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Pointwise ergodic theorem for unbounded operators in \(L_2\). (English) Zbl 1118.47006

Theory Probab. Appl. 50, No. 4, 662-676 (2006); translation from Teor. Veroyatn. Primen. 50, No. 4, 806-818 (2005).
The main purpose of the paper is to prove a kind of individual ergodic theorem for unbounded operators in \(L_2\) over a probability space. Instead of Cesàro means, Borel methods of summability are used. Recall the definitions of these methods. For \(\alpha >0\) and a sequence \(x=(\xi_n )\) of numbers (or vectors), set \(B_{\alpha }(t,x)=\alpha e^{-t}\sum_{n=0}^{\infty}\frac{t^{n\alpha }}{\Gamma (n\alpha +1)}\xi_n \), \(t>0\). If \((\xi)\) is summable to \(\xi\), we write \(B_{\alpha}\lim \xi_{\alpha }=\xi\).
The main results of the paper are as follows.
(1) Let \(A=\int_{\Delta}z E(dz)\) be the spectral representation of a normal unbounded operator in \(L_2 (\Omega,{\mathcal F},\mu )\). For \(\alpha =2^{-k}\), under some conditions, \(B_{\alpha }\lim A^n \xi = E\{ 1\} \xi\), \(\mu\)-a.e.
(2) For \(\alpha =2^{-k}\), under some hypotheses, the conditions \(B_{\alpha }\lim A^n \xi =E\{ 1\}\xi \) \(\mu\)-a.e.and \(E\{ z: 0<| 1-z| <2^{-n}\} \xi \rightarrow 0\) \(\mu \)-a.e.are equivalent.

MSC:

47A35 Ergodic theory of linear operators
60F15 Strong limit theorems
40G10 Abel, Borel and power series methods
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
46L55 Noncommutative dynamical systems
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