von Neumann mean ergodic theorem

Remarks

Proof idea: let $F = \set{v : Uv = v}$ and $B = \set{w - Uw : w \in H}$. For $\norm{U} \le 1$ one shows that $Uv = v$ if and only if $U^* v = v$, and this gives $H = F \oplus \overline{B}$, an orthogonal decomposition. On $F$ the averages are constantly $v$. On $B$ they telescope: $\frac{1}{n} \sum_{k=0}^{n-1} U^k (w - Uw) = \frac{1}{n}(w - U^n w)$, of norm at most $2\norm{w}/n$. Since all averaging operators have norm at most $1$, the convergence to $0$ extends from $B$ to its closure.

Main application: let $T$ be a measure-preserving transformation of a probability space $(X, \mathcal{A}, \mu)$, and let $U f = f \circ T$ on $L^2(\mu)$. Then $U$ is an isometry, and the theorem says that the Birkhoff averages of every $f \in L^2(\mu)$ converge in $L^2$ to the projection of $f$ onto the $T$-invariant functions. If $T$ is ergodic, the invariant functions are the constants, and the limit is the constant $\int_X f \, d\mu$.

Example: for $U = -\id$ on $H = \C$, the fixed space is $\set{0}$. The averages of $v$ equal $v/n$ for odd $n$ and $0$ for even $n$, so they tend to $0 = Pv$.

Going deeper

For a student

This is the Hilbert space form of the ergodic theorem, proved by von Neumann in 1932, and it gives convergence in mean rather than at almost every point. Through the composition operator $f \mapsto f \circ T$, it turns questions about a measure-preserving map into questions about a linear operator, which is the starting point of the spectral theory of dynamical systems. Birkhoff's theorem gives almost-everywhere convergence and needs a different, harder argument based on a maximal inequality.