This is a problem from Hoffman & Kunze book on linear algebra: Let V be a finite dimensional inner-product vector space. Let U be a self-adjoint unitary linear operator over V. Show that U(α)=β−γ, where α=β+γ, β is in W and γ is W⊥. Here, W is a subspace of V.
Sorry for my english.
Hint: $U$ is orthoganal diagoniazable as $U$ is selfadjoint. Every eigenvalue of $U$ must be real (by self-adjointness) and of modulus one (by unitarity). So the eigenspaces $W_+ = \ker (U-1)$ and $W_- = \ker (U +1)$ are orthogonal with sum $V$.