Operator norm limits of order continuous operators
type: Item
instance of: scholarly article
title: Operator norm limits of order continuous operators
zbMATH Open document ID: 1116.47032
author: N. E. Zubov
published in: Positivity
publication date: 2006-11-03T00:00:00Z
review text: Let \(X\) and \(Y\) be Banach lattices, and denote by \({\mathcal L}^b(X, Y)\) the space of order bounded linear operators from \(X\) into \(Y\) equipped with the order bound norm, a norm introduced by the second author [in: Functional analysis and economic theory, Samos, Greece, July 1996, 109--118 (1998; Zbl 0912.47018)]. Moreover, let \({\mathcal L}^r(X, Y)\) stand for the subspace of \({\mathcal L}^b(X, Y)\) consisting of regular operators equipped with the regular norm. Let \((T_n)\) be a sequence convergent to \(T\) in \({\mathcal L}^b(X, Y)\) [resp., \({\mathcal L}^r(X,Y)\)] or in the operator norm \(\|\cdot\|\). Assuming that \(T_n\), \(n= 1,2,\dots\), are continuous with respect to what the authors call weak order convergence (for order bounded nets), which amounts to order convergence if the vector lattice in question is Dedekind complete, they prove that \(T\) has the same continuity property. In the case of \(\|\cdot\|\), this is done under the extra condition that the Lorenz seminorm of \(Y\) be a norm. A counterexample is constructed to show that this condition may not be dispensed with entirely. In a note added in proof, the authors refer to an independent paper by \textit{Yu.\,Abramovich} and \textit{G.\,Sirotkin} [Positivity 9, No.\,3, 287--292 (2005; Zbl 1110.46002)], where some variants of the notion of order convergence are studied. Reviewer's remark: The presentation of the counterexample needs some corrections. In particular, in the definition of \(\psi\), one should replace ``\(2^t\)'' by ``\(2^n t\)''. Moreover, the description of \(\psi^{-1}(x)\) is wrong and the definition of \(S\) on p.\,353 is unclear.
reviewed by: Zbigniew Lipecki
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 06F20
zbMATH DE Number: 5070778
zbMATH Keywords: Banach lattice
zbMATH Keywords: order convergence
zbMATH Keywords: order bounded operator
zbMATH Keywords: order continuous operator
zbMATH Keywords: regular norm
MaRDI profile type: Publication
full work available at URL: s11117-005-8513-7
OpenAlex ID: W2076973227
cites work: Q4781239
cites work: Q4438875
cites work: Q5789040
cites work: Bands in lattices of operators
cites work: Q4213341
DOI: 10.1007/S11117-005-8513-7
Recommended article: On order convergence of nets
Recommended article: Q3972421
Recommended article: A GENERALIZATION OF ORDER CONVERGENCE IN THE VECTOR LATTICES
Recommended article: On the notion of stability of order convergence in vector lattices
Recommended article: Q3200099
Recommended article: Unbounded absolute weak convergence in Banach lattices
Recommended article: Q3011617
Recommended article: On the Brezis–Lieb Lemma and Its Extensions
Recommended article: A strong convergence in vector lattices with applications
Recommended article: Some loose ends on unbounded order convergence
description: scientific article; zbMATH DE number 5070778