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

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