Positive linear operator

In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space [math]\displaystyle{ (X, \leq) }[/math] into a preordered vector space [math]\displaystyle{ (Y, \leq) }[/math]is a linear operator [math]\displaystyle{ f }[/math] on [math]\displaystyle{ X }[/math] into [math]\displaystyle{ Y }[/math] such that for all positive elements [math]\displaystyle{ x }[/math] of [math]\displaystyle{ X, }[/math] that is [math]\displaystyle{ x \geq 0, }[/math] it holds that [math]\displaystyle{ f(x) \geq 0. }[/math] 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 [math]\displaystyle{ f }[/math] on a preordered vector space is called positive if it satisfies either of the following equivalent conditions:

1. [math]\displaystyle{ x \geq 0 }[/math] implies [math]\displaystyle{ f(x) \geq 0. }[/math]
2. if [math]\displaystyle{ x \leq y }[/math] then [math]\displaystyle{ f(x) \leq f(y). }[/math]

The set of all positive linear forms on a vector space with positive cone [math]\displaystyle{ C, }[/math] called the dual cone and denoted by [math]\displaystyle{ C^*, }[/math] is a cone equal to the polar of [math]\displaystyle{ -C. }[/math] The preorder induced by the dual cone on the space of linear functionals on [math]\displaystyle{ X }[/math] is called the dual preorder.

The order dual of an ordered vector space [math]\displaystyle{ X }[/math] is the set, denoted by [math]\displaystyle{ X^+, }[/math] defined by [math]\displaystyle{ X^+ := C^* - C^*. }[/math]

Canonical ordering

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

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

Properties

Proposition: Suppose that [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are ordered locally convex topological vector spaces with [math]\displaystyle{ X }[/math] being a Mackey space on which every positive linear functional is continuous. If the positive cone of [math]\displaystyle{ Y }[/math] is a weakly normal cone in [math]\displaystyle{ Y }[/math] then every positive linear operator from [math]\displaystyle{ X }[/math] into [math]\displaystyle{ Y }[/math] is continuous.

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

See also

• Cone-saturated
• Positive linear functional

References

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.