Positive linear operator

In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space ${\displaystyle (X,\leq )}$ into a preordered vector space ${\displaystyle (Y,\leq )}$ is a linear operator ${\displaystyle f}$ on ${\displaystyle X}$ into ${\displaystyle Y}$ such that for all positive elements ${\displaystyle x}$ of ${\displaystyle X,}$ that is ${\displaystyle x\geq 0,}$ it holds that ${\displaystyle f(x)\geq 0.}$ 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.

Definition

A linear function ${\displaystyle f}$ on a preordered vector space is called positive if it satisfies either of the following equivalent conditions:

1. ${\displaystyle x\geq 0}$ implies ${\displaystyle f(x)\geq 0.}$
2. if ${\displaystyle x\leq y}$ then ${\displaystyle f(x)\leq f(y).}$^[1]

The set of all positive linear forms on a vector space with positive cone ${\displaystyle C,}$ called the dual cone and denoted by ${\displaystyle C^{*},}$ is a cone equal to the polar of ${\displaystyle -C.}$ The preorder induced by the dual cone on the space of linear functionals on ${\displaystyle X}$ is called the dual preorder.^[1]

The order dual of an ordered vector space ${\displaystyle X}$ is the set, denoted by ${\displaystyle X^{+},}$ defined by ${\displaystyle X^{+}:=C^{*}-C^{*}.}$

Canonical ordering

Let ${\displaystyle (X,\leq )}$ and ${\displaystyle (Y,\leq )}$ be preordered vector spaces and let ${\displaystyle {\mathcal {L}}(X;Y)}$ be the space of all linear maps from ${\displaystyle X}$ into ${\displaystyle Y.}$ The set ${\displaystyle H}$ 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 ${\displaystyle M}$ is a vector subspace of ${\displaystyle {\mathcal {L}}(X;Y)}$ and if ${\displaystyle H\cap M}$ is a proper cone then this proper cone defines a canonical partial order on ${\displaystyle M}$ making ${\displaystyle M}$ into a partially ordered vector space.^[2]

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

Properties

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

Proposition: Suppose ${\displaystyle X}$ is a barreled ordered topological vector space (TVS) with positive cone ${\displaystyle C}$ that satisfies ${\displaystyle X=C-C}$ and ${\displaystyle Y}$ is a semi-reflexive ordered TVS with a positive cone ${\displaystyle D}$ that is a normal cone. Give ${\displaystyle L(X;Y)}$ its canonical order and let ${\displaystyle {\mathcal {U}}}$ be a subset of ${\displaystyle L(X;Y)}$ that is directed upward and either majorized (that is, bounded above by some element of ${\displaystyle L(X;Y)}$) or simply bounded. Then ${\displaystyle u=\sup {\mathcal {U}}}$ exists and the section filter ${\displaystyle {\mathcal {F}}({\mathcal {U}})}$ converges to ${\displaystyle u}$ uniformly on every precompact subset of ${\displaystyle X.}$^[2]

See also

• Cone-saturated
• Positive linear functional – ordered vector space with a partial orderPages displaying wikidata descriptions as a fallback
• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets

References

1. Narici & Beckenstein 2011, pp. 139–153.
2. 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.