Proposition 8

Suppose that $u \in \mathscr{L}(E)$ is normal. Let $N$ be the kernel of $u$ and $M$ the orthogonal of $N$ in $E$; let $m$ and $n$ be two positive integers such that $m + n \geq 1$. Then $N$ is the kernel of $u^m(u^*)^n$ and $M$ is both the initial and the final subspace of $u^m(u^*)^n$. In particular, $M$ is both the initial and the final subspace of $u$ and of $u^*$, and is stable under $u$ and $u^*$.

Prop. 7 shows that $u$ and $u^*$ have the same kernel $N$. By prop. 4, (ii) of V, p. 41, the subspace $M$ of $E$ is stable under $u$ and $u^*$ since this is so for $N = M^\circ$, since $M \cap N = \{0\}$, the endomorphisms of $M$ induced by $u$ and $u^*$ are injective. Let $v = u^m(u^*)^n$; the preceding argument shows that the restriction of $v$ to $M$ (resp. $N$) is injective (resp. null), hence $N$ is the kernel of $v$. Consequently, $M = N^\circ$ is the initial subspace of $v$. By prop. 4, (i) of V, p. 41, the final subspace of $v$ is equal to the initial subspace of $v^*$. But $v^* = u^n(u^*)^m$ and so the initial subspace of $v^*$ is equal to $M$ by the preceding.