Abstract
LetT be a Markov operator onL 1(X, Σ,m) withT*=P. We connect properties ofP with properties of all productsP ×Q, forQ in a certain class: (a) (Weak mixing theorem)P is ergodic and has no unimodular eigenvalues ≠ 1 ⇔ for everyQ ergodic with finite invariant measureP ×Q is ergodic ⇔ for everyu ∈L 1 with∝ udm=0 and everyf ∈L ∞ we haveN −1Σ ≠1/N n |<u, P nf>|→0. (b) For everyu ∈L 1 with∝ udm=0 we have ‖T nu‖1 → 0 ⇔ for every ergodicQ, P ×Q is ergodic. (c)P has a finite invariant measure equivalent tom ⇔ for every conservativeQ, P ×Q is conservative. The recent notion of mild mixing is also treated.
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Dedicated to the memory of Shlomo Horowitz
An erratum to this article is available at http://dx.doi.org/10.1007/BF02788933.
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Aaronson, J., Lin, M. & Weiss, B. Mixing properties of Markov operators and ergodic transformations, and ergodicity of cartesian products. Israel J. Math. 33, 198–224 (1979). https://doi.org/10.1007/BF02762161
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DOI: https://doi.org/10.1007/BF02762161