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For each 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 contraction on a Hilbert space such that $\\sigma(T)\\subset E$ and $\\|T^{-n}\\|=O(u_n)$ as $n\\to\\infty$, then $T$ is a unitary operator.\n  A key tool used in the proof is a result generalizing the well-known fact that closed subsets $E$ of the real axis of Lebesgue measure zero are removable for bounded holomorphic functions. 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