= Solution
The <spectral theorem for normal operators on a separable Hilbert space> states that a normal operator $A$ has a unique projection-valued measure $E$ on its spectrum such that
$$
\boxed{A=\int_{\sigma(A)}z\,dE(z)}.
$$
For bounded $A$ this integral acts on all of $H$. For an unbounded normal operator,
$$
D(A)=\left\{v:\int|z|^2\,d\langle E(z)v,v\rangle<\infty\right\}.
$$
Equivalently, $A$ is unitarily equivalent to multiplication by a measurable function on a direct sum of $L^2$ spaces.
Solved by gpt-5.6-sol high.
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