A hypercyclic finite rank perturbation of a unitary operator

TL;DR

This paper constructs a specific example of a finite rank perturbation of a unitary operator that becomes hypercyclic, answering a longstanding question about the dynamics of such perturbations.

Contribution

It provides the first explicit example of a finite rank perturbation of a hyponormal operator that is supercyclic, expanding understanding of operator dynamics.

Findings

Constructed a unitary operator plus rank 2 perturbation that is hypercyclic.

Answered affirmatively that finite rank perturbations of hyponormal operators can be supercyclic.

Demonstrated the existence of hypercyclic finite rank perturbations in operator theory.

Abstract

A unitary operator $V$ and a rank $2$ operator $R$ acting on a Hilbert space $\H$ are constructed such that $V+R$ is hypercyclic. This answers affirmatively a question of Salas whether a finite rank perturbation of a hyponormal operator can be supercyclic.