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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 603234, 7 pages
http://dx.doi.org/10.1155/2014/603234
Research Article

Chaos for Cosine Operator Functions on Groups

Department of Mathematics Education, National Taichung University of Education, Taichung 403, Taiwan

Received 18 December 2013; Revised 11 February 2014; Accepted 11 February 2014; Published 13 March 2014

Academic Editor: Wei-Shih Du

Copyright © 2014 Chung-Chuan Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let and be a locally compact group. We characterize chaotic cosine operator functions, generated by weighted translations on the Lebesgue space , in terms of the weight condition. In particular, chaotic cosine operator functions and chaotic weighted translations can only occur simultaneously. We also give a necessary and sufficient condition for the direct sum of a sequence of cosine operator functions to be chaotic.

1. Introduction

Let be a bounded linear operator on a Banach space . If is a fixed point of , then the orbit of under , denoted by , is . On the contrary, if there exists an element such that the orbit is dense in ; that is, , then is called hypercyclic and is a hypercyclic vector for . Hypercyclicity arose from the invariant subset problem in analysis, and was studied intensely during the last two decades. We refer to [1, 2] for recent books on this subject.

In the study of hypercyclicity, the weighted shifts on play an important role for researchers to demonstrate and construct the theories in [311]. Recently, we characterize chaotic, hypercyclic, and mixing translation operators on locally compact groups in [1215], which extends some results of weighted shifts on the discrete group in [5, 6, 811] and provides a class of hypercyclic operators on Banach spaces. In this note, we will continue our study in [16, 17] and determine when a cosine operator function, generated by such a weighted translation operator, is chaotic.

Let . According to the definition of Devaney chaos, a sequence of bounded linear operators on a Banach space is chaotic in the successive way in [18] if is topologically transitive and the set of periodic elements, denoted by = , is dense in . We recall that is topologically transitive if, given nonempty open subsets of , we have for some . If from some onwards, then is called topologically mixing. The notion of transitivity in topological dynamics is close to the notion of hypercyclicity in operator theory. Indeed, it is known in [19] that is transitive if, and only if, it is hypercyclic and has a dense set of hypercyclic vectors. In the more general setting, a sequence of operators is said to be hypercyclic if for some . If is generated by a single operator by its iterates, that is, , then hypercyclicity is equivalent to transitivity.

The interest to study cosine operator functions on groups is motivated by the work in [20, 21]. A cosine operator function on a Banach space is a mapping from the real line into the space of continuous operators on satisfying and the d’Alembert functional equation = + for all , which implies for all . In [20], Bonilla and Miana obtained a sufficient condition for a cosine operator function defined by to be transitive, where is a strongly continuous translation group on some weighted Lebesgue space . For a Borel measure and , Kalmes gave the characterization for cosine operator functions, generated by second order partial differential operators on , to be transitive and mixing in [21].

Throughout, let be a locally compact group with identity . Let be a right-invariant Haar measure on , and denote by the complex Lebesgue space with respect to .

A function is called a weight on . Let and let be the unit point mass at . A weighted translation on is a weighted convolution operator defined by where is a weight on and is the convolution: If , then the weighted translation operator is the inverse of . We write for to simplify notation.

In what follows, we assume and define a sequence of bounded linear operators by for all where . Then can be regarded as a cosine operator function by letting . Since for all , we will investigate the sequence of operators and give a necessary and sufficient condition for to be chaotic in terms of the weight function , the Haar measure , and the group element .

2. Chaotic Condition

In this section, we will show the main result and give some examples of chaotic cosine operator functions on various groups. Since is generated by some element , we first note that is never chaotic if is a torsion element by the fact in [17] that is not transitive when is torsion.

Lemma 1. Let be a locally compact group and let be a torsion element in . Let and be a weighted translation on with inverse . Let . Then is not chaotic.

An element in a group is called a torsion element if it is of finite order. In a locally compact group , an element is called periodic [22] (or compact [23]) if the closed subgroup generated by is compact. We call an element in aperiodic if it is not periodic. For discrete groups, periodic and torsion elements are identical; in other words, aperiodic elements are the nontorsion elements.

It has been shown in [15] that an element in a locally compact group is aperiodic if, and only if, for any compact subset , there exists such that for . We will make use of the aperiodic condition to obtain the result.

Now we turn our attention to the set of periodic elements of . Let be the set of periodic elements of a sequence of operator . By the d’Alembert functional equation and induction, we have a simple observation immediately.

Lemma 2. Let be the set of periodic elements of . Then for some and if, and only if, .

Proof. Let . By the d’Alembert functional equation, we have . Now assume and . Then, applying the d’Alembert functional equation again, we have which says .

Based on the work of characterizing transitive in [17], we are able to obtain the characterization for to be chaotic in this note. We state the result in [17] below.

Theorem 3 (see [17]). Let be a locally compact group and let be an aperiodic element in . Let and be a weighted translation on with inverse . Let . Then the following conditions are equivalent. (i) is topologically transitive.(ii)For each compact subset with , there are sequences of Borel sets , , and in such that , and both sequences admit, respectively, subsequences and satisfying

Now we are ready to show the main result.

Theorem 4. Let be a locally compact group and let be an aperiodic element in . Let and be a weighted translation on with inverse . Let , and let be the set of periodic elements. Then the following conditions are equivalent. (i) is chaotic.(ii) is dense in .(iii)For each compact subset with , there is a sequence of Borel sets in such that and both sequences admit, respectively, subsequences and satisfying

Proof. Since (i) (ii) is trivial, we only need to show (ii) (iii) and (iii) (i).
(ii) (iii). Let be dense in . Let be a compact subset of with . Then by the aperiodicity, there exists some such that for all . Let be the characteristic function of . By density of , we can find a sequence and a sequence of periodic points of such that and in which we may assume . Hence we have Let . Then we have Also, by the inequality below we have . Hence by letting . Moreover, using for , the right invariance of the Haar measure and for all and , we arrive at which proves condition (iii).
(iii) (i). The proof is similar to the proof of [12, Theorem 2.1]. We include the argument for completeness. By Theorem 3, a sequence of operators is topologically transitive. Hence we will show is dense in . It is known that the space of continuous functions on with compact support is dense in . Let with compact support . There is a sequence of Borel sets in such that and Similarly, Let Then by the weight assumption in the condition (iii). Also, using again, we have as which follows from On the other hand, is an element of by the equality Putting all these together, condition (iii) implies (i).

We note that [15] in many familiar nondiscrete groups, including the additive group , the Heisenberg group, and the affine group, all elements except the identity are aperiodic. On the other hand, if is discrete, then and for all in the proof of Theorem 4. Hence we have the characterization below for discrete groups.

Corollary 5. Let be a discrete group and let be a nontorsion element in . Let and be a weighted translation on . Let . Then the following conditions are equivalent. (i) is chaotic.(ii)For each finite subset , both sequences admit. Respectively, subsequences and satisfying

It is also interesting to know that condition (iii) in Theorem 4 is also the sufficient and necessary condition for to be chaotic in [12, Theorem 2.1]. In other words, is chaotic if, and only if, is chaotic. We conclude the result below.

Corollary 6. Let be a locally compact group and let be an aperiodic element in . Let and be a weighted translation on with inverse . Let . Then the following conditions are equivalent. (i) is chaotic.(ii) is chaotic.(iii) is chaotic.(iv) is chaotic for all .(v) is chaotic for all .

Proof. For , we denote the set of periodic elements of the operator by . We will show that conditions (ii) and (iii) are equivalent, and (ii) implies (iv).
(ii) (iii). It is known in [19] that an invertible operator is transitive, if and only if, its inverse is transitive. Also it is easy to see . Hence we prove the equivalence.
(ii) (iv). By [24], is transitive for all if is transitive. Moreover, we note that for all . Therefore condition (ii) implies (iv).

We end up this section with two examples on and , which says that one can construct many chaotic cosine operator functions on various groups.

Example 7. Let , which is nontorsion. Let be a weight on . Then the weighted translation operator on is the bilateral weighted forward shift , studied in [11] and given by with . Here is the canonical basis of and is a sequence of positive real numbers. Also, we have
Let and let where is the inverse of . Then by Corollary 5, and are chaotic if, given and , there exists an arbitrarily large such that In fact, there are many weight functions on satisfying the weight condition above. For example, one may define by

Example 8. Let , , and be a weight on . Then the weighted translation on is defined by
Let where . Then and are chaotic if, given a compact subset of , we have Similarly, one may choose by which is the required weight function in the above condition.

3. The Direct Sum of Cosine Operator Functions

Following the investigation on transitivity of the direct sum of a sequence of cosine operator functions in [17], we will give, in this section, the characterization for the direct sum of a sequence of cosine operator functions to be chaotic in terms of the similar weight condition in Theorem 4. The work on the direct sum of operators in linear dynamics has been studied by many authors, for example [5, 9, 11], where the notion of transitivity on direct sum of operators is related with another notion, namely, weak mixing, and hypercyclic criterion.

Given some , let be a sequence of weighted translation operators on , defined by sequences of aperiodic elements () in and positive weight functions () for . We write for for simplification. In [17], we have the result below.

Corollary 9 (see [17]). Let be a weighted translation defined above for . Let . Then the following conditions are equivalent. (i) is topologically transitive.(ii)For and each compact subset with , there are sequences of Borel sets , , and in such that , and both sequences admit, respectively, subsequences and satisfying

Using Corollary above, we characterize chaos of the direct sum of a sequence of cosine operator functions below.

Corollary 10. Let be a weighted translation for . Let . Then the following conditions are equivalent. (i) is chaotic.(ii)For and each compact subset with , there are sequences of Borel sets in such that and both sequences admit, respectively, subsequences and satisfying

Proof. The proof is similar to the proof of Theorem 4.
(i) (ii). By density of , we can find a sequence and sequences of periodic points of such that and . Let for each . Following the same estimation in the proof of Theorem 4, we can obtain the weight conditions.
(ii) (i). Let with compact support for , and let . Letting and repeating the argument in the proof of Theorem 4, one can show that as and is an element of . Hence is chaotic.

Remark 11. By Corollary 10 above, we note that is chaotic if, and only if, is chaotic. Similarly, and can be chaotic only at the same time.

Finally, we give an example of the direct sum of cosine operator functions on .

Example 12. Given some , let and for . We consider a sequence of weighted translations on , given by where is a sequence of positive weight functions. Then we define sequences of operators by where is the inverse of . By Corollary 10, is chaotic if, given and , there exists an arbitrarily large such that for all and , we have

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the reviewers for numerous helpful suggestions and remarks on this paper. The author is supported by NSC of Taiwan under Grant no. NSC 102-2115-M-142-001-.

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