On coincidence results for summing multilinear operators: interpolation, ${\ell }_1$-spaces and cotype

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.