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

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.

34 pages, to appear in Proceedings of the London Mathematical Society

References in corpus (3)

• The multiplicative Hilbert matrix
• Failure of Nehari's Theorem for Multiplicative Hankel Forms in Schatten Classes
• Multivariate Determinateness

Cited by in corpus (4)

• Contractive inequalities for Bergman spaces and multiplicative Hankel forms
• Norms of composition operators on the $H^2$ space of Dirichlet series
• On the spaces of bounded and compact multiplicative Hankel operators
• Projecting onto Helson matrices in Schatten classes