Saphar-Type Duality for Multilinear (p, S)-Summing Operators and Its Applications

Résumé

unifying framework that simultaneously generalizes several known classes of summing multilinear mappings. Our central result establishes a complete Saphar-type duality theorem: for $1\leq p<\infty$, the space of all $(p,S)$-summing $m$-linear operators from $E_1\times \dots \times E_m$ into the dual of a Banach space $Z$ is isometrically isomorphic to the dual of the tensor product $(E_1\tensor\dots\tensor E_m\tensor Z, d_{p^*}^S)$, where $d_{p^*}^S$ is a new generalized multilinear Saphar seminorm.

We prove that this class forms a Banach ideal of multilinear operators and establish a corresponding Pietsch-type factorization theorem through subspaces of vector-valued Lebesgue spaces. As concrete applications, we recover and clarify the known duality results for multilinear $(l_p^s,l_p)$-summing and op-dual $(l_p^s,l_p)$-summing operators. Furthermore, we provide a detailed analysis of inclusion relations between these ideals, constructing explicit counterexamples that demonstrate the strictness of these inclusionws.