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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","authors_text":"Thomas Ransford","cross_cats":["math.CV"],"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.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-05-15T14:35:33Z","title":"A proof of Esterle's conjecture on negative powers of Hilbert-space contractions"},"references":{"count":5,"internal_anchors":0,"resolved_work":5,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"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","year":1994},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"J. Esterle. 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