Exercise 2, § 5

Let E be a complex Hilbert space and let $u$ be a normal partial operator on E. Denote by $p_A$ the spectral projector of $u$ on A. Let $a < b$ be real numbers. One has

$\frac{1}{2}(p_{[a,b]}+p_{]a,b[}) =$ lim$_{\varepsilon\rightarrow 0}\frac{1}{2i\pi}\int_a^b(R(u, t-i\varepsilon )-R(u, t+i\varepsilon ))dt$,

in the space $\mathscr{L}(E)$ endowed with the topology of simple convergence ("Stone formula").