positive linear operator

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

1. $x\; \backslash geq\; 0$ implies $f(x)\; \backslash geq\; 0.$
2. if $x\; \backslash leq\; y$ then $f(x)\; \backslash leq\; f(y).${{sfn|Narici|Beckenstein|2011|pp=139-153}}

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 {{visible anchor|dual preorder}}.{{sfn|Narici|Beckenstein|2011|pp=139-153}}

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

Let $(X,\; \backslash leq)$ and $(Y,\; \backslash leq)$ be preordered vector spaces and let $\backslash mathcal\{L\}(X;\; Y)$ be the space of all linear maps from $X$ into $Y.$

The set $H$ of all positive linear operators in $\backslash mathcal\{L\}(X;\; Y)$ is a cone in $\backslash mathcal\{L\}(X;\; Y)$ that defines a preorder on $\backslash mathcal\{L\}(X;\; Y)$.

If $M$ is a vector subspace of $\backslash mathcal\{L\}(X;\; Y)$ and if $H\; \backslash cap\; M$ is a proper cone then this proper cone defines a {{visible anchor|canonical partial order}} on $M$ making $M$ into a partially ordered vector space.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

If $(X,\; \backslash leq)$ and $(Y,\; \backslash leq)$ are ordered topological vector spaces and if $\backslash mathcal\{G\}$ is a family of bounded subsets of $X$ whose union covers $X$ then the positive cone $\backslash mathcal\{H\}$ 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 $\backslash mathcal\{G\}$-topology.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

For $\backslash mathcal\{H\}$ 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.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

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 $\backslash mathcal\{H\}$ is a regular order.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

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.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

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 $\backslash mathcal\{U\}$ 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 $u\; =\; \backslash sup\; \backslash mathcal\{U\}$ exists and the section filter $\backslash mathcal\{F\}(\backslash mathcal\{U\})$ converges to $u$ uniformly on every precompact subset of $X.${{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

• {{annotated link|Cone-saturated}}
• {{annotated link|Positive linear functional}}
• {{annotated link|Vector lattice}}

{{reflist|group=note}}

{{reflist}}

• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}

{{Functional analysis}}

{{Ordered topological vector spaces}}

Category:Functional analysis

Category:Order theory