A Helson matrix with explicit eigenvalue asymptotics

Abstract

A Helson matrix (also known as a multiplicative Hankel matrix) is an infinite matrix with entries $\{a(jk)\}$ for $j,k\ge 1$. Here the $(j,k)$'th term depends on the product $jk$. We study a self-adjoint Helson matrix for a particular sequence $a(j)=(\sqrt{j}\log j(\log \log j{)}^{\alpha }){)}^{-1}$, $j\ge 3$, where $\alpha >0$, and prove that it is compact and that its eigenvalues obey the asymptotics ${\lambda }_n\sim \varkappa (\alpha )/n^{\alpha }$ as $n\to \infty$, with an explicit constant $\varkappa (\alpha )$. 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.