On S.\,Grivaux' example of a hypercyclic rank one perturbation of a unitary operator (Q2017426)

\textit{S. Grivaux} [Math. Nachr. 285, No. 5--6, 533--544 (2012; Zbl 1251.47007)], answering a question of \textit{S. Shkarin} [Math. Ann. 348, No.~2, 379--393 (2010; Zbl 1194.47009)], proved that there exists a unitary operator \(U\) in the Hilbert space \(\ell^2\) and a rank one operator \(R\) such that \(U+R\) is frequently hypercyclic, hence hypercyclic. The aim of the paper under review is to give a proof of Grivaux's theorem by function theoretic methods. The approach of the authors is based on a functional model for rank one perturbations of singular unitary operators that goes back to \textit{V. V. Kapustin} [J. Math. Sci., New York 92, No.1, 3619--3621 (1998); translation from Zap. Nauchn. Semin. POMI 232, 118--122 (1996; Zbl 0894.47011)]. This model translates any rank one perturbation of a unitary operator to some concrete operator in a model subspace \(K_\theta\) of the Hardy space \(H^2\) for some inner function \(\theta\) on the unit disk. In their main result, the authors show the existence of model spaces with a certain continuous family of vectors with similar properties to the ones constructed inductively by Grivaux in \(\ell^2\) to show the existence of rank one frequently hypercyclic perturbations of unitary operators. The properties of Herglotz functions play an important role in the present construction.