I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal direct sum (finite or countable) of cyclic sub-spaces (i.e. spaces of the form $\operatorname{cls}(\operatorname{span}\{U^nx/n\in\mathbb{Z}\})$ for some vector $x$).
I couldn't find a proof for that, so if someone could give me a reference or a sketch of the proof it would be great.
Best Answer
Take the following "typical" unitary operator: on the Hilbert space $L^2(\mathbb{T}, \mu)$ where $\mu$ is a finite Borel measure on the circle, define
$$ V f(z) = z f(z). $$
Then certainly the claim holds for $V$; the cyclic vector can be any trigonometric monomial and there is just one summand. (The trigonometric polynomials are dense by Stone-Weierstrass and regularity arguments.)
But the spectral theorem says an arbitrary unitary on a separable Hilbert space is unitarily equivalent to a countable direct sum of such $V$'s.