Exercise 13, § 4

Let $E$ be a hilbertian space, $G$ a subgroup of the group of all automorphisms of the hilbertian space structure of $E$. Let $E^G$ be the closed vector subspace of $E$ consisting of all vectors invariant under $G$, and $p$ the orthoprojector from $E$ 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 $s$ ranges over $G$ and $x \in E$. b) Let $H$ be a non-empty closed convex subset of $E$ which is stable under $G$. Show that the projection of 0 onto $H$ belongs to $E^G$. c) Suppose that $H$ is the closed convex envelope 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 $H \cap E^G$ reduces to the point $a$ (« Birkhoff-Alaoglu th. »). (Observe that $x - a$ is contained in the orthogonal complement of $E^G$.) d) Suppose $G$ is generated by an automorphism $u$ of $E$. Show that $p(x) = \lim_{n \to \infty} \frac{1}{n+1} \sum_{j=0}^n u^j(x)$ for all $x \in E$ (if $y_n = \frac{1}{n+1} \sum_{j=0}^n u^j(x)$, note that the sequence $(y_n)$ has a weak limit point $a$, and that $u(a) = a$, then use $c$). e) Suppose $G$ is the image of a homomorphism $t \mapsto u_t$ from $\mathbf{R}$ onto the group of automorphisms of $E$, such that for all $x \in E$, $t \mapsto u_t \cdot x$ is a continuous mapping from $\mathbf{R}$ into $E$. Show that $p(x) = \lim_{T \to \infty} \frac{1}{T} \int_0^T u_t \cdot x \, dt$ 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)).