On coincidence results for summing multilinear operators: interpolation, $\ell_1$-spaces and cotype

TL;DR

This paper explores coincidence results for multilinear operators, focusing on how space properties like cotype and the role of $\, ext{l}_1$ spaces influence these results, and solves an open problem on tensor product interpolation.

Contribution

It extends coincidence results to multilinear operators, analyzing the impact of cotype and $\, ext{l}_1$ spaces, and resolves an open problem in tensor product interpolation.

Findings

Cotype of spaces affects multilinear coincidence results.

$\, ext{l}_1$ spaces play a special role in interpolation.

An open problem on tensor product interpolation is solved.

Abstract

Grothendieck's theorem asserts that every continuous linear operator from $\ell_1$ to $\ell_2$ is absolutely $(1,1)$-summing. This kind of result is commonly called coincidence result. In this paper we investigate coincidence results in the multilinear setting, showing how the cotype of the spaces involved affect such results. The special role played by $\ell_1$ spaces is also investigated with relation to interpolation of tensor products. In particular, an open problem on the interpolation of $m$ injective tensor products is solved.