The Mean Ergodic Theorem, proved by John von Neumann in the early 1930s, is a foundational result of ergodic theory establishing that the time averages of a function along the orbits of a measure-preserving transformation converge, in the mean-square sense, to the function's average over the whole space. It is the L2 counterpart to the live Birkhoff Ergodic Theorem, which establishes the analogous convergence pointwise, almost everywhere, rather than in the mean.
Facts
StatementVon Neumann's mean ergodic theorem states that for a unitary operator U on a Hilbert space H, the time averages of the powers of U applied to any vector converge to the orthogonal projection of that vector onto the fixed points of U. 1 Sources
1. Wikipedia: Ergodic Theory
Wikimedia FoundationMean ergodic theorem subsection
Let U be a unitary operator on a Hilbert space H
Historical references section
von Neumann, John (1932), "Proof of the Quasi-ergodic Hypothesis", Proc. Natl. Acad. Sci. USA, vol. 18, no. 1, pp. 70-82
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