## Abstract

Let T be a Markov operator on L_{ 1}(X, Σ, m) with T*=P. We connect properties of P with properties of all products P ×Q, for Q in a certain class: (a) (Weak mixing theorem)P is ergodic and has no unimodular eigenvalues ≠ 1 ⇔ for every Q ergodic with finite invariant measure P ×Q is ergodic ⇔ for every u ∈L_{ 1} with ∝ udm=0 and every f ∈L_{ ∞} we have N^{ -1}Σ_{ n}^{ ≠1/N} |<u, P^{ n}f>|→0. (b) For every u ∈L_{ 1} with ∝ udm=0 we have {norm of matrix}T^{ n}u{norm of matrix}_{1} → 0 ⇔ for every ergodic Q, P ×Q is ergodic. (c)P has a finite invariant measure equivalent to m ⇔ for every conservative Q, P ×Q is conservative. The recent notion of mild mixing is also treated.

Original language | English |
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Pages (from-to) | 198-224 |

Number of pages | 27 |

Journal | Israel Journal of Mathematics |

Volume | 33 |

Issue number | 3-4 |

DOIs | |

State | Published - 1 Sep 1979 |

## ASJC Scopus subject areas

- General Mathematics