Riesz representation theorems for positive linear operators
We generalise the Riesz representation theorems for positive linear functionals on C c ( X ) and C 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 σ -al...
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| Vydané v: | Banach journal of mathematical analysis Ročník 16; číslo 3 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Cham
Springer International Publishing
01.07.2022
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| Predmet: | |
| ISSN: | 2662-2033, 1735-8787 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | We generalise the Riesz representation theorems for positive linear functionals on
C
c
(
X
)
and
C
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
σ
-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 example, for positive linear operators into Banach lattices with order continuous norms, into the regular operators on KB-spaces, into the self-adjoint linear operators on complex Hilbert spaces, and into JBW-algebras. |
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| ISSN: | 2662-2033 1735-8787 |
| DOI: | 10.1007/s43037-022-00177-7 |