Compact integration operators for Fréchet-space-valued measures.
type: Item
instance of: scholarly article
title: Compact integration operators for Fréchet-space-valued measures.
zbMATH Open document ID: 1044.28008
DOI: 10.1016/S0019-3577(02)80006-4
published in: Indagationes Mathematicae. New Series
publication date: 2002-01-01T00:00:00Z
review text: Let \(X\) be a Fréchet space with an increasing sequence of continuous seminorms \(p_1\leq p_2\leq\cdots\) determining the topology of \(X\), and for each \(k\geq 1\), let \(X_k\) be the Banach space obtained from the quotient normed space \(X/p_k^{-1}(\{0\})\) by completion. Let \(\Sigma\) be a \(\sigma\)-algebra of subsets of a non-empty set \(\Omega\), and let \(m:\Sigma\to X\) be a vector measure. Also, for each \(k\geq 1\), let \(\Pi_k\) be the canonical quotient map of \(X\) onto \(X_k\) and let \(m_k=\Pi_k{\circ}m\) be the Banach space valued measure with the variation measure \(| m_k| \). The integration operator \(I_m:L^1(m)\to X\) is defined by \(I_mf:=\smallint_{\Omega}f\,dm\) for \(f\in L^1(m)\). The authors prove that \(I_m:L^1(m)\to X\) is compact if and only if there exists a positive integer \(k\) and a Bochner \(| m_k| \)-integrable function \(G:\Omega\to X\) such that (i) \(L^1(m)=L^1(m_k)\); (ii) \(m(E)=(B)\smallint_EG\,d| m_k| \) for \(E\in\Sigma\) and (iii) \(G\) has \(| m_k| \)-essentially relatively compact range in \(X\). In this case, \(m\) necessarily has finite variation and \(I_mf=(B)\smallint_{\Omega}fG\,d| m_k|\) for \(f\in L^1(m)\).
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 47B07
zbMATH DE Number: 1893641
zbMATH Keywords: vector measure
zbMATH Keywords: Fréchet space
zbMATH Keywords: integration
zbMATH Keywords: compactness
author: S. Okada
author: Werner J. Ricker
reviewed by: Rimas Norvaiša
MaRDI profile type: Publication
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full work available at URL: s0019-3577(02)80006-4
OpenAlex ID: W2094649494
description: scientific article; zbMATH DE number 1893641