Ideal properties of the Dunford integration operator

Abstract

Let F be a Banach space. We establish necessary and sufficient conditions for the Dunford integration operator, from the space of F‐valued Dunford integrable functions to the bidual F′′ of F, to belong to a given operator ideal. We also show how this fact can be used to characterize important classes of Banach spaces, such as Banach spaces with the Banach‐Saks property, separable Banach spaces not containing c0, Banach spaces not containing c0 or ℓ1 and Asplund spaces not containing c0.

Ideals of polynomials between Banach spaces revisited

“…The natural extension to multilinear operators and polynomials was designed by Pietsch some years later in [25]. Nowadays, ideals of polynomials and multilinear operators are explored by several authors in different directions (see, for instance, [1,5,6,8,9,10,11,12,13,14,20]). In this paper we are mainly interested in the theory of ideals of polynomials and ideals of multilinear operators between Banach spaces.…”