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