Closed injective ideals of multilinear operators, related measures and interpolation

Abstract

We introduce and discuss several ways of extending the inner measure arisen from the closed injective hull of an ideal of linear operators to the multilinear case. In particular, we consider new measures that allow to characterize the operators that belong to a closed injective ideal of multilinear operators as those having measure equal to zero. Some interpolation formulas for these measures, and consequently interpolation results involving ideals of multilinear operators, are established. Examples and applications related to summing multilinear operators are also shown.

On interpolation of weakly compact bilinear operators

“…However, as far as we know, the research on interpolation properties of weakly compact bilinear operators has just started. The properties of these operators (see [5]) allow to apply the abstract results of Manzano, Rueda and Sánchez-Pérez [33,34] to derive interpolation results for weakly compact bilinear operators in certain degenerate cases. That is, when the couple in the target reduces to a single Banach space or both couples in the source reduce to single Banach spaces.…”