A proof of Esterle's conjecture on negative powers of Hilbert-space contractions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917500237643776 |
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| author | Ransford, Thomas |
| author_facet | Ransford, Thomas |
| contents | 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 $σ(T)\subset E$ and $\|T^{-n}\|=O(u_n)$ as $n\to\infty$, then $T$ is a unitary operator.
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 removable even for certain unbounded holomorphic functions of moderate growth near $E$, where the notion of `moderate' depends on $E$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16004 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A proof of Esterle's conjecture on negative powers of Hilbert-space contractions Ransford, Thomas Functional Analysis Complex Variables 30J05, 47A45, 47A56 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 $σ(T)\subset E$ and $\|T^{-n}\|=O(u_n)$ as $n\to\infty$, then $T$ is a unitary operator. 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 removable even for certain unbounded holomorphic functions of moderate growth near $E$, where the notion of `moderate' depends on $E$. |
| title | A proof of Esterle's conjecture on negative powers of Hilbert-space contractions |
| topic | Functional Analysis Complex Variables 30J05, 47A45, 47A56 |
| url | https://arxiv.org/abs/2605.16004 |