On invertible hypercyclic operators

Let $A$ be an invertible (bounded linear) operator acting on a complex Banach space $\mathcal{X}$. $A$ is called hypercyclic if there is a vector $y$ in $\mathcal{X}$ such that the orbit $\operatorname{Orb}(A;y): = \{ y,Ay,{A^2},y, \ldots \}$ is dense in $\mathcal{X}$. ( $\mathcal{X}$ is necessarily separable and infinite dimensional.)

Theorem 1. The following are equivalent for an invertible operator A acting on $\mathcal{X}:({\text{i}})A$ or ${A^{ - 1}}$ is hypercyclic; (ii) $A$ and ${A^{ - 1}}$ are hypercyclic; (iii) there is a vector $z$ such that $\operatorname{Orb}(A;z)^ - = \operatorname{Orb}({A^{ - 1}}{\text{;z}})^ - = \mathcal{X}$ (the upper bar denotes norm-closure); (iv) there is a vector $y$ in $\mathcal{X}$ such that

$$[\operatorname{Orb}(A;y) \cup \operatorname{Orb}({A^{ - 1}};y)]^ - = \mathcal{X}.$$

.

Theorem 2. If $A$ is not hypercyclic, then $A$ and ${A^{ - 1}}$ have a common nontrivial invariant closed subset.