Chaos for Cosine Operator Functions on Groups

and Applied Analysis 3 (iii) For each compact subset K ⊂ G with λ(K) > 0, there is a sequence of Borel sets (Ek) in K such that λ(K) = limk→∞λ(Ek) and both sequences


Introduction
Let  be a bounded linear operator on a Banach space .If V ∈  is a fixed point of , then the orbit of V under , denoted by Orb(, V), is Orb(, V) = {V, V,  2 V, . ..} = {V}.On the contrary, if there exists an element  ∈  such that the orbit is dense in ; that is, Orb(, ) = {, ,  2 , . ..} = , 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.
Let N 0 = N ∪ {0}.According to the definition of Devaney chaos, a sequence of bounded linear operators (  ) ∈N 0 on a Banach space  is chaotic in the successive way in [18] if (  ) ∈N 0 is topologically transitive and the set of periodic elements, denoted by P((  ) ∈N 0 ) = { ∈ ; ∃  ∈ N :    = ,  = 1, 2, 3, . ..}, is dense in .We recall that (  ) ∈N 0 is topologically transitive if, given nonempty open subsets ,  of , we have   () ∩  ̸ = 0 for some  ∈ N. If   () ∩  ̸ = 0 from some  onwards, then (  ) ∈N 0 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 (  ) ∈N 0 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 (  ) ∈N 0 is said to be hypercyclic if Orb(  , ) = {   :  ∈ N 0 } =  for some  ∈ .If (  ) ∈N 0 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 C from the real line into the space of continuous operators on  satisfying C(0) =  and the d' Alembert functional equation 2C()C() = C( + ) + C( − ) for all ,  ∈ R, which implies C() = C(−) for all  ∈ R. In [20], Bonilla and Miana obtained a sufficient condition for a cosine operator function C() defined by to be transitive, where  is a strongly continuous translation group on some weighted Lebesgue space   (R).For a Borel measure  and Ω ⊂ R  , Kalmes gave the characterization for cosine operator functions, generated by second order partial differential operators on   (Ω, ), 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   () (1 ≤  < ∞) the complex Lebesgue space with respect to .
A function  :  → (0, ∞) 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  −1 ∈  ∞ (), then the weighted translation operator In what follows, we assume ,  −1 ∈  ∞ () and define a sequence of bounded linear operators   :   () →   () by for all  ∈ Z where  − , := ( −1 , )  =   , .Then (  ) ∈Z can be regarded as a cosine operator function by letting C() =   .Since   =  − for all  ∈ Z, we will investigate the sequence of operators (  ) ∈N 0 and give a necessary and sufficient condition for (  ) ∈N 0 to be chaotic in terms of the weight function , the Haar measure , and the group element  ∈ .

Chaotic Condition
In this section, we will show the main result and give some examples of chaotic cosine operator functions on various groups.Since (  ) ∈N 0 is generated by some element  ∈ , we first note that (  ) ∈N 0 is never chaotic if  is a torsion element by the fact in [17] that (  ) ∈N 0 is not transitive when  is torsion.

Lemma 1.
Let  be a locally compact group and let  be a torsion element in .Let 1 ≤  < ∞ and  , be a weighted translation on   () with inverse  , .Let 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  ∈ N such that  ∩  ± = 0 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 (  ) ∈N 0 .Let P((  ) ∈N 0 ) be the set of periodic elements of a sequence of operator (  ) ∈N 0 .By the d' Alembert functional equation and induction, we have a simple observation immediately.
Based on the work of characterizing transitive (  ) ∈N 0 in [17], we are able to obtain the characterization for (  ) ∈N 0 to be chaotic in this note.We state the result in [17] below.
(ii) ⇒(iii).Let P((  ) ∈N 0 ) be dense in   ().Let  be a compact subset of  with () > 0. Then by the aperiodicity, there exists some  ∈ N such that  ∩  ± = 0 for all  > .Let   ∈   () be the characteristic function of .By density of P((  ) ∈N 0 ), we can find a sequence (  ) ⊂ N and a sequence (  ) of periodic points of (  ) ∈N 0 such that ‖   −   ‖  < 1/4  and      =   in which we may assume  +1 >   > .Hence we have which proves condition (iii).(iii) ⇒(i).The proof is similar to the proof of [ Let Then V  ∈   () by the weight assumption in the condition (iii).Also, using    ∩    = 0 again, we have V  →  as  → ∞ which follows from On the other hand, V  is an element of P((  ) ∈N 0 ) by the equality Putting all these together, condition (iii) implies (i).
We note that [15] in many familiar nondiscrete groups, including the additive group R  , the Heisenberg group, and the affine group, all elements except the identity are aperiodic.On the other hand, if  is discrete, then   = 0 and   =  for all  ∈ N in the proof of Theorem 4. Hence we have the characterization below for discrete groups.
admit.Respectively, subsequences (   ) and ( φ  ) 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, (  ) ∈N 0 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 1 ≤  < ∞ and  , be a weighted translation on   () with inverse  , .Let   = (1/2)(  , +   , ).Then the following conditions are equivalent.
(iv)   , is chaotic for all  ∈ N.
(v)   , is chaotic for all  ∈ N.
(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( , ) = P( , ).Hence we prove the equivalence.
We end up this section with two examples on  = Z and  = R, which says that one can construct many chaotic cosine operator functions on various groups.
Example 7. Let  = Z,  = 1 ∈ Z which is nontorsion.Let  *  1 be a weight on Z. Then the weighted translation operator  1, *  1 on ℓ  (Z) is the bilateral weighted forward shift , studied in [11] and given by   =    +1 with   = ().Here (  ) ∈Z is the canonical basis of ℓ  (Z) and (  ) ∈Z is a sequence of positive real numbers.Also, we have Let  −1 ∈ ℓ ∞ (Z) and let   = (1/2)(  +   ) where  is the inverse of  =  1, *  1 .Then by Corollary 5, (  ) ∈N 0 and  1, *  1 are chaotic if, given  > 0 and  ∈ N, there exists an arbitrarily large  ∈ N such that In fact, there are many weight functions  on Z satisfying the weight condition above.For example, one may define  : Z → (0, ∞) by Example 8. Let  = R,  = 2, and  be a weight on R. Then the weighted translation  2, on   (R) is defined by Let Similarly, one may choose  : R → (0, ∞) by which is the required weight function in the above condition.

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, let (   ,  ) be a sequence of weighted translation operators on   (), defined by sequences of aperiodic elements (  ) in  and positive weight functions (  ) for 1 ≤  ≤ .We write   for    ,  for simplification.In [17], we have the result below.