The Bartle bilinear integration and Carleman operators
type: Item
instance of: scholarly article
title: The Bartle bilinear integration and Carleman operators
zbMATH Open document ID: 0951.46021
author: Francisco J. Freniche
author: Juan Carlos García-Vázquez
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-08-15T00:00:00Z
review text: Let \(X\) and \(Y\) be Banach spaces, \(\Sigma\) a \(\sigma\)-algebra of subsets of a set \(S\) and \(v:\Sigma\to Y\) a countably additive vector measure. If \(f:S\to X\) is strongly measurable, the authors consider the bilinear Bartle integral of \(f\) with respect to \(v\) and the bilinear map \(X\times Y\to X\otimes_\varepsilon Y\), the inductive tensor product. They give several characterizatons of the integrability of \(f\), several which involve the compactness of the operator \(X'\to L^1(v)\), \(x'\to x'f\). The authors also relate the integrability of the function \(f\) with abstract Carleman operators. An operator \(u\) from \(X\) into an order complete Köthe function space \(L\) is a Carleman operator if there exists a strongly measurable function \(f: S\to X'\) such that for every \(x\in X\) we have \(u(x)(\cdot)= f(\cdot)x\) a.e. It is shown that the space of \(X\)-valued Bartle integrable functions can be identified with the space of compact Carleman operators from \(X'\) into \(L^1(v)\) which are weak\(^*\) weak continuous. A similar characterization for \(X'\)-valued strongly measurable functions is also given. Examples are given which show that analogous results fail if the inductive tensor product is replaced by the projective tensor product. The paper concludes with several interesting Fubini results for the injective tensor product of two vector measures.
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 1417567
zbMATH Keywords: Bartle integral
zbMATH Keywords: inductive tensor product
zbMATH Keywords: Carleman operators
zbMATH Keywords: projective tensor product
reviewed by: Charles Swartz
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6575
OpenAlex ID: W2092086719
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DOI: 10.1006/JMAA.1999.6575
description: scientific article; zbMATH DE number 1417567