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 un→∞ with the following property: if T is a Hilbert-space contraction such that σ(T)⊂E and ‖T−n‖=O(un) and rank(I−T⁎T)<∞, then T is a unitary operator. We further show that the condition of measure zero is sharp.