Algebrable sets of hypercyclic vectors for convolution operators
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ALGEBRABLE SETS OF HYPERCYCLIC VECTORS FOR CONVOLUTION OPERATORS∗
BY
Juan B` es Department of Mathematics and Statistics, Bowling Green State University Bowling Green, OH 43403, USA e-mail: [email protected] AND
Dimitris Papathanasiou Universit´e Clermont Auvergne, CNRS, LMBP, F-63000 Clermont Ferrand, France e-mail: [email protected] Dedicated to Joe Diestel and Victor Lomonosov
ABSTRACT
We show that several convolution operators on the space of entire functions, such as the MacLane operator, support a dense hypercyclic algebra that is not finitely generated. Birkhoff’s operator also has this property on the space of complex-valued smooth functions on the real line.
1. Introduction The search for large algebraic structures (e.g., linear spaces, closed subspaces, or infinitely generated algebras) in non-linear settings has drawn increasing interest over the past decade [1, 2, 9]. One such setting is given by the set HC(T ) = {f ∈ X : {f, T f, T 2f, . . . } is dense in X} ∗ This work is supported in part by MEC, Project MTM 2016-7963-P. We also thank
Fedor Nazarov and an anonymous referee for key observations for Section 2. Received May 25, 2018 and in revised form July 6, 2019
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` AND D. PAPATHANASIOU J. BES
Isr. J. Math.
of hypercyclic vectors for a given operator T on a topological vector space X. Whenever HC(T ) is non-empty we say that T is a hypercyclic operator, and in this case HC(T ) ∪ {0} always contains a dense linear subspace that is T invariant; see [24]. The question whether HC(T )∪{0} contains a closed and infinite dimensional subspace M (which in turn is called a hypercyclic subspace) has a negative answer for some hypercyclic operators T and a positive one for others [8, 18, 16], and many findings in this direction have been made; see, e.g., [7, Ch. 8] and [17, Ch. 10], and the more recent work [13, 19, 21, 20, 12, 6] The search for algebras of hypercyclic vectors, other perhaps than Read’s [22] construction of an operator on 1 (N) (endowed with any algebra structure) for which every non-zero vector is hypercyclic, may be traced back to the work of Aron et al. [3, 4] who showed that no translation operator can support a hypercyclic algebra on the space H(C) of entire functions, endowed with the compact open topology. They also showed that, in sharp contrast with the translation operators, the collection of entire functions f for which every power f n (n = 1, 2, . . . ) is hypercyclic for the operator D of complex differentiation is residual in H(C). A few years later Shkarin [23, Thm. 4.1] showed that HC(D) contains both a hypercyclic subspace and a hypercyclic algebra, and with a different approach Bayart and Matheron [7, Thm. 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). This prompted the following question, which also appeared in [9, p. 105], [2, p. 185]: Question 1 (Aron [7, p. 217]): Is it possible to find a hypercyclic algebra for D that is not finitely generated? By imitating Bayart an
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