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

For α ∈ ℝ, let ℓ²_α be the sequence space with squared norm ∑n≥0|x_n|²(n+1)^α. 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 |α| < 1. In that range, its spectrum is the closed lens whose boundary is traced by π/cos(π|α|/2 + iπt), t ∈ ℝ, together with 0. The boundary is the Fredholm essential and continuous spectrum. The lens interior is simple point spectrum when α < 0 and residual spectrum of defect one when α > 0; the interior Fredholm index is −sgn(α). The proof compares the diagonally conjugated matrix, modulo a Hilbert–Schmidt operator, with a half-line Wiener–Hopf operator whose symbol is π/cos(πα/2 − iπt). 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. Research status: Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-use disclosure: AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text.