Domination by positive narrow operators
type: Item
instance of: scholarly article
title: Domination by positive narrow operators
zbMATH Open document ID: 1051.47032
DOI: 10.1023/A:1026211909760
author: Julio Flores
author: César Ruiz
published in: Positivity
publication date: 2004-01-06T00:00:00Z
review text: Let \(E (\mu)\) be a Köthe function space on a probability space \((\Omega, \Sigma, \mu)\) with an atomless measure \(\mu\) and \(Y\) be a Banach space. An operator \(T: E (\mu) \to Y\) is called narrow if for each measurable set \(A \in \Sigma\) and very \(\varepsilon > 0\) there is a partition \(\{A_1, A_2\}\) of \(A\) with \(\mu (A_1) = \mu (A_2)\) with \(\| T (\chi_{A_1} - \chi_{A_2}) \| < \varepsilon\). \textit{A. M. Plichko} and \textit{M. M. Popov} asked the following question in [``Symmetric function spaces on atomless probability spaces'' (Diss. Math. 306) (1990; Zbl 0715.46011)]: Let \(1 < p < \infty\), \(p \neq 2\) and \(Y\) a Banach space. Is every strictly singular operator \(T: L^p (\Omega, \Sigma, \mu) \to Y\) narrow? Is every \(\ell^2\)-singular operator \(T: L^p (\Omega, \Sigma, \mu) \to Y\) narrow? The authors answer this question in the setting of regular operators between Banach lattices. More precisely, they obtain the following Theorem: Let \(F\) be an order continuous Banach lattice and \((\Omega, \Sigma, \mu)\) be a probability space with an atomless measure \(\mu\). Let \(T: L^p (\Omega, \Sigma, \mu) \to F\) be a regular operator, \(1 \leq p \leq \infty\). If \(T\) is \(\ell^2\)-singular, then \(T\) is narrow. Among other things, they note that the converse of this theorem is not true (Example 2.8). The authors also study the problem of domination by positive narrow operators. That is, if \(F\) is a Banach lattice and \(0 \leq S \leq T\): \(E (\mu) \to F\) are two positive operators with \(T\) narrow, which conditions on \(E (\mu)\) and \(F\) ensure that \(S\) is narrow? They show, in general, the domination problem in narrow operators has a negative answer (Example 3.3). They obtain a positive result when the norm on \(F\) is order continuous. Theorem: Let \((\Omega, \Sigma, \mu)\) be a probability space with an atomless measure \(\mu\) and \(E (\mu)\) a Köthe function space on \((\Omega, \Sigma, \mu)\). Let \(F\) be an order continuous Ban…
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 46A45
zbMATH DE Number: 2021995
zbMATH Keywords: Köthe function space
zbMATH Keywords: narrow operator
zbMATH Keywords: operators between Banach lattices
zbMATH Keywords: domination by narrow operators
reviewed by: Şafak Alpay
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2021995