Narrow orthogonally additive operators in lattice-normed spaces

We consider a new class of narrow orthogonally additive operators in lattice-normed spaces and prove the narrowness of every $C$-compact norm-laterally-continuous orthogonally additive operator from a Banach–Kantorovich space $V$ into a Banach space $Y$. Furthermore, every dominated Urysohn operator from $V$ into a Banach sequence lattice $Y$ is also narrow. We establish that the order narrowness of a dominated Urysohn operator from a Banach–Kantorovich space $V$ into a Banach space with mixed norm $W$ implies the order narrowness of the least dominant of the operator.