Chaotic differentiation operators on harmonic functions and simple connectivity
\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.
scientific article; zbMATH DE number 6483915
scientific article