Supercyclicity and Hypercyclicity of an Isometry Plus a Nilpotent

Abstract

Suppose that $X$ is a separable normed space and the operators $A$ and $Q$ are bounded on $X$. In this paper, it is shown that if $AQ=QA$, $A$ is an isometry, and $Q$ is a nilpotent then the operator $A+Q$ is neither supercyclic nor weakly hypercyclic. Moreover, if the underlying space is a Hilbert space and $A$ is a co-isometric operator, then we give sufficient conditions under which the operator $A+Q$ satisfies the supercyclicity criterion.