On cotype and a Grothendieck-type theorem for absolutely summing multilinear operators

Abstract

A famous result due to Grothendieck asserts that every continuous linear operator from ℓ 1 to ℓ 2 is absolutely (1, 1)-summing. If n ≥ 2, however, it is very simple to prove that every continuous n-linear operator from ℓ 1 × ··· × ℓ 1 to ℓ 2 is absolutely (1; 1,…, 1)-summing, and even absolutely -summing. In this note we deal with the following problem: Given a positive integer n ≥ 2, what is the best constant g n > 0 so that every n-linear operator from ℓ 1 × ··· × ℓ 1 to ℓ 2 is absolutely (g n 1,…, l)-summing? We prove that and also obtain an optimal improvement of previous recent results (due to Heinz Juenk et al. Geraldo Botelho et al. and Dumitru Popa) on inclusion theorems for absolutely summing multilinear operators.