Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. We can assume $\mathcal{H}$ is an $L^2$ space and $U$ acts as multiplication by a function $u$ with $|u(x)| = 1$ a.e (by the spectral theorem). A subspace $W \subset \mathcal{H}$ is reducing for $U$ if and only if the projection $P_W$ commutes with $U$. Given that these projections are self-adjoint, it's tempting to wonder whether one can use the spectral theorem for commuting normal operators to classify these projections, and hence the reducing subspaces of $U$.

Can this be done even with the (bilateral) shift operator $S$ on $L^2(T)$? Usually one classifies the reducing subspaces for $S$ by first establishing that the only operators commuting with $S$ are the multiplication operators, and of those the only idempotents are multiplications by characteristic functions. I'm wondering if there's a more general argument behind this, using spectral theory, which generalizes to other unitary operators.