The duality between ideals of multilinear operators and tensor norms

Abstract

In this article we introduce a new framework to study classes of multilinear operators. For doing this, we introduce the notion of ideals of multilinear operators derived from the geometrical approach of $$\varSigma $$-operators. We also introduce a new notion of tensor norms and establish a duality theory for these concepts. The main result of the article is the representation theorem for maximal ideals. Moreover, it is shown that the theory of multi-ideals is recovered as particular case. We prove that some recently introduced classes of multilinear operators, like those factoring through a Hilbert space and (p, q)-dominated multilinear operators, naturally fit in this framework.