On Helson matrices: moment problems, non-negativity, boundedness, and finite rank

TL;DR

This paper characterizes Helson matrices as moment matrices, linking their non-negativity and boundedness to measures on infinite-dimensional spaces, and classifies finite-rank Helson matrices similarly to classical Hankel matrices.

Contribution

It provides a novel characterization of non-negative and bounded Helson matrices via multivariate moment problems and describes finite-rank Helson matrices akin to Kronecker's theorem.

Findings

Helson matrices are non-negative iff they are moment matrices of measures on inity.

Bounded Helson matrices correspond to Carleson measures for Hardy spaces in countably many variables.

Finite-rank Helson matrices are fully characterized, paralleling classical Hankel matrix results.

Abstract

We study Helson matrices (also known as multiplicative Hankel matrices), i.e. infinite matrices of the form $M(\alpha) = \{\alpha(nm)\}_{n,m=1}^\infty$, where $\alpha$ is a sequence of complex numbers. Helson matrices are considered as linear operators on $\ell^2(\mathbb{N})$. By interpreting Helson matrices as Hankel matrices in countably many variables we use the theory of multivariate moment problems to show that $M(\alpha)$ is non-negative if and only if $\alpha$ is the moment sequence of a measure $\mu$ on $\mathbb{R}^\infty$, assuming that $\alpha$ does not grow too fast. We then characterize the non-negative bounded Helson matrices $M(\alpha)$ as those where the corresponding moment measures $\mu$ are Carleson measures for the Hardy space of countably many variables. Finally, we give a complete description of the Helson matrices of finite rank, in parallel with the classical Kronecker theorem on Hankel matrices.