How to prove the extension of a positive continuous linear operator is still positive? Let $E$ be a real ordered Banach space with the positive cone $P$ of $E$.
It is easy to see that the positive cone $P$ is the closed convex cone of $E$.
We assume that $E_0 := P - P $  is a dense linear subspace of $E$. This means that $P$ is a total cone of $E$ (i.e., $\overline{P-P} = E$).
We know that if there is a bounded linear operator $A \colon E_0 \to E_0$, by Continuous Linear extension theorem, then the operator $A$ has a unique extension
$\bar{A} \colon E \to E$ such that $\bar{A}$ is also a bounded linear operator with $\bar{A} v = Av$ for each $v \in E_0$ and $\| \bar{A} \| = \| A \|$.
My question is that if we suppose further that the operator $A$ is positive (i.e., $A(P) \subset P$), then is its extension $\bar{A}$ still positive (i.e., $\bar{A}(P) \subset P$)? If so, how to prove it?
Any ideas or suggestions are most welcome! Thanks in advance:)
 A: To answer the question asked in the OP:

Let $P$ be a cone in a Banach space $E$ so that $E_0=\mathrm{span}(P)$ is dense in $E$, if a continuous linear operator $A:E_0\to E$ is positive, does it follow that its closure $\overline{A}: E\to E$ is positive?

This is true for these special conditions, namely since $P\subset E_0$ we have $\overline A(P) = \overline A\lvert_{E_0}(P) = A(P)\subset P$ by the positivity of $A$.
Perhaps more natural is the question what happens when $P\not\subset E_0$. But we cannot be too general, for example if $E_0$ and $P$ have intersection $\{0\}$ then any map $E_0\to E$ is positive, even the restriction of non-positive maps will be so.
I think the clearest demands to make are that $E_0\cap P$ is dense in $P$ and that $P$ is a closed cone. Then for any $p\in P$ we have a sequence $p_n\in E_0\cap P$ with $p_n\to p$. So
$$\overline A(p) =\lim_n \overline A(p_n) =\lim_n\, A(p_n).$$
Now by positivity of $A$ we have that $A(p_n)$ is in $P$ and by closedness of $P$ the limit of this must lie in $P$. Thus $\overline A(p)\in P$. So $\overline A$ is a positive linear map.
