Finite-dimensional invariant subspaces for measurable semigroups of linear operators
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.
space of weakly left uniformly measurable functions
scientific article; zbMATH DE number 4034473