Mean ergodic theorem and unitary operators

Let U be a unitary operator on a real or complex Hilbert space. Average the first N powers of U, beginning with identity. The von Neumann mean ergodic theorem says these averages applied to any fixed vector converge in Hilbert norm to its orthogonal projection onto the fixed-vector subspace. This is strong operator convergence, meaning convergence for each vector, rather than convergence in operator norm. The theorem does not supply one uniform rate simultaneously for every unit vector.

For an invertible bimeasurable probability-preserving transformation, composing a square-integrable observable with T gives a unitary Koopman operator. Its fixed vectors are the square-integrable invariant observables. The orthogonal projection onto them is conditional expectation on the invariant sigma-algebra. A noninvertible preserving map still gives an isometry, but surjectivity and hence unitarity may fail. There is a broader mean ergodic theorem for contractions, but the unitary proof below is stated under the actual unitary hypothesis.

Call a vector of the form g minus Ug a coboundary. Its average equals g minus the Nth shift of g, divided by N, because all intermediate terms cancel. Unitarity bounds its norm by twice the norm of g divided by N. Every fixed vector is orthogonal to every coboundary. Conversely, a vector orthogonal to the range of identity minus U is fixed by the adjoint; for a unitary operator this is equivalent to being fixed by U.

Thus the Hilbert space splits orthogonally into fixed vectors and the closure of the coboundary range. The averaging operators have norm at most one, so convergence on actual coboundaries extends by approximation to that closure. This proves the theorem. Closure cannot generally be omitted: a vector can be approximated by coboundaries without being one with a controlled transfer vector g. Consequently the explicit inverse-N bound for one coboundary does not become a universal inverse-N rate for arbitrary observables.

If a nonzero vector is an eigenvector with a unit-modulus eigenvalue different from one, its average is a scalar geometric sum times that vector. The sum divided by N tends to zero, even though the individual shifted vectors keep their original norm forever. This is a clean example of norm convergence of Cesàro averages without convergence of individual iterates. A circle rotation supplies exactly such eigenmodes, and its preserved oscillations will obstruct weak mixing in a later lesson.

On a finite cycle, averaging over a full period projects exactly onto the constant functions on that cycle. With several cycles, the projection is constant separately on each one rather than globally. These finite examples also clarify why a spatial norm includes the probability weights: the squared L2 distance is an average over starting states, not an unweighted Euclidean sum unless the measure was chosen that way. A convergence statement should specify both its norm and its underlying invariant law.