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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 358 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-358/3/a/solution