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 ${\ell }_1$ to ${\ell }_2$ is absolutely $(1,1)$-summing. If $n\ge 2,$ however, it is very simple to prove that every continuous $n$-linear operator from ${\ell }_1\times ...\times {\ell }_1$ to ${\ell }_2$ is absolutely $(1;1,...,1)$-summing, and even absolutely $(\frac{2}% {n};1,...,1) $-summing$.$ In this note we deal with the following problem: Given a positive integer $n\ge 2$, what is the best constant $g_n>0$ so that every $n$-linear operator from ${\ell }_1\times ...\times {\ell }_1$ to ${\ell }_2$ is absolutely $(g_n;1,...,1)$-summing? We prove that $g_n\le \frac{2}{n+1}$ 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.

6 pages

References in corpus (2)

• Some remarks on absolutely summing multilinear operators
• Remarks on cotype absolutely summing multilinear operators