On the sum of narrow and $C$-compact operators

We consider narrow linear operators defined on a Banach–Kantorovich space and taking value in a Banach space. We prove that the sum $S+T$ of two operators is narrow whenever $S$ is a narrow operator and $T$ is a $(bo)$-continuous $C$-compact operator. For the proof of the main result we use the method of decomposition of an element of a lattice-normed space into a sum of disjoint fragments and an approximation of a $C$-compact operator by finite-rank operators.