## Arkiv för Matematik

• Ark. Mat.
• Volume 56, Number 1 (2018), 163-183.

### 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.

#### Article information

Source
Ark. Mat., Volume 56, Number 1 (2018), 163-183.

Dates
Revised: 31 July 2017
First available in Project Euclid: 19 June 2019

https://projecteuclid.org/euclid.afm/1560967225

Digital Object Identifier
doi:10.4310/ARKIV.2018.v56.n1.a10

Mathematical Reviews number (MathSciNet)
MR3800464

Zentralblatt MATH identifier
06869107

#### Citation

Perfekt, Karl-Mikael; Pushnitski, Alexander. On the spectrum of the multiplicative Hilbert matrix. Ark. Mat. 56 (2018), no. 1, 163--183. doi:10.4310/ARKIV.2018.v56.n1.a10. https://projecteuclid.org/euclid.afm/1560967225