Uniform Mean Ergodic Theorem

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 $f \in \HH$:

$\ds \lim_{N - M \mathop \to \infty} \dfrac 1 {N - M} \sum_{n \mathop = M}^{N - 1} \map {U^n} f = \map P f$

i.e.

$\ds \forall \epsilon \in \R_{>0}: \exists K \in \N: \forall M,N \in \N: N - M \ge K \implies \norm {\frac 1 {N - M} \sum_{n \mathop = M}^{N - 1} \map {U^n} f - \map P f} \le \epsilon$

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

\(\ds \norm {\frac 1 {N - M} \sum_{n \mathop = M}^{N - 1} \map {U^n} f - \map P f}\)

\(=\)

\(\ds \norm {\frac 1 {N - M} \sum_{n \mathop = 0}^{N - M - 1} \map {U^{M+n} } f - \map P f}\)

\(\ds \)

\(=\)

\(\ds \norm {\frac 1 {N - M} \sum_{n \mathop = M}^{N - 1} \map {U^{M+n} } f - \map {U^M} {\map P f} }\)

as $\map P f \in I$

\(\ds \)

\(=\)

\(\ds \norm {\map {U^M} {\frac 1 {N - M} \sum_{n \mathop = 0}^{N - M - 1} \map {U^n} f - \map P f} }\)

\(\ds \)

\(\le\)

\(\ds \norm {\frac 1 {N - M} \sum_{n \mathop = 0}^{N - M - 1} \map {U^n} f - \map P f}\)

by hypothesis

\(\ds \)

\(\to\)

\(\ds 0\)

as $N - M \to \infty$ due to Mean Ergodic Theorem

$\blacksquare$

Sources

• 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $2.5$ The Mean Ergodic Theorem