On two definitions of a narrow operator on Köthe-Bochner spaces
type: Item
instance of: scholarly article
title: On two definitions of a narrow operator on Köthe-Bochner spaces
zbMATH Open document ID: 1407.46031
author: Nariman Abasov
author: M. Pliev
published in: Archiv der Mathematik
publication date: 2018-08-10T00:00:00Z
review text: Let \((\Omega,\Sigma,\mu)\) be a finite atomless measure space, \(E\) be a Köthe-Banach space over \((\Omega,\Sigma,\mu)\), \(X\) be a Banach space and \(E(X)\) be the corresponding Köthe-Bochner space. Given a Banach space \(Y\), a continuous linear operator \(T : E(X) \to Y\) is called weakly functionally narrow if, for every \(A \in \Sigma\), \(x \in X\) and \(\varepsilon > 0\), there exists a decomposition of \(A\) into the disjoint union of \(B, C \in \Sigma\) in such a way that \(\|T(x \mathbf 1_B - x \mathbf 1_D)\| < \varepsilon\). The operator \(T\) is called narrow if, for every \(f \in E(X)\) and \(\varepsilon > 0\), there exist two mutually complemented fragments \(g\) and \(h\) of \(f\) such that \(\|T(g - h)\| < \varepsilon\). These definitions, given by the authors, generalize those ones that were previously known for \(X = \mathbb R\). It is demonstrated that, if the norm of \(E\) is order continuous, then the classes of narrow and weakly functionally narrow operators from \(E(X)\) to \(Y\) are the same. In the general case, without the assumption of order continuity, for given \(E(X)\) and \(Y\), the classes of narrow and weakly functionally narrow operators from \(E(X)\) to \(Y\) coincide if and only if the set of all simple elements is dense in \(E(X)\). The reader should take caution with the name ``narrow operator'', because in different papers it sometimes has similar but different meanings; for example, the paper [\textit{K. Boyko} et al., Zh. Mat. Fiz. Anal. Geom. 2, No. 4, 358--371 (2006; Zbl 1147.46006)] that addresses an analogous problem, works with a different non-equivalent definition.
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 47B99
zbMATH DE Number: 6916266
zbMATH Keywords: vector lattice
zbMATH Keywords: lattice-normed spaces
zbMATH Keywords: Köthe-Bochner space
zbMATH Keywords: narrow operator
MaRDI profile type: Publication
full work available at URL: s00013-018-1172-2
OpenAlex ID: W2792942864
cites work: Dominated operators from lattice-normed spaces to sequence Banach lattices
cites work: Q5491574
cites work: Q4148368
cites work: Narrow orthogonally additive operators in lattice-normed spaces
cites work: Narrow orthogonally additive operators
cites work: A lattice approach to narrow operators
cites work: Dividing measures and narrow operators
cites work: Q3202684
cites work: Domination problem for narrow orthogonally additive operators
cites work: Narrow operators on lattice-normed spaces
cites work: Narrow operators on function spaces and vector lattices
cites work: On the ``function'' and ``lattice'' definitions of a narrow operator
DOI: 10.1007/S00013-018-1172-2
description: scientific article; zbMATH DE number 6916266