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April 2018 On the spectrum of the multiplicative Hilbert matrix
Karl-Mikael Perfekt, Alexander Pushnitski
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Ark. Mat. 56(1): 163-183 (April 2018). DOI: 10.4310/ARKIV.2018.v56.n1.a10


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.


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Karl-Mikael Perfekt. Alexander Pushnitski. "On the spectrum of the multiplicative Hilbert matrix." Ark. Mat. 56 (1) 163 - 183, April 2018.


Received: 29 May 2017; Revised: 31 July 2017; Published: April 2018
First available in Project Euclid: 19 June 2019

zbMATH: 06869107
MathSciNet: MR3800464
Digital Object Identifier: 10.4310/ARKIV.2018.v56.n1.a10

Primary: 47B32 , 47B35

Keywords: absolutely continuous spectrum , Helson matrix , multiplicative Hilbert matrix

Rights: Copyright © 2018 Institut Mittag-Leffler


Vol.56 • No. 1 • April 2018
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