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For any isometric operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v0969101.png" /> on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v0969102.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v0969103.png" /> the limit
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v0969104.png" /></td> </tr></table>
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{{TEX|done}}
  
exists (in the sense of convergence in the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v0969105.png" />). For a continuous one-parameter group of unitary transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v0969106.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v0969107.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v0969108.png" />, the limit
+
{{MSC|47A35}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v0969109.png" /></td> </tr></table>
+
[[Category:General theory of linear operators]]
  
exists (in the same sense). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691010.png" /> is the orthogonal projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691011.png" /> onto the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691012.png" />- (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691013.png" />-) invariant elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691014.png" />.
+
For any isometric operator  $  U $
 +
on a Hilbert space  $  H $
 +
and for any  $  h \in H $
 +
the limit
  
J. von Neumann stated and proved this theorem in [[#References|[1]]], having in mind in the first instance its application in ergodic theory, when in a [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691015.png" /> an endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691016.png" /> is given (or a [[Measurable flow|measurable flow]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691017.png" />), when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691018.png" /> and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691019.png" /> is the shift operator:
+
$$
 +
\lim\limits _ {n \rightarrow \infty } 
 +
\frac{1}{n}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691020.png" /></td> </tr></table>
+
\sum _ { k=0} ^ { n-1} U  ^ {k} h  = \overline{h}\;
 +
$$
  
In this case von Neumann's theorem states that the time average of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691021.png" />, that is, the mean value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691022.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691023.png" />, on the time interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691024.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691025.png" />, when this interval is lengthened, converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691026.png" /> in mean square with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691027.png" /> (which is often emphasized by the term mean ergodic theorem). In particular, for a sufficiently long interval the averaged time mean of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691028.png" /> for the majority of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691029.png" /> is close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691030.png" />. 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|Birkhoff ergodic theorem]] (and its generalizations). From the latter (and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691031.png" />, from arguments used in its proof) one can in this case deduce von Neumann's ergodic theorem. However, in general, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691032.png" /> is not realized as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691033.png" /> and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691034.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691035.png" /> is not connected with any transformation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096910/v09691036.png" />, von Neumann's theorem does not follow from Birkhoff's.
+
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
  
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 [[#References|[2]]]) and it was generalized to wider classes of groups and semi-groups of operators on Banach spaces (see [[#References|[3]]], [[#References|[4]]]).
+
$$
 +
\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 {{Cite|N}}, having in mind in the first instance its application in ergodic theory, when in a [[Measure space|measure space]]  $  ( X , \mu ) $
 +
an endomorphism  $  T $
 +
is given (or a [[Measurable flow|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|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 {{Cite|H}}) and it was generalized to wider classes of groups and semi-groups of operators on Banach spaces (see {{Cite|VY}}, {{Cite|KSS}}).
  
 
Von Neumann's theorem, and its generalizations, is an [[Operator ergodic theorem|operator ergodic theorem]].
 
Von Neumann's theorem, and its generalizations, is an [[Operator ergodic theorem|operator ergodic theorem]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. von Neumann,   "Proof of the quasi-ergodic hypothesis" ''Proc. Nat. Acad. Sci. USA'' , '''18''' (1932) pp. 70–82</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.R. Halmos,   "Lectures on ergodic theory" , Math. Soc. Japan (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR></table>
+
{|
 
+
|valign="top"|{{Ref|N}}|| J. von Neumann, "Proof of the quasi-ergodic hypothesis" ''Proc. Nat. Acad. Sci. USA'' , '''18''' (1932) pp. 70–82 {{MR|}} {{ZBL|0004.31004}} {{ZBL|58.1271.03}}
 
+
|-
 +
|valign="top"|{{Ref|H}}|| P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) {{MR|0097489}} {{ZBL|0073.09302}}
 +
|-
 +
|valign="top"|{{Ref|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 {{MR|286981}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|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 {{MR|0584389}} {{ZBL|0399.28011}}
 +
|}
  
 
====Comments====
 
====Comments====
For a wider variety of ergodic theorems see [[#References|[a1]]].
+
For a wider variety of ergodic theorems see {{Cite|K}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  U. Krengel,   "Ergodic theorems" , de Gruyter (1985)</TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|K}}|| U. Krengel, "Ergodic theorems" , de Gruyter (1985) {{MR|0797411}} {{ZBL|0575.28009}}
 +
|}

Latest revision as of 17:38, 6 January 2024


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
How to Cite This Entry:
Von Neumann ergodic theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Von_Neumann_ergodic_theorem&oldid=11554
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article