View source for Von Neumann ergodic theorem

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[[Category:General theory of linear operators]]

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

<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>

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

<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>

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" />.

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:

<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>

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.

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]]]).

Von Neumann's theorem, and its generalizations, is an [[Operator ergodic theorem|operator ergodic theorem]].

====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 {{MR|}} {{ZBL|0004.31004}} {{ZBL|58.1271.03}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956) {{MR|0097489}} {{ZBL|0073.09302}} </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 {{MR|286981}} {{ZBL|}} </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 {{MR|0584389}} {{ZBL|0399.28011}} </TD></TR></table>



====Comments====
For a wider variety of ergodic theorems see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> U. Krengel, "Ergodic theorems" , de Gruyter (1985) {{MR|0797411}} {{ZBL|0575.28009}} </TD></TR></table>