Chaotic Banach algebras

TL;DR

This paper constructs a special infinite-dimensional Banach algebra with elements exhibiting dense orbits, and demonstrates a hypercyclic operator with unique spectral properties, advancing understanding of operator dynamics.

Contribution

It introduces a novel Banach algebra with elements having dense orbits and constructs a hypercyclic operator with a singleton spectrum where its direct sum is non-cyclic.

Findings

Existence of a Banach algebra with dense orbit sets for specific elements

Construction of a hypercyclic operator with spectrum {1}

Demonstration that T⊕T can be non-cyclic

Abstract

We construct an infinite dimensional non-unital Banach algebra $A$ and $a\in A$ such that the sets $\{za^n:z\in\C,\ n\in\N\}$ and $\{({\bf 1}+a)^na:n\in\N\}$ are both dense in $A$, where $\bf 1$ is the unity in the unitalization $A^{\#}=A\oplus \spann\{{\bf 1}\}$ of $A$. As a byproduct, we get a hypercyclic operator $T$ on a Banach space such that $T\oplus T$ is non-cyclic and $\sigma(T)=\{1\}$.