On hypercyclic rank one perturbations of unitary operators

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 countable union of perfect Carleson sets on the circle can be the spectrum of a perturbed (hypercyclic) operator. K E Y W O R D S functional model, hypercyclic operator, inner function, model space, rank one perturbation M S C ( 2 0 1 0 ) 30A76, 30H10, 47A16