Topologically transitive sequence of cosine operators on Orlicz spaces

I. AKBARBAGLU^∗, M. R. AZIMI AND V. KUMAR

Abstract. For a Young function φ and a locally compact second countable groupG,let L^φ(G) denote the Orlicz space onG.In this paper, we present a necessary and sufficient condition for the topological transitivity of a sequence of cosine operators {Cn}^∞n=1 :=

{¹₂(Tg,wⁿ +Sg,wⁿ )}^∞n=1, defined on L^φ(G). We investigate the conditions for a sequence of cosine operators to be topologically mixing. Further, we go on to prove a similar result for the direct sum of a sequence of cosine operators. Finally, we give an example of topologically transitive sequence of cosine operators.

1. Introduction and Preliminaries

A sequence of bounded linear operators{T_k}_k∈N0 acting on a Fr´echet spaceXis said to be topologically transitive if for any pair (U, V) consisting of two non-empty open subsetsU and V of X, there exists an n∈Nsuch thatT_n(U)∩V 6=∅. A bounded linear operatorT is said to be topologically transitive if the sequence{S_k}_k∈N0 withSk:=T^k,thek-iterates ofT with the convention thatT⁰ =I, the identity operator, is topologically transitive as a sequence of bounded linear operators. A bounded linear operatorT is called hypercyclic if there exists a vectorx∈X,calledhypercyclic vector, such that the orbit{T^kx:k= 0,1,2, ...}ofxis dense inX, whereT⁰ is the identity operator onX. It is worth mentioning that these two notions, namely, topological transitivity and hypercyclicity of an operator are more likely equivalent on a Fr´echet space X [3, 8]. An operator T is called topologically mixing whenever for any pair (U, V) of two non-empty open subsetsU and V of X, there exists an N ∈N such that Tⁿ(U)∩V 6=∅for everyn≥N. An operator of the formI+B, whereBdenotes the backward shift operator, is an example of topologically mixing operator which is also hypercyclic. We say a bounded linear operatorT on a Fr´echet spaceX isweakly mixing if and only if T⊕T is hypercyclic onX⊕X. Note that weakly mixing operators are topologically transitive but the converse is not true in general. For more details on dynamics of linear operators, we refer the interested reader to two excellent books [3] and [8].

2010Mathematics Subject Classification. Primary 47A16, 46E30 Secondary: 22D05.

Key words and phrases. Hypercyclicity; Topologically transitive; Topologically mixing; Weighted transla- tion operator; Orlicz space; Locally compact groups.

∗Corresponding author.

1

A convex and even function φ : R → [0,∞] with φ(0) = 0 is called a Young function if it possess one of the following conditions:

• φ is finite on [0,+∞) and is not identically to zero;

• There exists x₁ >0 such thatφis finite on [0, x₁] and φ(x) = +∞ forx > x₁;

• There exists x₁ > 0 such that φ is finite on [0, x₁) and lim_x→x⁻

1 φ(x) = +∞ and φ(x) = +∞ for each x≥x1.

Corresponding to each Young function φ, there is an associated Young function ψ : R → [0,+∞] defined byψ(y) := sup{x|y|−φ(x) : x≥0}, which is called thecomplementary Young functionofφ. We remind that for a Young functionφ, its inverse i.e.,φ⁻¹: [0,+∞)→[0,+∞]

is defined by φ⁻¹(y) := inf{x≥0 :φ(x)> y} with inf(∅) = +∞.

Let G be a second countable locally compact group with the identity element e and a right Haar measureλ. LetL^φ(G) denote theOrlicz space, consisting of all Borel measurable functions f :G→C such that

Z

G

φ(α|f(x)|)dλ(x)<+∞,

for some α > 0. It is well-known that the space L^φ(G) is a vector space. The Orlicz space L^φ(G) equipped with theLuxemburg-Nakano norm

N_φ(f) = inf

k >0 : Z

G

φ |f|

k

dλ≤1

,

is a Banach space [11]. Moreover, if we assume that the Young function φvanishes only at zero, then another equivalent norm called the Orlicz norm is defined onL^φ(G) by

kfk_φ:= sup Z

G

|f(x)ν(x)|dλ(x) : N_ψ(ν)≤1

.

The set of all Borel functions ν on Gsuch that Nψ(ν)≤1 will be denoted by Ω.

A Young function φ is said to satisfy ∆₂-regular condition, if there is a constant k > 0 such that φ(2t) ≤ kφ(t) for large values of t when λ(G) < ∞ and φ(2t) ≤ kφ(t) for each t > 0, whenever λ(G) = ∞. If φ is ∆₂-regular, then the space C_c(G) of all continuous functions on G with compact support is dense in L^φ(G), and the dual space (L^φ(G),k · k_φ) is (L^ψ(G), Nψ(·)). For further information, the interested reader is referred to [11].

It is well known that the hypercyclic phenomenon is occurred only on infinite-dimensional and separable spaces [3, 8]. For this reason, we assume that G is second countable and φ(x) = 0 if and only if x = 0 [11, p. 87, Theorem 1]. Throughout this paper, the Banach space of all essentially bounded and measurable functions on G is denoted by L^∞(G) and N(f, r) denotes a neighborhood of f ∈ L^φ(G) with radius r > 0. A bounded continuous function w :G → (0,∞) is called a weight. For g ∈G, let δg be the unit point mass at g.

Given a weightw on Gand g∈G, a weighted translation operator T_g,w :L^φ(G)→L^φ(G) is defined by

T_g,w(f) :=w·f∗δ_g, f ∈L^φ(G) wheref ∗δ_g is the convolution defined as

f∗δ_g(t) :=

Z

G

f(tx⁻¹)dδ_g(x) =f(tg⁻¹), t∈G.

Indeed, it is the right translation off byg⁻¹. Moreover, it is easy to check thatf∗δ_g ∈L^φ(G) whenever f ∈ L^φ(G). Recall that an element g ∈ G is called a torsion element if it is of finite order. An element g ∈ G is called periodic if the closed subgroup G(g) generated by g is compact. Further, an element in G is aperiodic if it is not periodic. Equivalently, g ∈ G is an aperiodic element, if and only if for any compact subset K ⊂ G, there exists an N ∈ N such that K∩Kg⁻ⁿ = ∅ for n > N [5, Lemma 2.1]. It is worth noting that a weighted translationT_g,w cannot be hypercyclic wheneverkwk∞≤1 orgis a torsion element [2, 5]. The hypercyclic weighted translations on locally compact groups was characterized by C. Chen [5] in details. In addition, he studied the hypercyclicity of weighted convolution operators.

Moreover, a study of the hypercyclic weighted translations on Orlicz spaces L^φ(G) was carried over by first two authors in [2].

Ifw⁻¹:= _w¹ ∈L^∞(G), the weighted translation operator Tg,w is invertible and its inverse is T_g⁻¹_,w⁻¹_∗δg which will be denoted by S_g,w throughout this paper. For each n ∈ Z, the cosine operator Cn:L^φ(G)→L^φ(G) is defined by

Cn:= 1

2(T_g,wⁿ +S_g,wⁿ ).

It follows that 2CnCm=Cn+m+Cn−m forn, m∈Z.

The study of cosine operator on Banach spaces is originally due to Bonilla and Miana [4].

They gave sufficient conditions for the hypercyclicity and topologically mixing of a strongly continuous cosine operator function. Afterwards, T. Kalmes in [9] characterized the hyper- cyclicity of cosine operator functions onL^p(Ω) (Ω is open subset ofR^d) generated by second order partial differential operators. He also showed that the hypercyclicity and weakly mixing of these type of operators are equivalent.

Furthermore, Chen [6] gave a necessary and sufficient condition for the topological transi- tivity of the cosine operatorC_nonL^p(G). In this article, we study the topological transitivity of the cosine operatorCnon a space which is a natural generalization ofL^p(G), namely, Orlicz space L^φ(G).

2. Topological transitivity of (Cn)n∈N

In this section we present our main results with some immediate consequences. We begin with the following theorem which give a necessary and sufficient condition on weight so that

cosine operator is topologically transitive on a second countable locally compact group G equipped with a right Haar measure λ.

Theorem 2.1. Let g ∈ G be an aperiodic element of G and let φ be a ∆₂-regular Young function. Letw, w⁻¹∈L^∞(G).If Cn:= ¹₂(T_g,wⁿ +S_g,wⁿ ) is a sequence of cosine operators on L^φ(G), then the following statements are equivalent.

(i) (C_n)n∈N is topologically transitive.

(ii) For each non-empty compact subset K ⊂G with λ(K) >0, there exist sequences of Borel sets (Ek),(E_k⁺) and (E_k⁻) in K, and a sequence (nk) of positive numbers such that for E_k=E_k⁺∪E_k⁻, we have

k→∞lim sup

ν∈Ω

Z

K\E_k

|ν(x)|dλ(x) = 0.

Moreover, the two sequence

ϕn=

n

Y

j=1

w∗δ_g^j−1 and ϕ˜n=





n−1

Y

j=0

w∗δ_g^j





−1

satisfy

k→∞lim sup

ν∈Ω

Z

Ek

ϕ_nk(x)|ν(xg^nk)|dλ(x) = 0,

k→∞lim sup

ν∈Ω

Z

Ek

˜

ϕ_nk(x)|ν(xg^nk)|dλ(x) = 0,

k→∞lim sup

ν∈Ω

Z

E_k⁺

ϕ2n_k(x)|ν(xg^2nk)|dλ(x) = 0,

k→∞lim sup

ν∈Ω

Z

E_k⁻

˜

ϕ2n_k(x)|ν(xg^2nk)|dλ(x) = 0.

Proof. (i) ⇒ (ii). In spite of being different underlying spaces, the approach of the proof is followed like as done in [6]. LetK be a compact subset ofGsuch thatλ(K)>0.Sinceg∈G is an aperiodic element, there existsN ∈Nsuch that K∩Kg^±n =∅ forn > N [5, Lemma 2.1]. Denote the characteristic function ofK defined onGbyχ_K.Clearlyχ_K ∈L^φ(G). Take ∈ (0,1) and U = N(χK, ²) and V = N(−χ_K, ²) in the definition of the topologically transitive for the sequence (Cn)n∈N. There exist f ∈L^φ(G) and m ∈N, of course, m > N such that

kf−χKk_φ< ² and kC_mf+χKk_φ< ². Hence, we can write that

kRe(f)−χKk_φ< ² and kRe(C_mf) +χKk_φ=kC_mRe(f) +χKk_φ< ²,

whereRe(f) is the real part of the complex valued functionf. Since the mapRe:L^φ(G,C)→ L^φ(G,R) is continuous and also commute with both Tg,w and Sg,w, hence without loss of

generality we may assume that the functionf is real valued. Therefore, for any Borel subset F ⊂G,we have

kC_mf⁺χFk_φ ≤ k(C_mf⁺)k_φ=k(C_mf+χK−χK)⁺k_φ

≤ k(C_mf+χK)⁺k_φ+k(−χ_K)⁺k_φ

= k(C_mf+χ_K)⁺k_φ≤ kC_mf+χ_Kk_φ< ², (2.1)

wheref⁺:= max{0, f} andf⁻:=f⁺−f. SetA={x∈K :|f(x)−1| ≥}.Then

² >kf −χ_Kk_φ = sup

ν∈Ω

Z

G

|f(x)−χ_K(x)||ν(x)|dλ(x)

≥ sup

ν∈Ω

Z

K

|f(x)−1||ν(x)|dλ(x)

≥ sup

ν∈Ω

Z

A

|ν(x)|dλ(x).

Therefore, we have

sup

ν∈Ω

Z

A

|ν(x)|dλ(x)< .

SetB_m={x∈K :|C_mf(x) + 1| ≥}.Then, by the similar argument, we get

sup

ν∈Ω

Z

Bm

|ν(x)|dλ(x)< .

Now, letEm:={x∈K:|f(x)−1|< } ∩ {x∈K:|C_mf(x) + 1|< }.Then, forx∈Em,we getf(x)>1− >0 andC_mf(x)< −1<0.Also,

sup

ν∈Ω

Z

K\Em

|ν(x)|dλ(x) = sup

ν∈Ω

Z

A∪Bm

|ν(x)|dλ(x)

= sup

ν∈Ω

Z

A

|ν(x)|dλ(x) + sup

ν∈Ω

Z

Bm

|ν(x)|dλ(x)

< += 2.

By keeping the facts that Haar measureλis right invariant, T_g,w^m f⁺ and S_g,w^m f⁺ are positive in the mind, with the aid of (2.1) we get that

2² > k2(C_mf⁺)χ_Emg^mk_φ=k(T_g,w^m f⁺+S_g,w^m f⁺)χ_Emg^mk_φ≥ kT_g,w^m f⁺χ_Emg^mk_φ

= sup

ν∈Ω

Z

Emg^m

|T_g,w^m f⁺(x)||ν(x)|dλ(x)

= sup

ν∈Ω

Z

Emg^m

|w(x)w(xg⁻¹). . . w(xg^−m+1)f⁺(xg^−m)||ν(x)|dλ(x)

= sup

ν∈Ω

Z

Em

w(xg^m)w(xg^m−1). . . w(xg)f⁺(x)|ν(xg^m)|dλ(x)

= sup

ν∈Ω

Z

Em

ϕm(x)f⁺(x)|ν(xg^m)|dλ(x)

> sup

ν∈Ω

Z

Em

(1−)ϕ_m(x)|ν(xg^m)|dλ(x).

Therefore,

sup

ν∈Ω

Z

Em

ϕ_m(x)|ν(xg^m)|dλ(x)< 2² 1−. By the similar argument, we get

2² >k(S_g,w^m f⁺)χ_Emg^mk_φ>(1−) sup

ν∈Ω

Z

Em

˜

ϕ_m(x)|ν(xg^m)|dλ(x)

and thus,

sup

ν∈Ω

Z

Em

˜

ϕm(x)|ν(xg^m)|dλ(x)< 2² 1−. Hence, the first part of Condition (ii) holds as is arbitrary.

Now, let E_m⁻ ={x∈ E_m :T_g,w^m f(x) < −1} and E_m⁺ =E_m\E_m⁻.Then, for x∈ E_m⁺,we have

−1> C_mf(x) = 1

2T_g,w^m f(x) + 1

2S_g,w^m f(x)≥ 1

2(−1) + 1

2S_g,w^m f(x) and therefore,

S_g,w^m f(x)< −1, x∈E_m⁺.

Now, consider the following (1−) sup

ν∈Ω

Z

E⁺m

ϕ_2m(x)|ν(xg^2m)|dλ(x)

< sup

ν∈Ω

Z

Em⁺

|w(xg^2m)w(xg^2m−1)w(xg^2m−2). . . w(xg)| |S_g,w^m f⁻(x)| |ν(xg^2m)|dλ(x)

= sup

ν∈Ω

Z

Em⁺g^2m

|w(x)w(xg)w(xg²). . . w(xg^−(2m−1))| |S_g,w^m f⁻(xg^−2m)| |ν(x)|dλ(x)

= sup

ν∈Ω

Z

Em⁺g^2m

|T_g,w^2mS_g,w^m f⁻(x)| |ν(x)|dλ(x)

= sup

ν∈Ω

Z

Em⁺g^2m

|T_g,w^m f⁻(x)| |ν(x)|dλ(x)

≤ 2 sup

ν∈Ω

Z

Em⁺g^2m

|C_mf⁻(x)| |ν(x)|dλ(x)

= 2 sup

ν∈Ω

Z

G

|C_mf⁻(x)χ_E⁺

mg^2m| |ν(x)|dλ(x)

= 2 sup

ν∈Ω

Z

G

|C_m(f⁺−f)(x)χ_E+

mg^2m| |ν(x)|dλ(x)

= 2 sup

ν∈Ω

Z

G

|(C_mf⁺)χ_E⁺

mg^2m−(Cmf+χK)χ_E⁺

mg^2m+χ_K∩E⁺

mg^2m| |ν(x)|dλ(x)

≤ 2 sup

ν∈Ω

Z

G

|(C_mf⁺)χ_E⁺

mg^2m||ν(x)|dλ(x) + 2 sup

ν∈Ω

Z

G

|(C_mf+χK)χ_E⁺

mg^2m| |ν(x)|dλ(x) + 2 sup

ν∈Ω

Z

K∩Em⁺g^2m

|ν(x)|dλ(x)

≤ 2k(C_mf⁺χ_E+

mg^2m)k_φ+ 2k(C_mf+χ_K)k_φ+ 2kχ_K∩E+ mg^2mk_φ

< 2²+ 2²+ 0 = 4².

In the last inequality, from the fact K∩Kg^±2m =∅, has been already used. Therefore, we get

sup

ν∈Ω

Z

Em⁺

ϕ_2m(x)|ν(xg^2m)|dλ(x)< 4² (1−). In similar lines, we also have

sup

ν∈Ω

Z

Em⁻

˜

ϕ2m(x)|ν(xg^2m)|dλ(x)< 4² (1−). Since is arbitrary, last two condition of (ii) part also fulfilled.

(ii) ⇒ (i). Let U and V be two non-empty open subsets ofL^φ(G). Since φ is ∆₂-regular we can choose two non-zero functions f and h in Cc(G) such that f ∈ U and h ∈ V. Set K = supp(f)∪supp(h), the supports of f and h respectively. Let E_k ⊂K and it satisfies condition (ii). But g ∈ G is an aperiodic element, hence there exists M ∈ N such that K∩Kg^±n=∅for all n > M. Subsequently, for a given >0, one can find N ∈Nsuch that

for each k > N,n_k > M and khk_∞·sup

ν∈Ω

Z

Ek

ϕn_k(x)|ν(xg^nk)|dλ(x)< , khk_∞sup

ν∈Ω

Z

K\E_k

|ν(x)|dλ(x)< .

Now, we have

kT_g,w^nk(hχEk)k_φ = sup

ν∈Ω

Z

G

|T_g,w^nk(hχEk)(x)ν(x)|dλ(x)

= sup

ν∈Ω

Z

G

|w(x)w(xg⁻¹). . . w(xg^−nk+1)h(xg^−nk)χ_Ek(xg^−nk)ν(x)|dλ(x)

= sup

ν∈Ω

Z

G

|w(xg^nk)w(xg^nk−1). . . w(xg)h(x)χ_Ek(x)ν(x)|dλ(x)

≤ khk_∞·sup

ν∈Ω

Z

Ek

ϕn_k(x)|ν(xg^nk)|dλ(x)< . Hence,

k→∞lim kT_g,w^nk(hχE_k)k_φ= 0.

Also,

kh−hχ_Ekk_φ = sup

ν∈Ω

Z

G

|h(x)−h(x)χ_Ek(x)||ν(x)|dλ(x)

= sup

ν∈Ω

Z

G

|h(x)χ_K\Ek(x)||ν(x)|dλ(x)

= Z

K\E_k

|h(x)||ν(x)|dλ(x)

≤ khk_∞ Z

K\E_k

|ν(x)|dλ(x)< .

By similar argument, using the conditions given in (ii) we get

k→∞lim kS_g,w^nk (hχ_Ek)k_φ= 0, lim

k→∞kS_g,w^2nk(hχ_E⁻

k

)k_φ= lim

k→∞kT_g,w^2nk(hχ_E⁺

k

)k_φ= 0.

In addition, we also have

k→∞lim kS_g,w^nk(f χEk)k_φ= lim

k→∞kT_g,w^nk(f χEk)k_φ= 0.

For eachk∈N,we set

v_k =f χ_Ek + 2T_g,w^nk(hχ_E⁺

k) + 2S_g,w^nk(hχ_E⁻

k).

In this stage, an application of the frequently used fact i.e.,K∩Kg^(m1−m2)nk =∅(m1, m2 ∈Z and m1 6=m2) with Minkowski inequality yield that

kv_k−fk_φ ≤ kf−f χE_kk_φ+ 2kT_g,w^nk(hχ_E⁺

k

)k_φ+ 2kS_g,w^nk(hχ_E⁻

k

)k_φ

≤ kfk∞

φ⁻¹(_λ(K\E¹

k)) + 2kT_g,w^nk(hχ_E⁺

k)k_φ+ 2kS_g,w^nk(hχ_E⁻

k)k_φ

and

kC_nkvk−hk_φ ≤ kh−hχEkk_φ+ 1

2kT_g,w^nk(f χEk)k_φ+1

2kS_g,w^nk (f χEk)k_φ + kT_g,w^2nk(hχ_E⁺

k

)k_φ+kS_g,w^2nk(hχ_E⁻

k

)k_φ

≤ khk∞

φ⁻¹(_λ(K\E¹

k)) +1

2kT_g,w^nk(f χEk)k_φ+1

2kS_g,w^nk(f χEk)k_φ + kT_g,w^2nk(hχ_E⁺

k

)k_φ+kS_g,w^nk (hχ_E⁻

k

)k_φ.

Therefore, we have limk→∞kv_k − fk_φ = 0 and limk→∞kC_nkv_k −hk_φ = 0, since λ(K \ E_k) → 0 and it is assumed that sup{x : φ(x) < +∞}= +∞. Eventually, lim_k→∞v_k = f and limk→∞Cnkvnk = h. So, Cnk(U)∩V 6= ∅ for some k. Hence, the sequence (Cn)n∈N is

topologically transitive.

The following corollary gives a characterization of topologically mixing property of cosine operators on Orlicz space L^φ(G). Since the proof is similar to above theorem therefore we will omit the proof.

Corollary 2.2. Let g ∈ G be an aperiodic element of G and let φ be a ∆₂-regular Young function. Let w, w⁻¹∈L^∞(G).If C_n:= ¹₂(T_g,wⁿ +S_g,wⁿ ) is cosine operator on L^φ(G) then the following statements are equivalent.

(i) (Cn)n∈N is topologically mixing.

(ii) For each non-empty compact subset K ⊂G with λ(K) >0, there exist sequences of Borel sets (E_n) such that

n→∞lim sup

ν∈Ω

Z

K\En

|ν(x)|dλ(x) = 0 and the two sequence

ϕn=

n

Y

j=1

w∗δ_g^j−1 and ϕ˜n=





n−1

Y

j=0

w∗δ_g^j





−1

satisfy

n→∞lim sup

ν∈Ω

Z

En

ϕn(x)|ν(xgⁿ)|dλ(x) = 0,

n→∞lim sup

ν∈Ω

Z

En

˜

ϕ_n(x)|ν(xgⁿ)|dλ(x) = 0.

We formulate the discrete version of Theorem2.1. IfGis discrete group with the counting measure as its Haar measure. Then the set Ek is nothing but set K itself. Therefore, we have the following result.

Corollary 2.3. Let g ∈G be a non-torsion element of G and let φ be a ∆₂-regular Young function. Let w, w⁻¹∈L^∞(G).If C_n:= ¹₂(T_g,wⁿ +S_g,wⁿ ) is cosine operator on L^φ(G) then the following statements are equivalent.

(i) (C_n)n∈N is topologically transitive.

(ii) For each non-empty finite subset K⊂G, there exist sequences of Borel sets (E_k⁺)and (E_k⁻) in K,and a sequence (nk) of positive numbers such that forK =E_k⁺∪E_k⁻, we have the two sequence

ϕn=

n

Y

j=1

w∗δ_g^j−1 and ϕ˜n=





n−1

Y

j=0

w∗δ_g^j





−1

satisfy

k→∞lim sup

ν∈Ω

X

x∈K

ϕnk(x)|ν(xg^nk)|= 0, lim

k→∞sup

ν∈Ω

X

x∈K

˜

ϕnk(x)|ν(xg^nk)|= 0,

k→∞lim sup

ν∈Ω

X

x∈E_k⁺

ϕ2n_k(x)|ν(xg^2nk)|= 0, and lim

k→∞sup

ν∈Ω

X

x∈E_k⁻

˜

ϕ2n_k(x)|ν(xg^2nk)|= 0.

Here we present a characterization of topological transitivity for a finite sequence of weighted cosine operators. We set the following notations for the sequence. For a fixM ∈N. Let{g_l}_1≤l≤M and{w_l}_1≤l≤M be the sequences of aperiodic elements of groupGand positive weights respectively. Then{T_gl,wl}_1≤l≤M is a sequence of weighted translation operators. We have the following characterization.

Theorem 2.4. Let{g_l}_land{w_l}_lbe the sequences of aperiodic elements and positive weights respectively such that w_l, w⁻¹_l ∈L^∞(G).Let C_l,n:= ¹₂(T_gⁿ

l,w_l+S_gⁿ

l,w_l) be the cosine operators onL^φ(G)for1≤l≤M, whereT_gl,wl is the weighted translation operator. Then the following statements are equivalent.

(i) (C1,n⊕C2,n⊕ · · · ⊕CM,n)n∈N0 is topologically transitive.

(ii) For each non-empty compact subset K ⊂G with λ(K) >0, there is some sequence (n_k) of positive integers such that for 1 ≤l ≤M, there exist sequences of Borel sets (E_l,k), (E_l,k⁺) and (E_l,k⁻) such that for E_l,k=E_l,k⁺ ∪E_l,k⁻, we have

k→∞lim sup

ν∈Ω

Z

K\E_l,k

|ν(x)|dλ(x) = 0 and the two sequence

ϕ_l,nk =

nk

Y

j=1

w_l∗δ^j

g⁻¹_l and ϕ˜_l,nk =





nk−1

Y

j=0

w_l∗δ_g^j

l





−1

satisfy

k→∞lim sup

ν∈Ω

Z

El,k

ϕl,n_k(x)|ν(xg^nk)|dλ(x) = 0,

k→∞lim sup

ν∈Ω

Z

El,k

˜

ϕ_l,nk(x)|ν(xg^nk)|dλ(x) = 0,

k→∞lim sup

ν∈Ω

Z

E_l,k⁺

ϕl,2nk(x)|ν(xg^2nk)|dλ(x) = 0,

k→∞lim sup

ν∈Ω

Z

E_l,k⁻

˜

ϕ_l,2nk(x)|ν(xg^2nk)|dλ(x) = 0.

Proof. (i)⇒(ii). Let K be a compact subset of Gsuch that λ(K) >0.Since (C1,n⊕C2,n⊕

· · · ⊕C_N,n)n∈M0 is topologically transitive, for ∈(0,1),there exist f_l ∈L^φ(G) and m∈N such that for 1≤l≤M, we have

kf_l−χKk_φ< ² and kC_l,mfl+χKk_φ< ².

Further, to complete the proof follow the proof of part (i) ⇒ (ii) of Theorem 2.1 to get desired conditions on weights w_l for each l.

(ii) ⇒ (i). Let Ul and Vl be non-empty open subsets of L^φ(G). Since φ is ∆2-regular we can choose two non-zero functions f_l and h_l in C_c(G) such that f_l ∈ U_l and h ∈ V_l. Set K =supp(f_l)∪supp(h_l)(the supports of functions f_l and h_l). Let E_l,k ⊂K and it satisfies condition (ii). Now, imitate the proof of (ii)⇒(i) of Theorem2.1to get thatCl,n_k(Ul)∩V_l 6=∅

for each l,1≤l≤M.

Remark 2.1. (a) As elementary example of∆₂-regular Young function let φ(x) =x^p/p and ψ(x) = x^q/q for 1 < p, q < ∞ with 1/p + 1/q = 1. Also, let φ1(x) =

|x|^α(1 +|log|x||) with α >1 and φ₂(x) =|x|^αln^β(|x|+e) with β ≥1 and α >1. By [11, Corollary 2.3.4] for i = 1,2, φ_i and its complementary function, ψ_i, satisfy the

∆2-regular condition.

(b) Theorems 2.1 and 2.4 extends [6, Theorem 2.1] and [6, Corollary 2.7] from weighted Lebesgue spaces to weighted Orlicz spaces.

Example 2.1. Let G= Z. Fix an aperiodic element g ∈Z with g ≥ 1. Define the Young functionφ(x) = (1 +|x|) ln(1 +|x|)− |x|, and consider the weight function

w(i) = ( ₁

2, i≥0,

3

2, i <0.

A direct computation gives the complementary of Young function ψ to φ which is given by ψ(x) = exp(|x|)− |x| −1.

Note that φis ∆₂-regular and vanishes only at zero.

Choose an arbitraryν∈Ω. Then,P+∞

n=−∞ψ(|ν(n)|)≤1if and only ifP+∞

n=−∞(exp(|ν(n)|)−

|ν(n)| −1)≤ 1. But the last is established only if |ν| ≤ 2. Each compact subset K ⊂ Z is a finite set, consisting of the integers a₁ ≤a₂ ≤ · · · ≤ a_m. Take E_i := {a₁,· · ·, a_i} and for each i ≥m, define Ei := Em. Put nk =i, E_k⁺ := Ei and E_k⁻ := ∅ in the statement (ii) of

Theorem 2.1. In this circumstances, we have

n→∞lim sup

ν∈Ω

X

j∈K\En

|ν(j)|= 0.

In addition, one may findn₀ ∈N such that a₁+n₀g≥0. Hence, for eachn≥n₀ we have, sup_ν∈ΩP

ai∈E_nϕ_n(a_i)|ν(a_i+ng)| ≤2P

ai∈E_nϕ_n(a_i)

= 2P

ai∈E_nw(a_i+g)w(a_i+ 2g)· · ·w(a_i+ng)

≤2P

ai∈Enw(a1+g)w(a1+ 2g)· · · w(a1+n0g)w(a1+ (n0+ 1)g)· · ·w(a1+ng)

≤2P

ai∈En(¹₂)ⁿ⁻ⁿ⁰w(a1)^|a1|

= 2card(E_n)(¹₂)ⁿ⁻ⁿ⁰w(a₁)^|a1|

≤2m(¹₂)ⁿ⁻ⁿ⁰w(a₁)^|a1|→0, as n→ ∞. Here,card(E_n) means the cardinality of the set E_n.

Similarly, there exists t₀∈Nsuch that a_m−t₀g≤0 and so for each n≥t₀, sup_ν∈ΩP

ai∈Enϕ˜_n(a_i)|ν(a_i+ng)| ≤2P

ai∈Enϕ˜_n(a_i)

= 2P

ai∈E_nw⁻¹(a_i−g)w⁻¹(a_i−2g)· · ·w⁻¹(a_i−ng)

≤2P

ai∈Enw⁻¹(am−g)w⁻¹(am−2g)· · · w⁻¹(am−t0g)w⁻¹(am−(t0+ 1)g)· · ·w⁻¹(am−ng)

≤2P

ai∈En(²₃)^n−t0w⁻¹(am)^|am|

≤2m(²₃)^n−t0w⁻¹(am)^|am|→0,

as n → ∞. The other statements of Theorem 2.1 can be verified in this way. Therefore, by Theorem 2.1, the corresponding sequence of cosine operators to the weight w and g, is topologically transitive.

Acknowledgment

Vishvesh Kumar is supported by FWO Odysseus 1 Grant G.0H94.18N: Analysis and Partial Differential Equations. We thank the referee for useful comments and remarks, which helped to improve the final version of the paper.

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Ibrahim Akbarbaglu

Email address: [email protected], [email protected]

Department of Mathematics, Farhangian University, Tehran, Iran.

Mohammad Reza Azimi

Email address: [email protected]

Department of Mathematics, Faculty of Sciences, University of Maragheh, 55181-83111, Maragheh, Iran.

Vishvesh Kumar

Email address: [email protected]

Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000, Ghent, Belgium