Mixing of Cartesian squares of positive operators

Let T be a power bounded positive operator in L ₁(X, Σ, m)of a probability space, given by a transition measure P (x, A). The Cartesian square S is the operator on L ₁ (X × X, Σ × Σ, m × m) induced by the transition measure Q((x, y), A × B)=P(x, A)P(y, B). T is completely mixing if ∝u e dm=0 implies T ⁿ u→0 weakly (where 0≦e ∈L _∞ with T ^* e=e). Theorem. If T has no fixed points, then T is completely mixing if and only if S is completely mixing.