A Helson matrix with explicit eigenvalue asymptotics

摘要

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) =(√jlogj(loglogj)~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.