On some convergence criteria for nets of positive operators on continuous function spaces

We establish a simple criterion concerning the convergence of nets (or generalized sequences) of positive linear operators on C(X)C(X), XX compact, toward a positive linear operator. As a consequence we discuss several criteria about the asymptotic behavior of the iterates of Markov operators. Finally, we present some applications concerning the asymptotic behavior of the iterates of Bernstein–Schnabl operators on a convex compact subset (not necessarily associated with a positive projection) and the iterates of a special linear operator generalizing the so-called Cesàro operator on C([0,1])C([0,1]).