Remark 23

Suppose that the conditions of No. 3 are fulfilled. Let $u$ and $v$ be two endomorphisms of $E$.

a) Suppose that $u$ and $uv$ are normal. In order that $vu$ be normal, it is necessary and sufficient that $v$ and $u^*u$ be commutable. (To see that the condition is necessary, use the relation $u(vu) = (uv)u$ and exerc. 22; to see that it is sufficient, use exerc. 14 a) and 10 d).)

b) Suppose that $u, v$ and $uv$ are normal; let $\beta$ be the greatest eigenvalue of $uu^*$, and let $F$ be the subspace of $E$ formed by the eigenvectors of $uu^*$ corresponding to the eigenvalue $\beta$. Now prove that $v(F) \subset F$. (Remark that for every normal endomorphism $u$ and every $x \in E$, one has

)

Reduce to the case where A is equal to K(i) or to the quaternion field over K; F then admits a basis formed by eigenvectors for u (and u*); remark that for such a vector z, one has $\Phi(u^*u v(z), v(z)) = \beta \Phi(v(z), v(z))$. Deduce that νu is normal (reason by induction on the number of distinct eigenvalues of $u^*u$). If h (resp. $h'$) is the positive hermitian endomorphism such that $h^2 = uu^*$ (resp. ${h'}^2 = \nu \nu^*$) and if one sets $u = hu_1, \nu = h'\nu_1$, where $u_1$ and $\nu_1$ are unitary, h permutable with $u_1$ and $h'$ with $\nu_1$ (exerc. 14 a)), prove that the pairs $(h, h'), (h, \nu_1)$ and $(h', u_1)$ are permutable; converse. Deduce that $u^m \nu^n, \nu^n u^m, u \nu^*$ and $\nu^* u$ are then normal ($m$ and $n$ arbitrary integers $> 0$).