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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Local ergodic theorems for noncommuting semigroups


Author: S. A. McGrath
Journal: Proc. Amer. Math. Soc. 79 (1980), 212-216
MSC: Primary 28D10; Secondary 47A35
DOI: https://doi.org/10.1090/S0002-9939-1980-0565341-3
MathSciNet review: 565341
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Abstract: Let $ (X,\mu )$ be a $ \sigma $-finite measure space and $ {L_p}(\mu ),1 \leqslant p \leqslant \infty $, the usual Banach spaces of complex-valued functions. For $ k = 1,2, \ldots ,n$, let $ \{ {T_k}(t):t \geqslant 0\} $ be a strongly continuous semigroup of Dunford-Schwartz operators. If

$\displaystyle f \in {R_{n - 1}} = \left\{ {g:\int_{\vert g\vert > t} {\vert g/t... ...g/t\vert)}^{n - 1}}d\mu < \infty {\text{for}}\;{\text{all}}\;t > 0} } \right\},$

then

$\displaystyle \frac{1}{{{\alpha _1}{\alpha _2} \cdots {\alpha _n}}}\int_0^{{\al... ...alpha _i}} {{T_n}({t_n}) \cdots {T_1}({t_1})f(x)d{t_1} \cdots d{t_n} \to f(x)} $

$ \mu $-a.e. as $ {\alpha _1} \searrow 0, \ldots ,{\alpha _n} \searrow 0$ independently. If $ f \in {L_p}(\mu ),1 < p < \infty $, then the limit exists in norm as well as pointwise.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0565341-3
Keywords: Local ergodic theorem, noncommuting semigroups, Dunford-Schwartz operators
Article copyright: © Copyright 1980 American Mathematical Society