On super-rigid and uniformly super-rigid operators

Abstract

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 $$\lambda T^n(U)\cap U\ne \emptyset $$ . 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 also investigate the basic properties of these two notions. In addition, we discuss the spectrum of these two operators. At the end, we study the super-recurrence, the super-rigidity and the uniform super-rigidity behaviors on finite-dimensional spaces.