Existence and extensions of positive linear operators (Q1007108)

This article studies the existence of the extension of positive linear operators on ordered vector spaces. Since in this setting one needs to have the extension preserve order, classical results of this nature, such as the Hahn-Banach or Mazur-Orlicz theorems, do not apply immediately. The authors observe that, if one assumes a monotonicity condition, then the Mazur-Orlicz theorem extends to the ordered vector space setting: Theorem. Let \(E\) be an ordered vector space, \(F\) be a Dedekind complete ordered vector space, and \(S:E\to F\) a monotone sublinear operator. Let \(K\) be a nonempty convex set, \(P:K\to E\) a convex operator, and \(Q:K\to F\) a concave operator. Then the following conditions are equivalent: {\parindent=7mm \begin{itemize}\item[(i)]There exists a positive linear operator \(L:E\to F\) with the properties (a) \(L\leq S\) on \(E\), and (b) \(Q\leq L\circ P \) on \(K\). \item[(ii)]The inequality \(Q\leq S\circ P \) holds on \(K\). \end{itemize}} As applications, they also obtain some new (and generalizations of some known) results on the extension of combinations of various operators and a minimax theorem. The main result of the paper is the following theorem that provides a sufficient condition for the existence of a monotone sublinear operator: Theorem. Let \(E_0\) be an ordered vector space, \(F\) a Dedekind complete ordered vector space, \(A\) and \(M\) nonempty sets. Consider the maps \(g:A\to E_0,\) \(h:M\to (E_0)_+\), \(f:A\to F\), \(r:M\to F\) and let \(E=\text{span}(g(A)\cup h(M)) \subset E_0\). Suppose that the following implication holds: \[ \sum_{i=1}^{n} \mu_i g(a_i) \leq \sum_{i=1}^{n} h(z_i) \Rightarrow \sum_{i=1}^{n} \mu_i f(a_i) \leq \sum_{i=1}^{n} r(z_i), \] for all finite sets \(\{a_i\} \subset A\), \(\{z_i\} \subset M\), \(\{m_i\} \subset \mathbb R\). Then there exists a monotone sublinear operator \(S:E\to F\) such that (a) \(S\circ g \leq f \) on \(A\) and (b) \(S\circ h \leq r \) on \(M\). The authors also study some particular cases when the condition in the main theorem also becomes necessary.