Vol. 14, No. 2, 2021

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Two families of hypercyclic nonconvolution operators

Alexander Myers, Muhammadyusuf Odinaev and David Walmsley

Vol. 14 (2021), No. 2, 349–360
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,b D 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,b φ(D) : φ(z)  is an entire function of exponential type with φ(0) = 0}

consists entirely of hypercyclic operators.

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