Dominated Operators from a Lattice-Normed Space to a Sequence Banach Lattice

TL;DR

This paper proves that dominated linear operators from Banach-Kantorovich spaces to certain sequence Banach lattices are narrow, revealing structural limitations of atomless Banach lattices and properties of order narrow operators.

Contribution

It establishes the narrowness of dominated operators to sequence Banach lattices and links order narrowness of operators to their exact dominants.

Findings

Dominated operators to l_p(Γ) and c_0(Γ) are narrow.

Atomless Banach lattices lack certain finite dimensional decompositions.

Order narrowness of an operator implies the same for its exact dominant.

Abstract

We show that every dominated linear operator from an Banach-Kantorovich space over atomless Dedekind complete vector lattice to a sequence Banach lattice $l_p({\Gamma})$ or $c_0({\Gamma})$ is narrow. As a conse- quence, we obtain that an atomless Banach lattice cannot have a finite dimensional decomposition of a certain kind. Finally we show that if a linear dominated operator T from lattice-normed space V to Banach- Kantorovich space W is order narrow then the same is its exact dominant $\ls T\rs$.