# Von Neumann ergodic theorem

2010 Mathematics Subject Classification: Primary: 47A35 [MSN][ZBL]

For any isometric operator $U$ on a Hilbert space $H$ and for any $h \in H$ the limit

$$\lim\limits _ {n \rightarrow \infty } \frac{1}{n} \sum _ { k= } 0 ^ { n- } 1 U ^ {k} h = \overline{h}\;$$

exists (in the sense of convergence in the norm of $H$). For a continuous one-parameter group of unitary transformations $\{ U _ {t} \}$ on $H$ and any $h \in H$, the limit

$$\lim\limits _ {T \rightarrow \infty } \frac{1}{T} \int\limits _ { 0 } ^ { T } U _ {t} h d t = \overline{h}\;$$

exists (in the same sense). Here $\overline{h}\;$ is the orthogonal projection of $h$ onto the space of $U$- (or $\{ U _ {t} \}$-) invariant elements of $H$.

J. von Neumann stated and proved this theorem in [N], having in mind in the first instance its application in ergodic theory, when in a measure space $( X , \mu )$ an endomorphism $T$ is given (or a measurable flow $\{ T _ {t} \}$), when $H = L _ {2} ( X , \mu )$ and where $U$ is the shift operator:

$$U h ( x) = h ( T x ) \ \ \textrm{ or } \ U _ {t} h ( x) = \ h ( T _ {t} ( x) ) .$$

In this case von Neumann's theorem states that the time average of $h ( x)$, that is, the mean value of $h ( T ^ {k} x )$, or $h ( T _ {t} x)$, on the time interval $0 \leq k < n$, or $0 \leq t \leq T$, when this interval is lengthened, converges to $\overline{h}\; ( x)$ in mean square with respect to $x$( which is often emphasized by the term mean ergodic theorem). In particular, for a sufficiently long interval the averaged time mean of $h ( x)$ for the majority of $x$ is close to $\overline{h}\; ( x)$. Therefore, von Neumann's theorem (and its generalizations) is frequently (especially when applied to a given case) called the statistical ergodic theorem, in contrast to the individual ergodic theorem, that is, the Birkhoff ergodic theorem (and its generalizations). From the latter (and for $\mu ( x) = \infty$, from arguments used in its proof) one can in this case deduce von Neumann's ergodic theorem. However, in general, when $H$ is not realized as $L _ {2} ( X , \mu )$ and the operator $U$ or $U _ {t}$ is not connected with any transformation in $X$, von Neumann's theorem does not follow from Birkhoff's.

Von Neumann's original proof was based on the spectral decomposition of unitary operators. Later a number of other proofs were published (the simplest is due to F. Riesz, see [H]) and it was generalized to wider classes of groups and semi-groups of operators on Banach spaces (see [VY], [KSS]).

Von Neumann's theorem, and its generalizations, is an operator ergodic theorem.

How to Cite This Entry:
Von Neumann ergodic theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Von_Neumann_ergodic_theorem&oldid=49163
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article