When $X$ is a vector subspace of an ordered vector space $A$, any positive linear functional $f: A \to R$ extends to all of $A$ as a positive linear functional provided one can find a nonvoid, absorbing, and convex subset of $A$, $A_{0}$, such that $f(x)\leq 1$ whenever $x \in X$ satisfies $x \leq y$ for some $y \in A_{0}$. The result is attributed to Baur and Namioka, and that condition is also necessary for an extension.

As far as positive linear operators are concerned, the only theorem I am aware of is due to Kantorovich, and gives only a necessary condition for extension. More precisely, if $F: X \to B$ is a positive linear operator with $X$ as above and $B$ also an ordered vector space, but of a special type (namely, a Dedekind complete Riesz space), then $F$ extends to a linear operator to all of $A$ provided $X$ majorizes $A$. (See for instance chapter one of this book.)

My question: Is there any full characterization (certainly after imposing some conditions on the codomain) for the extension of positive linear operators that resembles the Bauer-Namioka condition for positive linear functionals? I am sorry if this question is either trivial or doesn't make any sense, but I really cannot find any reference on that. Thanks.