Question on spectral theorem for unitary operator

Question

I am trying to understand the proof of spectral theorem for unitary operator $A$. In the proof from most materials, I noticed that people first show that $A$ is unitarily equivalent to some multiplication operator $M_\phi$, then they try to show that there exists a Borel measure for $M_\phi$ instead of $A$. So, why they can assmume $A=M_\phi$? Is it because $M_\phi$ and $A$ have the same spectrum?

Answer 1

"Unitarily equivalent" is much much stronger than "same spectrum". Two unitarily equivalent operators are the same in almost any respect. In particular, if $A$ and $B$ are unitarily equivalent, that is $A=U^*BU$, and $E(\Delta)$ is a spectral measure for $B$, then $U^*E(\Delta)U$ is a spectral measure for $A$.