Multilinear Operators Factoring through Hilbert Spaces

TL;DR

This paper extends the characterization of operators that factor through Hilbert spaces from linear to multilinear cases, providing new insights into their structure and properties.

Contribution

It generalizes a classical linear result to multilinear operators, introduces a tensor norm duality, and establishes the maximality of a class of such operators.

Findings

Multilinear operators factoring through Hilbert spaces are characterized by finite sequence behavior.

Hilbert-Schmidt and Lipschitz 2-summing multilinear operators naturally factor through Hilbert spaces.

The class of all such multilinear operators forms a maximal multi-ideal and has a dual tensor norm.

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 S. Kwapie\'{n}, from the linear to the multilinear setting. We prove that Hilbert-Schmidt and Lipschitz $2$-summing multilinear operators naturally factor through a Hilbert space. It is also proved 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 $\Gamma$.