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 $0\le t\le T$, 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

$$\bar{A}{}_N=\frac{1}{N}\sum _{n=0}^{N-1}{A}^n$$

or

$$\bar{A}{}_T=\frac{1}{T}\underset{0}{\overset{T}{\int }}{A}_{t}dt$$

and the Abel means, [HP],

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

or

$$\bar{A}{}_{\lambda }=\lambda \underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\lambda t}{A}_{t}dt.$$

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 $\theta \to 1$( or $\lambda \to 0$), play a major part. The limit of the means ( $\underset{N\to \mathrm{\infty }}{lim}\bar{A}{}_N$, 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 $\bar{A}{}_N\varphi$, etc., where $\varphi \in E$( 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.

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