Iterates of Markov Operators and Constructive Approximation of Semigroups

Abstract

In this paper we survey some recent results concerning the asymptotic behaviour of the iterates of asingle Markov operator or of a sequence of Markov operators. Among other things, a characterization of the convergence of the iterates of Markov operators toward a given Markov projection is discussed in terms of the involvedinterpolation sets.Constructive approximation problems for strongly continuous semigroups of operators in terms of iterates are alsodiscussed. In particular we present some simple criteria concerning their asymptotic behaviour.Finally, some applications are shown concerning Bernstein-Schnabl operators on convex compact sets andBernstein-Durrmeyer operators with Jacobi weights on the unit hypercube. A final section contains some suggestions for possible further researches.