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 $$\mathscr{OA}_r(E_{\mathbb C},F_{\mathbb C})$$ of regular orthogonally additive operators acting from the complexification $$E_{\mathbb C}$$ of a uniformly complete vector lattice $$E$$ to the complexification $$F_{\mathbb C}$$ of a Dedekind complete vector lattice $$F$$ is a complex vector lattice, and the modulus of an operator $$\mathscr T\colon E_{\mathbb C}\to F_{\mathbb C}$$ can be evaluated by the Riesz–Kantorovich formula. We also show that if an orthogonally additive operator $$T\colon E\to F$$ is $$\mathfrak A$$ -uniformly narrow, then so is its complex extension $$\mathscr T_{T}\colon E_{\mathbb C}\to F_{\mathbb C}$$ . In addition, if a regular orthogonally additive operator $$\mathscr T\colon E_{\mathbb C}\to F_{\mathbb C}$$ is narrow, then so is its modulus $$|\mathscr T|\colon E\to F$$ .