On weakly mixing Markov operators and non-singular transformations

Abstract

Let P be a Markov operator on L∞(X, Σ, m). Theorem 1: (i) P is weakly mixing ⇔ (ii) For every f∈L∞ there is a sequence {nt} of density 1 such that all w*-cluster points of \(\{ P^{n_i } f\} \) are constants ⇔ (iii) For every f∈L∞ there is a {kj} with \(P^{k_i } f\)w*-convergent to a constant. Theorem 2: If P is induced by a non-singular transformation θ, P is weakly mixing ⇔ For every AeΣ, {θ−n(A)} has a remotely trivial subsequence. The existence of a finite invariant measure is not required in these results.