$\newcommand{\mc}{\mathcal}$ $\newcommand{\ab}[1]{\langle #1\rangle}$
Theorem B.4 in Einsiedler and Ward's [EW] Ergodic Theory with a view towards Number Theory states the following:
Theorem 1. Spectral Theorem. Let $U $be a unitary operator on a complex Hilbert space $H$.
$(1)$ For each element $f\in H$, there is a unique Borel measure $\mu_f$ on $S^1$ with the property that $$ \ab{U^nf, f} = \int_{S^1} z^n\ d\mu_f(z) $$
(2) The map $$ \sum_{n=-N}^N c_nz^n \mapsto \sum_{n=-N}^N c_n U^n f $$ extends by continuity to a unitary isomorphism between $L^2(S^1, \mu_f)$ and the smallest $U$-invariant subspace in $H$ containing $f$.
(There was some text here. Thanks to @DisintegrationByParts for pointing our an error. I have thus removed the text.)
Question 1. I do not see any motivation for the theorem. The statement seems out of the blue. Can somebody provide some perspective here?
Lastly, let $(X, \mc F, \mu, T)$ be a measure-preserving system. Thus $U_T:L^2(X, \mu)\to L^2(X, \mu)$ defined as $f\mapsto f\circ T$ is a unitary operator.
Question 2. If $g\in L^2(X, \mu)$, what is the meaning of the phrase ``By item (2) of the theorem above, $L^2(S^1, \mu_g)$ is unitarily isomorphic to the cyclic sub-representation of $L^2(X, \mu)$ generated by $g$ under the unitary map $U_T$."
The statement in quotes appears on pg 184 of EW. I am not able to see what is the representation here. The group is probably $S^1$, and the vector space is $L^2(X, \mu)$. But the representation itself is not clear.
Also, how does item (2) of the theorem relate with representations?
Thank you.
You have a unitary representation $\pi:\mathbb{Z} \to Unit(H)$. This induces a *-homomorphism $\pi:L^1(\mathbb{Z}) \to Lin(H): (\pi(f)u,v):= \int f(n)(\pi(n)u,v)dn$ (Folland's harmonic analysis theorem $3.9$).
$L^1(\mathbb{Z})$ is a unital, commutative, Banach algebra whose spectrum can be identified with $S^1$, the dual of $\mathbb{Z}$ (Folland theorem 4.2).
Theorem $4.44$ of the same book then shows how to obtain part $(1)$ of the theorem in EW.
I'm guessing that for $(2)$, theorem $1.47$ of Folland's book is relevant.