Positive linear operator


In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space into a preordered vector space 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 ≥ 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 and be preordered vector spaces and let be the space of all linear maps from X into Y.
The set H of all positive linear operators in is a cone in that defines a preorder on.
If M is a vector subspace of 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.
If and are ordered topological vector spaces and if is a family of bounded subsets of X whose union covers X then the positive cone in, which is the space of all continuous linear maps from X into Y, is closed in when is endowed with the -topology.
For to be a proper cone in it is sufficient that the positive cone of X be total 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 defined by is a regular order.

Properties