On super-rigid and uniformly super-rigid operators

TL;DR

This paper introduces and explores the concepts of super-rigidity and uniform super-rigidity in operators on Banach spaces, analyzing their properties and relationships to super-recurrence, with special attention to finite-dimensional cases.

Contribution

It defines new classes of operators, studies their properties, and compares them to super-recurrent operators, extending the understanding of operator dynamics.

Findings

Super-rigid and uniformly super-rigid operators share properties with super-recurrent operators.

The paper characterizes these operators in finite-dimensional spaces.

Properties of these classes are analyzed in relation to existing operator classes.

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 $\lambda T^n(U)\cap U\neq\emptyset$. 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.