The lateral order on Köthe-Bochner spaces and orthogonally additive operators (Q6571086)

The present article introduces a novel concept of regular orthogonally additive operators defined on a lattice-normed space \((\mathcal{X},E)\) with values in a vector lattice \(F.\) Several important examples of these operators are presented. The authors demonstrate that an orthogonally additive operator \(T\) from a decomposable lattice-normed space \((\mathcal{X} ,E)\) to a Dedekind complete vector lattice \(F\) is dominated if and only if it is strongly $C$-bounded.\N\NLet \(E(X)\) be a Köthe-Bochner space and \(E\) be a Köthe-Banach space. Among other results, a criterion for the positivity of a nonlinear superposition operator \(T:E(X)\longrightarrow E\) is provided.