Existence and extensions of positive linear operators
type: Item
instance of: scholarly article
title: Existence and extensions of positive linear operators
zbMATH Open document ID: 1197.47053
author: Nicolae Dăneţ
author: Rodica-Mihaela Dăneţ
published in: Positivity
publication date: 2009-03-27T00:00:00Z
review text: This article studies the existence of the extension of positive linear operators on ordered vector spaces. Since in this setting one needs to have the extension preserve order, classical results of this nature, such as the Hahn-Banach or Mazur-Orlicz theorems, do not apply immediately. The authors observe that, if one assumes a monotonicity condition, then the Mazur-Orlicz theorem extends to the ordered vector space setting: Theorem. Let \(E\) be an ordered vector space, \(F\) be a Dedekind complete ordered vector space, and \(S:E\to F\) a monotone sublinear operator. Let \(K\) be a nonempty convex set, \(P:K\to E\) a convex operator, and \(Q:K\to F\) a concave operator. Then the following conditions are equivalent: {\parindent=7mm \begin{itemize}\item[(i)]There exists a positive linear operator \(L:E\to F\) with the properties (a) \(L\leq S\) on \(E\), and (b) \(Q\leq L\circ P \) on \(K\). \item[(ii)]The inequality \(Q\leq S\circ P \) holds on \(K\). \end{itemize}} As applications, they also obtain some new (and generalizations of some known) results on the extension of combinations of various operators and a minimax theorem. The main result of the paper is the following theorem that provides a sufficient condition for the existence of a monotone sublinear operator: Theorem. Let \(E_0\) be an ordered vector space, \(F\) a Dedekind complete ordered vector space, \(A\) and \(M\) nonempty sets. Consider the maps \(g:A\to E_0,\) \(h:M\to (E_0)_+\), \(f:A\to F\), \(r:M\to F\) and let \(E=\text{span}(g(A)\cup h(M)) \subset E_0\). Suppose that the following implication holds: \[ \sum_{i=1}^{n} \mu_i g(a_i) \leq \sum_{i=1}^{n} h(z_i) \Rightarrow \sum_{i=1}^{n} \mu_i f(a_i) \leq \sum_{i=1}^{n} r(z_i), \] for all finite sets \(\{a_i\} \subset A\), \(\{z_i\} \subset M\), \(\{m_i\} \subset \mathbb R\). Then there exists a monotone sublinear operator \(S:E\to F\) such that (a) \(S\circ g \leq f \) on \(A\) and (b) \(S\circ h \leq r \) on \(M\). The authors also study some parti…
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46A22
Mathematics Subject Classification ID: 47B60
zbMATH DE Number: 5534433
zbMATH Keywords: Mazur-Orlicz theorem
zbMATH Keywords: positive linear operators
zbMATH Keywords: sublinear operators
zbMATH Keywords: ordered vector spaces
MaRDI profile type: Publication
full work available at URL: s11117-008-2220-0
OpenAlex ID: W2314211085
cites work: Q4084773
cites work: Q4114319
cites work: Q3828504
cites work: Q3935671
cites work: Q3216033
cites work: Sur les espaces métriques linéaires (II)
cites work: Banach lattices
cites work: On the extension of positive linear functionals
cites work: On a theorem of Mazur and Orlicz
cites work: Q4243024
DOI: 10.1007/S11117-008-2220-0
description: scientific article; zbMATH DE number 5534433