Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators

We consider Markov operators L on C[0, 1] such that for a certain c is an element of [0, 1), parallel to(Lf)'parallel to <= c parallel to f 'parallel to for all f is an element of C-1 [0, 1]. It is shown that L has a unique invariant probability measure., and then. is used in order to characterize the limit of the iterates L-m of L. When L is a Kantorovich modification of a certain classical operator from approximation theory, the eigenstructure of this operator is used to give a precise description of the limit of Lm. This way we extend some known results; in particular, we extend the domain of convergence of the dual functionals associated with the classical Bernstein operator, which gives a partial answer to a problem raised in 2000 by Cooper and Waldron (JAT 105: 133-165, 2000, Remark after Theorem 4.20).