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 K\'erchy. Examples of stable subnormal contractions are given for which the range of $\varphi(T)$ is dense for every $\varphi\in H^\infty$ ($\varphi\not\equiv 0$).