Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space), let $K\subset\mathbb C$ be compact with the [subspace topology](/page/Subspace%20Topology) and Borel $\sigma$-algebra $\mathcal B(K)$, and let $E:\mathcal B(K)\to\mathcal L(H)$ be a projection-valued measure. Let $\iota:K\to\mathbb C$ be the coordinate function defined by $\iota(z)=z$ for every $z\in K$, and define $T:=\int_K \iota\,dE\in\mathcal L(H)$. Then $T$ is a bounded normal operator and $\|T\|_{\mathcal L(H)}\le \sup_{z\in K}|z|$. Moreover, for every bounded Borel function $f:K\to\mathbb C$, the operator $f(T;E):=\int_K f\,dE\in\mathcal L(H)$ commutes with $T$. If $K\subset\mathbb R$, then $T$ is self-adjoint.