Positive linear operator

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In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space ( X , ≤ ) {\displaystyle (X,\leq )} into a preordered vector space ( Y , ≤ ) {\displaystyle (Y,\leq )} is a linear operator f {\displaystyle f} on X {\displaystyle X} into Y {\displaystyle Y} such that for all positive elements x {\displaystyle x} of X , {\displaystyle X,} that is x ≥ 0 , {\displaystyle x\geq 0,} it holds that f ( x ) ≥ 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 f {\displaystyle f} on a preordered vector space is called positive if it satisfies either of the following equivalent conditions:

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

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

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

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

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

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

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

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

• 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.

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