Remark 2

Suppose that $G = H$ is quasi-complete and that $U$ satisfies the hypotheses of Prop. 16. Let M be a dense subset of the space $\mathscr{C}'(T)$, for the weak topology $\sigma(\mathscr{C}'(T), \mathscr{C}(T))$, and let X be a closed linear subspace of H such that $U(\lambda)(X) \subset X$ for every measure $\lambda \in M$. Then also $U(t)(X) \subset X$ for every $t \in T$: indeed, for every $x \in X$ and every $x' \in H'$ orthogonal to X, by hypothesis $\langle U(\lambda) \cdot x, x' \rangle = 0$ for all $\lambda \in M$, which may be written $\int \langle U(t) \cdot x, x' \rangle \, d\mu(t) = 0$. The continuous function $t \mapsto \langle U(t) \cdot x, x' \rangle$, being orthogonal to M, is therefore 0, which yields $\langle U(t) \cdot x, x' \rangle = 0$ for every $x' \in X^\circ$, whence $U(t) \cdot x \in X$ for all $t \in T$ and $x \in X$, and this proves our assertion.