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Narrow operators on lattice-normed spaces

100%
Pliev M.
Open Mathematics
|
2011
|
tom 9
|
nr 6
1276-1287
EN
The aim of this article is to extend results of Maslyuchenko, Mykhaylyuk and Popov about narrow operators on vector lattices. We give a new definition of a narrow operator, where a vector lattice as the domain space of a narrow operator is replaced with a lattice-normed space. We prove that every GAM-compact (bo)-norm continuous linear operator from a Banach-Kantorovich space V to a Banach lattice Y is narrow. Then we show that, under some mild conditions, a continuous dominated operator is narrow if and only if its exact dominant is so.
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Dividing measures and narrow operators

51%
Mykhaylyuk V., Pliev M., Popov M., Sobchuk O.
Studia Mathematica
|
2015
|
tom 231
|
nr 2
97-116
EN
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.
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