# Journal of Operator Theory

Volume 76, Issue 1, Summer 2016  pp. 141-158.

Hypercyclic convolution operators on spaces of entire functions

Authors:  Vinicius V. Favaro (1) and Jorge Mujica (2)
Author institution: (1) FAMAT, UFU, Av. Joao Naves de Avila, 2121, 38.400-902, Uberlandia, MG, Brazil
(2) IMECC, UNICAMP, Rua Sergio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil

Summary:  A classical result of Birkhoff states that every nontrivial translation operator on the space $\mathcal{H}(\mathbb{C})$ of entire functions of one complex variable is hypercyclic. Godefroy and Shapiro extended this result considerably by proving that every nontrivial convolution operator on the space $\mathcal{H}(\mathbb{C}^n)$ of entire functions of several complex variables is hypercyclic. In sharp contrast with these classical results we show that no convolution operator on the space $\mathcal{H}(\mathbb{C}^\mathbb{N})$ of entire functions of infinitely many complex variables is hypercyclic. On the positive side we obtain hypercyclicity results for convolution operators on spaces of entire functions on important locally convex spaces.

DOI: http://dx.doi.org/10.7900/jot.2015oct16.2084
Keywords:  hypercyclicity, convolution operators, entire functions, Banach spaces, locally convex spaces