Boundedness properties for Sobolev inner products

Sobolev orthogonal polynomials with respect to measures supported on subsets of the complex plane are considered. The connection between the following properties is studied: the multiplication operator M p(z) = zp(z) defined on the space P of algebraic polynomials with complex coefficients is bounded with respect to the norm defined by the Sobolev inner product, the supports of the measures are compact and the zeros of the orthogonal polynomials lie in a compact subset of the complex plane. As the main result we prove that the boundedness of the multiplication operator M always implies the compactness of the supports.