Operators on the space of vector-valued totally measurable functions
type: Item
instance of: scholarly article
title: Operators on the space of vector-valued totally measurable functions
zbMATH Open document ID: 1155.47038
author: Marian Nowak
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-11-06T00:00:00Z
review text: Several classes of linear operators from the space \(B(\Sigma, X)\) of \(X\)-valued totally measurable functions into a Banach space \(Y\) are characterized in this article. A bounded linear operator \(T:B(\Sigma, X) \to Y\) is called \(\sigma\)-smooth if \((T(f_n))\) converges to \(0\) in \(Y\) whenever \((f_n)\) is a sequence in \(B(\Sigma, X)\) with \((\| f_n(\cdot )\|_X)\) order convergent to \(0\) in \(B(\Sigma)\). The author proves that a bounded linear operator \(T:B(\Sigma, X) \to Y\) is \(\sigma\)-smooth exactly when its representing measure \(m\) is variationally semi-regular (that is, if \((\| m\| (A_n))\) converges to \(0\) whenever the sequence \((A_n)\) of \(\Sigma\)-measurable sets decreases to \(\varnothing\) where \(\| m\|\) is the semivariation of \(m\)). As a consequence, the space \(L_{\sigma s}(B(\Sigma,X), Y)\) of \(\sigma\)-smooth operators from \(B(\Sigma,X)\) to \(Y\) endowed with the strong operator topology is sequentially complete. With \(B(\Sigma,X)^{\widetilde{ }}_C\) denoting \(L_{\sigma s}(B(\Sigma,X), \mathbb{R})\) and \(\tau(B(\Sigma,X), B(\Sigma,X)^{\widetilde{ }}_C)\) denoting the Mackey topology on \(B(\Sigma,X)\), the author also proves that, if \(Y\) contains no isomorphic copy of \(c_0\), then \(T:B(\Sigma,X) \to Y\) is \(\sigma\)-smooth if and only if \(T\) is \((\tau(B(\Sigma,X), B(\Sigma,X)^{\widetilde{ }}_C), \|\cdot\|_Y)\)-continuous. More generally, the \((\tau(B(\Sigma,X), B(\Sigma,X)^{\widetilde{ }}_C), \|\cdot\|_Y)\)-continuous operators \(T:B(\Sigma,X) \to Y\) are characterized in terms of countable additivity of the measure \(m\) representing \(T\).
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 28B05
zbMATH DE Number: 5362161
zbMATH Keywords: \(\sigma\)-smooth operator
zbMATH Keywords: operator-valued measure
zbMATH Keywords: spaces of totally measurable functions
reviewed by: B. Turett
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.09.018
OpenAlex ID: W2083249200
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DOI: 10.1016/J.JMAA.2008.09.018
description: scientific article; zbMATH DE number 5362161