View source for Operator ergodic theorem

{{MSC|47A35}}

[[Category:General theory of linear operators]]

A general name for theorems on the limit of means along an unboundedly lengthening  "time interval"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o0683601.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o0683602.png" />, for the powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o0683603.png" /> of a [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o0683604.png" /> acting on a Banach space (or even on a topological vector space, see [[#References|[5]]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o0683605.png" />, or for a [[One-parameter semi-group|one-parameter semi-group]] of linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o0683606.png" /> acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o0683607.png" /> (cf. also [[Ergodic theorem|Ergodic theorem]]). In the latter case one can also examine the limit of means along an unboundedly diminishing time interval (local ergodic theorems, see [[#References|[5]]], [[#References|[6]]]; one also speaks of  "ergodicity at zero" , see [[#References|[1]]]). Means can be understood in various senses in the same way as in the theory of summation of series. The most frequently used means are the Cesàro means

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or

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and the Abel means, [[#References|[1]]],

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or

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The conditions of ergodic theorems automatically ensure the convergence of these infinite series or integrals; under these conditions, although the Abel means are formed by using all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o06836012.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o06836013.png" />, the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o06836014.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o06836015.png" /> in a finite period of time, unboundedly increasing when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o06836016.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o06836017.png" />), play a major part. The limit of the means (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o06836018.png" />, etc.) can be understood in various senses: In the strong or weak [[Operator topology|operator topology]] (statistical ergodic theorems, i.e. the [[Von Neumann ergodic theorem|von Neumann ergodic theorem]] — historically the first operator ergodic theorem — and its generalizations), in the uniform operator topology (uniform ergodic theorems, see [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]), while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o06836019.png" /> is a function space on a measure space, then also in the sense of almost-everywhere convergence of the means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o06836020.png" />, etc., where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o06836021.png" /> (individual ergodic theorems, i.e. the [[Birkhoff ergodic theorem|Birkhoff ergodic theorem]] and its generalizations; see, for example, the [[Ornstein–Chacon ergodic theorem|Ornstein–Chacon ergodic theorem]]; these are not always called operator ergodic theorems, however). Some operator ergodic theorems compare the force of various of the above-mentioned variants with each other, establishing that, from the existence of limits of means in one sense, it follows that limits exist in another sense [[#References|[1]]]. Some theorems speak not of the limit of means, but of the limit of the ratios of two means (e.g. the Ornstein–Chacon theorem).

There are also operator ergodic theorems for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068360/o06836022.png" />-parameter and even more general semi-groups.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hille,   R.S. Phillips,   "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Wiley  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Neveu,   "Mathematical foundations of the calculus of probabilities" , Holden-Day  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.M. Vershik,   S.A. Yuzvinskii,   "Dynamical systems with invariant measure"  ''Progress in Math.'' , '''8'''  (1970)  pp. 151–215  ''Itogi Nauk. Mat. Anal.'' , '''967'''  (1969)  pp. 133–187</TD></TR><TR><TD valign="top">[5]</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''' :  2  (1977)  pp. 974–1041  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''13'''  (1975)  pp. 129–262</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  U. Krengel,   "Recent progress in ergodic theorems"  ''Astérisque'' , '''50'''  (1977)  pp. 151–192</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  U. Krengel,   "Ergodic theorems" , de Gruyter  (1985)</TD></TR></table>