On the spectrum of the multiplicative Hilbert matrix

Abstract

We study the multiplicative Hilbert matrix, i.e. the infinite matrix with entries ${(\sqrt{mn} \log(mn))}^{-1}$ for $m, n \geq 2$. This matrix was recently introduced within the context of the theory of Dirichlet series, and it was shown that the multiplicative Hilbert matrix has no eigenvalues and that its continuous spectrum coincides with $[0, \pi]$. Here we prove that the multiplicative Hilbert matrix has no singular continuous spectrum and that its absolutely continuous spectrum has multiplicity one. Our argument relies on spectral perturbation theory and scattering theory. Finding an explicit diagonalisation of the multiplicative Hilbert matrix remains an interesting open problem.