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). X 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 Cesaro operator on C([0, 1]). (c) 2012 Elsevier Inc. All rights reserved.