Positive linear operator

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.

Canonical ordering

Throughout let (X, ≤) and (Y, ≤) be preordered vector spaces on X and let ${\mathcal {L}}(X;Y)$ be the space of all linear maps from X into Y. The set H of all positive linear operators in ${\mathcal {L}}(X;Y)$ is a cone in ${\mathcal {L}}(X;Y)$ that defines a preorder on ${\mathcal {L}}(X;Y)$. If M is a vector subspace of ${\mathcal {L}}(X;Y)$ 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.

If (X, ≤) and (Y, ≤) are ordered topological vector spaces (ordered TVSs) and if ${\mathcal {G}}$ is a family of bounded subsets of X whose union covers X then the positive cone ${\mathcal {H}}$ 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 ${\mathcal {G}}$-topology. For ${\mathcal {H}}$ to be a proper cone in $L(X;Y)$ 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. 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 ${\mathcal {H}}$ is a regular order.

Properties

Proposition:Suppose that X and Y are ordered locally convex topological vector spaces (TVSs) 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.

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 ${\mathcal {U}}$ 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 $u=\sup {\mathcal {U}}$ exists and the section filter ${\mathcal {F}}\left({\mathcal {U}}\right)$ converges to u uniformly on every precompact subset of X.

See also

• Cone-saturated
• Positive linear functional
• Vector lattice

References

• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. {{cite book}}: Invalid |ref=harv (help)