Monomial decompositions for homogeneous polynomials and tensor products

DANIEL CARANDO; SILVIA LASSALLE

Madrid

Congreso; Function theory on infinite dimensional spaces; 2007

We review some results on the existenceof atomic decomposition for tensor products of Banach spaces andspaces of homogeneous polynomials. First, we consider dualityproperties of atomic decompositions, showing how the concept ofshrinking Schauder bases can be adapted to the context of atomicdecompositions. Then, we show that if the Banach space $X$ admits anatomic decomposition of a certain kind, the symmetrized tensorproduct of the elements of the atomic decomposition is an atomicdecomposition for the symmetric tensor product $ f{n}_{s,mu} X$,for any symmetric tensor norm $mu$. Combined with our dualityresults, this allows us to establish the existence of monomialatomic decompositions for some usual ideals of polynomials on $X$.The reflexivity of spaces of polynomials is also studied.