Von Neumann ergodic theorem

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

Comments

For a wider variety of ergodic theorems see [K].

References

[K]	U. Krengel, "Ergodic theorems" , de Gruyter (1985) MR0797411 Zbl 0575.28009