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.