Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 358 / 3 / a / 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 thatFor bounded $A$ this integral acts on all of $H$. For an unbounded normal operator,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.