## When is the norm of all positive operators on an ordered Banach space determined by their values on the positive cone?

I'm trying to investigate the interplay between the norm and cone of positive elements in ordered Banach spaces. In particular, I would like a nice characterization of when the norm of a positive operator can be obtained as a supremum of norms over all norm 1 positive elements (see 1. below). It is easy to show that this always holds for Banach lattices, but gets tricky when trying to prove this more generally.

To be a bit more detailed, let $X$ be a Banach space with a cone (or wedge) of 'positive' elements, denoted by $X_{+}$; $X$ is then called an ordered Banach space and $x\leq y$ means $y-x\in X_{+}$. I will call $X_{+}$ generating if $X=X_{+}-X_{-}$; proper if $X_{+}\cap(-X_{+})={0}$; and normal if $0\leq x\leq y$ implies $\|x\|\leq\|y\|$. A bounded linear operator $T:X\to X$ is called positive, if $x\geq 0$ implies $Tx\geq 0$. An ordered Banach space with the property that every positive operator satisfies $$\|T\|=\sup\{\|Tx\|:x\in X_{+},\|x\|=1\}$$ I will say has the positive operator property, i.e., the norms of positive operators are completely determined by their behavior on the cone.

My question is:

Let $X$ be an ordered Banach space with a closed, proper, generating, normal cone. Is there a characterization of the positive operator property in terms of the cone-norm interaction?

Thus far I have found a few sufficient conditions:

1. For all $x\in X$ there exist $X\ni x_{1},x_{2}\geq0$ such that $x=x_{1}-x_{2}$ and $\|x_{j}\|\leq\|x\|$ for $j=1,2$.
2. For all $x\in X$ and $X\ni a,b\geq0$, $-a\leq x\leq b$ implies $\|x\|\leq\max \{ \|a\|,\|b\| \}$.

together imply the positive operator property.

Another sufficient condition is having the property that for any $x\in X$ it holds that $\|x\| = \inf \{\|z_+ + z_-\|:z_\pm \geq 0;x=z_+ -z_-\}$. Being a Banach lattice is sufficient to have this property (by invoking the property $\|x\|=\||x|\|$), but seems to be a bit more general since $\mathbb{R}^3$ with the Euclidean norm and the `ice-cream cone' $\{(x_1,x_2,x_3):x_1\geq (x_2^2+x_3^2)^{1/2} \}$ (which is not a lattice) also has this property (so being a lattice is not necessary for the positive operator property).

I've been unable to prove that any of these conditions are necessary. The last one had been my best bet so far, but I'm beginning to doubt the existence of a nice list of conditions on the norm and cone that together are necessary and sufficient.

Any suggestions, counterexamples or pointers to the literature that anyone may have will be greatly appreciated.

UPDATE: Cleaned it up a bit, and posted some new information I found since first posting. I found no mention in any literature to this question. I also asked around a bit with no hits from people who are a bit in the know, so it seems to be wide open. Gets the 'open-problem' tag.

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Can you explain why there are any non-zero positive operators? I presume we find a positive functional; and I presume we use Hahn-Banach for this, but I just don't see how right now... – Matthew Daws Mar 6 2012 at 7:43
Although I'm hardly an expert on the machinery, I think the reference* below answers your question, Matthew. As long as we assume the cone to be normal, we get "enough" positive functionals so that the dual cone/wedge is generating in the dual. From there we are able to make lots of positive of rank 1 operators. (Quickly browsing through the proof I see it does depend on some a separation theorem, so yes, does look like it depends on Hahn Banach.) *Cones & Duality by Aliprantis & Tourky books.google.nl/… – Miek Messerschmidt Mar 6 2012 at 9:20

After some digging, I found these two papers on the subject:

• Batty, Charles, and Derek Robinson. “Positive One-parameter Semigroups on Ordered Banach Spaces.” Acta Applicandae Mathematicae 2, no. 3 (1984): 221–296.

• Yamamuro, Sadayuki. “On Linear Operators on Ordered Banach Spaces.” Bulletin of the Australian Mathematical Society 27, no. 2 (1983): 285–305.

A characterization doesn't seem to exist, but Batty & Robinson does give one necessary condition to have this property.

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