Mean Ergodic Theorem (Hilbert Space)/Lemma

Lemma

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:

$\forall f \in \HH : \norm {\map U f} \le \norm f$

Let $B \subseteq \HH$ be the linear subspace defined as:

$B := \set {\map U h - h : h \in \HH }$

Then:

$I^\perp \subseteq \overline B$

Proof

It suffice to show $B^\perp \subseteq I$.

Indeed, then it follows:

\(\ds I ^\perp\)

\(\subseteq\)

\(\ds \paren {B^\perp} ^\perp\)

Orthocomplement Reverses Subset

\(\ds \)

\(=\)

\(\ds \overline B\)

Double Orthocomplement is Closed Linear Span

Recall that the adjoint $U^\ast$ of $U$ exists by Existence and Uniqueness of Adjoint.

Let $f \in B^\perp$.

Then $\map U f = f$, as:

\(\ds \norm {\map {U^\ast} f - f}^2\)

\(=\)

\(\ds \innerprod {\map {U^\ast} f - f} {\map {U^\ast} f - f}\)

Definition of Inner Product Norm

\(\ds \)

\(=\)

\(\ds \innerprod {\map {U^\ast} f} g - \innerprod f g\)

where $g := \map {U^\ast} f - f$

\(\ds \)

\(=\)

\(\ds \innerprod f {\map U g - g}\)

\(\ds \)

\(=\)

\(\ds 0\)

as $\map U g - g \in B$

Therefore $\map U f = f$, as:

\(\ds \norm {\map U f - f}^2\)

\(=\)

\(\ds \innerprod {\map U f - f} {\map U f - f}\)

Definition of Inner Product Norm

\(\ds \)

\(=\)

\(\ds \innerprod {\map U f} {\map U f} - \innerprod {\map U f} f - \innerprod f {\map U f} + \norm f^2\)

\(\ds \)

\(\le\)

\(\ds 2 \norm f^2 - \innerprod {\map U f} f - \innerprod f {\map U f}\)

by hypothesis

\(\ds \)

\(=\)

\(\ds 2 \norm f^2 - \innerprod f {\map {U^\ast} f} - \innerprod {\map {U^\ast} f} f\)

\(\ds \)

\(=\)

\(\ds 0\)

as $\map {U^\ast} f = f$

$\blacksquare$