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2021 Two families of hypercyclic nonconvolution operators
Alexander Myers, Muhammadyusuf Odinaev, David Walmsley
Involve 14(2): 349-360 (2021). DOI: 10.2140/involve.2021.14.349

Abstract

Let H() be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let λ,b, let Cλ,b:H()H() be the composition operator Cλ,bf(z)=f(λz+b), and let D be the derivative operator. We extend results on the hypercyclicity of the nonconvolution operators Tλ,b=Cλ,bD by showing that whenever |λ|1, the collection of operators

{ψ(Tλ,b):ψ(z)H(),ψ(0)=0  and ψ(Tλ,b)  is continuous}

forms an algebra under the usual addition and multiplication of operators which consists entirely of hypercyclic operators (i.e., each operator has a dense orbit). We also show that the collection of operators

{Cλ,bφ(D):φ(z)  is an entire function of exponential type with φ(0)=0}

consists entirely of hypercyclic operators.

Citation

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Alexander Myers. Muhammadyusuf Odinaev. David Walmsley. "Two families of hypercyclic nonconvolution operators." Involve 14 (2) 349 - 360, 2021. https://doi.org/10.2140/involve.2021.14.349

Information

Received: 29 November 2020; Revised: 8 December 2020; Accepted: 22 December 2020; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/involve.2021.14.349

Subjects:
Primary: 47A16

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.14 • No. 2 • 2021
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