Dividing measures and narrow operators

Volodymyr Mykhaylyuk, et al. "Dividing measures and narrow operators." Studia Mathematica 231.2 (2015): 97-116. <http://eudml.org/doc/285730>.

@article{VolodymyrMykhaylyuk2015,
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.},
author = {Volodymyr Mykhaylyuk, Marat Pliev, Mikhail Popov, Oleksandr Sobchuk},
journal = {Studia Mathematica},
keywords = {narrow operator; orthogonally additive operator; Urysohn operator; Köthe function space; vector lattice; Riesz space; Boolean algebra},
language = {eng},
number = {2},
pages = {97-116},
title = {Dividing measures and narrow operators},
url = {http://eudml.org/doc/285730},
volume = {231},
year = {2015},
}

TY - JOUR
AU - Volodymyr Mykhaylyuk
AU - Marat Pliev
AU - Mikhail Popov
AU - Oleksandr Sobchuk
TI - Dividing measures and narrow operators
JO - Studia Mathematica
PY - 2015
VL - 231
IS - 2
SP - 97
EP - 116
AB - 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.
LA - eng
KW - narrow operator; orthogonally additive operator; Urysohn operator; Köthe function space; vector lattice; Riesz space; Boolean algebra
UR - http://eudml.org/doc/285730
ER -