The Spectrum of the Hilbert Matrix on Power-Weighted ℓ² Spaces

For \(\alpha\in\mathbb R\) , let \(\ell^2_\alpha\) be the sequence space with squared norm \(\sum_{n\geq0}|x_n|^2(n+1)^\alpha\) . We determine the spectrum of the classical Hilbert matrix \(H=(m+n+1)^{-1}\) on these spaces, answering a problem from the 2024 AIM workshop on Riemann–Hilbert problems and Toeplitz matrices. The matrix is bounded exactly when \(|\alpha|<1\) . In that range its spectrum is the closed lens bounded by \[ \left\{\frac{\pi}{\cos(\pi|\alpha|/2+i\pi t)}:t\in\mathbb R\right\} \cup\{0\}. \] The boundary is the Fredholm essential and continuous spectrum. The lens interior is simple point spectrum when \(\alpha<0\) and residual spectrum of defect one when \(\alpha>0\) ; the interior Fredholm index is \(-\operatorname{sgn}\alpha\) . The proof compares the diagonally conjugated matrix, modulo a Hilbert–Schmidt operator, with a half-line Wiener–Hopf operator whose symbol is \(\pi/\cos(\pi\alpha/2-i\pi t)\) . Hill's complete classification of the latent eigenvectors of the Hilbert matrix then fixes the point and residual parts. This manuscript is unrefereed and makes no absolute priority claim.