Chaotic differentiation operators on harmonic functions and simple connectivity
type: Item
instance of: scholarly article
title: Chaotic differentiation operators on harmonic functions and simple connectivity
zbMATH Open document ID: 1334.47012
author: Kit C. Chan
author: Gokul R. Kadel
author: Leonardo Pinheiro
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2015-09-21T00:00:00Z
review text: \textit{J. H. Shapiro} proved in [Suppl. Rend. Circ. Mat. Palermo, II. Ser. 56, 27--48 (1998; Zbl 0996.47009)] that a nonconstant differential operator \(P(D)\) is hypercyclic (or chaotic) on the space of holomorphic functions \(H(G)\) on a region \(G\) of the complex plane if and only if the region \(G\) is simply connected. The authors obtain characterizations of simple connectedness of a planar region \(G\) in terms of the hypercyclicity of differentiation type operators \(p(\partial, \overline{\partial})\) on the space \(h(G)\) of complex valued harmonic functions on \(G\) endowed with the topology of uniform convergence on compact subsets of \(G\). Some examples of universal or chaotic operators on \(h(G)\) are included.
reviewed by: José Bonet
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 46E10
zbMATH DE Number: 6483915
zbMATH Keywords: chaotic operator
zbMATH Keywords: hypercyclic operator
zbMATH Keywords: differentiation operator
zbMATH Keywords: simply connected
zbMATH Keywords: spaces of harmonic functions
MaRDI profile type: Publication
full work available at URL: s13398-014-0188-0
OpenAlex ID: W1994279876
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DOI: 10.1007/S13398-014-0188-0
description: scientific article; zbMATH DE number 6483915