Hypercyclic algebras for convolution operators of unimodular constant term

Abstract

We study the existence of hypercyclic algebras for convolution operators Φ(D) on the space of entire functions whose symbol Φ has unimodular constant term. In particular, we provide new eigenvalue criteria for the existence of densely strongly algebrable sets of hypercyclic vectors. Particular attention has been given to this question for the case where X = H(C) is the algebra of entire functions on the complex plane, endowed with the compact-open topology, and where T is a convolution operator, that is, an operator that commutes with all translations. Godefroy and Shapiro [13] showed that such operators are precisely the ones that commute with the operator D of complex differentiation, that they coincide with the operators of the form T = Φ(D) with Φ ∈ H(C) of exponential type, and that a convolution operator supports hypercyclic vectors precisely when it is not a scalar multiple of the identity. Aron et al [2] first noted that any translation τ a , which is the operator Φ(D) with Φ(z) = e az , fails to support a hypercyclic algebra in a dramatic way: given any f ∈ H(C), the multiplicity of any zero of an element in the orbit of f p must be divisible by p. They also stopped short from giving a positive answer for D, showing that for the generic element f of H(C), each power f n (n ∈ N) is a hypercyclic vector for D. Shkarin [17], and independently Bayart and Matheron [4] finally showed that D supports a hypercyclic algebra, prompting the question: Question 1. Which convolution operators support a hypercyclic algebra? In other words, for which Φ ∈ H(C) of exponential type does Φ(D) support a hypercyclic algebra?Several new positive examples were obtained since then by various authors [8,9,6], including for instance when Φ is of subexponential growth having zero constant

Baire theorem and hypercyclic algebras

“…When |φ(0)| < 1, the existence of hypercyclic algebras is well-understood since [6]: such an algebra does exist if and only if φ is not a multiple of an exponential function. When |φ(0)| = 1, sufficient conditions are given in [6] or in [17] but almost nothing, except a very specific example, is known…”

Linear Dynamical Systems

“…Sufficient conditions for convolution operators in the case |Φ(0)| = 1 were also given in [12], and this was extended by Bès et al [39] with the following theorem. Theorem 6.7 (Bès, Ernst and Prieto [39]). Let Φ be a nonconstant entire function of exponential type with Φ(0) = 1.…”

“…In [39] they augment Theorem 6.7 by giving sufficient conditions for convolution operators to admit a hypercyclic algebra in the case when Φ is not of subexponential growth.…”