{\Gamma}-supercyclicity

TL;DR

This paper characterizes subsets of complex numbers for which the concepts of b3-supercyclicity, supercyclicity, and hypercyclicity of operators on Banach spaces coincide, extending previous results in operator theory.

Contribution

It provides new characterizations of sets b3 for which b3-supercyclicity matches hypercyclicity and supercyclicity, broadening understanding of operator dynamics.

Findings

Identifies sets b3 where b3-supercyclicity equals hypercyclicity.

Characterizes sets b3 for which hypercyclicity implies b3-supercyclicity.

Extends previous results by Lef3n-Mfcller and Bourdon-Feldman.

Abstract

We characterize the subsets $\Gamma$ of $\C$ for which the notion of $\Gamma$-supercyclicity coincides with the notion of hypercyclicity, where an operator $T$ on a Banach space $X$ is said to be $\Gamma$-supercyclic if there exists $x\in X$ such that $\overline{\text{Orb}}(\Gamma x, T)=X$. In addition we characterize the sets $\Gamma \subset \C$ for which, for every operator $T$ on $X$, $T$ is hypercyclic if and only if there exists a vector $x\in X$ such that the set $\text{Orb}(\Gamma x, T)$ is somewhere dense in $X$. This extends results by Le\'on-M\"uller and Bourdon-Feldman respectively. We are also interested in the description of those sets $\Gamma \subset \C$ for which $\Gamma$-supercyclicity is equivalent to supercyclicity.