Exercise 4, § 4

Let $E$ be a complex finite-dimensional Hilbert space of dimension $d$.

a) For every $u \geq 0$ in $\mathscr{L}(E)$ (V, p. 45), proving that there exists a unique $v \geq 0$ in $\mathscr{L}(E)$ such that $u = v^2$ (diagonalize $u$); it is denoted by $v = u^{1/2}$.

b) For every $u \in \mathscr{L}(E)$, put $\operatorname{abs}(u) = (u^* u)^{1/2}$. Now prove that $u$ and $\operatorname{abs}(u)$ have the same norm and that one has $\operatorname{abs}(\Lambda^n(u)) = \Lambda^n(\operatorname{abs}(u))$ for every integer $n \leq d$.

c) Let $s_1(u) \geq s_2(u) \geq \cdots \geq s_d(u) \geq 0$ be the sequence of eigenvalues of $\operatorname{abs}(u)$ counted with their order of multiplicity. Now prove that $\|u\| = s_1(u)$ and that for every integer $n \leq d$ one has $\|\Lambda^n(u)\| = s_1(u) s_2(u) \ldots s_n(u)$. In order that $\|\Lambda^n(u)\| = \|u\|^n$ for every $n$ such that $1 \leq n \leq d$, it is necessary and sufficient that $u^* u$ be a homothety, in other words that $u$ be a scalar multiple of a unitary operator.