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For every densely defined self-adjoint linear operator $ A : \mathcal D(A) \subset H \to H $ there is a unique spectral representation $$ A = \int t \, dE_A(t) $$ where $E_A$ is a spectral measure on $\mathbb R$. Now let $ B : \mathcal D(B) \subset H \to H $ be another densely defined self-adjoint operator on $H$ with spectral measure $E_B$ which is unitarly equivalent to $A$ with unitary map $U$: $$ AU=UB $$

This page is stating that the spectral measures are unitarily equivalent as well but I don't know how to prove it. Can someone give me a hint?

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The spectral measure of $A$ is uniquely determined by Stone's Formula: $$ \frac{1}{2}(E[a,b]+E(a,b))x \\= \lim_{\epsilon\downarrow 0}\frac{1}{2\pi i}\int_{a}^{b}\{(A-(u+i\epsilon)I)^{-1}x-(A-(u-i\epsilon)I)^{-1}x\} du $$ If you replace $A$ by $UBU^{-1}$, you can work out how $U$ intertwines with $E_A$ and $E_B$.

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