Corollary 1

a) For every $f\in \mathscr{L}_u$(Sp($u$)), the partial operator $f(u)$ is normal and $f(u)^*=\overline{f}(u)$. Moreover, $f(u)$ is positive if $f\geqslant 0$, and self-adjoint if $f$ is real-valued;

b) Let $k\in \mathbf{N}$ and $f(z) =z^k$ for $z\in$ Sp($u$). Then $f(u) =u^k$;

c) Let $\lambda \in \mathbf{C}-$ Sp($u$) and $f(z) = (\lambda -z)^{-1}$ for $z\in$ Sp($u$). Then $f(u) = R(u, \lambda )$;

d) One has $\beta (u) =b(u)$;

e) Let $f\in \mathscr{L}_u$(Sp($u$)) and $g\in \mathscr{L}_u$(Sp($u$)) be such that $|g|\leqslant 1 +|f|$. The domain of $g(u)$ is a core for $f(u)$.

In view of the spectral theorem (th. 1 of IV, p. 266), this follows from the preceding proposition combined, respectively, with:

a) lemma 2, a) of IV, p. 266, prop. 23 of IV, p. 253, and its corollary;

b) prop. 24, b) of IV, p. 255;

c) prop. 22, b) of IV, p. 252;

d) lemma 2, a) of IV, p. 266;

e) prop. 6, b) of IV, p. 232.