I really need your help in solving the following problem: Prove that a unitary operator $U$ acting on a Hilbert space $H$ is the Cayley transform of some self-adjoint operator if and only if $1$ is not an eigenvalue of $U$.
At the first direction, I have tried assuming that 1 is an eigenvalue for the eigenfunction $f$. This implies $Uf=f$ ,where $U=(L-i)(L+i)^{-1}$ for some $L$. But since $U^* U=1$ we get that also $U^* f =f $ , and thus: $
As for the other direction, I need some detailed guidance.
Hope someone will be able to help
Thanks !