Proposition 9

Let $E$ be a complex Hilbert space and let $u$ be a hermitian element of $\mathscr{L}(E)$.

a) One has $m(u) =$ inf Sp($u$) and $M(u) =$ sup Sp($u$) ;

b) If $E$ is nonzero, one has $\|u\|=$ sup($|m(u)|,|M(u)|$).

Let $\lambda \in \mathbf{R}$. In order that $\lambda$ be a lower bound of the spectrum of $u$, it is necessary and sufficient that $u-\lambda \geqslant 0$. This is equivalent (Prop. 8, (ii)) to the condition $\langle x|u(x)\rangle \geqslant \lambda \|x\|^2$ for every $x\in E$, that is, to $m(u)\geqslant \lambda$. This proves that $m(u)$ is the greatest lower bound of Sp($u$). Similarly, one verifies that $M(u)$ is the least upper bound of Sp($u$).

Since $u$ is normal, one has $\varrho (u) =\|u\|$ (Cor. 1 of I, p. 108). Since $E\not=\{0\}$, the spectrum of $u$ is not empty (Cor. 1 of I, p. 26) and $\varrho (u)$ is the radius of the smallest disk with center 0 which contains Sp($u$) (Theorem 1 of I, p. 24), hence b) follows from a).