  For $\alpha > 0$ we consider the operator $K_\alpha \colon \ell^2 \to \ell^2$
corresponding to the matrix
\[\left(\frac{(nm)^{-\frac{1}{2}+\alpha}}{[\max(n,m)]^{2\alpha}}\right)_{n,m=1}^\infty.\]
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 $\mathscr{H}^2$.
