Dividing measures and narrow operators

Volodymyr Mykhaylyuk ¹   ; Marat Pliev ²   ; Mikhail Popov ³   ; Oleksandr Sobchuk ¹

¹ Department of Mathematics and Informatics Chernivtsi National University Kotsiubyns’koho 2 Chernivtsi 58012, Ukraine
² South Mathematical Institute Vladikavkaz Scientific Center Russian Academy of Sciences Vladikavkaz 362027, Russian Federation
³ Institute 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

Studia Mathematica, Tome 231 (2015) no. 2, pp. 97-116

Résumé

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.

Détail

DOI : 10.4064/sm7878-2-2016

Keywords: technique measures boolean algebras investigate narrow operators vector lattices first prove under mild assumptions every finite rank operator strictly narrow before known operators narrow every order continuous operator atomless vector lattice purely atomic order narrow explains what sense vector lattice structure atomless vector lattice given unconditional basis far its original vector lattice structure third main result asserts every operator density range space density domain space strictly narrow gives positive answer problem narrow operators function spaces vector lattices randrianantoanina third named author reals results obtained general setting nonlinear orthogonally additive operators