# Algebras of frequently hypercyclic vectors

## TL;DR

This paper investigates the algebraic structures of frequently hypercyclic vectors for certain operators, showing non-existence of such algebras in some cases and algebrability in others, revealing nuanced algebraic properties of hypercyclic vectors.

## Contribution

It characterizes when algebras of hypercyclic vectors exist for specific operators and demonstrates algebrability of these vectors under suitable products.

## Key findings

- Backward shift operators lack frequently hypercyclic algebras.
- Differentiation operator's hypercyclic vectors do not form frequently hypercyclic algebras.
- Hypercyclic vectors form algebrable sets under certain conditions.

## Abstract

We show that the multiples of the backward shift operator on the spaces $\ell_{p}$, $1\leq p<\infty$, or $c_{0}$, when endowed with coordinatewise multiplication, do not possess frequently hypercyclic algebras. More generally, we characterize the existence of algebras of $\mathcal{A}$-hypercyclic vectors for these operators. We also show that the differentiation operator on the space of entire functions, when endowed with the Hadamard product, does not possess frequently hypercyclic algebras. On the other hand, we show that for any frequently hypercyclic operator $T$ on any Banach space, $FHC(T)$ is algebrable for a suitable product, and in some cases it is even strongly algebrable.