Positive linear operator

Short description: Concept in functional analysis

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In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space (X,≤) into a preordered vector space (Y,≤) is a linear operator f on X into Y such that for all positive elements x of X, that is x≥0, it holds that f(x)≥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.

## Contents

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

## Definition

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

1. x≥0 implies f(x)≥0.
2. if x≤y then f(x)≤f(y).[1]

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.[1]

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

## Canonical ordering

Let (X,≤) and (Y,≤) 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∩M is a proper cone then this proper cone defines a canonical partial order on M making M into a partially ordered vector space.[2]

If (X,≤) and (Y,≤) are ordered topological vector spaces and if 𝒢 is a family of bounded subsets of X whose union covers X then the positive cone 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.[2] For 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.[2] 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 ℋ is a regular order.[2]

## 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.[2]

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.[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.

## External links

* Positive linear operators and approximation properties

* A Class Of Positive Linear Operators And Applications To Nonlinear Boundary Value Problems

* Note On Positive Linear Operators

* 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

Template:Ordered topological vector spaces

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