# Unitarily equivalent operators have unitarily equivalent spectral measures

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?

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 $E_B$.