Mean Ergodic Theorem (Hilbert Space)

Theorem
Let $\GF \in \set {\R, \C}$.

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\mathbb F$.

Let $U : \HH \to \HH$ be a bounded linear operator such that:
 * $\forall f \in \HH : \norm {\map U f} \le \norm f$

Then for each $f \in \HH$:
 * $\ds \lim_{N \mathop \to \infty} \dfrac 1 N \sum_{n \mathop = 0}^{N - 1} \map {U^n} f = \map P f$

where:
 * $U^n$ denotes the $n$ times composition of $U$
 * $I := \set {f \in \HH : \map U f = f}$
 * $P : \HH \to I$ denotes the orthogonal projection on $I$

Proof
Note that $I$ is a closed linear subspace of $\HH$, since $U$ is bounded.

Especially, $P : \HH \to I$ is well-defined.

Moreover, by Direct Sum of Subspace and Orthocomplement:
 * $\HH = I \oplus I^\perp$

Let $f \in \HH$.

We can write:
 * $ f = \map P f + f^\perp$

where $f^\perp \in I^\perp$.

Then we have:

Thus we need to show:
 * $\ds \lim_{N \mathop \to \infty} \dfrac 1 N \sum_{n \mathop = 0}^{N - 1} \map {U^n} {f^\perp} = 0$

Lemma
Let $\epsilon > 0$ be arbitrary.

Since $f^\perp \in \overline B$, there is a $g \in B$ such that:
 * $\norm {f^\perp - g} < \epsilon$

where $g = \map U h - h$ for an $h \in \HH$.

Thus for all $N \ge 2 \norm h \epsilon^{-1}$: