Baire theorem and hypercyclic algebras

TL;DR

This paper investigates conditions under which hypercyclic operators on Fréchet algebras admit hypercyclic algebras, providing new criteria for convolution operators and backward shifts, including frequent hypercyclicity and hypercyclic algebra existence.

Contribution

It introduces new criteria and characterizations for hypercyclic algebras in the context of convolution operators and backward shifts, including conditions for frequent hypercyclicity.

Findings

Criteria for hypercyclic algebras in convolution operators

Conditions for frequent hypercyclic algebras in weighted backward shifts

Existence of hypercyclic algebras in specific operator settings

Abstract

The question of whether a hypercyclic operator $T$ acting on a Fr{\'e}chet algebra $X$ admits or not an algebra of hypercyclic vectors (but 0) has been addressed in the recent literature. In this paper we give new criteria and characterizations in the context of convolution operators acting on $H(\mathbb C)$ and backward shifts acting on a general Fr{\'e}chet sequence algebra.Analogous questions arise for stronger properties like frequent hypercyclicity. In this trend we give a sufficient condition for a weighted backward shift to admit an upper frequently hypercyclic algebra and we find a weighted backward shift acting on $c_0$ admitting a frequently hypercyclic algebra for the coordinatewise product. The closed hypercyclic algebra problem is also covered.