  The aim of this article is to extend results of Maslyuchenko O., Mykhaylyuk
V. Popov M. about narrow operators on vector lattices. We give a new definition
of a narrow operator where a vector lattice as the domain space of a narrow
operator is replaced with a lattice-normed space. We prove that every
GAM-compact (bo)-norm continuous linear operator from a Banach-Kantorovich
space V to a Banach lattice Y is narrow. Then we show that, under some mild
conditions, a continuous dominated operator is narrow if and only if its exact
dominant is narrow.
