Generalized Kantorovich Operators on Convex Compact Subsets and Their Application to Evolution Problems

In this short survey paper, we review some of recent results contained in Altomare et al. (Banach J Math Anal 11:591–614, 2017; J Math Anal Appl 458:153–173, 2018) and concerning with the generalized Kantorovich operators C n defined on convex compact subsets of ℝ d (d ≥ 1). Such operators constitute a positive approximation process for continuous functions and, in some cases, for integrable functions. Moreover, an asymptotic formula for such approximating operators leads to a differential operator which pregenerates a Markov semigroup on C(K) for which we obtain an approximation formula, in terms of suitable powers of C n , useful to infer some preservation properties of it and, as a consequence, of solutions to evolution problems associated with the generators.