryson operators Banach lattices 

Mathematics Subject Classification (1991)

Primary 47H30 Secondary 47H99 


  1. 1.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Dordrecht (2006)zbMATHGoogle Scholar
  2. 2.
    Flores, J., Hernández, F.L., Tradacete, P.: Domination problem for strictly singular operators and other related classes. Positivity 15(4), 595–616 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Flores, J., Ruiz, C.: Domination by positive narrow operators. Positivity 7, 303–321 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Kadets, V.M., Kadets, M.I.: Rearrangements of series in Banach spaces. Transl. Math. Mon., vol. 86, AMS, Providence (1991)Google Scholar
  5. 5.
    Kusraev, A.G.: Dominated Operators. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Kusraev, A.G., Pliev, M.A.: Orthogonally additive operators on lattice-normed spaces. Vladikavkaz Math. J. 3, 33–43 (1999)MathSciNetGoogle Scholar
  7. 7.
    Kusraev, A.G., Pliev, M.A.: Weak integral representation of the dominated orthogonally additive operators. Vladikavkaz Math. J. 4, 22–39 (1999)Google Scholar
  8. 8.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. In: Sequence Spaces, vol. 1. Springer, Berlin (1977)Google Scholar
  9. 9.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. In: Function Spaces, vol. 2. Springer, Berlin (1979)Google Scholar
  10. 10.
    Maslyuchenko, O.V., Mykhaylyuk, V.V., Popov, M.M.: A lattice approach to narrow operators. Positivity 13, 459–495 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Mykhaylyuk, V.V., Popov, M.M.: On sums of narrow operators on Köthe function spaces. J. Math. Anal. Appl. 404, 554–561 (2013)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Mazón, J.M., Segura de León, S.: Order bounded ortogonally additive operators. Rev. Roumane Math. Pures Appl. 35(4), 329–353 (1990)zbMATHGoogle Scholar
  13. 13.
    Mazón, J.M., Segura de León, S.: Uryson operators. Rev. Roumane Math. Pures Appl. 35(5), 431–449 (1990)zbMATHGoogle Scholar
  14. 14.
    Plichko, A.M., Popov, M.M.: Symmetric function spaces on atomless probability spaces. Dissertationes Math. (Rozprawy Mat.) 306, 1–85 (1990)MathSciNetGoogle Scholar
  15. 15.
    Pliev, M.: Uryson operators on the spaces with mixed norm. Vladikavkaz Math. J. 3, 47–57 (2007)MathSciNetGoogle Scholar
  16. 16.
    Pliev, M.: Narrow operators on lattice-normed spaces. Cent. Eur. J. Math. 9(6), 1276–1287 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Popov, M.M., Semenov, E.M., Vatsek D.O.: Some problems on narrow operators on function spaces. Cent. Eur. J. Math.Google Scholar
  18. 18.
    Popov, M., Randrianantoanina, B.: Narrow operators on function spaces and vector lattices. In: De Gruyter Studies in Mathematics, vol. 45. De Gruyter, Berlin (2013)Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.South Mathematical InstituteRussian Academy of SciencesVladikavkazRussia
  2. 2.Department of Applied MathematicsChernivtsi National UniversityChernivtsiUkraine

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