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Optimal growth of harmonic functions frequently hypercyclic for the partial differentiation operator

Part of: General theory of linear operators Higher-dimensional theory

Published online by Cambridge University Press:  18 January 2019

Clifford Gilmore ,
Eero Saksman  and
Hans-Olav Tylli
Clifford Gilmore
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland (clifford.gilmore@helsinki.fi; eero.saksman@helsinki.fi; hans-olav.tylli@helsinki.fi)
Eero Saksman
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland (clifford.gilmore@helsinki.fi; eero.saksman@helsinki.fi; hans-olav.tylli@helsinki.fi)
Hans-Olav Tylli
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland (clifford.gilmore@helsinki.fi; eero.saksman@helsinki.fi; hans-olav.tylli@helsinki.fi)
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Abstract

We solve a problem posed by Blasco, Bonilla and Grosse-Erdmann in 2010 by constructing a harmonic function on ℝN, that is frequently hypercyclic with respect to the partial differentiation operator ∂/∂xk and which has a minimal growth rate in terms of the average L2-norm on spheres of radius r > 0 as r → ∞.

Keywords

Frequent hypercyclicitypartial differentiation operatorharmonic functionsgrowth rate

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators
Secondary: 31B05: Harmonic, subharmonic, superharmonic functions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1577 - 1594
Copyright
Copyright © Royal Society of Edinburgh 2019 

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