Extensões ideais de classes de operadores multilineares e polinômios homogêneos entre espaços de Banach

Abstract

The main goal of this work is to establish necessary and su cient conditions for a class of continuous multilinear operators taking values in Banach spaces belonging to a certain class to be extendible to an ideal of multilinear operators (multi-ideal) and, moreover, to show how to extend an extendible class. Three applications of this extension result are provided: (i) It is proved that the class of quasi- (p)-summing multilinear operators taking values in dual spaces is extendible if and only if this class coincides isometrically with the class of (p)-summing multilinear operators; (ii) Starting with an ideal of (n + 1)-linear functionals, we construct an ideal of n-linear operators and study the main properties of the resulting ideal; (iii) The class of w -sequentially compact multilinear operators taking values in dual spaces is proved not to be extendible to a multi-ideal. Finally, almost the same de nitions, results and proofs are obtained for the case of classes of continuous homogeneous polynomials.