Grothendieck's theorem for absolutely summing multilinear operators is optimal

Abstract

Grothendieck's theorem asserts that every continuous linear operator from ${\ell }_1$ to ${\ell }_2$ is absolutely $\left(1;1\right)$-summing. In this note we prove that the optimal constant $g_m$ so that every continuous $m$-linear operator from ${\ell }_1\times \cdots \times {\ell }_1$ to ${\ell }_2$ is absolutely $\left(g_m;1\right)$-summing is $\frac{2}{m+1}$. We also show that if $g_m<\frac{2}{m+1}$ there is $\mathfrak{c}$ dimensional linear space composed by continuous non absolutely $\left(g_m;1\right)$-summing $m$-linear operators from ${\ell }_1\times \cdots \times {\ell }_1$ to ${\ell }_2.$ In particular, our result solves (in the positive) a conjecture posed by A.T. Bernardino in 2011.