Spectral measures and automatic continuity
type: Item
instance of: scholarly article
title: Spectral measures and automatic continuity
zbMATH Open document ID: 0890.46030
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 1998-07-08T00:00:00Z
full work available at URL: https://eudml.org/doc/119452
review text: Let \(\Sigma\) be a \(\sigma\)-algebra of subset of a set \(S\), \(X\) be a locally convex Hausdorff space and \(L(X)\) the space of all continuous linear operators from \(X\) into itself equipped with the strong operator topology. Let \(P:\Sigma\to L(X)\) be countably additive such that \(P(E\cap F)= P(E)P(F)\) holds for every \(E,F\in\Sigma\) and \(P(S)=I\), the identity operator (i.e., \(P\) is a spectral measure). For each \(x\in X\) let \(P_x\) be the vector measure on \(\Sigma\) defined by \(P_x(E)= P(E)x\). If \(f\) is \(P\)-integrable, then \(f\) is \(P_x\)-integrable for every \(x\in X\) and the integration map \(\int f dP\) is continuous on \(X\) and \(\int f dP\) satisfies \((\int f dP)x=\int f dP_x\) for every \(x\in X\). The authors address the converse. Namely, if \(f\) is \(P_x\) integrable for every \(x\in X\) when is the integration map \(P_f: X\to X\), \(P_f(x)=\int f dP_x\), continuous. The authors establish the automatic continuity of the map \(P_f\) under 3 separate hypotheses; for example, they show this is the case if \(X\) is barrelled. These results improve results of Dodds and de Pagter. Examples are given relevant to the various hypotheses in the theorems.
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 47B40
Mathematics Subject Classification ID: 46A08
zbMATH DE Number: 942753
zbMATH Keywords: spectral measure
zbMATH Keywords: vector measure
zbMATH Keywords: automatic continuity
zbMATH Keywords: barrelled
author: S. Okada
author: Werner J. Ricker
reviewed by: Charles Swartz
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 942753