The problem statement, all variables and given/known data Let W be a complex finite dimensional vector space with a hermitian scalar product and let T: W -> W be linear and normal. Prove that U is a T-invariant subspace of W if and only if V is a T*-invariant subspace, where V is the orthogonal complement of U. The attempt at a solution Let u in U and v in V. If U is T-invariant, then (Tu, v) = 0. But (Tu, v) = (u, T*v) so T*v is in V and V is T*-invariant. Similarly, if V is T*-invariant, then U is T-invariant. Note that I haven't used the fact that T is normal. It seems to me that T doesn't have to be normal. Am I wrong?