On a Class of Operators on Complex Vector Lattices

We propose a new approach to the study of orthogonally additive operators acting in complex vector lattices. This approach is different from that presented in a recent work by Pliev and Sukochev. We show that the vector space OAᵣ(EC,FC) of regular orthogonally additive operators acting from the complexification EC of a uniformly complete vector lattice E to the complexification FC of a Dedekind complete vector lattice F is a complex vector lattice, and the modulus of an operator T EC→ FC can be evaluated by the Riesz–Kantorovich formula. We also show that if an orthogonally additive operator T E→ F is A -uniformly narrow, then so is its complex extension TT EC→ FC . In addition, if a regular orthogonally additive operator T EC→ FC is narrow, then so is its modulus | T| E→ F .