On Markov operators preserving polynomials

The paper is concerned with a special class of positive linear operators acting on the space C(K) of all continuous functions defined on a convex compact subset K of R^d, d \geq 1, having non-empty interior. Actually, this class consists of all positive linear operators T on C(K) which leave invariant the polynomials of degree at most 1 and which, in addition, map polynomials into polynomials of the same degree. Among other things, we discuss the existence of such operators in the special case where K is strictly convex by also characterizing them within the class of positive projections. In particular we show that such operators exist if and only if ∂K is an ellipsoid. Furthermore, a characterization of balls of R^d in terms of a special class of them is furnished. Additional results and illustrative examples are presented as well.