Two-sided polynomial ideals on Banach spaces

Classes of homogeneous polynomials between Banach spaces have been studied in the last three decades from the perspective of the stability with respect to the composition of a homogeneous polynomial with linear operators. In an attempt to explore the underlying nonlinearity in a more consistent way and to bring the subject closer to its roots in Complex Analysis - and, also, taking into account recent results in the field - in this paper we propose the study of classes of homogeneous polynomials that are stable under the composition with homogeneous polynomials.