Unified Grothendieck's and Kwapie\'{n}'s theorems for multilinear operators

TL;DR

This paper extends classical theorems by Kwapień and Grothendieck to multilinear operators, unifying their results and exploring the properties of multiple summing and absolutely summing multilinear operators.

Contribution

It provides a unified framework for Kwapień's and Grothendieck's theorems in the multilinear setting, covering multiple summing and absolutely summing operators.

Findings

Unified version of Kwapień's and Grothendieck's theorems for multilinear operators

Characterization of multiple summing multilinear operators

Extension of absolute summing properties to multilinear context

Abstract

Kwapie\'{n}'s theorem asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{p}$ is absolutely $\left( r,1\right) $-summing for $1/r=1-\left\vert 1/p-1/2\right\vert .$ When $p=2$ it recovers the famous Grothendieck's theorem. In this paper investigate multilinear variants of these theorems and related issues. Among other results we present a unified version of Kwapie\'{n}'s and Grothendieck's results that encompasses the cases of multiple summing and absolutely summing multilinear operators.