Let $H$ be a seperable komplex Hilbert space and $A:D(A)\subset H \rightarrow H$ self adjoint then $C_A:=(A-i)(A+i)^{-1} \in L(H)$ is unitary and known as the Cayley transform of $A$.
Since the Calyey transform is a bijection from the set of all self adjoint operators $A:D(A)\subset H \rightarrow H$ to the set of all unitary operator $U$ satisfying $\mathrm{ran}(I-U)\subset H$ is dense, I wonder if from there it would be correct to just say that any unitary operator $U \in L(H)$ is unitary equivalent to a multiplication operator using the above bijection and the fact that $A:D(A)\subset H \rightarrow H$ self adjoint on a seperable Hilbert space is unitary equivalent to a multiplication operator (spectral theorem for unbounded selfadjoint operators).