Completely additive and C-compact operators in lattice-normed spaces

In this article, we investigate some classes of dominated orthogonally additive operators in lattice-normed spaces. We say that an orthogonally additive operator T from a lattice-normed space (V, E) to a lattice-normed space (W, F) is completely additive if, for every order summable family of pairwise disjoint elements (v ) .. of V, the family (Tv) .. is order summable in W. The first part of the article is devoted to completely additive operators. We prove that a dominated orthogonally additive operator T : V. W from a Banach-Kantorovich space V to a Banach-Kantorovich space W is completely additive if and only if so is its exact dominant.T. : E. F. One of the our main results asserts that an orthogonally additive map defined on a lateral ideal and dominated by a positive completely additive operator can be extended to a dominated completely additive operator defined on the whole space. In the second part of the article, we consider C-compact dominated orthogonally additive operators. We show that the C-compactness of a dominated orthogonally additive operator T : V. W from a decomposable lattice-normed space (V, E) to an order continuous Banach lattice W implies the C-compactness of its exact dominant T : E -> W.