On sums of narrow operators on Köthe function spaces
type: Item
instance of: scholarly article
title: On sums of narrow operators on Köthe function spaces
zbMATH Open document ID: 1316.47034
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-01-26T00:00:00Z
review text: The paper deals with narrow operators in the sense of \textit{A. M. Plichko} and the second author [Diss. Math. 306, 85 p. (1990; Zbl 0715.46011)] defined on a space of functions on an atomless finite measure space \((\Omega, \Sigma, \mu)\) (see [the second author and \textit{B. Randrianantoanina}, Narrow operators on function spaces and vector lattices. Berlin: Walter de Gruyter (2013; Zbl 1258.47002)] for a detailed exposition of the subject). The authors solve two open problems. First, they answer in the negative the long standing question whether for every Banach space \(X\) a sum of two narrow operators from \(L_1[0, 1]\) to \(X\) must be narrow. Secondly, they demonstrate the existence of regular narrow operators \(T_1,T_2: L_\infty[0,1] \to L_\infty[0,1]\) with non-narrow sum \(T=T_1+T_2\). They also prove some interesting general results related to these problems.
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 6390625
zbMATH Keywords: narrow operator
zbMATH Keywords: Köthe function space
zbMATH Keywords: Banach space \(L_1\)
zbMATH Keywords: regular operator
author: V. V. Mykhaylyuk
author: M. M. Popov
MaRDI profile type: Publication
full work available at URL: j.jmaa.2013.03.008
OpenAlex ID: W2011815833
cites work: UNCONDITIONALLY CONVERGENT SERIES OF OPERATORS AND NARROW OPERATORS ON $L_1$
cites work: Q4653284
cites work: A note on narrow operators in \(L_{\infty}\)
cites work: Q4188284
cites work: Theorems on decomposition of operators in L 1 and their generalization to vector lattices
cites work: A lattice approach to narrow operators
cites work: Q3202684
cites work: Narrow operators (a survey)
cites work: Narrow operators on function spaces and vector lattices
cites work: Q4049667
DOI: 10.1016/J.JMAA.2013.03.008
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description: scientific article; zbMATH DE number 6390625