Narrow and C-compact Orthogonally Additive Operators in Lattice-Normed Spaces

Abstract

In this article we consider orthogonally additive operators on lattice-normed spaces. In the first part of the article we present some examples of narrow, laterally-to-norm continuous and C-compact operators defined on a lattice-normed space and taking value in a Banach space. We show that any laterally-to-norm continuous narrow orthogonally additive operator defined on a decomposable lattice-normed space (V, E) over an atomic vector lattice E with the projection property is equal to zero. In the second part we prove that the sum of two orthogonally additive operators $$T+S$$ defined on a order complete, decomposable lattice-normed space V and taking value in Banach space X, where $$T:V\rightarrow X$$ is a laterally-to-norm continuous C-compact operator and $$S:V\rightarrow X$$ is a narrow operator, is a narrow operator as well.