Grothendieck's theorem for absolutely summing multilinear operators is optimal

TL;DR

This paper determines the optimal constant for absolutely summing multilinear operators from ℓ₁ to ℓ₂, confirming a conjecture and revealing the precise bounds for such operators.

Contribution

It establishes the exact optimal constant for absolutely summing multilinear operators from ℓ₁ spaces, solving a conjecture by Bernardino from 2011.

Findings

Optimal constant g_m = 2/(m+1) for multilinear operators

Existence of non-absolutely summing operators below this constant

Resolution of Bernardino's 2011 conjecture

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.