$U=I+C$ is a unitary operator on a Hilbert space $H$, where $C$ is compact operator. Then eigenvectors of operator $U$ creates a orthonormal basis for the Hilbert space $H$.
It is classical theorem that the above fact is correct for the normal compact operators.
If $U$ is unitary then $U^*U=I$, then we conclude: $C+C^*+CC^*=0$. Therefore, $CC^*=C^*C$. It means that $C$ is compact and normal. Therefore there is a complete eigenvectors of operator $C$ for Hilbert space $H$. Eigenvectors of $U$ and $C$ are the same. Hence the proof completes.
Is there other approches?