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

TL;DR

This paper proves Esterle's conjecture by showing that for certain measure-zero subsets of the unit circle, specific growth conditions on the inverse powers of a Hilbert-space contraction imply the operator is unitary.

Contribution

It confirms Esterle's conjecture by establishing conditions under which a contraction with spectrum in a measure-zero set must be unitary, extending classical removability results.

Findings

Confirmed Esterle's conjecture for measure-zero spectra

Established new removability results for unbounded holomorphic functions

Linked spectral properties to operator unitarity

Abstract

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. 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 $E$.