Exercise 4, § 4

Let $E$ be a complex hilbertian space of finite dimension $d$. a) For every $u \geq 0$ in $\mathscr{L}(E)$ (V, p. 45), prove that there exists a unique $v \geq 0$ in $\mathscr{L}(E)$ such that $u = v^2$ (diagonalize $u$); we write $v = u^{1/2}$.

b) For every $u \in \mathscr{L}(E)$, put $\operatorname{abs}(u) = (u^*u)^{1/2}$. Show that $u$ and $\operatorname{abs}(u)$ have the same norm and that we have $\operatorname{abs}(\Lambda^n(u)) = \Lambda^n(\operatorname{abs}(u))$ for every integer $n \leq d$. c) Let $s_1(u) \geq s_2(u) \geq \ldots \geq s_d(u) \geq 0$ be the sequence of eigenvalues of $\operatorname{abs}(u)$ counted with their order of multiplicity. Show that $\|u\| = s_1(u)$ and that for every integer $n \leq d$, $\|\Lambda^n(u)\| = s_1(u) s_2(u) \ldots s_n(u)$. In order that $\|\Lambda^n(u)\| = \|u\|^n$ for all $n$ such that $1 \leq n \leq d$, it is necessary and sufficient that $u^*u$ is a homothety, in other words, that $u$ is a scalar multiple of a unitary operator.