Finite-dimensional invariant subspaces for measurable semigroups of linear operators
type: Item
instance of: scholarly article
title: Finite-dimensional invariant subspaces for measurable semigroups of linear operators
zbMATH Open document ID: 0635.47037
DOI: 10.1016/0022-247X(87)90129-6
author: A. T.-M. Lau
author: James C. S. Wong
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: The authors show that on a measurable semigroup S, if the space of weakly left uniformly measurable functions on S (denoted by WLUM(S)) has a left invariant mean, then S satisfies F(n) for every \(n=1,2,...\), where F(n) is the following property: Let E be a separated locally convex space, let T:S\(\times E\to E\) be a weakly measurable action of S on E and let \(X\subset E\) be a subset such that there exists a closed S-invariant subspace H of E with codimension n for which \((x+H)\cap X\) is compact and convex for each \(x\in E\). If \({\mathcal L}_ n(X)\) is the family of all n-dimensional subspaces of E contained in X and \({\mathcal L}_ n(X)\) is S-invariant, then there exists \(L_ 0\in {\mathcal L}_ n(X)\) such that \(T_ s(L_ 0)=L_ 0\) for each \(s\in S.\) Conversely, if S satisfies F(1), then WLUM(S) has a left invariant mean. Versions of this result were previously obtained for left amenable (discrete) semigroups and for topological semigroups by K. Fan and, respectively, by the first named author. The proof of this result requires a fixed point property which characterized the existence of a left invariant mean on a space of measurable functions, which, in turn, implies an older fixed point theorem due to B. Simon.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 4034473
zbMATH Keywords: space of weakly left uniformly measurable functions
zbMATH Keywords: left invariant mean
zbMATH Keywords: fixed point property
reviewed by: Florian-Horia Vasilescu
MaRDI profile type: Publication
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full work available at URL: 0022-247x(87)90129-6
OpenAlex ID: W2045980317
description: scientific article; zbMATH DE number 4034473