Chaos for Cosine Operator Functions on Groups

Abstract

Let 1≤𝑝<∞ and 𝐺 be a locally compact group. We characterize chaotic cosine operator functions, generated by weighted translations on the Lebesgue space 𝐿^𝑝(𝐺), 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 𝑇 be a bounded linear operator on a Banach space 𝑋. If 𝑣∈𝑋 is a fixed point of 𝑇, then the orbit of 𝑣 under 𝑇, denoted by Orb(𝑇,𝑣), is Orb(𝑇,𝑣)={𝑣,𝑇𝑣,𝑇²𝑣,…}={𝑣}. On the contrary, if there exists an element 𝑥∈𝑋 such that the orbit is dense in 𝑋; that is, Orb(𝑇,𝑥)={𝑥,𝑇𝑥,𝑇²𝑥,…}=𝑋, then 𝑇 is called hypercyclic and 𝑥 is a hypercyclic vector for 𝑇. 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 ℓ^𝑝(ℤ) play an important role for researchers to demonstrate and construct the theories in [3–11]. Recently, we characterize chaotic, hypercyclic, and mixing translation operators on locally compact groups in [12–15], which extends some results of weighted shifts on the discrete group ℤ in [5, 6, 8–11] 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 ℕ₀=ℕ∪{0}. According to the definition of Devaney chaos, a sequence of bounded linear operators (𝑇_𝑛)_𝑛∈ℕ0 on a Banach space 𝑋 is chaotic in the successive way in [18] if (𝑇_𝑛)_𝑛∈ℕ0 is topologically transitive and the set of periodic elements, denoted by 𝒫((𝑇_𝑛)_𝑛∈ℕ0) = {𝑥∈𝑋;∃𝑚∈ℕ∶𝑇_𝑘𝑚𝑥=𝑥,𝑘=1,2,3,…}, is dense in 𝑋. We recall that (𝑇_𝑛)_𝑛∈ℕ0 is topologically transitive if, given nonempty open subsets 𝑈,𝑉 of 𝑋, we have 𝑇_𝑛(𝑈)∩𝑉≠∅ for some 𝑛∈ℕ. If 𝑇_𝑛(𝑈)∩𝑉≠∅ from some 𝑛 onwards, then (𝑇_𝑛)_𝑛∈ℕ0 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 (𝑇_𝑛)_𝑛∈ℕ0 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 (𝑇_𝑛)_𝑛∈ℕ0 is said to be hypercyclic if Orb(𝑇_𝑛,𝑥)={𝑇_𝑛𝑥∶𝑛∈ℕ₀}=𝑋 for some 𝑥∈𝑋. If (𝑇_𝑛)_𝑛∈ℕ0 is generated by a single operator 𝑇 by its iterates, that is, 𝑇_𝑛∶=𝑇^𝑛, 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 𝑋 is a mapping 𝒞 from the real line into the space of continuous operators on 𝑋 satisfying 𝒞(0)=𝐼 and the d’Alembert functional equation 2𝒞(𝑡)𝒞(𝑠) = 𝒞(𝑡+𝑠 + 𝒞(𝑡−𝑠) for all 𝑠,𝑡∈ℝ, which implies 𝒞(𝑡)=𝒞(−𝑡) for all 𝑡∈ℝ. In [20], Bonilla and Miana obtained a sufficient condition for a cosine operator function 𝒞(𝑡) defined by 𝒞(𝑡)=12(𝑇(𝑡)+𝑇(−𝑡)), (1) to be transitive, where 𝑇 is a strongly continuous translation group on some weighted Lebesgue space 𝐿^𝑝(ℝ). For a Borel measure 𝜇 and Ω⊂ℝ^𝑑, Kalmes gave the characterization for cosine operator functions, generated by second order partial differential operators on 𝐿^𝑝(Ω,𝜇), to be transitive and mixing in [21].

Throughout, let 𝐺 be a locally compact group with identity 𝑒. Let 𝜆 be a right-invariant Haar measure on 𝐺, and denote by 𝐿^𝑝(𝐺)(1≤𝑝<∞) the complex Lebesgue space with respect to 𝜆.

A function 𝑤∶𝐺→(0,∞) is called a weight on 𝐺. Let 𝑎∈𝐺 and let 𝛿_𝑎 be the unit point mass at 𝑎. A weighted translation on 𝐺 is a weighted convolution operator 𝑇_𝑎,𝑤∶𝐿^𝑝(𝐺)→𝐿^𝑝(𝐺) defined by 𝑇_𝑎,𝑤𝑓∶=𝑤𝑇_𝑎𝑓𝑓∈𝐿^𝑝(𝐺), (2) where 𝑤 is a weight on 𝐺 and 𝑇_𝑎(𝑓)=𝑓∗𝛿_𝑎∈𝐿^𝑝(𝐺) is the convolution: 𝑓∗𝛿_𝑎(𝑥)∶=_𝐺𝑓𝑥𝑦⁻¹𝑑𝛿_𝑎𝑦=𝑓𝑥𝑎⁻¹(𝑥∈𝐺). (3) If 𝑤⁻¹∈𝐿^∞(𝐺), then the weighted translation operator 𝑇_{𝑎−1,𝑤−1∗𝛿𝑎}⁻¹ is the inverse of 𝑇_𝑎,𝑤. We write 𝑆_𝑎,𝑤 for 𝑇_{𝑎−1,𝑤−1∗𝛿𝑎}⁻¹ to simplify notation.

In what follows, we assume 𝑤,𝑤⁻¹∈𝐿^∞(𝐺) and define a sequence of bounded linear operators 𝐶_𝑛∶𝐿^𝑝(𝐺)→𝐿^𝑝(𝐺) by 𝐶_𝑛=12𝑇^𝑛_𝑎,𝑤+𝑆^𝑛_𝑎,𝑤 (4) for all 𝑛∈ℤ where 𝑇^−𝑛_𝑎,𝑤∶=(𝑇⁻¹_𝑎,𝑤)^𝑛=𝑆^𝑛_𝑎,𝑤. Then (𝐶_𝑛)_𝑛∈ℤ can be regarded as a cosine operator function by letting 𝒞(𝑛)=𝐶_𝑛. Since 𝐶_𝑛=𝐶_−𝑛 for all 𝑛∈ℤ, we will investigate the sequence of operators (𝐶_𝑛)_𝑛∈ℕ0 and give a necessary and sufficient condition for (𝐶_𝑛)_𝑛∈ℕ0 to be chaotic in terms of the weight function 𝑤, the Haar measure 𝜆, and the group element 𝑎∈𝐺.

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 (𝐶_𝑛)_𝑛∈ℕ0 is generated by some element 𝑎∈𝐺, we first note that (𝐶_𝑛)_𝑛∈ℕ0 is never chaotic if 𝑎 is a torsion element by the fact in [17] that (𝐶_𝑛)_𝑛∈ℕ0 is not transitive when 𝑎 is torsion.

Lemma 1. Let 𝐺 be a locally compact group and let 𝑎 be a torsion element in 𝐺. Let 1≤𝑝<∞ and 𝑇_𝑎,𝑤 be a weighted translation on 𝐿^𝑝(𝐺) with inverse 𝑆_𝑎,𝑤. Let 𝐶_𝑛=(1/2)(𝑇^𝑛_𝑎,𝑤+𝑆^𝑛_𝑎,𝑤). Then (𝐶_𝑛)_𝑛∈ℕ0 is not chaotic.

An element 𝑎 in a group 𝐺 is called a torsion element if it is of finite order. In a locally compact group 𝐺, an element 𝑎∈𝐺 is called periodic [22] (or compact [23]) if the closed subgroup 𝐺(𝑎) generated by 𝑎 is compact. We call an element in 𝐺 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 𝑎 in a locally compact group 𝐺 is aperiodic if, and only if, for any compact subset 𝐾⊂𝐺, there exists 𝑁∈ℕ such that 𝐾∩𝐾𝑎^±𝑛=∅ for 𝑛>𝑁. We will make use of the aperiodic condition to obtain the result.

Now we turn our attention to the set of periodic elements of (𝐶_𝑛)_𝑛∈ℕ0. Let 𝒫((𝐶_𝑛)_𝑛∈ℕ0) be the set of periodic elements of a sequence of operator (𝐶_𝑛)_𝑛∈ℕ0. By the d’Alembert functional equation and induction, we have a simple observation immediately.

Lemma 2. Let 𝒫((𝐶_𝑛)_𝑛∈ℕ0) be the set of periodic elements of (𝐶_𝑛)_𝑛∈ℕ0. Then 𝐶_𝑚𝑓=𝑓 for some 𝑚∈ℕ and 𝑓∈ℓ^𝑝(ℤ) if, and only if, 𝑓∈𝒫((𝐶_𝑛)_𝑛∈ℕ0).

Proof. Let 𝐶_𝑚𝑓=𝑓. By the d’Alembert functional equation, we have 𝐶_2𝑚𝑓=𝑓. Now assume 𝐶_𝑘𝑚𝑓=𝑓 and 𝐶_(𝑘+1)𝑚𝑓=𝑓. Then, applying the d’Alembert functional equation again, we have 2𝑓=2𝐶_𝑚𝑓=2𝐶_𝑚𝐶_(𝑘+1)𝑚𝑓=𝐶_(𝑘+2)𝑚𝑓+𝐶_𝑘𝑚𝑓=𝐶_(𝑘+2)𝑚𝑓+𝑓 (5) which says 𝐶_(𝑘+2)𝑚𝑓=𝑓.

Based on the work of characterizing transitive (𝐶_𝑛)_𝑛∈ℕ0 in [17], we are able to obtain the characterization for (𝐶_𝑛)_𝑛∈ℕ0 to be chaotic in this note. We state the result in [17] below.

Theorem 3 (see [17]). Let 𝐺 be a locally compact group and let 𝑎 be an aperiodic element in 𝐺. Let 1≤𝑝<∞ and 𝑇_𝑎,𝑤 be a weighted translation on 𝐿^𝑝(𝐺) with inverse 𝑆_𝑎,𝑤. Let 𝐶_𝑛=(1/2)(𝑇^𝑛_𝑎,𝑤+𝑆^𝑛_𝑎,𝑤). Then the following conditions are equivalent. (i)(𝐶_𝑛)_𝑛∈ℕ0 is topologically transitive.(ii)For each compact subset 𝐾⊂𝐺 with 𝜆(𝐾)>0, there are sequences of Borel sets (𝐸_𝑘), (𝐸⁺_𝑘), and (𝐸⁻_𝑘) in 𝐾 such that 𝐸_𝑘=𝐸⁺_𝑘∪𝐸⁻_𝑘, 𝜆(𝐾)=lim_𝑘→∞𝜆(𝐸_𝑘) and both sequences 𝜑_𝑛∶=^𝑛_𝑠=1𝑤∗𝛿^𝑠_𝑎−1,𝜑_𝑛∶=^𝑛−1_𝑠=0𝑤∗𝛿^𝑠_𝑎⁻¹ (6) admit, respectively, subsequences (𝜑_𝑛𝑘) and (𝜑_𝑛𝑘) satisfying lim_𝑘→∞𝜑_𝑛𝑘∣_𝐸𝑘∞=lim_𝑘→∞𝜑_𝑛𝑘∣_𝐸𝑘∞=0,lim_𝑘→∞𝜑_2𝑛𝑘∣_𝐸⁺_𝑘∞=lim_𝑘→∞𝜑_2𝑛𝑘∣_𝐸⁻_𝑘∞=0. (7)

Now we are ready to show the main result.

Theorem 4. Let 𝐺 be a locally compact group and let 𝑎 be an aperiodic element in 𝐺. Let 1≤𝑝<∞ and 𝑇_𝑎,𝑤 be a weighted translation on 𝐿^𝑝(𝐺) with inverse 𝑆_𝑎,𝑤. Let 𝐶_𝑛=(1/2)(𝑇^𝑛_𝑎,𝑤+𝑆^𝑛_𝑎,𝑤), and let 𝒫((𝐶_𝑛)_𝑛∈ℕ0) be the set of periodic elements. Then the following conditions are equivalent. (i)(𝐶_𝑛)_𝑛∈ℕ0 is chaotic.(ii)𝒫((𝐶_𝑛)_𝑛∈ℕ0) is dense in 𝐿^𝑝(𝐺).(iii)For each compact subset 𝐾⊂𝐺 with 𝜆(𝐾)>0, there is a sequence of Borel sets (𝐸_𝑘) in 𝐾 such that 𝜆(𝐾)=lim_𝑘→∞𝜆(𝐸_𝑘) and both sequences 𝜑_𝑛∶=^𝑛_𝑠=1𝑤∗𝛿^𝑠_𝑎−1,𝜑_𝑛∶=^𝑛−1_𝑠=0𝑤∗𝛿^𝑠_𝑎⁻¹ (8) admit, respectively, subsequences (𝜑_𝑛𝑘) and (𝜑_𝑛𝑘) satisfying lim_𝑘→∞^∞_{𝑙=1𝐸𝑘}𝜑^𝑝_𝑙𝑛𝑘(𝑥)𝑑𝜆(𝑥)+^∞_{𝑙=1𝐸𝑘}𝜑^𝑝_𝑙𝑛𝑘(𝑥)𝑑𝜆(𝑥)=0. (9)

Proof. Since (i) ⇒(ii) is trivial, we only need to show (ii) ⇒(iii) and (iii) ⇒(i).
(ii) ⇒(iii). Let 𝒫((𝐶_𝑛)_𝑛∈ℕ0) be dense in 𝐿^𝑝(𝐺). Let 𝐾 be a compact subset of 𝐺 with 𝜆(𝐾)>0. Then by the aperiodicity, there exists some 𝑁∈ℕ such that 𝐾∩𝐾𝑎^±𝑚=∅ for all 𝑚>𝑁. Let 𝜒_𝐾∈𝐿^𝑝(𝐺) be the characteristic function of 𝐾. By density of 𝒫((𝐶_𝑛)_𝑛∈ℕ0), we can find a sequence (𝑛_𝑘)⊂ℕ and a sequence (𝑓_𝑘) of periodic points of (𝐶_𝑛)_𝑛∈ℕ0 such that ‖𝑓_𝑘−𝜒_𝐾‖_𝑝<1/4^𝑘 and 𝐶_𝑛𝑘𝑓_𝑘=𝑓_𝑘 in which we may assume 𝑛_𝑘+1>𝑛_𝑘>𝑁. Hence we have 14^𝑘>𝑓_𝑘−𝜒_𝐾𝑝≥𝑓_𝑘𝜒_𝐾−𝜒_𝐾𝑝. (10) Let 𝐴_𝑘={𝑥∈𝐾∶|𝑓_𝑘𝜒_𝐾(𝑥)−1|>1/2^𝑘}. Then we have 𝑓_𝑘𝜒_𝐾(𝑥1)≥1−2^𝑘𝑥∈𝐾⧵𝐴_𝑘. (11) Also, by the inequality below 14^𝑝𝑘>𝑓_𝑘𝜒_𝐾−𝜒_𝐾^𝑝_𝑝=_𝐺𝑓_𝑘𝜒_𝐾(𝑥)−𝜒_𝐾(𝑥)^𝑝(𝑥)≥𝑑𝜆_𝐾𝑓_𝑘𝜒_𝐾(𝑥)−1^𝑝𝑑𝜆(𝑥)≥_𝐴𝑘𝑓_𝑘𝜒_𝐾(𝑥)−1^𝑝(𝑥)>1𝑑𝜆2^𝑝𝑘𝜆𝐴_𝑘, (12) we have 𝜆(𝐴_𝑘)<1/2^𝑝𝑘. Hence 𝜆(𝐾⧵𝐸_𝑘)<1/2^𝑝𝑘 by letting 𝐸_𝑘=𝐾⧵𝐴_𝑘. Moreover, using 𝑓_𝑘=𝐶_𝑙𝑛𝑘𝑓_𝑘 for 𝑙∈ℕ, the right invariance of the Haar measure 𝜆 and 𝐾𝑎^𝑟𝑛𝑘∩𝐾𝑎^𝑠𝑛𝑘=∅ for all 𝑟,𝑠∈ℤ and 𝑟≠𝑠, we arrive at 14^𝑝𝑘>𝑓_𝑘𝜒_𝐾−𝜒_𝐾^𝑝_𝑝=_𝐺𝑓_𝑘𝜒_𝐾(𝑥)−𝜒_𝐾(𝑥)^𝑝(𝑥)≥𝑑𝜆_𝐺⧵𝐾𝑓_𝑘𝜒_𝐾(𝑥)^𝑝(𝑥)≥𝑑𝜆_{∞𝑙=1𝐾𝑎𝑘𝑙𝑛}𝑓_𝑘𝜒_𝐾(𝑥)^𝑝(𝑥)+𝑑𝜆_{∞𝑙=1𝐾𝑎𝑘−𝑙𝑛}𝑓_𝑘𝜒_𝐾(𝑥)^𝑝(𝑥)=𝑑𝜆_{∞𝑙=1𝐾}𝑓_𝑘𝜒_𝐾𝑥𝑎^{𝑙𝑛𝑘𝑝}(𝑥)+𝑑𝜆_{∞𝑙=1𝐾}𝑓_𝑘𝜒_𝐾𝑥𝑎^{−𝑙𝑛𝑘𝑝}(𝑥)=1𝑑𝜆2^𝑝∞_𝑙=1𝐾𝑇^𝑙𝑛𝑘_𝑎,𝑤𝑓_𝑘𝜒_𝐾𝑥𝑎^𝑙𝑛𝑘+𝑆^𝑙𝑛𝑘_𝑎,𝑤𝑓_𝑘𝜒_𝐾𝑥𝑎^{𝑙𝑛𝑘𝑝}(𝑥)+1𝑑𝜆2^𝑝∞_𝑙=1𝐾𝑇^𝑙𝑛𝑘_𝑎,𝑤𝑓_𝑘𝜒_𝐾𝑥𝑎^{−𝑙𝑛𝑘}+𝑆^𝑙𝑛𝑘_𝑎,𝑤𝑓_𝑘𝜒_𝐾𝑥𝑎^{−𝑙𝑛𝑘𝑝}(𝑥)≥1𝑑𝜆2^𝑝∞_{𝑙=1𝐸𝑘}𝜑_𝑙𝑛𝑘(𝑥)𝑓_𝑘𝜒_𝐾(𝑥)+𝜑⁻¹_𝑙𝑛𝑘𝑥𝑎^𝑙𝑛𝑘𝑓_𝑘𝜒_𝐾𝑥𝑎^{2𝑙𝑛𝑘𝑝}(𝑥)+1𝑑𝜆2^𝑝∞_{𝑙=1𝐸𝑘}𝜑⁻¹_𝑙𝑛𝑘𝑥𝑎^{−𝑙𝑛𝑘}𝑓_𝑘𝜒_𝐾𝑥𝑎^{−2𝑙𝑛𝑘}+𝜑_𝑙𝑛𝑘(𝑥)𝑓_𝑘𝜒_𝐾(𝑥)^𝑝(𝑥)=1𝑑𝜆2^𝑝∞_{𝑙=1𝐸𝑘}𝜑_𝑙𝑛𝑘(𝑥)𝑓_𝑘𝜒_𝐾(𝑥)^𝑝(𝑥)+1𝑑𝜆2^𝑝∞_{𝑙=1𝐸𝑘}𝜑_𝑙𝑛𝑘(𝑥)𝑓_𝑘𝜒_𝐾(𝑥)^𝑝(𝑥)≥1𝑑𝜆2^𝑝11−2^𝑘𝑝∞_{𝑙=1𝐸𝑘}𝜑^𝑝_𝑙𝑛𝑘(𝑥)(𝑥)+1𝑑𝜆2^𝑝11−2^𝑘𝑝∞_{𝑙=1𝐸𝑘}𝜑^𝑝_𝑙𝑛𝑘(𝑥)𝑑𝜆(𝑥) (13) 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 (𝐶_𝑛)_𝑛∈ℕ0 is topologically transitive. Hence we will show 𝒫((𝐶_𝑛)_𝑛∈ℕ0) is dense in 𝐿^𝑝(𝐺). It is known that the space 𝐶_𝑐(𝐺) of continuous functions on 𝐺 with compact support is dense in 𝐿^𝑝(𝐺). Let 𝑓∈𝐶_𝑐(𝐺) with compact support 𝐾⊂𝐺. There is a sequence of Borel sets (𝐸_𝑘) in 𝐾 such that 𝜆(𝐾)=lim_𝑘→∞𝜆(𝐸_𝑘) and 𝑇^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘^𝑝_𝑝=_{𝐸𝑘𝑎𝑘𝑙𝑛}𝑤(𝑥)𝑤𝑥𝑎⁻¹⋅⋅⋅𝑤𝑥𝑎^{−(𝑙𝑛𝑘−1)𝑝}×𝑓𝑥𝑎^{−𝑙𝑛𝑘𝑝}(𝑥)=𝑑𝜆_𝐸𝑘𝑤𝑥𝑎^𝑙𝑛𝑘𝑤𝑥𝑎^{𝑙𝑛𝑘−1}⋅⋅⋅𝑤(𝑥𝑎)^𝑝𝑓(𝑥)^𝑝(𝑥)=𝑑𝜆_𝐸𝑘𝜑^𝑝_𝑙𝑛𝑘(𝑥)𝑓(𝑥)^𝑝𝑑𝜆(𝑥)≤𝑓^𝑝_∞𝐸𝑘𝜑^𝑝_𝑙𝑛𝑘(𝑥)𝑑𝜆(𝑥). (14) Similarly, 𝑆^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘^𝑝_𝑝≤𝑓^𝑝_∞𝐸𝑘𝜑^𝑝_𝑙𝑛𝑘(𝑥)𝑑𝜆(𝑥). (15) Let 𝑣_𝑘=𝑓𝜒_𝐸𝑘+^∞_𝑙=1𝑇^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘+^∞_𝑙=1𝑆^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘. (16) Then 𝑣_𝑘∈𝐿^𝑝(𝐺) by the weight assumption in the condition (iii). Also, using 𝐾𝑎^𝑟𝑛𝑘∩𝐾𝑎^𝑠𝑛𝑘=∅ again, we have 𝑣_𝑘→𝑓 as 𝑘→∞ which follows from 𝑣_𝑘−𝑓^𝑝_𝑝≤𝑓^𝑝_∞𝜆𝐾⧵𝐸_𝑘+^∞_𝑙=1𝑇^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘^𝑝_𝑝+^∞_𝑙=1𝑆^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘^𝑝_𝑝≤𝑓^𝑝_∞𝜆𝐾⧵𝐸_𝑘+𝑓^𝑝_∞^∞_{𝑙=1𝐸𝑘}𝜑^𝑝_𝑙𝑛𝑘(𝑥)(𝑥)+𝑑𝜆_{∞𝑙=1𝐸𝑘}𝜑^𝑝_𝑙𝑛𝑘(𝑥)𝑑𝜆(𝑥). (17) On the other hand, 𝑣_𝑘 is an element of 𝒫((𝐶_𝑛)_𝑛∈ℕ0) by the equality 2𝐶_𝑛𝑘𝑣_𝑘=𝑇^𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘+^∞_𝑙=1𝑇^𝑛𝑘_𝑎,𝑤𝑇^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘+^∞_𝑙=1𝑇^𝑛𝑘_𝑎,𝑤𝑆^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘+𝑆^𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘+^∞_𝑙=1𝑆^𝑛𝑘_𝑎,𝑤𝑇^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘+^∞_𝑙=1𝑆^𝑛𝑘_𝑎,𝑤𝑆^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘=^∞_𝑙=1𝑇^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘+𝑓𝜒_𝐸𝑘+^∞_𝑙=1𝑆^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘+𝑓𝜒_𝐸𝑘+^∞_𝑙=1𝑇^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘+^∞_𝑙=1𝑆^𝑙𝑛𝑘_𝑎,𝑤𝑓𝜒_𝐸𝑘=𝑣_𝑘+𝑣_𝑘=2𝑣_𝑘. (18) Putting all these together, condition (iii) implies (i).

We note that [15] in many familiar nondiscrete groups, including the additive group ℝ^𝑑, the Heisenberg group, and the affine group, all elements except the identity are aperiodic. On the other hand, if 𝐺 is discrete, then 𝐴_𝑘=∅ and 𝐸_𝑘=𝐾 for all 𝑘∈ℕ in the proof of Theorem 4. Hence we have the characterization below for discrete groups.

Corollary 5. Let 𝐺 be a discrete group and let 𝑎 be a nontorsion element in 𝐺. Let 1≤𝑝<∞ and 𝑇_𝑎,𝑤 be a weighted translation on ℓ^𝑝(𝐺). Let 𝐶_𝑛=(1/2)(𝑇^𝑛_𝑎,𝑤+𝑆^𝑛_𝑎,𝑤). Then the following conditions are equivalent. (i)(𝐶_𝑛)_𝑛∈ℕ0 is chaotic.(ii)For each finite subset 𝐾⊂𝐺, both sequences 𝜑_𝑛=^𝑛_𝑠=1𝑤∗𝛿^𝑠_𝑎−1,𝜑_𝑛=^𝑛−1_𝑠=0𝑤∗𝛿^𝑠_𝑎⁻¹ (19) admit. Respectively, subsequences (𝜑_𝑛𝑘) and (𝜑_𝑛𝑘) satisfying lim_𝑘→∞^∞_𝑙=1𝐾𝜑^𝑝_𝑙𝑛𝑘(𝑥)+^∞_𝑙=1𝐾𝜑^𝑝_𝑙𝑛𝑘(𝑥)=0. (20)

It is also interesting to know that condition (iii) in Theorem 4 is also the sufficient and necessary condition for 𝑇_𝑎,𝑤 to be chaotic in [12, Theorem 2.1]. In other words, (𝐶_𝑛)_𝑛∈ℕ0 is chaotic if, and only if, 𝑇_𝑎,𝑤 is chaotic. We conclude the result below.

Corollary 6. Let 𝐺 be a locally compact group and let 𝑎 be an aperiodic element in 𝐺. Let 1≤𝑝<∞ and 𝑇_𝑎,𝑤 be a weighted translation on 𝐿^𝑝(𝐺) with inverse 𝑆_𝑎,𝑤. Let 𝐶_𝑛=(1/2)(𝑇^𝑛_𝑎,𝑤+𝑆^𝑛_𝑎,𝑤). Then the following conditions are equivalent. (i)(𝐶_𝑛)_𝑛∈ℕ0 is chaotic.(ii)𝑇_𝑎,𝑤 is chaotic.(iii)𝑆_𝑎,𝑤 is chaotic.(iv)𝑇^𝑛_𝑎,𝑤 is chaotic for all 𝑛∈ℕ.(v)𝑆^𝑛_𝑎,𝑤 is chaotic for all 𝑛∈ℕ.

Proof. For 𝑛∈ℕ, we denote the set of periodic elements of the operator 𝑇^𝑛_𝑎,𝑤 by 𝒫(𝑇^𝑛_𝑎,𝑤)={𝑓∈𝐿^𝑝(𝐺);∃𝑚∈ℕ∶(𝑇^𝑛_𝑎,𝑤)^𝑚𝑓=𝑓}. 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 𝒫(𝑇_𝑎,𝑤)=𝒫(𝑆_𝑎,𝑤). Hence we prove the equivalence.
(ii) ⇒(iv). By [24], 𝑇^𝑛_𝑎,𝑤 is transitive for all 𝑛∈ℕ if 𝑇_𝑎,𝑤 is transitive. Moreover, we note that 𝒫(𝑇_𝑎,𝑤)=𝒫(𝑇^𝑛_𝑎,𝑤) for all 𝑛∈ℕ. Therefore condition (ii) implies (iv).

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

Example 7. Let 𝐺=ℤ, 𝑎=1∈ℤ which is nontorsion. Let 𝑤∗𝛿₁ be a weight on ℤ. Then the weighted translation operator 𝑇_{1,𝑤∗𝛿1} on ℓ^𝑝(ℤ) is the bilateral weighted forward shift 𝑇, studied in [11] and given by 𝑇𝑒_𝑗=𝑤_𝑗𝑒_𝑗+1 with 𝑤_𝑗=𝑤(𝑗). Here (𝑒_𝑗)_𝑗∈ℤ is the canonical basis of ℓ^𝑝(ℤ) and (𝑤_𝑗)_𝑗∈ℤ is a sequence of positive real numbers. Also, we have 𝜑_𝑙𝑛𝑗=^𝑙𝑛_𝑠=1𝑤∗𝛿₁∗𝛿^𝑠₋₁𝑗=^𝑙𝑛−1_𝑠=0𝑤,𝜑𝑗+𝑠_−1𝑙𝑛𝑗=^𝑙𝑛−1_𝑠=0𝑤∗𝛿₁∗𝛿^𝑠₁𝑗=^𝑙𝑛_𝑠=1𝑤𝑗−𝑠. (21)
Let 𝑤⁻¹∈ℓ^∞(ℤ) and let 𝐶_𝑛=(1/2)(𝑇^𝑛+𝑆^𝑛) where 𝑆 is the inverse of 𝑇=𝑇_{1,𝑤∗𝛿1}. Then by Corollary 5, (𝐶_𝑛)_𝑛∈ℕ0 and 𝑇_{1,𝑤∗𝛿1} are chaotic if, given 𝜀>0 and 𝑞∈ℕ, there exists an arbitrarily large 𝑛∈ℕ such that ^∞_{𝑙=1|𝑗|≤𝑞}^𝑙𝑛−1_𝑠=0𝑤𝑗+𝑠^𝑝+^∞_{𝑙=1|𝑗|≤𝑞}1∏^𝑙𝑛_𝑠=1𝑤𝑗−𝑠^𝑝<𝜀. (22) In fact, there are many weight functions 𝑤 on ℤ satisfying the weight condition above. For example, one may define 𝑤∶ℤ→(0,∞) by 𝑤𝑗=12if𝑗≥02if𝑗<0. (23)

Example 8. Let 𝐺=ℝ, 𝑎=2, and 𝑤 be a weight on ℝ. Then the weighted translation 𝑇_2,𝑤 on 𝐿^𝑝(ℝ) is defined by 𝑇_2,𝑤𝑓(𝑥)=𝑤(𝑥)𝑓(𝑥−2)𝑓∈𝐿^𝑝(ℝ),𝜑_𝑙𝑛𝑘(𝑥)=^𝑙𝑛𝑘_𝑠=1𝑤∗𝛿^𝑠₋₂(𝑥)=^𝑙𝑛𝑘_𝑠=1𝑤(),𝜑𝑥+2𝑠_{𝑙𝑛𝑘−1}(𝑥)=^{𝑙𝑛𝑘−1}_𝑠=0𝑤∗𝛿^𝑠₂(𝑥)=^{𝑙𝑛𝑘−1}_𝑠=0𝑤(𝑥−2𝑠). (24)
Let 𝐶_𝑛=(1/2)(𝑇^𝑛_2,𝑤+𝑆^𝑛_2,𝑤) where 𝑆_2,𝑤=𝑇⁻¹_2,𝑤. Then (𝐶_𝑛)_𝑛∈ℕ0 and 𝑇_2,𝑤 are chaotic if, given a compact subset 𝐾 of ℝ, we have lim_𝑘→∞^∞_𝑙=1𝐾^𝑙𝑛𝑘_𝑠=1𝑤(𝑥+2𝑠)^𝑝(𝑥)+𝑑𝜆_{∞𝑙=1𝐾}1∏^{𝑙𝑛𝑘−1}_𝑠=0𝑤(𝑥−2𝑠)^𝑝𝑑𝜆(𝑥)=0. (25) Similarly, one may choose 𝑤∶ℝ→(0,∞) by 𝑤(𝑥)=121if𝑥≥12^𝑥if−1<𝑥<12if𝑥≤−1, (26) 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 𝑁∈ℕ, let (𝑇_{𝑎𝑚,𝑤𝑚}) be a sequence of weighted translation operators on 𝐿^𝑝(𝐺), defined by sequences of aperiodic elements (𝑎_𝑚) in 𝐺 and positive weight functions (𝑤_𝑚) for 1≤𝑚≤𝑁. We write 𝑇_𝑚 for 𝑇_{𝑎𝑚,𝑤𝑚} for simplification. In [17], we have the result below.

Corollary 9 (see [17]). Let 𝑇_𝑚 be a weighted translation defined above for 1≤𝑚≤𝑁. Let 𝐶_𝑚,𝑛=(1/2)(𝑇^𝑛_𝑚+𝑆^𝑛_𝑚). Then the following conditions are equivalent. (i)(𝐶_1,𝑛⊕𝐶_2,𝑛⊕⋅⋅⋅⊕𝐶_𝑁,𝑛)_𝑛∈ℕ0 is topologically transitive.(ii)For 1≤𝑚≤𝑁 and each compact subset 𝐾⊂𝐺 with 𝜆(𝐾)>0, there are sequences of Borel sets (𝐸_𝑚,𝑘), (𝐸⁺_𝑚,𝑘), and (𝐸⁻_𝑚,𝑘) in 𝐾 such that 𝐸_𝑚,𝑘=𝐸⁺_𝑚,𝑘∪𝐸⁻_𝑚,𝑘, 𝜆(𝐾)=lim_𝑘→∞𝜆(𝐸_𝑚,𝑘) and both sequences 𝜑_𝑚,𝑛∶=^𝑛_𝑠=1𝑤_𝑚∗𝛿^𝑠_𝑎𝑚−1,𝜑_𝑚,𝑛∶=^𝑛−1_𝑠=0𝑤_𝑚∗𝛿^𝑠_𝑎𝑚⁻¹ (27) admit, respectively, subsequences (𝜑_{𝑚,𝑛𝑘}) and (𝜑_{𝑚,𝑛𝑘}) satisfying lim_𝑘→∞𝜑_{𝑚,𝑛𝑘}|_{𝐸𝑚,𝑘∞}=lim_𝑘→∞𝜑_{𝑚,𝑛𝑘}|_{𝐸𝑚,𝑘∞}=0,lim_𝑘→∞𝜑_{𝑚,2𝑛𝑘}|_𝐸⁺_𝑚,𝑘∞=lim_𝑘→∞𝜑_{𝑚,2𝑛𝑘}|_𝐸⁻_𝑚,𝑘∞=0. (28)

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

Corollary 10. Let 𝑇_𝑚 be a weighted translation for 1≤𝑚≤𝑁. Let 𝐶_𝑚,𝑛=(1/2)(𝑇^𝑛_𝑚+𝑆^𝑛_𝑚). Then the following conditions are equivalent. (i)(𝐶_1,𝑛⊕𝐶_2,𝑛⊕⋅⋅⋅⊕𝐶_𝑁,𝑛)_𝑛∈ℕ0 is chaotic.(ii)For 1≤𝑚≤𝑁 and each compact subset 𝐾⊂𝐺 with 𝜆(𝐾)>0, there are sequences of Borel sets (𝐸_𝑚,𝑘) in 𝐾 such that 𝜆(𝐾)=lim_𝑘→∞𝜆(𝐸_𝑚,𝑘) and both sequences 𝜑_𝑚,𝑛∶=^𝑛_𝑠=1𝑤_𝑚∗𝛿^𝑠_𝑎𝑚−1,𝜑_𝑚,𝑛∶=^𝑛−1_𝑠=0𝑤_𝑚∗𝛿^𝑠_𝑎𝑚⁻¹ (29) admit, respectively, subsequences (𝜑_{𝑚,𝑛𝑘}) and (𝜑_{𝑚,𝑛𝑘}) satisfying lim_𝑘→∞^∞_{𝑙=1𝐸𝑚,𝑘}𝜑^𝑝_{𝑚,𝑙𝑛𝑘}(𝑥)(𝑥)+𝑑𝜆_{∞𝑙=1𝐸𝑚,𝑘}𝜑^𝑝_{𝑚,𝑙𝑛𝑘}(𝑥)𝑑𝜆(𝑥)=0. (30)

Proof. The proof is similar to the proof of Theorem 4.
(i) ⇒(ii). By density of 𝒫((𝐶_𝑚,𝑛)_𝑛∈ℕ0), we can find a sequence (𝑛_𝑘)⊂ℕ and sequences (𝑓_𝑚,𝑘) of periodic points of (𝐶_𝑚,𝑛)_𝑛∈ℕ0 such that ‖𝑓_𝑚,𝑘−𝜒_𝐾‖_𝑝<1/4^𝑘 and 𝐶_{𝑚,𝑛𝑘}𝑓_𝑚,𝑘=𝑓_𝑚,𝑘. Let 𝐸_𝑚,𝑘=𝐾⧵𝐴_𝑚,𝑘 for each 𝑚. Following the same estimation in the proof of Theorem 4, we can obtain the weight conditions.
(ii) ⇒(i). Let 𝑓_𝑚∈𝐶_𝑐(𝐺) with compact support 𝐾_𝑚⊂𝐺 for 1≤𝑚≤𝑁, and let 𝐾=∪^𝑁_𝑚=1𝐾_𝑚. Letting 𝑣_𝑚,𝑘∶=𝑓_𝑚𝜒_{𝐸𝑚,𝑘}+^∞_𝑙=1𝑇^𝑙𝑛𝑘_𝑚𝑓_𝑚𝜒_{𝐸𝑚,𝑘}+^∞_𝑙=1𝑆^𝑙𝑛𝑘_𝑚𝑓_𝑚𝜒_{𝐸𝑚,𝑘}. (31) and repeating the argument in the proof of Theorem 4, one can show that 𝑣_𝑚,𝑘→𝑓_𝑚 as 𝑘→∞ and 𝑣_𝑚,𝑘 is an element of 𝒫((𝐶_𝑚,𝑛)_𝑛∈ℕ0). Hence (𝐶_1,𝑛⊕𝐶_2,𝑛⊕⋯⊕𝐶_𝑁,𝑛)_𝑛∈ℕ0 is chaotic.

Remark 11. By Corollary 10 above, we note that (𝐶_𝑛)_𝑛∈ℕ0 is chaotic if, and only if, (𝐶_𝑛⊕𝐶_𝑛)_𝑛∈ℕ0 is chaotic. Similarly, 𝑇_𝑎,𝑤 and 𝑇_𝑎,𝑤⊕𝑇_𝑎,𝑤 can be chaotic only at the same time.

Finally, we give an example of the direct sum of cosine operator functions on ℓ^𝑝(ℤ).

Example 12. Given some 𝑁∈ℕ, let 𝐺=ℤ and 𝑎_𝑚=1 for 1≤𝑚≤𝑁. We consider a sequence of weighted translations (𝑇_𝑚) on ℓ²(ℤ), given by 𝑇_𝑚∶=𝑇_{1,𝑤𝑚∗𝛿1} where (𝑤_𝑚) is a sequence of positive weight functions. Then we define sequences of operators (𝐶_𝑚,𝑛)_𝑛∈ℕ0 by 𝐶_𝑚,𝑛=(1/2)(𝑇^𝑛_𝑚+𝑆^𝑛_𝑚) where 𝑆_𝑚 is the inverse of 𝑇_𝑚. By Corollary 10, (𝐶_1,𝑛⊕𝐶_2,𝑛⊕⋯⊕𝐶_𝑁,𝑛)_𝑛∈ℕ0 is chaotic if, given 𝜀>0 and 𝑞∈ℕ, there exists an arbitrarily large 𝑛 such that for all |𝑗|≤𝑞 and 1≤𝑚≤𝑁, we have ^∞_{𝑙=1|𝑗|≤𝑞}^𝑙𝑛−1_𝑠=0𝑤_𝑚𝑗+𝑠^𝑝+^∞_{𝑙=1|𝑗|≤𝑞}1∏^𝑙𝑛_𝑠=1𝑤_𝑚𝑗−𝑠^𝑝<𝜀. (32)

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-.