The spectrum of some Hardy kernel matrices

Abstract

For $\alpha >0$ we consider the operator $K_{\alpha }:{\ell }^2\to {\ell }^2$ corresponding to the matrix \[\left(\frac{(nm)^{-\frac{1}{2}+α}}{[\max(n,m)]^{2α}}\right)_{n,m=1}^\infty.\] By interpreting $K_{\alpha }$ as the inverse of an unbounded Jacobi matrix, we show that the absolutely continuous spectrum coincides with $[0,2/\alpha ]$ (multiplicity one), and that there is no singular continuous spectrum. There is a finite number of eigenvalues above the continuous spectrum. We apply our results to demonstrate that the reproducing kernel thesis does not hold for composition operators on the Hardy space of Dirichlet series ${\mathcal{H}}^2$.

This paper has been accepted for publication in Annales de l'Institut Fourier

References in corpus (4)

• The multiplicative Hilbert matrix
• Sharp norm estimates for composition operators and Hilbert-type inequalities
• Family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix
• The spectral density of Hardy kernel matrices

Cited by in corpus (2)

• The best constant in a Hilbert-type inequality
• The spectral density of Hardy kernel matrices