Jump to content

Positive linear operator

From Wikipedia, the free encyclopedia

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

[edit]

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

[edit]

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

[edit]

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

[edit]

References

[edit]
  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.