Sur les idéaux d’opérateurs sommants

Abstract

The present thesis is devoted to summing non linear operators. We focus our attention on introducing and studying polynomials and multilinear mappings that share good properties of summability with distinguished classes of summing linear operators. In the second chapter we introduce the class of Cohen p-nuclear m-linear operators between Banach spaces. This is the multilinear version of p-nuclear operators. The polynomial variant is obtained thanks to consider the symmetric multilinear mapping associated to the polynomial. This polynomial variant forms the p-nuclear polynomial, and it is used as an illustrative example also in Chapter IV. The main results proved in Chapter II are: a characterization in terms of Pietsch's domination theorem and the related factorization theorem, which is an extension to the multilinear setting of Kwapien's factorization theorem for dominated linear operators. Connections with the theory of absolutely summing m-linear operators are also established. It is worth mentioning that, as a consequence of our results, we show that every Cohen p-nuclear m-linear mapping on arbitrary Banach spaces is weakly compact. The third chapter deals with transformations of sequences via summing nonlinear operators. Operators Tthat belong to some summing operator ideal can be characterized by means of the continuity of an associated tensor operator Tthat is defined between tensor products of sequences spaces. Our aim is to provide a unifying treatment of these tensor product characterizations of summing operators. We work in the more general frame, provided by homogeneous polynomials, where an associated tensor polynomial which plays the role of T, needs to be determined first. Examples of applications are shown. In Chapter IV we characterize in terms of summability those homogeneous polynomials whose linearization is p-nuclear. This characterization provides a strong link between the theory of p-nuclear linear operators and the (non linear) homogeneous p-nuclear polynomials that significantly improves former approaches. The deep connection with Grothendieck integral polynomials is also analyzed.