Positive linear operator

In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space into a preordered vector space is a linear operator on into such that for all positive elements of that is it holds that 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

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

  1.   implies  
  2. if   then  [1]

The set of all positive linear forms on a vector space with positive cone   called the dual cone and denoted by   is a cone equal to the polar of   The preorder induced by the dual cone on the space of linear functionals on   is called the dual preorder.[1]

The order dual of an ordered vector space   is the set, denoted by   defined by  

Canonical ordering

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Let   and   be preordered vector spaces and let   be the space of all linear maps from   into   The set   of all positive linear operators in   is a cone in   that defines a preorder on  . If   is a vector subspace of   and if   is a proper cone then this proper cone defines a canonical partial order on   making   into a partially ordered vector space.[2]

If   and   are ordered topological vector spaces and if   is a family of bounded subsets of   whose union covers   then the positive cone   in  , which is the space of all continuous linear maps from   into   is closed in   when   is endowed with the  -topology.[2] For   to be a proper cone in   it is sufficient that the positive cone of   be total in   (that is, the span of the positive cone of   be dense in  ). If   is a locally convex space of dimension greater than 0 then this condition is also necessary.[2] Thus, if the positive cone of   is total in   and if   is a locally convex space, then the canonical ordering of   defined by   is a regular order.[2]

Properties

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Proposition: Suppose that   and   are ordered locally convex topological vector spaces with   being a Mackey space on which every positive linear functional is continuous. If the positive cone of   is a weakly normal cone in   then every positive linear operator from   into   is continuous.[2]

Proposition: Suppose   is a barreled ordered topological vector space (TVS) with positive cone   that satisfies   and   is a semi-reflexive ordered TVS with a positive cone   that is a normal cone. Give   its canonical order and let   be a subset of   that is directed upward and either majorized (that is, bounded above by some element of  ) or simply bounded. Then   exists and the section filter   converges to   uniformly on every precompact subset of  [2]

See also

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References

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  1. ^ a b Narici & Beckenstein 2011, pp. 139–153.
  2. ^ a b c d e f 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.