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Abstract. We show that the set of those Markov operators on the Schatten class C 1 such that limn→∞ P n − Q = 0, where Q is one-dimensional projection, is norm open and dense. If we require that the limit projections must be on strictly positive states, then such operators P form a norm dense G δ . Surprisingly, for the strong operator topology operators the situation is quite the opposite.

We study the convergence of iterates of quadratic stochastic operators that are mean monotonic. They are defined on the convex set of probability measures concentrated on a weakly compact order interval S = [0, f ] of a fixed Banach lattice F. We study their regularity and identify the limits of trajectories either as the "infimum" or "supremum" of the support of initial distributions.