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On $ d$-parameter pointwise ergodic theorems in $ L\sb 1$


Authors: Shigeru Hasegawa and Ryotaro Sato
Journal: Proc. Amer. Math. Soc. 123 (1995), 3455-3465
MSC: Primary 47A35
DOI: https://doi.org/10.1090/S0002-9939-1995-1249881-0
MathSciNet review: 1249881
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Abstract: Let $ {P_1}, \ldots ,{P_d}$ be commuting positive linear contractions on $ {L_1}$ and let $ {T_1}, \ldots ,{T_d}$ be (not necessarily commuting) linear contractions on $ {L_1}$ such that $ \vert{T_i}f\vert \leq {P_i}\vert f\vert$ for $ 1 \leq i \leq d$ and $ f \in {L_1}$. In this paper we prove that if each $ {P_i},1 \leq i \leq d$, satisfies the mean ergodic theorem, then the averages $ {A_n}({T_1}, \ldots ,{T_d})f = {A_n}({T_1}) \cdots {A_n}({T_d})f$, where $ {A_n}({T_i}) = {n^{ - 1}}\sum\nolimits_{k = 0}^{n - 1} {T_i^k} $, converge a.e. for every $ f \in {L_1}$. When $ {T_1}, \ldots ,{T_d}$ commute, we further prove that the $ {L_1}$-norm convergence of the averages $ {A_n}({P_1}, \ldots ,{P_d})f$ for every $ f \in {L_1}$ implies the a.e. convergence of the averages $ {A_n}({T_1}, \ldots ,{T_d})f$ for every $ f \in {L_1}$. These improve Çömez and Lin's ergodic theorem.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1249881-0
Keywords: Mean and pointwise ergodic theorems, linear contraction, linear modulus, Brunel operator
Article copyright: © Copyright 1995 American Mathematical Society

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