Linear operators preserving polynomials, barycenter and difference estimates

We obtain estimates for the difference of linear operators that reproduce all the polynomials (equivalently, all the monomials) up to a fixed degree. The operators under consideration go beyond the class of positive linear operators and are typically characterized by higher rates of convergence. The estimates are based on a notion of barycenter for the linear operators that replace the positive linear functionals in previous settings. For this concept of barycenter, we provide different formulations and geometrical interpretations. Moreover, we explicitly determine it for several examples of operators.