Proposition 4

Let E be a complex Hilbert space and let $u$ be a normal endomorphism of E. For every function $f\in \mathscr{C}$ (Sp($u$)) and every $\lambda \in \mathbf{C}$, one has

Ker($u-\lambda 1_E$)$\subset$ Ker($f(u)-f(\lambda )1_E$).

The proof is analogous to that of proposition 1 of I, p. 128; let us take up its arguments again. The algebra A introduced in loc. cit. is here a unital involutive subalgebra of $\mathscr{L}(E)$ (EVT, V, p. 43, cor.). It therefore contains the unital involutive subalgebra B generated by $u$, which is commutative. The mapping $\chi : B\rightarrow \mathbf{C}$ which, to $v$, assigns the eigenvalue of $v$ relative to $x$ is a character of B such that $\chi (u) =\lambda$. For every $f\in \mathscr{C}$ (Sp($u$))), one has $f(u)\in B$ and $\chi (f(u)) =f(\chi (u)) =f(\lambda )$ by prop. 8 of I, p. 112, whence the assertion.