On S. Grivaux’ example of a hypercyclic rank one perturbation of a unitary operator

Abstract

Recently, Sophie Grivaux showed that there exists a rank one perturbation of a unitary operator in a Hilbert space which is hypercyclic. We give a similar construction using a functional model for rank one perturbations of singular unitary operators.Mathematics Subject Classification. 47A16, 30A76, 30H10.

On hypercyclic rank one perturbations of unitary operators

“…It is clear that the set E in Theorem can be chosen to be everywhere dense on (or on some arc) and so it is possible that . This answers a question posed in .…”

“…In 2015 A. Baranov and A. Lishanskii gave another proof of this theorem using a functional model of rank one perturbations of unitary operators due to V. V. Kapustin and A. Baranov and D. Yakubovich . The constructions of and were both based on the following interesting theorem (also due to S. Grivaux) which gives a condition sufficient for hypercyclicity in the case when the operator has sufficiently many eigenvectors corresponding to unimodular eigenvalues with a certain “continuity property”. Theorem Let X be a complex separable infinite‐dimensional Banach space, and let T be a bounded operator on X .…”

“…In 2015 A. Baranov and A. Lishanskii [1] gave another proof of this theorem using a functional model of rank one perturbations of unitary operators due to V. Kapustin [11] and A.D. Baranov and D.V. Yakubovich [2].…”

Rank One Perturbations of Diagonal Operators Without Eigenvalues

“…We have that m(P n,p ) = (p+1) 2 2 2n . Moreover L n,p (z) is a subset of the following bigger square…”

“…On the other hand, the opposite phenomenon can happen. Indeed it is proven in [Gri12] that there exists a rank one perturbation of a unitary diagonal operator which has uncountably many eigenvalues (see also [BL15] for an alternate proof).…”