Proposition 15

Let $u$ be a self-adjoint partial operator on E. Suppose that $u$ is bounded below by $c\in \mathbf{R}$ (Def. 7 of IV, p. 237). Then

inf(Sp($u$)) $=$ inf$_{x\in dom(u)-\{0\}}\frac{\langle x|u(x)\rangle}{\|x\|^2}\in [c,+\infty [$.

Let $m$ be the right-hand side of the equality to be proved. Let $x\in E$ and let $\mu_x$ be the spectral measure of $x$ relative to $u$. We have

Sp($u$)

$\geqslant$ inf(Sp($u$))$\int_{Sp(u)}d\mu_x(t) =$ inf(Sp($u$))$\|x\|^2$,

(formula (6), p. 271), hence inf(Sp($u$))$\leqslant m$. Conversely, the partial operator $u-m$ is positive, hence inf(Sp($u$))$-m=$ inf(Sp($u-m\cdot 1_E$))$\geqslant 0$ (Prop. 17 of IV, p. 248).

Prop. 9 of I, p. 139 corresponds to the particular case of this proposition when $u$ is a hermitian element of $\mathscr{L}$ (E), in which case it is necessarily bounded below. In this case, one also has

sup(Sp($u$)) $=$ sup$_{x\in dom(u)-\{0\}}\frac{\langle x|u(x)\rangle}{\|x\|^2}$

(loc. cit.); if $u$ does not belong to $\mathscr{L}$ (E), then this least upper bound is $+\infty$.