# Von Neumann ergodic theorem

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2020 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.

#### References

 [N] J. von Neumann, "Proof of the quasi-ergodic hypothesis" Proc. Nat. Acad. Sci. USA , 18 (1932) pp. 70–82 Zbl 0004.31004 Zbl 58.1271.03 [H] P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) MR0097489 Zbl 0073.09302 [VY] A.M. Vershik, S.A. Yuzvinskii, "Dynamical systems with an invariant measure" Progress in Math. , 8 (1970) pp. 151–215 Itogi Nauk. Mat. Anal. 1967 (1969) pp. 133–187 MR286981 [KSS] A.B. Katok, Ya.G. Sinai, A.M. Stepin, "Theory of dynamical systems and general transformation groups with invariant measure" J. Soviet Math. , 7 : 6 (1977) pp. 964–1065 Itogi Nauk. i Tekhn. Mat. Anal. , 13 (1975) pp. 129–262 MR0584389 Zbl 0399.28011