Order-continuous component of a majorized operator
type: Item
instance of: scholarly article
title: Order-continuous component of a majorized operator
zbMATH Open document ID: 0643.47043
DOI: 10.1007/BF00973851
published in: Siberian Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: A real vector space X equipped with a map \(| \cdot |:X\to E\), where E is a K-space \((=\) order complete vector lattice), such that (1) \(| x| \geq 0\); \(| x| =0\) if and only if \(x=0\) (x\(\in X)\); (2) \(| x+y| \leq | x| +| y|\) (x,y\(\in X)\); (3) \(| \lambda x| =| \lambda | | x|\) (x\(\in X,\lambda \in R)\); is called a K-normed space. The norm \(| \cdot |\) is called decomposable if, for every \(x\in X\) and \(e_ 1,e_ 2\in E\) with \(e_ 1\wedge e_ 2=0\) and \(| x| =e_ 1+e_ 2\), there exist \(x_ 1,x_ 2\in X\) such that \(x=x_ 1+x_ 2\) and \(| x_ i| =e_ i\), \(i=1,2\). The K-normed space X is called bo-complete if, for every net \((x_{\alpha})\) in X with \(o-\lim | x_{\alpha}- x_{\beta}| =0\), there exists \(x\in X\) with \(o-\lim | x- x_{\alpha}| =0\). Let Y be another K-normed space, the norm of which takes values in a K-space F. A linear operator T:X\(\to Y\) is called bo- continuous if, for every net \((x_{\alpha})\) in X with \(o-\lim | x_{\alpha}| =0\), we have \(o-\lim | Tx_{\alpha}| =0\). We call T majorized if there exists a positive operator S:E\(\to F\) such that \(| Tx| \leq S(| x|)\) for all \(x\in X\). The vector space of such operators is denoted by M(X,Y). (All these notions go back to L. V. Kantorovich.) Generalizing the Yosida-Hewitt decomposition theorem, the first author and \textit{V. Z. Strizhevskij} [Trudy Inst. Mat. 7, 132-158 (1987)] proved that if X and Y are bo-complete, then every \(T\in M(X,Y)\) can be uniquely represented as follows: \(T=T_ n+T_ s\), where \(T_ n\), \(T_ s\in M(X,Y)\) and \(T_ n\) is bo-continuous and \(T_ s\) is singular in an appropriate sense. In the present paper this result is generalized in two directions by relaxing the assumptions on X and Y and introducing a more general notion of continuity. Furthermore, two formulas for the bo-continuous part of T, both involving iterated limits, are exhibited. Those formulas have origin in measure theory, and the authors' approach uses some ideas of T. Traynor. The d…
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 28B05
zbMATH DE Number: 4048455
zbMATH Keywords: K-space
zbMATH Keywords: order complete vector lattice
zbMATH Keywords: K-normed space
zbMATH Keywords: bo-complete
zbMATH Keywords: bo-continuous
zbMATH Keywords: Yosida-Hewitt decomposition theorem
author: Anatoly G. Kusraev
author: Sergey Artem'evich Malyugin
reviewed by: Zbigniew Lipecki
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00973851
OpenAlex ID: W2019489737
description: scientific article; zbMATH DE number 4048455