Supercyclic vectors of operators on normed linear spaces

TL;DR

This paper investigates supercyclic vectors in infinite-dimensional normed spaces, proving that for such vectors, certain subsequences of their orbits do not span the entire space, addressing a question in operator theory.

Contribution

It provides an affirmative answer to a question about the structure of supercyclic vectors and their orbits in infinite-dimensional normed spaces.

Findings

Existence of subsequences of orbits with non-dense span

Supercyclic vectors do not generate the whole space via certain subsequences

Addresses a previously open question in operator theory

Abstract

We give an affirmative answer to a question asked by Faghih-Ahmadi and Hedayatian regarding supercyclic vectors. We show that if $\mathcal X$ is an infinite-dimensional normed linear space and $T$ is a supercyclic operator on $\mathcal X$, then for any supercyclic vector $x$ for $T$, there exists a strictly increasing sequence $(n_k)_k$ of positive integers such that the closed linear span of the set $\{T^{n_k}x: k\ge 1\}$ is not the whole $\mathcal X$.