A proof of Esterle's conjecture on negative powers of Hilbert-space contractions

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 $u_n\to \mathrm{\infty }$ with the following property: if $T$ is a contraction on a Hilbert space such that $\sigma (T)\subset E$ and $\mathrm{\parallel }{T}^{-n}\mathrm{\parallel }=O(u_n)$ as $n\to \mathrm{\infty }$, 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 $E$.