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{{MSC|47A35}}

[[Category:General theory of linear operators]]

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|linear operator]]  $  A $
acting on a Banach space (or even on a topological vector space, see {{Cite|KSS}})  $  E $, 
or for a [[One-parameter semi-group|one-parameter semi-group]] of linear operators  $  \{ A _ {t} \} $
acting on  $  E $(
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 {{Cite|KSS}}, {{Cite|K}}; one also speaks of "ergodicity at zero" , see {{Cite|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, {{Cite|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|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 {{Cite|HP}}, {{Cite|DS}}, {{Cite|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|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 {{Cite|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====
{|
|valign="top"|{{Ref|HP}}|| E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) {{MR|0089373}} {{ZBL|0392.46001}} {{ZBL|0033.06501}}
|-
|valign="top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Wiley (1988) {{MR|1009164}} {{MR|1009163}} {{MR|1009162}} {{ZBL|}}
|-
|valign="top"|{{Ref|N}}|| J. Neveu, "Mathematical foundations of the calculus of probabilities" , Holden-Day (1965) (Translated from French) {{MR|0198505}} {{ZBL|}}
|-
|valign="top"|{{Ref|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 {{MR|0286981}} {{ZBL|0252.28006}}
|-
|valign="top"|{{Ref|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 {{MR|0584389}} {{ZBL|0399.28011}}
|-
|valign="top"|{{Ref|K}}|| U. Krengel, "Recent progress in ergodic theorems" ''Astérisque'' , '''50''' (1977) pp. 151–192 {{MR|486418}} {{ZBL|}}
|-
|valign="top"|{{Ref|K2}}|| U. Krengel, "Ergodic theorems" , de Gruyter (1985) {{MR|0797411}} {{ZBL|0575.28009}}
|}