Definition 5

The mapping $f\mapsto f(u)$ of $\mathscr{L}_u$(Sp($u$)) into the set of normal partial operators on E defined by prop. 3 of IV, p. 271 is called the universally measurable functional calculus mapping associated with $u$.

More generally, let T be a set and $g:$ Sp($u$)$\times T\rightarrow \mathbf{C}$ a mapping; let $t\in T$ be such that the function $g_t:z\mapsto g(z, t)$ is universally measurable on Sp($u$). We then write $g(u, t) =g_t(u)$.

Formula (7) implies in particular that (8) $\|f(u)(y)\|^2=\int|f|^2\mu_y$

Sp($u$)

for every $f\in \mathscr{L}_u$(Sp($u$)) and every $y\in D_f$. Taking $f= 1$, we deduce that $\mu_y$ is a positive measure of total mass $\|y\|^2$.

If $u$ is a normal partial operator on a Hilbert space E and $f\in \mathscr{K}$ (Sp($u$)), then $D_f= E$ and the partial operator $f(u)$ is then continuous, since

for every $y\in E$ by (8). The endomorphism $f(u)$ coincides with that of Definition 3 of IV, p. 268.