Positive linear operator

In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space $$(X, \leq)$$ into a preordered vector space $$(Y, \leq)$$ is a linear operator $$f$$ on $$X$$ into $$Y$$ such that for all positive elements $$x$$ of $$X,$$ that is $$x \geq 0,$$ it holds that $$f(x) \geq 0.$$ In other words, a positive linear operator maps the positive cone of the domain into the positive cone of the codomain.

Every positive linear functional is a type of positive linear operator. The significance of positive linear operators lies in results such as Riesz–Markov–Kakutani representation theorem.

Definition
A linear function $$f$$ on a preordered vector space is called positive if it satisfies either of the following equivalent conditions:


 * 1) $$x \geq 0$$ implies $$f(x) \geq 0.$$
 * 2) if $$x \leq y$$ then $$f(x) \leq f(y).$$

The set of all positive linear forms on a vector space with positive cone $$C,$$ called the dual cone and denoted by $$C^*,$$ is a cone equal to the polar of $$-C.$$ The preorder induced by the dual cone on the space of linear functionals on $$X$$ is called the .

The order dual of an ordered vector space $$X$$ is the set, denoted by $$X^+,$$ defined by $$X^+ := C^* - C^*.$$

Canonical ordering
Let $$(X, \leq)$$ and $$(Y, \leq)$$ be preordered vector spaces and let $$\mathcal{L}(X; Y)$$ be the space of all linear maps from $$X$$ into $$Y.$$ The set $$H$$ of all positive linear operators in $$\mathcal{L}(X; Y)$$ is a cone in $$\mathcal{L}(X; Y)$$ that defines a preorder on $$\mathcal{L}(X; Y)$$. If $$M$$ is a vector subspace of $$\mathcal{L}(X; Y)$$ and if $$H \cap M$$ is a proper cone then this proper cone defines a  on $$M$$ making $$M$$ into a partially ordered vector space.

If $$(X, \leq)$$ and $$(Y, \leq)$$ are ordered topological vector spaces and if $$\mathcal{G}$$ is a family of bounded subsets of $$X$$ whose union covers $$X$$ then the positive cone $$\mathcal{H}$$ in $$L(X; Y)$$, which is the space of all continuous linear maps from $$X$$ into $$Y,$$ is closed in $$L(X; Y)$$ when $$L(X; Y)$$ is endowed with the $\mathcal{G}$-topology. For $$\mathcal{H}$$ to be a proper cone in $$L(X; Y)$$ it is sufficient that the positive cone of $$X$$ be total in $$X$$ (that is, the span of the positive cone of $$X$$ be dense in $$X$$). If $$Y$$ is a locally convex space of dimension greater than 0 then this condition is also necessary. Thus, if the positive cone of $$X$$ is total in $$X$$ and if $$Y$$ is a locally convex space, then the canonical ordering of $$L(X; Y)$$ defined by $$\mathcal{H}$$ is a regular order.

Properties
Proposition: Suppose that $$X$$ and $$Y$$ are ordered locally convex topological vector spaces with $$X$$ being a Mackey space on which every positive linear functional is continuous. If the positive cone of $$Y$$ is a weakly normal cone in $$Y$$ then every positive linear operator from $$X$$ into $$Y$$ is continuous.

Proposition: Suppose $$X$$ is a barreled ordered topological vector space (TVS) with positive cone $$C$$ that satisfies $$X = C - C$$ and $$Y$$ is a semi-reflexive ordered TVS with a positive cone $$D$$ that is a normal cone. Give $$L(X; Y)$$ its canonical order and let $$\mathcal{U}$$ be a subset of $$L(X; Y)$$ that is directed upward and either majorized (that is, bounded above by some element of $$L(X; Y)$$) or simply bounded. Then $$u = \sup \mathcal{U}$$ exists and the section filter $$\mathcal{F}(\mathcal{U})$$ converges to $$u$$ uniformly on every precompact subset of $$X.$$