Exercise 8, § 4

a) Let E be a hilbertian space, and let T be a continuous linear mapping from E into itself such that $\|T\| \leq 1$. Show that the relations $T.x = x, \langle T.x|x \rangle = \|x\|, T^*.x = x$ are equivalent, and that the kernel of $1_E - T$ and the closure of the image of $1_E - T$ are the orthogonal complements. b) Let T be a continuous linear mapping from E into itself, satisfying the inequality

for all $x \in E$. Then $\|T.x\| < \|x\|$ for all $x$ such that $T.x \neq x$; for every $x \in E$, the sequence $(T^n.x)$ converges to a point $P.x$, and $P$ is the orthogonal projector onto the kernel of $1_E - T$. c) Let $P_1, ..., P_r$ be orthoprojectors on E. Show that the product $T = P_1P_2 ... P_r$ satisfies relation (1) (argue by induction on r); hence the orthoprojector $P$ is the orthogonal projector onto the intersection of the images of the projectors $P_j$ (note that if $\|P_j.x\| < \|x\|$ for an index j, then $\|T.x\| < \|x\|$).