Von Neumann mean ergodic theorem

Statement

Let $U$ be a linear isometry on a Hilbert space $H$ , and let $P$ be the orthogonal projection onto $\ker(I-U)$ . The mean ergodic theorem states

\frac1N\sum_{n=0}^{N-1}U^nf\longrightarrow Pf \quad\text{in the norm of }H\qquad(f\in H).

In particular it applies to unitary operators and to Koopman isometries of noninvertible probability-preserving maps.

Proof by telescoping

For an isometry, $\ker(I-U^*)=\ker(I-U)$ , so

H=\ker(I-U)\oplus\overline{\operatorname{ran}(I-U)}.

The averages fix the first summand. On $f=(I-U)h$ , they equal $(h-U^Nh)/N$ , whose norm is at most $2\|h\|/N$ . The averages have norm at most one; approximation therefore gives convergence to zero on the second summand.

Conditional expectation

For $U=U_T$ on a probability space, $P f=\mathbb E[f\mid\mathcal I_T]$ , with $\mathcal I_T$ the invariant sigma-algebra. This is conditional expectation, which is orthogonal projection in $L^2$ . If $T$ is ergodic, $Pf=(\int f\,d\mu)1$ .

Mode of convergence

The conclusion is convergence in $L^2$ for each observable, not operator-norm convergence of the averaging operators. It alone does not imply pointwise convergence along trajectories.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §1.6.1.