Positive linear operator

In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space $(X,\le )$ into a preordered vector space $(Y,\le )$ is a linear operator $f$ on $X$ into $Y$ such that for all positive elements $x$ of $X,$ that is $x\ge 0,$ it holds that $f(x)\ge 0.$ 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. $x\ge 0$ implies $f(x)\ge 0.$
2. if $x\le y$ then $f(x)\le f(y).$

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 dual preorder.

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

Canonical ordering

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

If $(X,\le )$ and $(Y,\le )$ are ordered topological vector spaces and if $𝒢$ is a family of bounded subsets of $X$ whose union covers $X$ then the positive cone $\mathscr{H}$ 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 $𝒢$-topology. For $\mathscr{H}$ 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 $\mathscr{H}$ 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 $𝒰$ 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 $u=\sup 𝒰$ exists and the section filter $ℱ(𝒰)$ converges to $u$ uniformly on every precompact subset of $X.$

See also

• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets