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Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators

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Abstract

We consider Markov operators L on C[0, 1] such that for a certain \(c \in [0,1)\), \(\Vert (Lf)' \Vert \le c \Vert f' \Vert \) for all \( f \in C^1[0,1]\). It is shown that L has a unique invariant probability measure \(\nu \), and then \(\nu \) 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).

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Authors and Affiliations

  1. School of Mathematics and Natural Sciences, University of Wuppertal, Gaußstraße 20, 42119, Wuppertal, Germany

    Margareta Heilmann

  2. Department of Mathematics, Technical University, Str. Memorandumului 28, 400114, Cluj-Napoca, Romania

    Ioan Raşa

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  1. Margareta Heilmann
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  2. Ioan Raşa
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Correspondence to Margareta Heilmann.

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Heilmann, M., Raşa, I. Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators. Positivity 21, 897–910 (2017). https://doi.org/10.1007/s11117-016-0441-1

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  • DOI: https://doi.org/10.1007/s11117-016-0441-1

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