Riesz representation theorems for positive linear operators

TL;DR

This paper extends Riesz representation theorems to positive linear operators into various ordered spaces, providing explicit formulas for representing measures and broadening applicability beyond classical settings.

Contribution

It generalizes Riesz representation theorems to positive linear operators into partially ordered vector spaces, including non-lattice and non-normed spaces, with explicit measure formulas.

Findings

Representing measures exist for operators into Banach lattices with order continuous norms.

Explicit formulas for measures on open and compact sets are provided.

Results apply to operators into JBW-algebras and other advanced ordered spaces.

Abstract

We generalise the Riesz representation theorems for positive linear functionals on $\mathrm{C}_{\mathrm c}(X)$ and $\mathrm{C}_{\mathrm 0}(X)$, where $X$ is a locally compact Hausdorff space, to positive linear operators from these spaces into a partially ordered vector space $E$. The representing measures are defined on the Borel $\sigma$-algebra of $X$ and take their values in the extended positive cone of $E$; the corresponding integrals are order integrals. We give explicit formulas for the values of the representing measures at open and at compact subsets of $X$. Results are included where the space $E$ need not be a vector lattice, nor a normed space. Representing measures exist for positive linear operators into Banach lattices with order continuous norms, into the regular operators on a KB-space, into the self-adjoint linear operators in a weakly closed complex linear subspace of the bounded linear operators on a complex Hilbert space, and into JBW-algebras.