Exercise 8, § 4

a) Let E be a Hilbert space, $T$ a continuous linear mapping of E into itself such that $\|T\|\leq 1$. Prove that the relations $T.x=x,\ \langle T.x|x\rangle=\|x\|^2,\ T^*.x=x$ are equivalent, and that the kernel of $1_E-T$ and the closure of the image of $1_E-T$ are orthogonal supplementary subspaces. b) Let $T$ be a continuous linear mapping of E into itself, satisfying the inequality

for all $x\in E$. One then has $\|T.x\|<\|x\|$ for all $x$ such that $T.x\neq x$; for all $x\in E$, the sequence $(T^n.x)$ converges towards a point $P.x$, and $P$ is the orthogonal projector onto the kernel of $1_E-T$. c) Let $P_1,\ldots,P_r$ be orthogonal projectors in E. Prove that the product $T=P_1P_2\ldots P_r$ satisfies the relation (1) (reason by induction on $r$); then the orthogonal projector $P$ 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 $j$, one has $\|T.x\|<\|x\|$).