Approximation theorems for positive operators on \(L^ p\)-spaces
type: Item
instance of: scholarly article
title: Approximation theorems for positive operators on \(L^ p\)-spaces
zbMATH Open document ID: 0727.41034
DOI: 10.1016/0021-9045(90)90097-A
author: Ryszard Grząślewicz
published in: Journal of Approximation Theory
publication date: 1990-01-01T00:00:00Z
review text: For \(T\in {\mathcal L}(L^ p(m),L^ r(n))\), where (X,\({\mathcal A},m)\) and (Y,\({\mathcal B},n)\) are \(\sigma\)-finite mesure spaces, let \(M(T)=\{f\in L^ p(m):\| Tf\| =\| T\| \| f\| \}\) and \(J(T)=\{\sup p f: f\in M(T)\}\). A linear bounded operator T is said to be positive (T\(\geq 0)\) if Tf\(\geq 0\) whenever \(f\geq 0\). The set of all positive operators is denoted by \({\mathcal L}_+(L^ p(m),L^ r(n))\) and the positive part of the unit ball of \({\mathcal L}(L^ p(m),L^ r(n))\) is denoted by \({\mathcal P}\). Let \(1<r\leq p<\infty\) and \({\mathcal N}=\{T\in {\mathcal L}_+(L^ p(m),L^ r(n):\) \(X\in J(T),Y\in J(T^*)\}\), then the author proves that \({\mathcal N}\) is norm dense in \({\mathcal L}_+(L^ p(m),L^ r(n))\) and the convex hull conv(\({\mathcal N}\cap ext {\mathcal P}\}\) is strong operator dense in the positive part of the unit sphere of \({\mathcal L}(L^ p(m),L^ r(n))\). The author also considers extreme points of the positive part of the unit ball of \({\mathcal L}(\ell^ p,\ell^ r)\), and proves that, if \(0\neq T\in {\mathcal L}_+(\ell^ p,\ell^ r)\cap {\mathcal N}\), and \(1<r\leq p<\infty\), \(T\in {\mathcal N}\cap ext{\mathcal P}\) if and only if \(\| T\| =1\) and the connected components of the graph G(T) are trees. Furthermore, he proves that, when \(1<r\leq p<\infty\) and \(T\in {\mathcal L}(\ell^ p,\ell^ r)\) if \(T\in {\mathcal N}\cap ext{\mathcal P}\), then T is exposed but T is not a strongly exposed point of \({\mathcal P}\).
reviewed by: Zhang Guanglu
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 4200876
MaRDI profile type: Publication
cites work: Doubly stochastic measures and Markov operators
cites work: Approximation theorems for Markov operators
cites work: On extreme doubly stochastic measures
cites work: Q5834711
cites work: Convex Sets of Non-Negative Matrices
cites work: Q3844775
cites work: The support of discrete extremal measures with given marginals
cites work: On extremal measures and subspace density
cites work: Q3254057
cites work: Q3784675
cites work: Extreme Positive Contractions on Finite Dimensional lp-Spaces
cites work: Exposed points of the unit ball of \({\mathcal L}(H)\)
cites work: On extreme infinite doubly stochastic matrices
cites work: On isometric domains of positive operators on $L^p$-spaces
cites work: Compact extremal operators
cites work: Q3930254
cites work: Term ranks and permanents of nonnegative matrices
cites work: Approximations of doubly substochastic operators
cites work: Approximations of positive contractions onL? [0,1]
cites work: Approximations of positive contractions on \(L^\infty\). II
cites work: Extreme contraction operators on \(\ell_\infty\)
cites work: Représentation des mesures de probabilité sur le produit de deux espaces denombrables, de marges données
cites work: A Remark on Extreme Doubly Stochastic Measures
cites work: On operators which attain their norm
cites work: Counter-examples to some conjectures about doubly stochastic measures
cites work: Supports of extremal measures with given marginals
cites work: Q4183861
cites work: On the representation of doubly stochastic operators
cites work: Doubly stochastic measures with prescribed support
full work available at URL: 0021-9045(90)90097-a
OpenAlex ID: W1969751999
description: scientific article; zbMATH DE number 4200876