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arXiv · 2606.13495

A proof of conjectures of Esterle and Ransford on negative powers of contractions

William Verreault
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Abstract

Building on work of Ransford, we prove that whenever $E$ is a closed subset of the unit circle of Lebesgue measure zero, there exists a positive sequence $u_n\to\infty$ such that if $T$ is a contraction on a Hilbert space with $\sigma(T)\subset E$ and $\|T^{-n}\|=O(u_n)$, then $T$ is unitary. This confirms conjectures of Esterle and Ransford. Our main new idea is a spikes-in-collars principle for positive subharmonic functions.

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William Verreault. 2026-06-11. A proof of conjectures of Esterle and Ransford on negative powers of contractions. https://arxiv.org/abs/2606.13495

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