On the spectrum of the multiplicative Hilbert matrix

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.

Keywords

multiplicative Hilbert matrix, Helson matrix, absolutely continuous spectrum

2010 Mathematics Subject Classification

47B32, 47B35

Full Text (PDF format)

Received 29 May 2017

Received revised 31 July 2017

Accepted 30 August 2017

Published 30 April 2018