# The cyclic behavior of translation
operators on Hilbert spaces of entire functions

#

Kit C. Chan and Joel H. Shapiro
##

*Indiana Univ. Math. J. 40
(1991) 1421--1449*

**Abstract: **We
show that the translation operator *Tf(z) --> f(z+1),*
acting on certain Hilbert spaces consisting of entire functions
of slow growth, is *hypercyclic* in thesense that for some
function f in the space, the orbit *{T^n f: n >=0}* is
dense. We further show that the operator* T-I* can
be made compact, with approximation numbers decreasing as quickly
as desired, simply by choosing the underlying Hilbert space to
be sufficiently small. This shows that hypercyclic operators can
arise as perturbations of the identity by ``arbitrarily compact''
operators. Our work extends that of G.D. Birkhoff (1929), who
showed that *T* is hypercyclic on the Fréchet
space of *all *entire functions, and it complements recent
work of Herrero and Wang, who were the first to discover that
perturbations of the identity by compacts could be hypercyclic.

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