Topological Modules/Orthogonal projection

Theorem (Von Neumann ergodic theorem):

Let be Hilbert space, and let be a unitary operator. Further, let the orthogonal projection onto the space be given by . Then


where the limit is taken with respect to the operator norm on , the space of bounded operators on . Moreover, the inequality

is a valid estimate for the convergence rate.

Proof: Suppose first that and . Then

Further, if we set


we obtain

If now the sequence is convergent, we see that its limit is indeed contained within . From the respective former consideration, we may hence infer that the sequence does in fact converge to . We are thus reduced to proving the convergence of the sequence in operator norm. Since is Hilbert space, proving that is a Cauchy sequence will be sufficient. But since

for this is the case; the gaps are closed using that

Taking in the next to last computation yields the desired rate of convergence. These computations also reveal the underlying cause of convergence: The sequence becomes more and more uniform, since applying to it does not change it by a large amount.