On positive-off-diagonal operators on ordered normed spaces
type: Item
instance of: scholarly article
title: On positive-off-diagonal operators on ordered normed spaces
zbMATH Open document ID: 1055.47035
DOI: 10.4171/ZAA/1142
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2003-01-01T00:00:00Z
review text: Let \((X,K, \| . \|)\) be an ordered normed space, where \(X\) is a real vector space, \(\| . \|\) is a norm on \(X\) and \(K\) is a cone in \(X\) with \(K \cap (-K)= \{0\}\). Let \(X'\) denote the vector space of all continuous linear functionals on \(X\) and \(L(X)\) the vector space of all continuous linear operators on \(X\) and \(I \in L(X)\) the identity operator. \(K'\) denotes the wedge of all positive functionals in \(X'\). An operator \(A \in L(X)\) is called positive-off-diagonal if \(x \in K\) and \(f \in K'\) with \(f(x)=0\) imply \(f(Ax) \geq 0.\) In the paper under review, the main result is that if \((X,K, \| .\|)\) is an ordered normed space that satisfies the Riesz decomposition property and \(K\) is a closed normal non-flat \(b\)-generating cone and there exists a total set of extremals of \(K'\), then for any operator \(A \in L(X)\) the following statements are equivalent: (i) \(A\) is a positive-off-diagonal operator; (ii) \(\| A \| I + A \geq 0 .\) Among other things, the author gives two examples. The first example shows that a positive-off-diagonal operator \(A\) in general cannot be represented as \(A=sI + B\) with a positive operator \(B\) and a real number \(s\), even if \(A\) operates on a finite-dimensional space. In the second example, an ordered normed space that is not a vector lattice satisfying all assumptions of the main result is given.
reviewed by: Ömer Gök
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46B40
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 1895124
zbMATH Keywords: positive-off-diagonal operators
zbMATH Keywords: ordered normed spaces
zbMATH Keywords: Riesz decomposition property
zbMATH Keywords: non-flat cone
zbMATH Keywords: normal cone
zbMATH Keywords: b-generating cone
zbMATH Keywords: Archimedean vector lattice
zbMATH Keywords: generator of a semigroup
author: Anke Kalauch
MaRDI profile type: Publication
cites work: Q3748810
cites work: The Riesz-Kantorovich formula and general equilibrium theory
cites work: One-parameter semigroups of positive operators
full work available at URL: https://doi.org/10.4171/zaa/1142
OpenAlex ID: W2070024116
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description: scientific article; zbMATH DE number 1895124