Positive linear operator explained

(X, \leq) into a preordered vector space (Y, \leq) is a linear operator f on X into Y such that for all positive elements x of X, that is x \geq 0 , it holds that 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

f on a preordered vector space is called positive if it satisfies either of the following equivalent conditions: x \geq 0 implies f(x) \geq 0 .

1. if

x \leq y then f(x) \leq f(y).

The set of all positive linear forms on a vector space with positive cone

C, called the dual cone and denoted by C ^{*,} is a cone equal to the polar of - C. The preorder induced by the dual cone on the space of linear functionals on X is called the .

The order dual of an ordered vector space

X is the set, denoted by X ^{+,} defined by X ^{+} : = C ^{*} - C ^{*.}

Canonical ordering

Let

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

If

(X, \leq) and (Y, \leq) are ordered topological vector spaces and if l{G} is a family of bounded subsets of X whose union covers X then the positive cone l{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 l{G} -topology. For l{H} to be a proper cone in L(X; Y) it is sufficient that the positive cone of X be total in X (that is, 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 l{H} is a regular order.

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.

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 l{U} be a subset of L(X; Y) that is directed upward and either majorized (that is, bounded above by some element of L(X; Y) ) or simply bounded. Then u = \sup l{U} exists and the section filter l{F}( l{U}) converges to u uniformly on every precompact subset of X.