 Research
 Open access
 Published:
Fixed points and orbits of nonconvolution operators
Fixed Point Theory and Applications volume 2014, Article number: 221 (2014)
Abstract
A continuous linear operator T on a Fréchet space F is hypercyclic if there exists a vector f\in F (which is called hypercyclic for T) such that the orbit \{{T}^{n}f:n\in \mathbb{N}\} is dense in F. A subset M of a vector space F is spaceable if M\cup \{0\} contains an infinitedimensional closed vector space. In this paper note we study the orbits of the operators {T}_{\lambda ,b}f={f}^{\prime}(\lambda z+b) (\lambda ,b\in \mathbb{C}) defined on the space of entire functions and introduced by Aron and Markose (J. Korean Math. Soc. 41(1):6576, 2004). We complete the results in Aron and Markose (J. Korean Math. Soc. 41(1):6576, 2004), characterizing when {T}_{\lambda ,b} is hypercyclic on H(\mathbb{C}). We characterize also when the set of hypercyclic vectors for {T}_{\lambda ,b} is spaceable. The fixed point of the map z\to \lambda z+b (in the case \lambda \ne 1) plays a central role in the proofs.
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\in F (called hypercyclic vector for T) such that the orbit (\{{T}^{n}f:n\in \mathbb{N}\}) is dense in F. We refer to the books [1, 2] 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\cup \{0\} contains an infinitedimensional closed vector space. The study of spaceability of (usually pathological) subsets is a natural question which has been studied extensively (see [1] Chapter 8 or the recent survey [3] and the references therein).
In 1991, Godefroy and Shapiro [4] showed that every continuous linear operator L:H(\mathbb{C})\to H(\mathbb{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 [5]).
In [5], Aron and Markose introduced new examples of hypercyclic operators on H(\mathbb{C}) which are not convolution operators. Namely, {T}_{\lambda ,b}f={f}^{\prime}(\lambda z+b), \lambda ,b\in \mathbb{C}. In the first section we show that if \lambda \in \mathbb{D} and b\in \mathbb{C} then {T}_{\lambda ,b} is not hypercyclic on H(\mathbb{C}). This result together with the results in [5] and [6] shows the following characterization: {T}_{\lambda ,b} is hypercyclic on H(\mathbb{C}) if and only if \lambda \ge 1. Thus, we complete the results of Aron and Markose [5] and Fernández and Hallack [6] characterizing when {T}_{\lambda ,b} (\lambda ,b\in \mathbb{C}) is hypercyclic. Let us denote by HC(T) the set of hypercyclic vectors for T. In Section 3 we characterize when HC({T}_{\lambda ,b}) is spaceable. Namely HC({T}_{\lambda ,b}) is spaceable if and only if \lambda =1. During the proofs, it is essential to take into account the fixed point of the map z\to \lambda z+b (\lambda \ne 1).
2 Characterizing the hypercyclicity of {T}_{\lambda ,b}
The proof of this result follows the ideas of the proof of Proposition 14 in [5].
Theorem 2.1 For any \lambda \in \mathbb{D} and b\in \mathbb{C} and for any f\in H(\mathbb{C}), the sequence {T}_{\lambda ,b}^{n}f\to 0 uniformly on compact subsets of ℂ. Therefore {T}_{\lambda ,b} is not hypercyclic on H(\mathbb{C}).
Proof Set \phi (z)=\lambda z+b, \lambda \in \mathbb{D} and b\in \mathbb{C}. Since \lambda \ne 1, \phi (z) has a fixed point {z}_{0}. Indeed, {z}_{0}=\frac{b}{1\lambda}. We denote by {\phi}_{n}(z) the sequence of the iterates defined by
an easy computation yields
Let us observe that the iterates of the operator {T}_{\lambda ,b} have the form
where {f}^{(n)} denotes the n th derivative of f. It is well known that if \lambda \in \mathbb{D} then {z}_{0} is an attractive fixed point, that is, {\phi}_{n}(z) converges to the fixed point {z}_{0} uniformly on compact subsets. Indeed, let R>0. If z\le R, then
as n\to \mathrm{\infty}. Thus, there exists {n}_{0} such that if z\le R then {\phi}_{n}(z){z}_{0}<1/2 for all n\ge {n}_{0}.
If n\ge {n}_{0} and z\le R, we have by the Cauchy inequality
Now, it follows from Stirling’s formula that n!\le e{n}^{n+1/2}{e}^{n}. Hence, if z\le R and n\ge {n}_{0}, then
and since 2n{\lambda }^{(n1)/2}\to 0 as n\to \mathrm{\infty}, we conclude that {max}_{z\le R}{T}_{\lambda ,b}^{n}f(z)\to 0, as n\to \mathrm{\infty}, as desired. We point out that this is a refinement of the argument by Aron and Markose. One of the referees chased the constants and recovered the factor {n}^{1/2} that was missing but that does not break the argument. □
Theorem 13 in [5] and Theorem 2.1 give the following characterization.
Theorem 2.2 For any \lambda \in \mathbb{C} and b\in \mathbb{C}, the operator {T}_{\lambda ,b} is hypercyclic in H(\mathbb{C}) if and only if \lambda \ge 1.
3 Spaceability of the set of hypercyclic vectors for {T}_{\lambda ,b}
As stated in [3], there are few nontrivial examples of subsets M which are lineable (that is, M\cup \{0\} contains an infinitedimensional vector space) and are not spaceable. The following result provides the following examples: for \lambda >1, the set HC({T}_{\lambda ,b}) is lineable but it is not spaceable.
Shkarin [7] showed that for the derivative operator D, the set of hypercyclic vectors HC(D) is spaceable.
Theorem 3.1 For any \lambda \in \mathbb{C} and b\in \mathbb{C}, HC({T}_{\lambda ,b}) is spaceable if and only if \lambda =1.
Proof Firstly, let us suppose that \lambda >1, and let us prove that HC({T}_{\lambda ,b}) does not contain a closed infinite dimensional subspace. Let {z}_{0} be the fixed point of \phi (z)=\lambda z+b. Then we consider a sequence of norms defining the topology of H(\mathbb{C}). Namely, for n\in \mathbb{N} and f\in H(\mathbb{C}), we write
It is easy to see that the above sequence of seminorms is increasing and defines the original topology on H(\mathbb{C}).
Given the sequence of increasing seminorms \{{p}_{n}\}, according to Theorem 10.25 in [2], it is sufficient to find a sequence of subspaces {M}_{n}\subset H(\mathbb{C}) of finite codimension, positive numbers {C}_{n}\to \mathrm{\infty} and N\ge 1 satisfying the following:

(a)
{p}_{N}(f)>0, \mathrm{\forall}f\in HC({T}_{\lambda ,b}).

(b)
{p}_{N}({T}_{\lambda ,b}^{n}f)\ge {C}_{n}{p}_{n}(f), \mathrm{\forall}f\in {M}_{n}.
Indeed, let us consider the subspaces
which are clearly of finite codimension.
Notice that {\phi}_{n}(z){z}_{0}={\lambda}^{n}(z{z}_{0}), so that {\phi}_{n}(z) maps the disk D({z}_{0},1)=\{z{z}_{0}\le 1\} onto D({z}_{0},\parallel \lambda {}^{n}). Hence,
If f\in {M}_{1} then f({z}_{0})=0, so that f(z)={\int}_{[{z}_{0},z]}{f}^{\prime}(\xi )d\xi. Therefore we have
and it follows easily by induction that if f\in {M}_{n} then
Thus,
and it follows that condition (b) is satisfied with N=0 and {C}_{n}={\lambda }^{\frac{{n}^{2}2n}{4}}\to \mathrm{\infty} as n\to \mathrm{\infty}, and therefore HC({T}_{\lambda ,b}) is not spaceable.
Now, let us suppose that \lambda =1, and let us prove that HC({T}_{\lambda ,b}) is spaceable. Indeed, let us suppose first that \lambda =1. If b=0 then {T}_{1,0}=D, and it was proved by Shkarin [7] that HC(D) is spaceable. If b\ne 0 then {T}_{1,b}=D{e}^{bD}, so that {T}_{1,b}=\psi (D), where \psi (z)=z{e}^{bz} is an entire function of exponential type that is not a polynomial, and according to Example 10.12 in [[2], p.275], the space HC({T}_{1,b}) is spaceable.
Now let us consider the case \lambda \in \partial \mathbb{D}\setminus \{1\}. Set {z}_{0}=\frac{b}{1\lambda} the fixed point of \phi (z)=\lambda z+b. According to Theorem 10.2 in [2], since {T}_{\lambda ,b} satisfies the hypercyclicity criterion for the full sequence of natural numbers, it suffices to exhibit an infinite dimensional closed subspace {M}_{0} of H(\mathbb{C}) on which suitable powers of {T}_{\lambda ,b} tend to 0. Now the proof mimics some ideas contained in Example 10.13 in [2]. Indeed, for any n\ge 1, there is some {C}_{n}>0 such that
Let us consider a strictly increasing sequence of positive integers {({n}_{k})}_{k} satisfying {n}_{k+1}\ge {C}_{{n}_{k}}. If j\ge k+1, then {n}_{j}\ge {n}_{k+1}\ge {C}_{{n}_{k}}, therefore by (1) we have
Let us consider {M}_{0} the closed subspace of H(\mathbb{C}) of all entire functions f of the form
and let us prove that {T}_{\lambda ,b}^{{n}_{k}}f\to 0 uniformly on compact subsets as k\to \mathrm{\infty}.
We have
Notice that \lambda =1 and the map {\phi}_{{n}_{k}} takes the disc D({z}_{0},R) onto itself, so that
Finally, we have
In the last step we used inequality (2). This completes the proof of Theorem 3.1. □
References
Bayart F, Matheron É Cambridge Tracts in Mathematics 179. In Dynamics of Linear Operators. Cambridge University Press, Cambridge; 2009. p. xiv+337
GrosseErdmann KG, Peris Manguillot A Universitext. In Linear Chaos. Springer, London; 2011. p. xii+386
BernalGonzález L, Pellegrino D, SeoaneSepúlveda JB: Linear subsets of nonlinear sets in topological vector spaces. Bull. Am. Math. Soc. (N.S.) 2014, 51(1):71–130. 10.1090/S027309792013014216
Godefroy G, Shapiro JH: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 1991, 98(2):229–269. 10.1016/00221236(91)90078J
Aron R, Markose D: On universal functions. J. Korean Math. Soc. 2004, 41(1):65–76. Satellite Conference on Infinite Dimensional Function Theory 10.4134/JKMS.2004.41.1.065
Fernández G, Hallack AA: Remarks on a result about hypercyclic nonconvolution operators. J. Math. Anal. Appl. 2005, 309(1):52–55. 10.1016/j.jmaa.2004.12.006
Shkarin S: On the set of hypercyclic vectors for the differentiation operator. Isr. J. Math. 2010, 180: 271–283. 10.1007/s118560100104z
Acknowledgements
The research was supported by Junta de Andalucía FQM257. The authors would like to thank the referee for reading our manuscript carefully and for giving such constructive comments, which helped improving the quality of the paper substantially.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally in this article. They read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
LeónSaavedra, F., Romerode la Rosa, P. Fixed points and orbits of nonconvolution operators. Fixed Point Theory Appl 2014, 221 (2014). https://doi.org/10.1186/168718122014221
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122014221