Ergodic Theory - Mean Ergodic Theorem

Mean Ergodic Theorem

Von Neumann's mean ergodic theorem, holds in Hilbert spaces.

Let U be a unitary operator on a Hilbert space H; more generally, an isometric linear operator (that is, a not necessarily surjective linear operator satisfying ‖Ux‖ = ‖x‖ for all x in H, or equivalently, satisfying U*U = I, but not necessarily UU* = I). Let P be the orthogonal projection onto {ψ ∈ H| Uψ = ψ} = Ker(I - U).

Then, for any x in H, we have:

where the limit is with respect to the norm on H. In other words, the sequence of averages

converges to P in the strong operator topology.

This theorem specializes to the case in which the Hilbert space H consists of L2 functions on a measure space and U is an operator of the form

where T is a measure-preserving endomorphism of X, thought of in applications as representing a time-step of a discrete dynamical system. The ergodic theorem then asserts that the average behavior of a function ƒ over sufficiently large time-scales is approximated by the orthogonal component of ƒ which is time-invariant.

In another form of the mean ergodic theorem, let U_t be a strongly continuous one-parameter group of unitary operators on H. Then the operator

converges in the strong operator topology as T → ∞. In fact, this result also extends to the case of strongly continuous one-parameter semigroup of contractive operators on a reflexive space.

Remark: Some intuition for the mean ergodic theorem can be developed by considering the case where complex numbers of unit length are regarded as unitary transformations on the complex plane (by left multiplication). If we pick a single complex number of unit length (which we think of as U), it is intuitive that its powers will fill up the circle. Since the circle is symmetric around 0, it makes sense that the averages of the powers of U will converge to 0. Also, 0 is the only fixed point of U, and so the projection onto the space of fixed points must be the zero operator (which agrees with the limit just described).