Spectral theorem for a unitary operator

Statement

For a unitary operator $U$ on a complex Hilbert space $H$ , there is a unique normalized projection-valued measure $E$ on the Borel sets of $\mathbb S^1=\{z:|z|=1\}$ such that

U=\int_{\mathbb S^1}z\,dE(z),\qquad U^n=\int_{\mathbb S^1}z^n\,dE(z)\quad(n\in\mathbb Z).

Moreover, $E(\{\lambda\})$ projects onto $\ker(U-\lambda I)$ .

Eigenvectors need not span

The measure may have a continuous part. Thus spectral resolution is more general than diagonalization in an eigenvector basis. Applied to a Koopman operator, it resolves the possible temporal frequencies of observables.

Averaging

The mean ergodic projection is $E(\{1\})$ : the scalar averages $N^{-1}\sum_{n=0}^{N-1}z^n$ converge to $1$ at $z=1$ and to $0$ otherwise. Dominated convergence for the scalar spectral measures gives the Hilbert-space limit.

References

The Spectral Theorem, UC Berkeley-hosted notes, p. 1, spectral theorem (ii)–(iii). Apply the normal-operator theorem to a unitary, whose spectrum lies on the unit circle.