Mixing properties of Markov operators and ergodic transformations, and ergodicity of cartesian products

Abstract

LetT be a Markov operator onL1(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 ∈L1 with∝ udm=0 and everyf ∈L∞ we haveN−1Σn≠1/N| |→0. (b) For everyu ∈L1 with∝ udm=0 we have ‖Tnu‖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.