Let $\mathcal{H}$ be a Hilbert space and $E:\Sigma\to\mathcal{L}(\mathcal{H})$ be a compactly supported spectral on the Borel $\sigma$algebra $\Sigma$ of $\mathbb{C}$. Then we can form the bounded, normal operator $$A=\int \operatorname{id}_\mathbb{C}\;dE\in\mathcal{L}(\mathcal{H})$$ Do you know a proof for the fact, that $E(\operatorname{spec} A)=\operatorname{id}_H$?
