Francesco Altomare, Mirella Cappelletti Montano, Vita Leonessa, Ioan Rasa; Markov operators, positive semigroups and approximation processes, De Gruyter Studies in Mathematics, vol. 61, Walter de Gruyter, Berlin, 2014, xi+ 313 pp. ISBN 978-3-11-037274-8/

Abstract

Let C(X);C(Y ) be the Banach spaces (with respect to the uniform norm k k1) of real- or complex-valued continuous functions on compact Hausdor spaces X; Y , respectively. A positive linear operator T : C(X) ! C(Y ) is called a Markov operator if T1X = 1Y , where 1Z denotes the function identically equal to 1 on Z. It follows kTk = kT1Xk1 = 1. As a special class of positive linear operators, the Markov operators inherit their properties. For reader's convenience, the authors present in the rst chapter, Positive linear operators and approximation problems, the main notions, tools and results from the theory of linear operators { positive Radon measures, Choquet boundaries, Bauer simplices, Korovkin-type approximation, etc. Good sources for results of this kind are the book by F. Altomare and M. Campiti, Korovkin-type approximation theory and its applications, de Gruyter Studies in Mathematics, vol. 17, W. de Gruyter, Berlin, 1994, and the survey paper by F. Altomare, Korovkin-type theorems and approximation by positive linear operators, Surv. Approx. Theory vol. 5 (2010), 92-164.