On super-rigid and uniformly super-rigid operators

An operator $T$ acting on a Banach space $X$ is said to be super-recurrent if for each open subset $U$ of $X$, there exist $\lambda\in\mathbb{K}$ and $n\in \mathbb{N}$ such that $\lambda T^n(U)\cap U\neq\emptyset$. In this paper, we introduce and study the notions of super-rigidity and uniform super-rigidity which are related to the notion of super-recurrence. We investigate some properties of these classes of operators and show that they share some properties with super-recurrent operators. At the end, we study the case of finite-dimensional spaces.