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 $2\times 2$ operator matrices.Moreover, for each integer number $n \geq 2$, 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.