# Operator ergodic theorem

2010 Mathematics Subject Classification: Primary: 47A35 [MSN][ZBL]

A general name for theorems on the limit of means along an unboundedly lengthening "time interval" $n = 0 \dots N$, or $0 \leq t \leq T$, for the powers $\{ A ^ {n} \}$ of a linear operator $A$ acting on a Banach space (or even on a topological vector space, see [KSS]) $E$, or for a one-parameter semi-group of linear operators $\{ A _ {t} \}$ acting on $E$( cf. also Ergodic theorem). In the latter case one can also examine the limit of means along an unboundedly diminishing time interval (local ergodic theorems, see [KSS], [K]; one also speaks of "ergodicity at zero" , see [HP]). 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

$$\overline{A}\; _ {N} = \frac{1}{N} \sum _ { n= } 0 ^ { N- } 1 A ^ {n}$$

or

$$\overline{A}\; _ {T} = \frac{1}{T} \int\limits _ { 0 } ^ { T } A _ {t} dt$$

and the Abel means, [HP],

$$\overline{A}\; _ \theta = ( 1- \theta ) \sum _ { n= } 0 ^ \infty \theta ^ {n} A ^ {n} ,\ \ | \theta | < 1 ,$$

or

$$\overline{A}\; _ \lambda = \lambda \int\limits _ { 0 } ^ \infty e ^ {- \lambda t } A _ {t} dt.$$

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 $A ^ {n}$ or $A _ {t}$, the values of $A ^ {n}$ or $A _ {t}$ in a finite period of time, unboundedly increasing when $\theta \rightarrow 1$( or $\lambda \rightarrow 0$), play a major part. The limit of the means ( $\lim\limits _ {N \rightarrow \infty } \overline{A}\; _ {N}$, etc.) can be understood in various senses: In the strong or weak operator topology (statistical ergodic theorems, i.e. the von Neumann ergodic theorem — historically the first operator ergodic theorem — and its generalizations), in the uniform operator topology (uniform ergodic theorems, see [HP], [DS], [N]), while if $E$ is a function space on a measure space, then also in the sense of almost-everywhere convergence of the means $\overline{A}\; _ {N} \phi$, etc., where $\phi \in E$( individual ergodic theorems, i.e. the Birkhoff ergodic theorem and its generalizations; see, for example, the 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 [HP]. 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 $n$- parameter and even more general semi-groups.

How to Cite This Entry:
Operator ergodic theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Operator_ergodic_theorem&oldid=48048
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article