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 $\mathfrak{A}$ is an ideal of $n$-linear mappings we give conditions for which the following equality $\mathfrak{A}(E_1,\dots ,E_n;F)={\mathfrak{A}}^{min}(E_1,\dots ,E_n;F)$ holds isometrically. As an application, we obtain in many cases that the monomials form a Schauder basis on the space $\mathfrak{A}(E_1,\dots ,E_n;F)$. Several structural and geometric properties are also derived using this equality. We apply our results to the particular case where $\mathfrak{A}$ is the classical ideal of extendible or Pietsch-integral multilinear operators. Similar statements are given for ideals of vector-valued homogeneous polynomials.

29 pages. Accepted for publication in Journal of Mathematical Analysis and Applications