Operator ergodic theorem

2020 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.

References

[HP]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) MR0089373 Zbl 0392.46001 Zbl 0033.06501
[DS]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Wiley (1988) MR1009164 MR1009163 MR1009162
[N]	J. Neveu, "Mathematical foundations of the calculus of probabilities" , Holden-Day (1965) (Translated from French) MR0198505
[VY]	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 MR0286981 Zbl 0252.28006
[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 : 2 (1977) pp. 974–1041 Itogi Nauk. i Tekhn. Mat. Anal. , 13 (1975) pp. 129–262 MR0584389 Zbl 0399.28011
[K]	U. Krengel, "Recent progress in ergodic theorems" Astérisque , 50 (1977) pp. 151–192 MR486418
[K2]	U. Krengel, "Ergodic theorems" , de Gruyter (1985) MR0797411 Zbl 0575.28009