We show that every dominated linear operator from a Banach–Kantorovich space over an atomless Dedekind-complete vector lattice to a sequence Banach lattice or 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 from a lattice-normed space to the Banach space with a mixed norm over an order-continuous Banach lattice implies the order-narrowness of its exact dominant ||.
"Dominated operators from lattice-normed spaces to sequence Banach lattices." Ann. Funct. Anal. 7 (4) 646 - 655, November 2016. https://doi.org/10.1215/20088752-3660990