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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Weighted ergodic theorems
for mean ergodic $L_{1}$-contractions


Authors: Dogan Çömez, Michael Lin and James Olsen
Journal: Trans. Amer. Math. Soc. 350 (1998), 101-117
MSC (1991): Primary 47A35, 28D99
MathSciNet review: 1433114
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Abstract: It is shown that any bounded weight sequence which is good for all probability preserving transformations (a universally good weight) is also a good weight for any $L_{1}$-contraction with mean ergodic (ME) modulus, and for any positive contraction of $L_{p}$ with $1 < p <\infty $. We extend the return times theorem by proving that if $S$ is a Dunford-Schwartz operator (not necessarily positive) on a Lebesgue space, then for any $g$ bounded measurable $\{S^{n} g(\omega )\}$ is a universally good weight for a.e. $\omega .$ We prove that if a bounded sequence has "Fourier coefficents", then its weighted averages for any $L_{1}$-contraction with mean ergodic modulus converge in $L_{1}$-norm. In order to produce weights, good for weighted ergodic theorems for $L_{1}$-contractions with quasi-ME modulus (i.e., so that the modulus has a positive fixed point supported on its conservative part), we show that the modulus of the tensor product of $L_{1}$-contractions is the product of their moduli, and that the tensor product of positive quasi-ME $L_{1}$-contractions is quasi-ME.


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Additional Information

Dogan Çömez
Affiliation: Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105

Michael Lin
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel

James Olsen
Affiliation: Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01986-2
PII: S 0002-9947(98)01986-2
Received by editor(s): October 9, 1995
Dedicated: Dedicated to the Memory of Professor Robert Sine
Article copyright: © Copyright 1998 American Mathematical Society