Coincidence of extendible vector-valued ideals with their minimal kernel

Abstract

We provide coincidence results for vector-valued ideals of multilinear operators. More precisely, if A is an ideal of n-linear mappings we give conditions for which the equality A(E1, ..., En; F) = Amin(E1, ..., En; F) holds isometrically. As an application, we obtain in many cases that the monomials form a Schauder basis of the space A(E1, ..., En; F). Several structural and geometric properties are also derived using this equality. We apply our results to the particular case where A is the classical ideal of extendible or Pietsch-integral multilinear operators. Similar statements are given for ideals of vector-valued homogeneous polynomials.