{"paper":{"title":"A proof of Esterle's conjecture on negative powers of Hilbert-space contractions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"For any closed zero-measure set E on the unit circle, there is a growth sequence u_n such that a contraction with spectrum in E and ||T^{-n}||=O(u_n) must be unitary.","cross_cats":["math.CV"],"primary_cat":"math.FA","authors_text":"Thomas Ransford","submitted_at":"2026-05-15T14:35:33Z","abstract_excerpt":"We establish the following result, confirming a conjecture of Jean Esterle. For each closed subset $E$ of the unit circle of Lebesgue measure zero, there exists a positive sequence $u_n\\to\\infty$ with the following property: if $T$ is a contraction on a Hilbert space such that $\\sigma(T)\\subset E$ and $\\|T^{-n}\\|=O(u_n)$ as $n\\to\\infty$, then $T$ is a unitary operator.\n  A key tool used in the proof is a result generalizing the well-known fact that closed subsets $E$ of the real axis of Lebesgue measure zero are removable for bounded holomorphic functions. We show that such sets remain removab"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For each closed subset E of the unit circle of Lebesgue measure zero, there exists a positive sequence u_n→∞ with the following property: if T is a contraction on a Hilbert space such that σ(T)⊂E and ||T^{-n}||=O(u_n) as n→∞, then T is a unitary operator.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The generalization that closed subsets E of Lebesgue measure zero remain removable for certain unbounded holomorphic functions of moderate growth near E (with the notion of moderate growth depending on E). This is the key tool invoked to handle the operator-theoretic conclusion, as described in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves that for every closed zero-measure subset E of the unit circle there exists u_n to infinity making contractions T with σ(T) ⊂ E and ||T^{-n}|| = O(u_n) unitary.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For any closed zero-measure set E on the unit circle, there is a growth sequence u_n such that a contraction with spectrum in E and ||T^{-n}||=O(u_n) must be unitary.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"795e9687b73c58400c224edab59d426d8ca1575fcfbac57d8f2c8ff4a116f866"},"source":{"id":"2605.16004","kind":"arxiv","version":1},"verdict":{"id":"51bef880-1841-48ee-b805-74d050adbf25","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:47:31.450903Z","strongest_claim":"For each closed subset E of the unit circle of Lebesgue measure zero, there exists a positive sequence u_n→∞ with the following property: if T is a contraction on a Hilbert space such that σ(T)⊂E and ||T^{-n}||=O(u_n) as n→∞, then T is a unitary operator.","one_line_summary":"Proves that for every closed zero-measure subset E of the unit circle there exists u_n to infinity making contractions T with σ(T) ⊂ E and ||T^{-n}|| = O(u_n) unitary.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The generalization that closed subsets E of Lebesgue measure zero remain removable for certain unbounded holomorphic functions of moderate growth near E (with the notion of moderate growth depending on E). This is the key tool invoked to handle the operator-theoretic conclusion, as described in the abstract.","pith_extraction_headline":"For any closed zero-measure set E on the unit circle, there is a growth sequence u_n such that a contraction with spectrum in E and ||T^{-n}||=O(u_n) must be unitary."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16004/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T19:01:34.348836Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T19:01:18.993184Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:42.172106Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:01:55.653800Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"af6c52e9d2b64808758473f69cab6f2f4f112af856e22975aaae3516ffa1c93a"},"references":{"count":5,"sample":[{"doi":"","year":1994,"title":"J. Esterle. Distributions on Kronecker sets, strong forms of uniqueness, and closed ideals ofA +.J. Reine Angew. Math., 450:43–82, 1994","work_id":"fdcb6a87-8e68-4eb4-b9d1-f6a0425a556d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1992,"title":"J. Esterle. Uniqueness, strong forms of uniqueness and negative powers of contractions. InFunctional analysis and operator theory (Warsaw, 1992), volume 30 ofBanach Center Publ., pages 127–145. Polish","work_id":"437b9a90-52ce-42ea-8593-0580c6a24abb","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"K. Kellay. Contractions et hyperdistributions ` a spectre de Carleson.J. London Math. Soc. (2), 58(1):185–196, 1998","work_id":"1678b66f-6852-4aaa-9355-a5afb4eba84c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"T. Ransford. Negative powers of Hilbert-space contractions.J. Funct. Anal., 286(10):Paper No. 110397, 21, 2024","work_id":"885ea56c-c43d-4824-ae35-8712832812bf","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1993,"title":"M. Zarrabi. Contractions ` a spectre d´ enombrable et propri´ et´ es d’unicit´ e des ferm´ es d´ enombrables du cercle.Ann. Inst. Fourier (Grenoble), 43(1):251–263, 1993. D´epartement de math´ematique","work_id":"c0e08639-c46a-463a-91aa-e8c60fdad189","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":5,"snapshot_sha256":"697173eb75472229d7af06d7b72ac0e6f95262807e7e879da4ff0bb355f95062","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d41c0d722012e4e77625778b9d2dad97a3bff4cb162330c538a27227facb9747"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}