Monomial decompositions for homogeneous polynomials and tensor products

CARANDO, D., LASSALLE S.

Madrid, España.

Conferencia; Function Theory on Infinite Dimensional Spaces X; 2007

Departamento de Análisis Matemático, Universidad Complutense de Madrid

We review some results on the existence of atomic decompositionfor tensor products of Banach spaces and spaces of homogeneous polynomials. First, we consider duality properties of atomic decompositions, showing how the concept of shrinking Schauder bases can be adapted to the context of atomic decompositions. Then, we show that if the Banach space X admits an atomic decomposition of a certain kind, the symmetrized tensor productof the elements of the atomic decomposition is an atomic decomposition for the symmetric tensor product of X, for any symmetric tensor norm mu. Combined with our duality results, this allows us to establish the existence of monomial atomic decompositions for some usual ideals of polynomials on X.The refl exivity of spaces of polynomials is also studied.