Absolutely simple p-summing multilinear operators and applications

Résumé

Building upon the recent work of Saleh and Shaakir [7] on absolutely simple p-summing linear operators, we extend this novel theory to the multilinear setting. We introduce and study the class of absolutely simple p-summing multilinear operators between arbitrary real Banach spaces. We demonstrate that this class forms a normed weak multi-ideal and establish its fundamental properties, including injectivity and an integral characterization via a Pietsch-type domination theorem. A significant advantage of this approach is the ability to compute the simple p-summing norms exactly for any multilinear operator between finite-dimensional normed spaces through linear programming techniques, addressing notable difficulties present in the computation of standard p-summing multilinear norms. We provide a detailed analysis of the duality and factorization properties of this class, establishing several new characterizations. Finally, we conclude with some open problems that naturally arise from this generalization