Limits of recurrent operators

Abstract

An operator $T$ acting on a complex, infinite-dimensional Hilbert space $\mathcal{H}$ is deemed recurrent (or super-recurrent) if, for each open subset $U \subset \mathcal{H}$, there exists an integer $n$ (alternatively, there exist $n\in \mathbb{N}$ and $\lambda\in \mathbb{C}$) such that $T^n U\cap U\neq\emptyset,$ (or $\lambda T^n U\cap U\neq\emptyset$ for the super-recurrent case). It is known that if $T$ is recurrent, then the set of eigenvalues of $T^*$, the adjoint of $T^*$, is contained in the unit circle $\mathbb{T}$ and that the union of the spectrum of $T$ and $\mathbb{T}$ is a connected set. By these results, we gave a complete spectral characterization of the norm closure of the class $REC(\mathcal{H})$, which consists of all recurrent operators acting on $\mathcal{H}$. Furthermore, analogous results are obtained for the closely related class $SREC(\mathcal{H})$, the set of all super-recurrent operators on $\mathcal{H}$.