A Helson matrix with explicit eigenvalue asymptotics

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 $a(j)=(\sqrt{j}\log j(\log \log j{)}^{\alpha }){)}^{-1}$, $j\geq 3$,where$\alpha>0$, and prove that it is compact and that its eigenvalues obey the asymptotics ${\lambda }_n\sim \mathrm{\varkappa }(\alpha )\mathrm{/}{n}^{\alpha }$ as $n\to \mathrm{\infty }$, with an explicit constant $\mathrm{\varkappa }(\alpha )$. 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.