Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space), let $I\in\mathcal L(H)$ denote the identity operator on $H$, let $T \in \mathcal L(H)$ be a bounded normal operator, and set $K:=\sigma(T)$. Let $C^*(T,I)\subset\mathcal L(H)$ denote the unital $C^*$-subalgebra generated by $T$ and $I$, and let $\Phi:C(K)\to C^*(T,I)$ be the continuous functional calculus isometric unital $*$-isomorphism for $T$. For a [continuous function](/page/Continuous%20Function) $f:K\to\mathbb C$, define $f(T):=\Phi(f)$. Then

\begin{align*} \sigma(f(T)) = f(\sigma(T)). \end{align*}