Exercise 4, § 5

Let E be a Hilbert space and $u$ a self-adjoint partial operator on E. The mapping $\sigma :f\mapsto f(u)$ of $\mathscr{L}_u^{\infty}$(Sp($u$)) into $\mathscr{L}(E)$ is the unique morphism of involutive algebras satisfying the following three conditions:

(i) One has $\|\sigma (f)\|\leqslant \|f\|_{\infty}$ for every $f\in \mathscr{L}_u^{\infty}$(Sp($u$));

(ii) If $(f_n)_{n\in\mathbf{N}}$ is a bounded sequence in $\mathscr{L}_u^{\infty}$(Sp($u$)) and $f_n(t)\rightarrow f(t)$ for every $t\in$ Sp($u$), then $\sigma (f_n)\rightarrow \sigma (f)$ in $\mathscr{L}(E)$ endowed with the topology of simple convergence.

(iii) If $(f_n)_{n\in\mathbf{N}}$ is a sequence in $\mathscr{L}_u^{\infty}$(Sp($u$)) such that $|f_n(t)|\leqslant |t|$ and $f_n(t)\rightarrow t$ for every $t\in$ Sp($u$), then one has $u(x) =$ lim $\sigma (f_n)(x)$ for every $x\in$ dom($u$).