Narrow Operators on Lattice-normed Spaces and Vector Measures

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.

Dedicated to the memory of Adriaan Cornelis Zaanen, one of the great founders of the theory of vector lattices