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Dividing measures and narrow operators

Volume 231 / 2015

Volodymyr Mykhaylyuk, Marat Pliev, Mikhail Popov, Oleksandr Sobchuk Studia Mathematica 231 (2015), 97-116 MSC: Primary 47H30; 47B60; Secondary 28A60. DOI: 10.4064/sm7878-2-2016 Published online: 17 February 2016

Abstract

We use a new technique of measures on Boolean algebras to investigate narrow operators on vector lattices. First we prove that, under mild assumptions, every finite rank operator is strictly narrow (before it was known that such operators are narrow). Then we show that every order continuous operator from an atomless vector lattice to a purely atomic one is order narrow. This explains in what sense the vector lattice structure of an atomless vector lattice given by an unconditional basis is far from its original vector lattice structure. Our third main result asserts that every operator such that the density of the range space is less than the density of the domain space, is strictly narrow. This gives a positive answer to Problem 2.17 from “Narrow Operators on Function Spaces and Vector Lattices” by B. Randrianantoanina and the third named author for the case of reals. All the results are obtained for a more general setting of (nonlinear) orthogonally additive operators.

Authors

  • Volodymyr MykhaylyukDepartment of Mathematics and Informatics
    Chernivtsi National University
    Kotsiubyns’koho 2
    Chernivtsi 58012, Ukraine
    e-mail
  • Marat PlievSouth Mathematical Institute
    Vladikavkaz Scientific Center
    Russian Academy of Sciences
    Vladikavkaz 362027, Russian Federation
    e-mail
  • Mikhail PopovInstitute of Mathematics
    Pomeranian University in Słupsk
    Arciszewskiego 22d
    76-200 Słupsk, Poland
    and
    Department of Mathematics and Informatics
    Chernivtsi National University
    Kotsiubyns’koho 2
    Chernivtsi 58012, Ukraine
    e-mail
  • Oleksandr SobchukDepartment of Mathematics and Informatics
    Chernivtsi National University
    Kotsiubyns’koho 2
    Chernivtsi 58012, Ukraine
    e-mail

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