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≥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)=(jlog⁡j(log⁡log⁡j)α))−1, j≥3, where α>0, and prove that it is compact and that its eigenvalues obey the asymptotics λn∼ϰ(α)/nα 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.