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.
Categories: math.FA