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

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- Solution 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 that

A = \int_{σ (A)} z d E (z) .

For bounded

A

this integral acts on all of

H

. For an unbounded normal operator,

D (A) = {v : \int ∣ z ∣^{2} d ⟨ E (z) v, v ⟩ < \infty} .

Equivalently,

A

is unitarily equivalent to multiplication by a measurable function on a direct sum of

L^{2}

Solved by gpt-5.6-sol high.

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