A Helson matrix with explicit eigenvalue asymptotics

Abstract

Abstract. A Helson matrix (also known as a multiplicative Hankel matrix) is an infinite matrix with entries {a(jk)} for j, k ≥ 1. Here the (j, k)'th term depends on the product jk. We study a self-adjoint Helson matrix for a particular sequence a(j) = ( √ j log j(log log j) α )) −1 , j ≥ 3, where α > 0, and prove that it is compact and that its eigenvalues obey the asymptotics λ n ∼ κ(α)/n α as n → ∞, with an explicit constant κ(α). We also establish some intermediate results (of an independent interest) which give a connection between the spectral properties of a Helson matrix and those of its continuous analogue, which we call the integral Helson operator.

Generalized multiplicative Hilbert operators

“…The multiplicative Hilbert matrix M p is related to the study of the theory of Hardy spaces of Dirichlet series and Riemann zeta function. See [2], [3] , [4], [6] for some recent results of this topic. We will call H p multiplicative Hilbert operator.…”

The spectrum of some Hardy kernel matrices

“…In [11], some norm bounds relating K and K were considered for Hankel kernels, i.e. k(x, y) = h(x + y).…”

Restriction theorems for Hankel operators

“…We also relate this result to the well known unitary equivalence between Hankel matrices and integral Hankel operators. This paper appeared as an attempt to consider one of the technical ingredients of [5] on a more systematic basis. Theorem 1.1 and its proof is based on the same set of ideas as [5,Theorem 3.2].…”

“…This paper appeared as an attempt to consider one of the technical ingredients of [5] on a more systematic basis. Theorem 1.1 and its proof is based on the same set of ideas as [5,Theorem 3.2].…”