Negative powers of Hilbert-space contractions

Abstract

We show that, given a 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 Hilbert-space contraction such that $\sigma (T)\subset E$ and $\Vert T^{-n}\Vert =O(u_n)$ and rank$(I-T^{\ast }T)<\infty$, then $T$ is a unitary operator. We further show that the condition of measure zero is sharp.

arXiv comment18 pages. Corrects a typo in the previous version