Exercise 13, § 4

Let $E$ be a Hilbert space, $G$ a subgroup of the automorphism group of the Hilbert space structure of $E$. Let $E^G$ be the closed vector subspace of $E$ constituted by the vectors invariant under $G$, and $p$ the orthogonal projector of $E$ onto $E^G$. a) Show that the orthogonal complement of $E^G$ in $E$ is the closed vector subspace generated by the vectors $s.x - x$, where $s$ runs through $G$ and $x \in E$. b) Let $H$ be a nonempty convex, closed and stable subset of $E$ under $G$. Show that the projection of 0 onto $H$ belongs to $E^G$. c) Suppose that $H$ is the closed convex hull of the orbit of a point $x$ of $E$, and let $a$ be the projection of 0 onto $H$. Show that $a = p(x)$ and that $H \cap E^G$ is reduced to the point $a$ (“th. of Birkhoff-Alaoglu”). (Note that $x - a$ is contained in the orthogonal complement of $E^G$.) d) Suppose that $G$ is generated by an automorphism $u$ of $E$. Show that one has

e) Suppose that $G$ is the image of a homomorphism $t \mapsto u_t$ of $\mathbf{R}$ into the group of automorphisms of $E$, such that for every $x \in E$, $t \mapsto u_t.x$ is a continuous mapping of $\mathbf{R}$ into $E$. Show that one has $p(x) = \lim_{T \to \infty} \frac{1}{T} \int_0^T u_t.x \, dt$ for all $x \in E$. f) Suppose that there exists an element $x \neq 0$ of $E$ and a number $\alpha$ such that $0 < \alpha < 1$ and $\|s.x - x\| \leq \alpha \|x\|$ for all $s \in G$. Show that $E^G \neq \{0\}$ (use c)).