On ideal operators (Q1412303)

Let \(E, F\) be Riesz spaces. \(T: E \to F\) is called an ideal (inverse ideal) operator if \(T (I) (T^{-1} (J))\) is an order ideal in \(E (F)\) for each order ideal \(I (J)\) in \(E (F)\). It is shown that these operators can be characterized by their action on principal order ideals. \(E\) is said to have the \(Z (E)\)-extension property if each central operator on an order ideal can be extended to a central operator on the whole space \(E\). If \(E, F\) are uniformly complete and \(E\) has the \(Z (E)\)-extension property, then \(T\) is an ideal operator if and only if for any \(\pi \in Z (F)\) and \(u \in E\), there exists \(S \in Z(E)\) with \(\pi (Tu) = T (Su)\). There is also a similar characterization of inverse ideal operators. The relation between ideal (inverse ideal) and disjointness preserving operators is studied and a partial converse is given to a theorem of \textit{D. R. Hart} [Indag. Math. 47, 183--197 (1985; Zbl 0576.47021)]. Examples are provided to distinguish these operators from other well-known classes of operators.