Two families of hypercyclic nonconvolution operators

Myers, Alexander; Odinaev, Muhammadyusuf; Walmsley, David

doi:10.2140/involve.2021.14.349

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Two families of hypercyclic nonconvolution operators

Alexander Myers, Muhammadyusuf Odinaev and David Walmsley

Vol. 14 (2021), No. 2, 349–360

DOI: 10.2140/involve.2021.14.349

Abstract

Let $H (ℂ)$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $λ, b \in ℂ$ , let $C_{λ, b} : H (ℂ) \to H (ℂ)$ be the composition operator $C_{λ, b} f (z) = f (λ z + b)$ , and let $D$ be the derivative operator. We extend results on the hypercyclicity of the nonconvolution operators $T_{λ, b} = C_{λ, b} \circ D$ by showing that whenever $| λ | \geq 1$ , the collection of operators

{ψ (T_{λ, b}) : ψ (z) \in H (ℂ), ψ (0) = 0 and ψ (T_{λ, 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

{C_{λ, b} \circ φ (D) : φ (z) is an entire function of exponential type with φ (0) = 0}

consists entirely of hypercyclic operators.

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Keywords

hypercyclic operators, nonconvolution operators

Mathematical Subject Classification

Primary: 47A16

Milestones

Received: 29 November 2020

Revised: 8 December 2020

Accepted: 22 December 2020

Published: 6 April 2021

Communicated by Stephan Garcia

Authors

Alexander Myers
St. Olaf College Northfield, MN United States

Muhammadyusuf Odinaev
St. Olaf College Northfield, MN United States

David Walmsley
Department of Mathematics, Statistics, and Computer Science St. Olaf College Northfield, MN United States

Vol. 14, No. 2, 2021