Narrow operators on lattice-normed spaces and vector measures

TL;DR

This paper investigates properties of narrow operators on lattice-normed spaces, proving that finite rank operators are strictly narrow and that certain dominated, order continuous operators are order narrow, expanding understanding of their structure.

Contribution

It establishes that finite rank linear operators are strictly narrow and characterizes order narrowness of dominated, order continuous operators between specific lattice-normed spaces.

Findings

Finite rank operators are strictly narrow under mild conditions.

Dominated, order continuous operators from atomless to atomic lattice-normed spaces are order narrow.

Extends the theory of narrow operators in lattice-normed spaces.

Abstract

We consider linear narrow operators on lattice-normed spaces. We prove that, under mild assumptions, every finite rank linear operator is strictly narrow (before it was known that such operators are narrow). Then we show that every dominated, order continuous linear operator from a lattice-normed space over atomless vector lattice to an atomic lattice-normed space is order narrow.