Algebrability of the set of hypercyclic vectors for backward shift operators

TL;DR

This paper investigates the algebraic structure of hypercyclic vectors for weighted backward shift operators on sequence spaces, demonstrating their algebrability under certain multiplications, including for classical operators like Rolewicz's and MacLane's.

Contribution

It establishes the algebrability of hypercyclic vectors for weighted backward shifts, including classical operators, under coordinatewise and Cauchy product multiplications.

Findings

Hypercyclic vectors form algebras under coordinatewise multiplication.

Hypercyclic vectors form algebras under Cauchy product.

Sets of hypercyclic vectors for Rolewicz's and MacLane's operators are algebrable.

Abstract

We study the existence of algebras of hypercyclic vectors for weighted backward shifts on Fr\'echet sequence spaces that are algebras when endowed with coordinatewise multiplication or with the Cauchy product. As a particular case we obtain that the sets of hypercyclic vectors for Rolewicz's and MacLane's operators are algebrable.