Commutant hypercyclicity of Hilbert space operators

Abstract

An operator T on a Hilbert space H is commutant hypercyclic if there is a vector x in H such that the set {Sx : TS = ST} is dense in H. We prove that operators on finite dimensional Hilbert space, a rich class of weighted shift operators, isometries, exponentially isometries and idempotents are all commutant hypercyclic. Then we discuss on commutant hypercyclicity of 2 ? 2 operator matrices. Moreover, for each integer number n ? 2, we give a commutant hypercyclic nilpotent operator of order n on an infinite dimensional Hilbert space. Finally, we study commutant transitivity of operators and give necessary and sufficient conditions for a vector to be a commutant hypercyclic vector.