{
  "schema_version": 1,
  "problem_number": "AIM-ANALYSIS-0015",
  "title": "The Spectrum of the Hilbert Matrix on Power-Weighted ℓ² Spaces",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "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.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.FA",
    "math.SP"
  ],
  "keywords": [
    "Hilbert matrix",
    "weighted ℓ² spaces",
    "Wiener–Hopf operators",
    "spectral theory",
    "essential spectrum",
    "AIM-ANALYSIS-0015",
    "open mathematics",
    "mathematical proof",
    "math.FA",
    "math.SP"
  ],
  "manuscript_version_date": "2026-08-29",
  "publication_date": "2026-09-02",
  "publication_date_kind": "first public online release",
  "version": "1.0 (typesetting revision 2026-09-05)",
  "date_modified": "2026-09-05",
  "presentation_revision_only": true,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-analysis-0015/",
  "pdf_url": "https://eulersolve.org/papers/aim-analysis-0015/paper.pdf?v=73a10ba6bf22",
  "doi": "10.5281/zenodo.22245611",
  "zenodo_record_url": "https://zenodo.org/records/22245611",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
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  "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."
}