Multilinear operators factoring through Hilbert spaces

Abstract

We characterize those bounded multilinear operators that factor through a Hilbert space in terms of its behavior in finite sequences. This extends a result, essentially due to Kwapień, from the linear to the multilinear setting. We prove that Hilbert–Schmidt and Lipschitz $2$-summing multilinear operators naturally factor through a Hilbert space. We also prove that the class $\Gamma$ of all multilinear operators that factor through a Hilbert space is a maximal multi-ideal; moreover, we give an explicit formulation of a finitely generated tensor norm $\gamma$ which is in duality with $\Gamma$.