Fixed points and orbits of non-convolution operators
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Fixed points and orbits of non-convolution operators Fernando León-Saavedra1* and Pilar Romero-de la Rosa2 *
Correspondence: [email protected] Department of Mathematics, University of Cádiz, Avda. de la Universidad s/n, Jerez de la Frontera, Cádiz 11405, Spain Full list of author information is available at the end of the article 1
Abstract A continuous linear operator T on a Fréchet space F is hypercyclic if there exists a vector f ∈ F (which is called hypercyclic for T) such that the orbit {T n f : n ∈ N} is dense in F. A subset M of a vector space F is spaceable if M ∪ {0} contains an infinite-dimensional closed vector space. In this paper note we study the orbits of the operators Tλ,b f = f (λz + b) (λ, b ∈ C) defined on the space of entire functions and introduced by Aron and Markose (J. Korean Math. Soc. 41(1):65-76, 2004). We complete the results in Aron and Markose (J. Korean Math. Soc. 41(1):65-76, 2004), characterizing when Tλ,b is hypercyclic on H(C). We characterize also when the set of hypercyclic vectors for Tλ,b is spaceable. The fixed point of the map z → λz + b (in the case λ = 1) plays a central role in the proofs. Keywords: fixed point; Denjoy-Wolf theorem; non-convolution operator; hypercyclic operator; spaceability
1 Introduction Let us denote by F a complex infinite dimensional Fréchet space. A continuous linear operator T defined on F is said to be hypercyclic if there exists a vector f ∈ F (called hypercyclic vector for T) such that the orbit ({T n f : n ∈ N}) is dense in F. We refer to the books [, ] and the references therein for further information on hypercyclic operators. From a modern terminology, a subset M of a vector space F is said to be spaceable if M ∪ {} contains an infinite-dimensional closed vector space. The study of spaceability of (usually pathological) subsets is a natural question which has been studied extensively (see [] Chapter or the recent survey [] and the references therein). In , Godefroy and Shapiro [] showed that every continuous linear operator L : H(C) → H(C) which commutes with translations (these operators are called convolution operators) and which is not a multiple of the identity is hypercyclic. This result unifies two classical results by Birkhoff and MacLane (see the survey []). In [], Aron and Markose introduced new examples of hypercyclic operators on H(C) which are not convolution operators. Namely, Tλ,b f = f (λz+b), λ, b ∈ C. In the first section we show that if λ ∈ D and b ∈ C then Tλ,b is not hypercyclic on H(C). This result together with the results in [] and [] shows the following characterization: Tλ,b is hypercyclic on H(C) if and only if |λ| ≥ . Thus, we complete the results of Aron and Markose [] and Fernández and Hallack [] characterizing when Tλ,b (λ, b ∈ C) is hypercyclic. Let us denote by HC(T) the set of hypercyclic vectors for T. In Section we characterize when HC(Tλ,b ) © 2014 León-Saavedra and Romero-de la Rosa; licensee Springer. This is an Open Access article distributed under the terms of
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