Von Neumann ergodic theorem
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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 , 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 ) and it was generalized to wider classes of groups and semi-groups of operators on Banach spaces (see , ).
Von Neumann's theorem, and its generalizations, is an operator ergodic theorem.
References
| [1] | J. von Neumann, "Proof of the quasi-ergodic hypothesis" Proc. Nat. Acad. Sci. USA , 18 (1932) pp. 70–82 MR Template:ZBL Template:ZBL |
| [2] | P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) MR0097489 Template:ZBL |
| [3] | 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 Template:ZBL |
| [4] | 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 Template:ZBL |
Comments
For a wider variety of ergodic theorems see .
References
| [5] | U. Krengel, "Ergodic theorems" , de Gruyter (1985) MR0797411 Template:ZBL |
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