Riesz representation theorems for positive linear operators

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 repres...

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Veröffentlicht in:arXiv.org
Hauptverfasser: de Jeu, Marcel, Jiang, Xingni
Format: Paper
Sprache:Englisch
Veröffentlicht: Ithaca Cornell University Library, arXiv.org 05.01.2022
Schlagworte:
Hilbert space
Integrals
Lattices (mathematics)
Linear operators
Norms
Operators (mathematics)
Representations
Theorems
ISSN:2331-8422
Online-Zugang:Volltext
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Zusammenfassung: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.
Bibliographie:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
content type line 50
ISSN:2331-8422
DOI:10.48550/arxiv.2104.12153