Grothendieck-type characterization of representable multilinear mappings

Let μ be a finite measure, T:L1(μ)→X be an operator into a Banach space X, and I:L∞(μ)→L1(μ) be the natural inclusion. A well-known theorem due to Grothendieck states that T is representable if and only T∘I is nuclear. In a 2022 paper we gave a multilinear version of this theorem replacing nuclearity by the weaker property that we call the left integral ℓ1-factorization property. Here we prove the rather surprising result that the multilinear version is also true for nuclearity. As an application, we show that the composition P∘T of an integral operator T with a polynomial P whose Aron–Berner extension to the bidual takes values in the original range space is nuclear. This extends (and strengthens) to the polynomial setting a result on composition of operators also due to Grothendieck.