Weighted ergodic theorems for mean ergodic $L_1$-contractions

Authors:
Doğan Çömez, Michael Lin and James Olsen

Journal:
Trans. Amer. Math. Soc. **350** (1998), 101-117

MSC (1991):
Primary 47A35, 28D99

DOI:
https://doi.org/10.1090/S0002-9947-98-01986-2

MathSciNet review:
1433114

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Abstract | References | Similar Articles | Additional Information

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

**Doğan 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

Received by editor(s):
October 9, 1995

Dedicated:
Dedicated to the Memory of Professor Robert Sine

Article copyright:
© Copyright 1998
American Mathematical Society