Positive linear operator

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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.

Canonical ordering[]

Let (X, ≤) and (Y, ≤) be preordered vector spaces and let ${\displaystyle {\mathcal {L}}(X;Y)}$ be the space of all linear maps from X into Y. The set H of all positive linear operators in ${\displaystyle {\mathcal {L}}(X;Y)}$ is a cone in ${\displaystyle {\mathcal {L}}(X;Y)}$ that defines a preorder on ${\displaystyle {\mathcal {L}}(X;Y)}$. If M is a vector subspace of ${\displaystyle {\mathcal {L}}(X;Y)}$ 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.^[1]

If (X, ≤) and (Y, ≤) are ordered topological vector spaces and if ${\displaystyle {\mathcal {G}}}$ is a family of bounded subsets of X whose union covers X then the positive cone ${\displaystyle {\mathcal {H}}}$ in ${\displaystyle L(X;Y)}$, which is the space of all continuous linear maps from X into Y, is closed in ${\displaystyle L(X;Y)}$ when ${\displaystyle L(X;Y)}$ is endowed with the ${\displaystyle {\mathcal {G}}}$-topology.^[1] For ${\displaystyle {\mathcal {H}}}$ to be a proper cone in ${\displaystyle L(X;Y)}$ 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.^[1] Thus, if the positive cone of X is total in X and if Y is a locally convex space, then the canonical ordering of ${\displaystyle L(X;Y)}$ defined by ${\displaystyle {\mathcal {H}}}$ is a regular order.^[1]

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.^[1]

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 ${\displaystyle {\mathcal {U}}}$ 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 ${\displaystyle u=\sup {\mathcal {U}}}$ exists and the section filter ${\displaystyle {\mathcal {F}}\left({\mathcal {U}}\right)}$ converges to u uniformly on every precompact subset of X.^[1]

See also[]

• Cone-saturated
• Positive linear functional
• Vector lattice

References[]

• 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.