Unified Grothendieck’s and Kwapień’s Theorems for Multilinear Operators

Abstract

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