Convolution operators supporting hypercyclic algebras

Abstract

We show that any convolution operator induced by a non-constant polynomial that vanishes at zero supports a hypercyclic algebra. This partially solves a question raised by R. Aron.

Linear Dynamical Systems

“…On the other hand, it was independently demonstrated (via different approaches) by Bayart and Matheron [22, Section 8.5], and Shkarin [129] that the differentiation operator D : H(C) → H(C) admits a hypercyclic algebra. Bès et al [37] subsequently extended this for convolution operators Φ(D) induced by the differentiation operator. Theorem 6.5 (Bès, Conejero and Papathanasiou [37]).…”

Hypercyclic algebras for convolution and composition operators

“…The geometric assumption (1.1) in Theorem 3 does not seem to be a necessary one, as the following example by Félix Martínez suggests. The polynomial P (z) := 9 9/8 8 z(z 8 − 1) vanishes at zero, so P (D) supports a hypercyclic algebra by [12,Thm. 1].…”

“…8.26] also showed that the set of f ∈ H(C) that generate an algebra consisting entirely (but the origin) of hypercyclic vectors for D is residual in H(C), and by using the latter approach we now know the following: Theorem 2. (Shkarin [25], Bayart and Matheron [7], Bès, Conejero, Papathanasiou [12]) Let P be a non-constant polynomial with P (0) = 0. Then the set of functions f ∈ H(C) that generate a hypercyclic algebra for P (D) is residual in H(C).…”

Multiplicative structures of hypercyclic functions for convolution operators

“…Recall that the algebra generated by a function f is nothing but the set {P •f : P polynomial, P (0) = 0}. The existence of (one-generated) algebras of hypercyclic functions for D was also independently proved by Shkarin in [32] and extended by Bès et al in [12] to operators of the form P (D), where P is a nonconstant polynomial with P (0) = 0. The existence of hypercyclic algebras for many convolution operators not induced by polynomials, such as cos(D), De D , or e D − aI, where 0 < a ≤ 1 was recently shown in [13].…”