THE CYCLIC BEHAVIOR OF TRANSLATION OPERATORS ON HILBERT-SPACES OF ENTIRE-FUNCTIONS

We show that the translation operator T: f(z) --> f(z + 1), acting on certain Hilbert spaces consisting of entire functions of slow growth, is hypercyclic in the sense that for some function f in the space, the orbit {T(n)f}0 infinity 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 "axbitrarily compact" operators. Our work extends that of G. D. Birkhoff (1929), who showed that T is hypercyclic on the Frechet 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.