Typicality of operators on Fréchet algebras admitting a hypercyclic algebra

2023/12/13 by William Alexandre, Alexandre, William, Clifford Gilmore +3
Computer Science · #46B87 #46E05 #54E52 #7A16 #Advanced Algebra and Logic #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2312.08321

openalex publication_date 2023/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

This paper is devoted to the study of typical properties (in the Baire Category sense) of certain classes of continuous linear operators acting on Fréchet algebras, endowed with the topology of pointwise convergence. Our main results show that within natural Polish spaces of continuous operators acting on the algebra H(ℂ) of entire functions on ℂ, a typical operator supports a hypercyclic algebra. We also investigate the case of the complex Fréchet algebras X=ℓ_p(ℕ), 1≤ p<+∞, or X=c₀(ℕ) endowed with the coordinatewise product, and show that whenever M>1, a typical operator on X of norm less than or equal to M admits a hypercyclic algebra.

Citations

• Linear dynamical systems on Hilbert spaces: typical properties and\n explicit examples

Related

• Baire theorem and hypercyclic algebras
• Hypercyclic operators on countably dimensional spaces
• A review of some recent work on hypercyclicity
• Linear Factorization of Hypercyclic Functions for Differential Operators
• \mathcal F-Hypercyclic operators on Fr' echet spaces