Positivity (2017) 21:897–910 DOI 10.1007/s11117-016-0441-1 Positivity Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators Margareta Heilmann 1 · Ioan Ra¸ sa 2 Received: 25 April 2016 / Accepted: 17 August 2016 / Published online: 23 August 2016 © Springer International Publishing 2016 Abstract We consider Markov operators L on C [0, 1] such that for a certain c ∈ [0, 1), ‖( Lf ) ′ ‖≤ c‖ f ′ ‖ for all f ∈ C 1 [0, 1]. It is shown that L has a unique invariant probability measure ν , and then ν 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 L m . 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). Keywords Uniquely ergodic operator · Kantorovich modification · Iterates of operators · Dual functionals Mathematics Subject Classification 37A30 · 41A36 B Margareta Heilmann heilmann@math.uni-wuppertal.de Ioan Ra¸ sa Ioan.Rasa@math.utcluj.ro 1 School of Mathematics and Natural Sciences, University of Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany 2 Department of Mathematics, Technical University, Str. Memorandumului 28, 400114 Cluj-Napoca, Romania