Vector measures and Mackey topologies
type: Item
instance of: scholarly article
title: Vector measures and Mackey topologies
zbMATH Open document ID: 1247.46035
author: Marian Nowak
published in: Indagationes Mathematicae. New Series
publication date: 2012-02-25T00:00:00Z
review text: Let \((\Omega,\Sigma)\) be a measurable space, and consider the Banach lattice \(B(\Sigma)\) of all bounded and \(\Sigma\)-measurable real functions endowed with the Mackey topology \(\tau=\tau(B(\Sigma),ca(\Sigma))\). In this paper, \((\tau, \xi)\)-continuous linear operators from \(B(\Sigma)\) to a quasi complete locally convex space \((E,\xi)\) are studied. In Section 2, the interesting class of the \(\sigma\)-smooth operators, which has been already studied by the author in other papers, is introduced in order to find a useful characterization of the continuity of linear maps quoted above (Proposition 2.1 and also Proposition 3.1). After that, in Theorem 2.3, a characterization of relatively compact sets in the space of \((\tau,\xi)\)-continuous linear maps is given. In Section 3, integral representation of operators by means of vector measures provides a convenient tool for analysing \((\| \cdot\|,\tau)\)-continuous operators. This allows to prove a Vitali-Hahn-Saks theorem for operators from \(B(\Sigma)\) to \(E\) (Theorem 3.2) and a Nikodym type theorem (Theorem 3.3). As an application it is shown that the space \((B(\Sigma),\tau)\) has the strict Dunford-Pettis property. Finally, in Section 4, a Yosida-Hewitt type decomposition theorem for weakly compact operators on \(B(\Sigma)\) is shown (Theorem 4.3).
reviewed by: E. A. Sánchez-Pérez
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46A70
Mathematics Subject Classification ID: 46A32
zbMATH DE Number: 6009291
zbMATH Keywords: bounded measurable functions
zbMATH Keywords: \(\sigma\)-smooth operators
zbMATH Keywords: Mackey topologies
zbMATH Keywords: strongly Mackey space
zbMATH Keywords: weakly compact operators
zbMATH Keywords: countably additive measures
zbMATH Keywords: strict Dunford-Pettis property
MaRDI profile type: Publication
full work available at URL: j.indag.2011.11.001
OpenAlex ID: W2072468895
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DOI: 10.1016/J.INDAG.2011.11.001
description: scientific article; zbMATH DE number 6009291