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

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 ^-1Σ _n ^≠1/N |<u, P ⁿf>|→0. (b) For every u ∈L ₁ with ∝ udm=0 we have {norm of matrix}T ⁿu{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.