Remark 16

Suppose that the conditions of No. 3 are satisfied and in addition that $A = K(i)$. Let $u$ be an endomorphism of $E$.

a) The set of eigenvalues of $u$ is contained in the set $U$ of the values of $\Phi(x, u(x))$ when $x$ runs through the set of elements of $E$ such that $\Phi(x, x) = 1$.

b) A subset $C$ of $A = K(i)$ is said to be convex if, for every pair of elements $(\xi, \eta) \in C^2$ and every $\tau \in K$ such that $0 \leqslant \tau \leqslant 1$ one has $\tau \xi + (1-\tau)\eta \in C$. Show that the set $U$ is convex. (Reduce to the case where $n = 2$; by writing $u$ in the form $\nu + i\omega$, where $\nu$ and $\omega$ are hermitian, and by replacing if necessary $u$ by $\lambda u$, where $\lambda \in A$ and $\lambda \overline{\lambda} = 1$, show that everything reduces to proving the following property. Let $f(\xi_1, \xi_2) = \xi_1 \overline{\xi}_1 + \xi_2 \overline{\xi}_2$, $g(\xi_1, \xi_2) = a \xi_1 \overline{\xi}_1 + b \xi_2 \overline{\xi}_2$ ($a \in K, b \in K$), $h(\xi_1, \xi_2) = \alpha \xi_1 \overline{\xi}_1 + \beta \xi_1 \overline{\xi}_2 + \beta \xi_2 \overline{\xi}_1 + \gamma \xi_2 \overline{\xi}_2$ ($\alpha \in K, \gamma \in K, \beta \in A$); let $(\eta_1, \eta_2) \in A^2, (\zeta_1, \zeta_2) \in A^2$ be such that $f(\eta_1, \eta_2) = f(\zeta_1, \zeta_2) = 1$, $g(\eta_1, \eta_2) = g(\zeta_1, \zeta_2) = 1$, $h(\eta_1, \eta_2) > 0$, $h(\zeta_1, \zeta_2) < 0$; there then exists $(theta_1, theta_2) \in A^2$ such that $f(\theta_1, \theta_2) = 1$, $g(\theta_1, \theta_2) = 1$ and $h(\theta_1, \theta_2) = 0$. One will begin by observing that for every pair $(\xi_1, \xi_2)$, there exists $\mu \in A$ such that $\mu \overline{\mu} = 1$ and $\beta \mu \xi_1 \overline{\xi}_2 + \beta \mu \xi_2 \overline{\xi}_1 = 0$, which will make it possible to reduce to the case where $\beta = 0$; use prop. 5 of Chap. VI, § 2, No. 5.)

c) Show that if $u$ is normal, $U$ is the smallest convex set containing all the eigenvalues of $u$. Give an example where $u$ is not normal but where $U$ still has the preceding property. (Take as eigenvalues of $u$ the elements $\pm 1 \pm i$ as simple eigenvalues, and 0 as a double eigenvalue.)