Positive linear operator

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

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

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

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

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

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

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

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

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

• Cone-saturated
• Positive linear functional
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• 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.