Fixed points and orbits of non-convolution operators
Fixed Point Theory and Applications volume 2014, Article number: 221 (2014)
A continuous linear operator T on a Fréchet space F is hypercyclic if there exists a vector (which is called hypercyclic for T) such that the orbit is dense in F. A subset M of a vector space F is spaceable if contains an infinite-dimensional closed vector space. In this paper note we study the orbits of the operators () 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 is hypercyclic on . We characterize also when the set of hypercyclic vectors for is spaceable. The fixed point of the map (in the case ) plays a central role in the proofs.
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 (called hypercyclic vector for T) such that the orbit 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 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 8 or the recent survey  and the references therein).
In 1991, Godefroy and Shapiro  showed that every continuous linear operator 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 which are not convolution operators. Namely, , . In the first section we show that if and then is not hypercyclic on . This result together with the results in  and  shows the following characterization: is hypercyclic on if and only if . Thus, we complete the results of Aron and Markose  and Fernández and Hallack  characterizing when () is hypercyclic. Let us denote by the set of hypercyclic vectors for T. In Section 3 we characterize when is spaceable. Namely is spaceable if and only if . During the proofs, it is essential to take into account the fixed point of the map ().
2 Characterizing the hypercyclicity of
The proof of this result follows the ideas of the proof of Proposition 14 in .
Theorem 2.1 For any and and for any , the sequence uniformly on compact subsets of ℂ. Therefore is not hypercyclic on .
Proof Set , and . Since , has a fixed point . Indeed, . We denote by the sequence of the iterates defined by
an easy computation yields
Let us observe that the iterates of the operator have the form
where denotes the n th derivative of f. It is well known that if then is an attractive fixed point, that is, converges to the fixed point uniformly on compact subsets. Indeed, let . If , then
as . Thus, there exists such that if then for all .
If and , we have by the Cauchy inequality
Now, it follows from Stirling’s formula that . Hence, if and , then
and since as , we conclude that , as , 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 that was missing but that does not break the argument. □
Theorem 13 in  and Theorem 2.1 give the following characterization.
Theorem 2.2 For any and , the operator is hypercyclic in if and only if .
3 Spaceability of the set of hypercyclic vectors for
As stated in , there are few non-trivial examples of subsets M which are lineable (that is, contains an infinite-dimensional vector space) and are not spaceable. The following result provides the following examples: for , the set is lineable but it is not spaceable.
Shkarin  showed that for the derivative operator D, the set of hypercyclic vectors is spaceable.
Theorem 3.1 For any and , is spaceable if and only if .
Proof Firstly, let us suppose that , and let us prove that does not contain a closed infinite dimensional subspace. Let be the fixed point of . Then we consider a sequence of norms defining the topology of . Namely, for and , we write
It is easy to see that the above sequence of semi-norms is increasing and defines the original topology on .
Given the sequence of increasing semi-norms , according to Theorem 10.25 in , it is sufficient to find a sequence of subspaces of finite codimension, positive numbers and satisfying the following:
Indeed, let us consider the subspaces
which are clearly of finite codimension.
Notice that , so that maps the disk onto . Hence,
If then , so that . Therefore we have
and it follows easily by induction that if then
and it follows that condition (b) is satisfied with and as , and therefore is not spaceable.
Now, let us suppose that , and let us prove that is spaceable. Indeed, let us suppose first that . If then , and it was proved by Shkarin  that is spaceable. If then , so that , where is an entire function of exponential type that is not a polynomial, and according to Example 10.12 in [, p.275], the space is spaceable.
Now let us consider the case . Set the fixed point of . According to Theorem 10.2 in , since satisfies the hypercyclicity criterion for the full sequence of natural numbers, it suffices to exhibit an infinite dimensional closed subspace of on which suitable powers of tend to 0. Now the proof mimics some ideas contained in Example 10.13 in . Indeed, for any , there is some such that
Let us consider a strictly increasing sequence of positive integers satisfying . If , then , therefore by (1) we have
Let us consider the closed subspace of of all entire functions f of the form
and let us prove that uniformly on compact subsets as .
Notice that and the map takes the disc onto itself, so that
Finally, we have
In the last step we used inequality (2). This completes the proof of Theorem 3.1. □
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The research was supported by Junta de Andalucía FQM-257. 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.
The authors declare that they have no competing interests.
Both authors contributed equally in this article. They read and approved the final manuscript.
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León-Saavedra, F., Romero-de la Rosa, P. Fixed points and orbits of non-convolution operators. Fixed Point Theory Appl 2014, 221 (2014). https://doi.org/10.1186/1687-1812-2014-221