Operator ergodic theorem

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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  ) $ 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  ,  ; one also speaks of "ergodicity at zero" , see  ). 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,  ,

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

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  ,  ,  ), 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  . 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

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