Von Neumann’s ergodic theorem
This general theorem for Hilbert spaces can be used to obtain an ergodic theorem for the space by taking to be the space, and to be the composition operator (also called Koopman operator) associated to a transformation that preserves a measure , i.e., , where . The space of invariant functions is the set of functions such that almost everywhere. For any , the sequence:
converges in to the orthogonal projection of the function over the space of invariant functions.
The 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 . Actually, from Birkhoff ergodic theorem one can derive a version of the ergodic theorem where convergence in holds, for any .
|Title||Von Neumann’s ergodic theorem|
|Date of creation||2014-03-18 14:02:09|
|Last modified on||2014-03-18 14:02:09|
|Last modified by||Filipe (28191)|
|Related topic||Birkhoff ergodic theorem|