Dominated operators from lattice-normed spaces to sequence Banach lattices

Abstract

We show that every dominated linear operator from a Banach–Kantorovich space over an atomless Dedekind-complete vector lattice to a sequence Banach lattice ${\ell }_p(\Gamma )$ or $c_0(\Gamma )$ is narrow. As a consequence, we obtain that an atomless Banach lattice cannot have a finite-dimensional decomposition of a certain kind. Finally, we show that the order-narrowness of a linear dominated operator $T$ from a lattice-normed space $V$ to the Banach space with a mixed norm $(W,F)$ over an order-continuous Banach lattice $F$ implies the order-narrowness of its exact dominant |$T$|.