On hypercyclic rank one perturbations of unitary operators

TL;DR

This paper presents a new construction method for hypercyclic rank one perturbations of unitary operators, demonstrating that any Carleson set can serve as the spectrum of such a perturbed operator.

Contribution

It introduces a novel functional model approach for constructing hypercyclic rank one perturbations of singular unitary operators.

Findings

Any Carleson set can be the spectrum of a hypercyclic perturbed operator.

Provides a new construction method for hypercyclic rank one perturbations.

Extends previous results with a different functional model approach.

Abstract

Recently, S. Grivaux showed that there exists a rank one perturbation of a unitary operator in a Hilbert space which is hypercyclic. Another construction was suggested later by the first and the third authors. Here, using a functional model for rank one perturbations of singular unitary operators, we give yet another construction of hypercyclic rank one perturbation of a unitary operator. In particular, we show that any Carleson set on the circle can be the spectrum of a perturbed (hypercyclic) operator.