Limits of hypercyclic operators on Hilbert spaces

Abstract

This article concerns the operators $T\in L\left(H\right)$, defined on a separable Hilbert space H, that belong to the norm closure $\overline{HC\left(H\right)}$ in $L\left(H\right)$ of the set $HC\left(H\right)$ of all hypercyclic operators. Starting from a Herrero's characterization of these operators [11] we deduce some criteria that are very useful in many concrete cases. We also show that if $T\in L\left(H\right)$ is invertible then $T\in \overline{HC\left(H\right)}$ if and only if $T^{-1}\in \overline{HC\left(H\right)}$. This result extends to $\overline{HC\left(H\right)}$ a known result of Kitai and Herrero established for hypercyclic operators, ([13]).