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Narrow operators on lattice-normed spaces

Marat Pliev — 2011

Open Mathematics

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...

Dividing measures and narrow operators

Volodymyr Mykhaylyuk;  Marat Pliev;  Mikhail Popov;  Oleksandr Sobchuk — 2015

Studia Mathematica

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...

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