The spectrum of some Hardy kernel matrices

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