When is the norm of all positive operators on an ordered Banach space determined by their values on the positive cone? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T06:44:45Z http://mathoverflow.net/feeds/question/90277 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90277/when-is-the-norm-of-all-positive-operators-on-an-ordered-banach-space-determined When is the norm of all positive operators on an ordered Banach space determined by their values on the positive cone? Miek Messerschmidt 2012-03-05T13:33:46Z 2012-11-03T11:14:53Z <p>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 <em>positive</em> 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.</p> <p>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 $X_{+}$ <em>generating</em> if $X=X_{+}-X_{-}$; <em>proper</em> if $X_{+}\cap(-X_{+})={0}$; and <em>normal</em> if $0\leq x\leq y$ implies $\|x\|\leq\|y\|$. A bounded linear operator $T:X\to X$ is called <em>positive</em>, 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 <em>the positive operator property</em>, <strong>i.e., the norms of positive operators are completely determined by their behavior on the cone</strong>.</p> <p>My question is:</p> <blockquote> <p>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?</p> </blockquote> <p>Thus far I have found a few sufficient conditions:</p> <ol> <li>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 $j=1,2$.</li> <li>For all $x\in X$ and $X\ni a,b\geq0$, $-a\leq x\leq b$ implies $\|x\|\leq\max \{ \|a\|,\|b\| \}$.</li> </ol> <p>together imply the positive operator property.</p> <p>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). </p> <p>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.</p> <p>Any suggestions, counterexamples or pointers to the literature that anyone may have will be greatly appreciated.</p> <p>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.</p> http://mathoverflow.net/questions/90277/when-is-the-norm-of-all-positive-operators-on-an-ordered-banach-space-determined/111364#111364 Answer by Miek Messerschmidt for When is the norm of all positive operators on an ordered Banach space determined by their values on the positive cone? Miek Messerschmidt 2012-11-03T11:14:53Z 2012-11-03T11:14:53Z <p>After some digging, I found these two papers on the subject: </p> <ul> <li><p>Batty, Charles, and Derek Robinson. “Positive One-parameter Semigroups on Ordered Banach Spaces.” Acta Applicandae Mathematicae 2, no. 3 (1984): 221–296.</p></li> <li><p>Yamamuro, Sadayuki. “On Linear Operators on Ordered Banach Spaces.” Bulletin of the Australian Mathematical Society 27, no. 2 (1983): 285–305.</p></li> </ul> <p>A characterization doesn't seem to exist, but Batty &amp; Robinson does give one necessary condition to have this property. </p>