Exercise 5, § 2

Let $u\in \mathscr{L}(E)$ such that $\|u\|\leqslant 1$. For every $\lambda \in \mathbf{C}$, let $p_{\lambda}$ be the orthogonal projection of E onto the eigenspace of $u$ corresponding to $\lambda$. For all $x$ and $y$ in E, one has

1 N $n$ 2 2 2

lim $\sum|\langle x|u(y)\rangle |=\sum|\langle p_{\lambda}(y)|p_{\lambda}(x)\rangle |=\sum|\langle y|p_{\lambda}(x)\rangle |$

$n=1\lambda \in \mathbf{C}\lambda \in \mathbf{C}$

(one may use exercise 64 of II, p. 300).