It is known that for a unitary operator $U$: $\mathcal{H}\rightarrow \mathcal{H}$, where $\mathcal{H}$ is a separable Hilbert space, there exist a probability measure $\mu$ on the boundary of the unit disk in $\mathbb{C}$, $\partial\mathbb{D}$, and a linear mapping $V: \mathcal{H}\rightarrow L^2(\partial\mathbb{D}, d\mu)$ (which universally exists and does not depend on $U$?), such that
a. $\forall x\in\mathcal{H}$, $V(Ux)=zV(x)$;
b. $\forall x, y\in\mathcal{H}$, $(x, y)_{\mathcal{H}}=(Vx, Vy)_{L^2(\partial\mathbb{D}, d\mu)}$
For this statement, I am wondering if it is true that two unitary operators $U_1, U_2$ have the same spectral measure $\mu$ if and only if there exists a unitary operator $W$ such that $U_1=W^*U_2W$.