Von Neumann’s ergodic theorem

Let U:HH be an isometry in a Hilbert space H. Consider the subspace I(U)={vH:Uv=v}, called the space of invariant vectors. Denote by P the orthogonal projection over the subspace I(U). Then,


This general theorem for Hilbert spaces can be used to obtain an ergodic theorem for the L2(μ) space by taking H to be the L2(μ) space, and U to be the composition operator (also called Koopman operator) associated to a transformationMathworldPlanetmath f:MM that preserves a measure μ, i.e., Uf(ψ)=ψf, where ψ:M𝐑. The space of invariant functions is the set of functions ψ such that ψf=ψ almost everywhere. For any ψL2(μ), the sequence:


converges in L2(μ) to the orthogonal projection ψ~ of the function ψ over the space of invariant functions.

The L2(μ) version of the ergodic theorem for Hilbert spaces can be derived directly from the more general Birkhoff ergodic theorem, which asserts pointwise convergence instead of convergence in L2(μ). Actually, from Birkhoff ergodic theorem one can derive a version of the ergodic theorem where convergence in Lp(μ) holds, for any p>1.

Title Von Neumann’s ergodic theorem
Canonical name VonNeumannsErgodicTheorem
Date of creation 2014-03-18 14:02:09
Last modified on 2014-03-18 14:02:09
Owner Filipe (28191)
Last modified by Filipe (28191)
Numerical id 6
Author Filipe (28191)
Entry type Theorem
Related topic Birkhoff ergodic theorem