Weighted ergodic theorems for mean ergodic 𝐿₁-contractions

Çömez, Doğan; Lin, Michael; Olsen, James

doi:10.1090/S0002-9947-98-01986-2

Weighted ergodic theorems for mean ergodic $L_1$-contractions
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by Doğan Çömez, Michael Lin and James Olsen

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

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

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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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