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On local ergodic convergence of semi-groups and additive processes. (English) Zbl 0545.47016

Let \(\{T_ t\}_{t>0}\) be a strongly continuous (at \(t>0)\) semigroup of bounded linear operators in the \(L_ p\) space \((1\leq p<\infty)\) of a probability space \((X,\Sigma\),m). The question considered is: under what conditions does \(\lim_{\epsilon \to 0+}\frac{1}{\epsilon}\int^{\epsilon}_{0}T_ tf(x)dt\) exist a.e. for every \(f\in L_ p?\)
Many authors have considered this question with various restrictions on p and \(\{T_ t\}\). Here the author considers positive operators \(T_ t\) on \(L_ 1\) and shows that the above limit exists for every \(f\in L_{\infty}\) and \(\sup_{0<t\leq 1}\| T_ t\|<\infty\) if and only if \(\{T_ t\}\) is strongly continuous at zero (in \(L_ 1).\)
The method used in the proof is then used to prove various other results in this area and some results for n-parameter semigroups.
Reviewer: D.Newton

MSC:

47A35 Ergodic theory of linear operators
47D03 Groups and semigroups of linear operators
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