Bilinear integration in tensor products
type: Item
instance of: scholarly article
title: Bilinear integration in tensor products
zbMATH Open document ID: 0936.46035
author: Brian Jefferies
author: S. Okada
published in: Rocky Mountain Journal of Mathematics
publication date: 2000-05-18T00:00:00Z
full work available at URL: cont282.html
review text: Let \(X\), \(Y\) be Banach spaces with \(X\otimes Y\) their tensor product. If \(\tau\) is a norm on \(X\otimes Y\), denote the space \(X\otimes Y\) with this norm by \(X\otimes_\tau Y\) and its completion by \(X\widehat\otimes_\tau Y\). Call \(\tau\) a norm tensor product if there exists \(C>0\) such that (i) \(\tau(x\otimes y)\leq C\|x\|\|y\|\) for all \(x\in X\), \(y\in Y\) and (ii) \(X'\otimes Y'\) can be identified with a linear subspace of \((X\otimes_\tau y)'\) with \(\|x'\otimes y'\|\leq C\|x'\|\|y'\|\) for all \(x'\in X'\), \(y'\in Y'\). If \(\tau\) is a norm tensor product, then \(\tau\) is completely separated if \(X'\times Y'\) separates \(X\widehat\otimes_\tau Y\); for example, if \(\tau\) is the injective tensor product, this always holds, and if \(\tau\) is the projective tensor product, this holds if \(X\) or \(Y\) has the approximation property. Let \(\Sigma\) be a \(\sigma\)-algebra of subsets of a set \(S\) and \(m:\Sigma\to Y\) be countably additive. If \(\varphi= \sum^n_{j=1} x_j\chi_{A_j}: S\to X\) is an \(X\)-valued \(\Sigma\)-simple function, the integral of \(\varphi\) with respect to \(m\) over \(A\in\Sigma\) is defined by \[ \int_A \varphi\otimes dm= \sum^n_{j= 1}x_j\otimes m(A\cap A_j). \] Under the assumption that \(\tau\) is completely separating, a function \(f:S\to X\) is \(m\)-integrable if there exist \(X\)-valued \(\Sigma\)-simple functions \(\{\varphi_k\}\) such that \(\varphi_k\to f\) \(m\)-a.e. and \(\left\{\int_A \varphi_k\otimes dm\right\}\) converges in \(X\widehat\otimes_\tau Y\) for every \(A\in\Sigma\). The integral of \(f\) with respect to \(m\) over \(A\in \Sigma\) is defined to be \(\int_A f\otimes dm= \lim\int_A \varphi_k\otimes dm\). The authors show that the integral is well-defined and then very thoroughly investigate its important properties and compare it with other integrals. For example, if \(m\) is scalar-valued, the integral is equivalent to the Pettis integral for strongly measurable functions. More generally…
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 28B05
zbMATH DE Number: 1309729
zbMATH Keywords: bilinear integration
zbMATH Keywords: norm tensor product
zbMATH Keywords: completely separated
zbMATH Keywords: injective tensor product
zbMATH Keywords: projective tensor product
zbMATH Keywords: approximation property
zbMATH Keywords: Pettis integral for strongly measurable functions
zbMATH Keywords: integrals of Bartle and Dobrakov
zbMATH Keywords: integration of vector-valued functions with respect to operator-valued measures
reviewed by: Charles Swartz
MaRDI profile type: Publication
OpenAlex ID: W2069977424
DOI: 10.1216/RMJM/1181071785
description: scientific article; zbMATH DE number 1309729