Non-stable subnormal contractions have nontrivial hyperinvariant subspaces

A contraction $T$ on a (complex, separable) Hilbert space is stable, or of class $C_{0\cdot }$, if $T^n\to 0$ in the strong operator topology. It is proved that for a non-stable pure subnormal contraction $T$ there exists a singular inner function $\theta$ such that the range of $\theta (T)$ is not dense. Consequently, $T$ has nontrivial hyperinvariant subspaces. The proof is based on results by Esterle and Kérchy. Examples of stable subnormal contractions are given for which the range of $\phi (T)$ is dense for every $\phi \in H^{\mathrm{\infty }}$ ($\phi \equiv ̸0$).