# Von Neumann’s ergodic theorem

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

 $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{n-1}U^{j}(v)=P(v),\forall v\in H$

This general theorem for Hilbert spaces can be used to obtain an ergodic theorem for the $L^{2}(\mu)$ space by taking $H$ to be the $L^{2}(\mu)$ space, and $U$ to be the composition operator (also called Koopman operator) associated to a transformation  $f:M\rightarrow M$ that preserves a measure $\mu$, i.e., $U_{f}(\psi)=\psi\circ f$, where $\psi:M\rightarrow\textbf{R}$. The space of invariant functions is the set of functions $\psi$ such that $\psi\circ f=\psi$ almost everywhere. For any $\psi\in L^{2}(\mu)$, the sequence:

 $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{n-1}\psi\circ f^{j}$

converges in $L^{2}(\mu)$ to the orthogonal projection $\tilde{\psi}$ of the function $\psi$ over the space of invariant functions.

The $L^{2}(\mu)$ 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 $L^{2}(\mu)$. Actually, from Birkhoff ergodic theorem one can derive a version of the ergodic theorem where convergence in $L^{p}(\mu)$ holds, for any $p>1$.

Title Von Neumann’s ergodic theorem VonNeumannsErgodicTheorem 2014-03-18 14:02:09 2014-03-18 14:02:09 Filipe (28191) Filipe (28191) 6 Filipe (28191) Theorem Birkhoff ergodic theorem