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
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