# Von Neumann’s ergodic theorem

Let $U:H\to H$ 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,

$$\underset{n\to \mathrm{\infty}}{lim}\frac{1}{n}\sum _{j=0}^{n-1}{U}^{j}(v)=P(v),\forall v\in H$$ |

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^{} $f:M\to M$ that preserves a measure $\mu $, i.e., ${U}_{f}(\psi )=\psi \circ f$, where $\psi :M\to \text{\mathbf{R}}$. 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:

$$\underset{n\to \mathrm{\infty}}{lim}\frac{1}{n}\sum _{j=0}^{n-1}\psi \circ {f}^{j}$$ |

converges in ${L}^{2}(\mu )$ to the orthogonal projection $\stackrel{~}{\psi}$ 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 |
---|---|

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 |