Positive linear operator: Difference between revisions

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{{Short description|Concept in functional analysis}}

{{Multiple issues|{{refimprove|date=June 2020}}{{lead rewrite|date=June 2020|reason=The lead should be a summary of the body of the article.}}}}

In [[mathematics]], more specifically in [[functional analysis]], a '''positive linear operator''' from an [[orderedOrdered vector space|preordered vector space]] <math>(''X'', ≤\leq)</math> into a preordered vector space <math>(''Y'', ≤\leq)</math> is a [[linear operator]] ''<math>f''</math> on ''<math>X''</math> into ''<math>Y''</math> such that for all [[positivePositive element (ordered group)|positive element]]s ''<math>x''</math> of ''<math>X'',</math> that is ''<math>x'' ≥\geq 0,</math> it holds that ''<math>f''(''x'') ≥\geq 0.</math>

In other words, a positive linear operator maps the positive cone of the [[___domainDomain of a function|___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]].

== Canonical orderingDefinition ==

Let (''X'', ≤) and (''Y'', ≤) be preordered vector spaces and let <math>\mathcal{L}(X; Y)</math> be the space of all linear maps from ''X'' into ''Y''.

The set ''H'' of all positive linear operators in <math>\mathcal{L}(X; Y)</math> is a cone in <math>\mathcal{L}(X; Y)</math> that defines a preorder on <math>\mathcal{L}(X; Y)</math>.

If ''M'' is a vector subspace of <math>\mathcal{L}(X; Y)</math> 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.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

A [[linear function]] <math>f</math> on a [[Ordered vector space|preordered vector space]] is called '''positive''' if it satisfies either of the following equivalent conditions:

If (''X'', ≤) and (''Y'', ≤) are [[ordered topological vector space]]s and if <math>\mathcal{G}</math> is a family of bounded subsets of ''X'' whose union covers ''X'' then the [[positive cone]] <math>\mathcal{H}</math> in <math>L(X; Y)</math>, which is the space of all continuous linear maps from ''X'' into ''Y'', is closed in <math>L(X; Y)</math> when <math>L(X; Y)</math> is endowed with the [[topology of uniform convergence|<math>\mathcal{G}</math>-topology]].{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

For <math>\mathcal{H}</math> to be a proper cone in <math>L(X; Y)</math> it is sufficient that the positive cone of ''X'' be total in ''X'' (i.e. 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 <math>L(X; Y)</math> defined by <math>\mathcal{H}</math> is a regular order.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

# <math>x \geq 0</math> implies <math>f(x) \geq 0.</math>

== Properties ==

# if <math>x \leq y</math> then <math>f(x) \leq f(y).</math>{{sfn|Narici|Beckenstein|2011|pp=139-153}}

The set of all positive linear forms on a vector space with positive cone <math>C,</math> called the '''[[Dual cone and polar cone|dual cone]]''' and denoted by <math>C^*,</math> is a cone equal to the [[Polar set|polar]] of <math>-C.</math>

:'''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 [[normal cone (functional analysis)|weakly normal cone]] in ''Y'' then every positive linear operator from ''X'' into ''Y'' is continuous.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

The preorder induced by the dual cone on the space of linear functionals on <math>X</math> is called the '''{{visible anchor|dual preorder}}'''.{{sfn|Narici|Beckenstein|2011|pp=139-153}}

The '''[[Order dual (functional analysis)|order dual]]''' of an ordered vector space <math>X</math> is the set, denoted by <math>X^+,</math> defined by <math>X^+ := C^* - C^*.</math>

:'''Proposition''': Suppose ''X'' is a [[barreled space|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 (functional analysis)|normal cone]]. Give ''L''(''X''; ''Y'') its canonical order and let <math>\mathcal{U}</math> be a subset of ''L''(''X''; ''Y'') that is directed upward and either majorized (i.e. bounded above by some element of ''L''(''X''; ''Y'')) or simply bounded. Then <math>u = \sup \mathcal{U}</math> exists and the section filter <math>\mathcal{F}\left( \mathcal{U} \right)</math> converges to ''u'' uniformly on every precompact subset of ''X''.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

==Canonical See also ordering==

Let <math>(''X'', ≤\leq)</math> and <math>(''Y'', ≤\leq)</math> be preordered vector spaces and let <math>\mathcal{L}(X; Y)</math> be the space of all linear maps from ''<math>X''</math> into ''<math>Y''.</math>

* [[Cone-saturated]]

The set ''<math>H''</math> of all positive linear operators in <math>\mathcal{L}(X; Y)</math> is a cone in <math>\mathcal{L}(X; Y)</math> that defines a preorder on <math>\mathcal{L}(X; Y)</math>.

* [[Positive linear functional]]

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

* [[Vector lattice]]

If <math>(''X'', ≤\leq)</math> and <math>(''Y'', ≤\leq)</math> are [[ordered topological vector space]]s and if <math>\mathcal{G}</math> is a family of bounded subsets of ''<math>X''</math> whose union covers ''<math>X''</math> then the [[positive cone of an ordered vector space|positive cone]] <math>\mathcal{H}</math> in <math>L(X; Y)</math>, which is the space of all continuous linear maps from ''<math>X''</math> into ''<math>Y'',</math> is closed in <math>L(X; Y)</math> when <math>L(X; Y)</math> is endowed with the [[topologyTopology of uniform convergence|<math>\mathcal{G}</math>-topology]].{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

== References ==

For <math>\mathcal{H}</math> to be a proper cone in <math>L(X; Y)</math> it is sufficient that the positive cone of ''<math>X''</math> be total in ''<math>X''</math> (i.e.that is, the span of the positive cone of ''<math>X''</math> be dense in ''<math>X''</math>).

If ''<math>Y''</math> 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 ''<math>X''</math> is total in ''<math>X''</math> and if ''<math>Y''</math> is a locally convex space, then the canonical ordering of <math>L(X; Y)</math> defined by <math>\mathcal{H}</math> is a regular order.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

== Properties ==

{{Reflist}}

* {{cite book | last=Schaefer | first=Helmut H. | authorlink=Helmut H. Schaefer | last2=Wolff | first2=Manfred P. | title=Topological Vector Spaces | publisher=Springer New York Imprint Springer | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | publication-place=New York, NY | year=1999 | isbn=978-1-4612-7155-0 | oclc=840278135 | ref=harv}} <!-- {{sfn | Schaefer | Wolff | 1999 | p=}} -->

:'''Proposition''': Suppose that ''<math>X''</math> and ''<math>Y''</math> are ordered [[locally convex]] topological vector spaces with ''<math>X''</math> being a [[Mackey space]] on which every [[positive linear functional]] is continuous. If the positive cone of ''<math>Y''</math> is a [[normalNormal cone (functional analysis)|weakly normal cone]] in ''<math>Y''</math> then every positive linear operator from ''<math>X''</math> into ''<math>Y''</math> is continuous.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

{{Functional analysis}}

:'''Proposition''': Suppose ''<math>X''</math> is a [[barreledBarreled space|barreled]] [[ordered topological vector space]] (TVS) with positive cone ''<math>C''</math> that satisfies ''<math>X'' = ''C'' - ''C''</math> and ''<math>Y''</math> is a [[semi-reflexive]] ordered TVS with a positive cone ''<math>D''</math> that is a [[normalNormal cone (functional analysis)|normal cone]]. Give ''<math>L''(''X''; ''Y'')</math> its canonical order and let <math>\mathcal{U}</math> be a subset of ''<math>L''(''X''; ''Y'')</math> that is directed upward and either majorized (i.e.that is, bounded above by some element of ''<math>L''(''X''; ''Y'')</math>) or simply bounded. Then <math>u = \sup \mathcal{U}</math> exists and the section filter <math>\mathcal{F}\left( \mathcal{U} \right)</math> converges to ''<math>u''</math> uniformly on every precompact subset of ''<math>X''.</math>{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}

<!--- Categories --->

==See also==

* [[{{annotated link|Cone-saturated]]}}

* [[{{annotated link|Positive linear functional]]}}

* [[{{annotated link|Vector lattice]]}}

== References ==

{{reflist|group=note}}

{{reflist}}

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | Beckenstein | 2011 | p=}} -->

* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | Wolff | 1999 | p=}} -->

{{Functional analysis}}

{{Ordered topological vector spaces}}

<!--- Categories --->

[[Category:Functional analysis]]

[[Category:Order theory]]