Exercise 24, § 1

Let E be a real or complex Banach space and $u\in \mathscr{L}(E)$. Suppose that the sequence $(u^n)_{n\in\mathbf{N}}$ is bounded and that $u$ is compact from E into E endowed with the weak topology $\sigma (E,E')$. For every integer $N\geqslant 1$, put

a) Let F be the closure of the image of $1_E-u$. For every $x\in F$, one has $v_N(x)\rightarrow 0$ in E when $N\rightarrow +\infty$.

b) Let $x\in E$. The sequence $(v_N(x))$ converges to an element $y\in E$ such that $u(y) =y$. (There exists a subsequence of $(v_N(x))$ which converges weakly to $y\in E$; show that $x-y\in F$ using the Hahn–Banach theorem.)

c) The mapping $w: E\rightarrow E$ such that $w(x)$ is the element $y$ given by the preceding question is a continuous projector with image the proper space of $u$ relative to the eigenvalue 1.

d) Let $z\in \mathbf{C}$ such that $|z|= 1$. The sequence in $\mathscr{L}(E)$ defined by

converges in $\mathscr{L}(E)$ endowed with the topology of simple convergence. Let $w_z$ be its limit. Then $w_z$ is a projector, one has $w_zw=ww_z=zw_z$ and $w_{z_1}w_{z_2}= 0$ if $z_1\not =z_2$.