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

We study Helson matrices (also known as multiplicative Hankel matrices), that is, infinitematrices of the form M(α)={α(nm)}∞n,m=1,whereα is a sequence of complex numbers. Helsonmatrices are considered as linear operators on ℓ2(N). By interpreting Helson matrices as Hankelmatrices in countably many variables we use the theory of multivariate moment problems toshow that M(α) is non-negative if and only if α is the moment sequence of a measure μ on R∞,assuming that α does not grow too fast. We then characterize the non-negative bounded Helsonmatrices M (α) as those where the corresponding moment measures μ are Carleson measuresfor the Hardy space of countably many variables. Finally, we give a complete description ofthe Helson matrices of finite rank, in parallel with the classical Kronecker theorem on Hankelmatrices.