  For a power bounded or polynomially bounded operator $T$ sufficient
conditions for the existence of a nontrivial hyperinvariant subspace are given.
The obtained hyperinvariant subspaces of $T$ have the form of the closure of
the range of $\varphi(T)$. Here $\varphi$ is a singular inner function, if $T$
is polynomially bounded, or $\varphi$ is an analytic in the unit disc function
with absolutely summable Taylor coefficients and singular inner part, if $T$ is
supposed to be power bounded only.
  Also, an example of a quasianalytic contraction $T$ is given. The
quasianalytic spectral set of $T$ is not the whole unit circle $\mathbb T$,
while $\sigma(T)=\mathbb T$. Proofs are based on results by Esterle, Kellay,
Borichev and Volberg.
