$(p,q)$-Dominated Multilinear Operators and Laprest\'e tensor norms

Abstract

We introduce a notion of $(p,q)$-dominated multilinear operators which stems from the geometrical approach provided by $\Sigma$-operators. We prove that $(p,q)$-dominated multilinear operators can be characterized in terms of their behavior on finite sequences and in terms of their relation with a Laprest\'e tensor norm. We also prove that they verify a generalization of the Pietsch's Domination Theorem and Kwapie\'n's Factorization Theorem. Also, we study the collection $\mathcal{D}_{p,q}$ of all $(p,q)$-dominated multilinear operators showing that $\mathcal{D}_{p,q}$ has a maximal ideal demeanor and that the Laprest\'e norm has a finitely generated behavior.