Negative powers of Hilbert-space contractions

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