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

Abstract

We study Helson matrices (also known as multiplicative Hankel matrices), that is, infinite matrices of the form M (α) = {α(nm)} ∞ n,m=1 , where α is a sequence of complex numbers. Helson matrices are considered as linear operators on 2 (N). By interpreting Helson matrices as Hankel matrices in countably many variables we use the theory of multivariate moment problems to show 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 Helson matrices M (α) as those where the corresponding moment measures μ 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.

A Helson matrix with explicit eigenvalue asymptotics

“…Proof. This was proven in [9], but for completeness we repeat the proof. First let us check that w 1/2 H(ζ 1 )w 1/2 is bounded.…”

“…Reduction to weighted integral Hankel operator. We start by recalling a theorem from [9] which establishes a unitary equivalence modulo kernels between a Helson matrix M(a), where a has an integral representation of the type (2.1), and a weighted integral Hankel type operator w 1/2 H(ζ(·+1))w 1/2 with the integral kernel…”

Contractive inequalities for Bergman spaces and multiplicative Hankel forms

“…In other words, he asked whether there is an analogue of Nehari's theorem [32] in this context. Helson's question inspired several papers [9,11,25,26,35,36]. Following the program outlined in [26], it was established in [35] that there are bounded Hankel forms that do not extend to bounded functionals on H 1 (D ∞ ).…”

“…In the positive direction, it was proved in [25] that if the Hankel form (1) instead satisfies the stronger property of being Hilbert-Schmidt, then its symbol does extend to a bounded functional on H 1 (D ∞ ). Briefly summarizing the most recent development, the result of [35] was generalized in [9], in [11] an analogue of the classical Hilbert matrix was introduced and studied, and in [36] the boundedness of the Hankel form (1) was characterized in terms of Carleson measures in the special case that the form is positive semi-definite.…”

“…A positive answer is only known for even integers. Additionally, the boundedness of positive definite Hankel forms (1) can be formulated in terms of Carleson measures on D ∞ [36], and the same is true for the Volterra operators studied in [10]. From [15], it is known that a measure µ on D d , for d < ∞, is a H p -Carleson measure for one 0 < p < ∞ if and only if it is a Carleson measure for every 0 < p < ∞.…”