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.

## Contents

* 1 Definition
* 2 Canonical ordering
* 3 Properties
* 4 See also
* 5 References

## 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][1]

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.[1]

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.[2]

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.[2] 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.[2] 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.[2]

## 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.[2]

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][2]

## See also

* Cone-saturated
* Positive linear functional

## References

1. ↑ 1.0 1.1 Narici & Beckenstein 2011, pp. 139-153.
2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 Schaefer & Wolff 1999, pp. 225–229.

* 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.

* v
* t
* e

Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
* Montel
* Nuclear
* Normed (norm)
* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
* Strong (polar
* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
* Bilinear (form
* operator
* sesquilinear)
* (Un)Bounded
* Closed
* Compact (on Hilbert spaces)
* (Dis)Continuous
* Densely defined
* Fredholm
* Hilbert–Schmidt
* Functionals (positive)
* Normal
* Nuclear
* Self-adjoint
* Strictly singular
* Trace class
* Transpose
* Unitary

Operator theory|

* Banach algebras
* C*-algebras
* Spectrum (C*-algebra
* radius)
* Spectral theory (of ODEs
* Spectral theorem)
* Polar decomposition
* Singular value decomposition

Theorems|

* Banach–Alaoglu
* Banach–Mazur
* Banach–Saks
* Banach–Schauder (open mapping)
* Banach–Steinhaus (Uniform boundedness)
* Bessel's inequality
* Cauchy–Schwarz inequality
* Closed graph
* Closed range
* Eberlein–Šmulian
* Freudenthal spectral
* Gelfand–Mazur
* Gelfand–Naimark
* Goldstine
* Hahn–Banach (hyperplane separation)
* Kakutani fixed-point
* Krein–Milman
* Lomonosov's invariant subspace
* Mackey–Arens
* Mazur's lemma
* M. Riesz extension
* Riesz representation
* Parseval's identity
* Schauder fixed-point

Analysis|

* Abstract Wiener space
* Bochner space
* Differentiation in Fréchet spaces
* Derivatives (Fréchet
* Gateaux
* functional
* holomorphic)
* Integrals (Bochner
* Dunford
* Gelfand–Pettis
* regulated
* Paley–Wiener
* weak)
* Functional calculus (Borel
* continuous
* holomorphic)
* Inverse function theorem (Nash–Moser theorem)
* Measures (Lebesgue
* Projection-valued
* Vector)
* Weakly measurable function

Types of sets|

* Absolutely convex
* Absorbing
* Balanced
* Bounded
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Radial
* Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Algebraic interior (core)
* Bounding points
* Convex hull
* Extreme point
* Interior
* Minkowski addition
* Polar

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Positive linear operator. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Positive_linear_operator&oldid=2645275"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template