2 The Sequence Spaces `

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www.i-csrs.org

Available free online at http://www.geman.in

Some Generalized Difference

Sequence Spaces of Non-Absolute Type

Sinan Ercan¹ and C¸ i˘gdem A. Bekta¸s²

1Department of Mathematics, Firat University, Elazi˘g, Turkey E-mail: [email protected]

2Department of Mathematics, Firat University, Elazi˘g, Turkey E-mail: [email protected]

(Received: 4-3-15 / Accepted: 12-4-15) Abstract

In this paper, we introduce the spaces `∞(∆^m_λ), c(∆^m_λ) and c₀(∆^m_λ), which areBK-spaces of non-absolute type and we prove that these spaces are linearly isomorphic to the spaces `∞, c and c₀, respectively. Moreover, we give some inclusion relations and compute the α−, β− andγ−duals of these spaces. We also determine the Schauder basis of the c(∆^m_λ) and c₀(∆^m_λ).

Keywords: Sequence spaces of non-absolute type, BK-spaces, Difference Sequence Spaces.

1 Introduction

A sequence space is defined to be a linear space of real or complex sequences.

Let w denote the spaces of all complex sequences. If x ∈ w, then we simply writex= (x_k) instead of x= (x_k)^∞_k=0.

LetX be a sequence space. If X is a Banach space and τ_k :X →C, τ_k(x) = x_k (k = 1,2, ...) is a continuous for allk,X is called a BK−space.

We shall write `_∞, c and c₀ for the sequence spaces of all bounded, convergent and null sequences, respectively, which areBK−spaces with the norm given by kxk_∞= sup_k|x_k| for all k∈N.

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For a sequence space X, the matrix domain X_A of an infinite matrix A defined by

XA={x= (xk)∈w:Ax∈X} (1) which is a sequence space.

We shall denote the collection of all finite subsets of N byF.

M. Mursaleen and A. K. Noman [9] introduced the sequence spaces`^λ_∞, c^λ and c^λ₀ as the sets of all λ−bounded, λ−convergent and λ−null sequences, respectively, that is

`^λ_∞ = {x∈w: sup

n |Λ_n(x)|<∞}

c^λ = {x∈w: lim

n→∞Λ_n(x) exists}

c^λ₀ = {x∈w: lim

n→∞Λ_n(x) = 0}

where Λ_n(x) = _λ¹

n

Pn k=0

(λ_k−λ_k−1)x_k, k ∈N.

M. Mursaleen and A. K. Noman [10] also introduced the sequence spaces c^λ(∆) and c^λ₀(∆), respectively, that is

c^λ(∆) ={x∈w: lim

n→∞

Λ¯_n(x) exists}

c^λ₀(∆) ={x∈w : lim

n→∞

Λ¯_n(x) = 0}.

where ¯Λ_n(x) = _λ¹

n

Pn k=0

(λ_k−λ_k−1) (x_k−x_k−1), k∈N.

H. Ganie and N. A. Sheikh [2] introduced the spaces c₀(∆^λ_u) and c(∆^λ_u) as follows:

c₀(∆^λ_u) = {x∈w: lim

n→∞Λb_n(x) = 0}

c(∆^λ_u) = {x∈w: lim

n→∞Λb_n(x) exists}

whereΛ^b_n(x) = _λ¹

n

Pn

k=0(λ_k−λk−1)u_k(x_k−xk−1), k∈N.

2 The Sequence Spaces `

_∞

(∆

^m_λ

) , c(∆

^m_λ

) and c

₀

(∆

^m_λ

) of Non-Absolute Type

We define the sequence spaces`∞(∆^m_λ) ,c(∆^m_λ) and c₀(∆^m_λ) as follows;

`∞(∆^m_λ) =

x∈w: sup

n

Λ˜_n(x)<∞

c(∆^m_λ) =

x∈w: lim

n→∞

Λ˜_n(x) exists

c₀(∆^m_λ) =

x∈w: lim

n→∞

Λ˜_n(x) = 0

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where ˜Λ_n(x) = _λ¹

n

Pn

k=0(λ_k−λk−1)∆^mx_k, k, m ∈N. ∆ denotes the difference operator. i.e., ∆⁰x_k=x_k, ∆x_k =x_k−xk−1 and ∆^mx_k=^P^m_v=0(−1)^v m

v

!

xk−v. λ= (λ_k)^∞_k=0is a strictly increasing sequence of positive reals tending to infinity, that is 0< λ₀ < λ₁ < ... and λ_k → ∞as k → ∞.

Here and in sequel, we use the convention that any term with a negative subscript is equal to naught. e.g. λ−1 = 0 and x−1 = 0.

If we takem= 1 sequence spaces which we defined reduces to`^λ_∞(∆), c^λ(∆) and c^λ₀(∆).

We define the matrix ˜Λ =λ˜nk

for all n, k ∈N by

λ˜_nk =







n

P

i=k

m i−k

!

(−1)^{i−k λ}ⁱ^−λ_λ ⁱ⁻¹

n , k≤n

0, n < k

.

Λ =˜ λ˜_nk equality can be eaisly seen from Λ˜n(x) = 1

λ_n

n

X

k=0

(λk−λk−1) ∆^mxk (2) for all m, n ∈ N and every x = (x_k) ∈ w. Then it leads us together with (1) to the fact that

`∞(∆^m_λ) = (`∞)Λ˜, c₀(∆^m_λ) = (c₀)Λ˜, c(∆^m_λ) = (c)Λ˜.

The matrix ˜Λ = λ˜_nk is a triangle, i.e., ˜λ_nn 6= 0 and ˜λ_nk = 0 (k > n) for all n, k ∈ N. Further, for any sequence x = (x_k) we define the sequence y(λ) ={y_k(λ)}as the ˜Λ-transform ofx, i.e.,y(λ) = ˜Λ(x) and so we have that

y(λ) =

k

X

j=0 k

X

i=j

(−1)^i−j m i−j

! λ_i−λi−1

λk

!

x_j (3)

for k ∈N. Here and in what follows, the summation running from 0 tok−1 is equal to zero whenk = 0.

Theorem 2.1 `_∞(∆^m_λ), c₀(∆^m_λ) and c(∆^m_λ) are BK-spaces with the norm kxk_(`_∞₎

Λ˜ =Λ˜n(x)

∞ = sup

n

Λ˜n(x). (4) Proof: We know that c and c₀ are BK−spaces with their natural norms from [5]. (3) holds and ˜Λ =˜λ_nkis a triangle matrix and from Theorem 4.3.12 of Wilansky [1], we derive that`∞(∆^m_λ), c₀(∆^m_λ) and c(∆^m_λ) are BK−spaces.

This completes the proof.

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Remark 2.2 The absolute property does not hold on the`∞(∆^m_λ), c₀(∆^m_λ)and c(∆^m_λ) spaces. For instance, if we take |x| = (|x_k|) we hold kxk_(`_∞₎

Λ˜ 6=

k|x|k_(`

∞)Λ˜ .Thus, the space `∞(∆^m_λ), c₀(∆^m_λ) and c(∆^m_λ) are BK-space of non- absolute type.

Theorem 2.3 The sequence spaces `∞(∆^m_λ), c₀(∆^m_λ) and c(∆^m_λ) of non- absolute type are linearly isomorphic to the spaces `∞, c0 and c, respectively, that is `∞(∆^m_λ)∼=`∞ , c₀(∆^m_λ)∼=c₀ and c(∆^m_λ)∼=c.

Proof: We only consider c₀(∆^m_λ) ∼= c₀ and others will prove similarly.

To prove the theorem we must show the existence of linear bijection operator betweenc₀(∆^m_λ) andc₀.Hence, let define the linear operator with the notation (3), fromc₀(∆^m_λ) and c₀ by x→y(λ) =T x.

Then T x = y(λ) = ˜Λ (x) ∈ c₀ for every x ∈ c₀(∆^m_λ). Also, the linearity of T is clear. Further, it is trivial that x = 0 whenever T x = 0. Hence T is injective.

Lety = (y_k)∈c₀ and define the sequence x={x(λ)} by x_k(λ) =

k

X

j=0

m+k−j −1 k−j

! _j X

i=j−1

(−1)^j−i λ_i λ_j −λj−1

y_i. (5) and we have

∆^mx_k =

k

X

i=k−1

(−1)^k−i λi

λ_k−λk−1

y_i. (6)

Thus, for everyk ∈N, we have by (2) that Λ˜_n(x) = 1

λn n

X

k=0 k

X

i=k−1

(−1)^k−iλ_iy_i = 1 λn

n

X

k=0

(λ_ky_k−λk−1yk−1) =y_n (7) This shows that ˜Λ(x) = yand sincey∈c₀,we obtain that ˜Λ(x)∈c₀.Thus we deduce thatx∈c0(∆^m_λ) and T x=y. Hence T is surjective.

Further, we have for every x∈c₀(∆^m_λ) that kT xk_c

0 =kT xk_`_∞ =ky(λ)k

`∞ =Λ(x)˜

`∞

=kxk

(c0) ˜Λ

(8) which means that c₀(∆^m_λ) andc₀ is linearly isomorphic.

3 The Inclusion Relations

Theorem 3.1 The inclusion c₀(∆^m_λ)⊂c(∆^m_λ) strictly holds.

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Proof: It is clear that c₀(∆^m_λ) ⊂ c(∆^m_λ). To show strict, consider the sequencex= (x_k) defined by x_k =k^m for all k ∈N. Then we obtain that

Λ˜n(x) = 1 λ_n

n

X

k=0

(λk−λk−1) ∆^mxk=m! (9) forn ∈N which shows that ˜Λ(x)∈c−c₀. Thus, the sequencex is in c(∆^m_λ) but not in c0(∆^m_λ). Hence the inclusion c0(∆^m_λ) ⊂ c(∆^m_λ) is strict and this completes the proof.

Theorem 3.2 The inclusion c⊂c₀(∆^m_λ) strictly holds.

Proof: Let x ∈ c. Then ˜Λ(x) ∈ c₀. This shows that x ∈ c₀(∆^m_λ). Hence, the inclusion c⊂c₀(∆^m_λ) holds. Then, consider the sequence y= (y_k) defined byy_k = √

k+ 1 for k ∈N. It is trivial that y /∈ c. On the other hand, it can easily be seen that ˜Λ(y)∈c₀ and y∈c₀(∆^m_λ).Consequently, the sequencey is inc₀(∆^m_λ) but not in c. We therefore deduce that the inclusion c⊂c₀(∆^m_λ) is strict. This concludes proof.

Theorem 3.3 The inclusion c∆^m−1_λ ⊂c(∆^m_λ) holds.

Proof: Let x∈c∆^m−1_λ . Then we have Λ˜_n(x) = 1

λ_n

n

X

k=0

(λ_k−λk−1) ∆^m−1x_k →l (k→ ∞). (10) Furthermore, we obtain that x ∈ c(∆^m_λ) from the following inequality, hence the inclusion c∆^m−1_λ ⊂c(∆^m_λ) holds.

1 λn

Pn

k=0(λ_k−λk−1) ∆^mx_k≤_λ¹

n

Pn

k=0(λ_k−λk−1) ∆^m−1x_k−l +_λ¹

n

Pn

k=0(λ_k−λk−1) ∆^m−1xk−1−l→0. (11) Theorem 3.4 The inclusion `_∞(∆^m−1_λ )⊂`_∞(∆^m_λ) strictly holds.

Proof: Let x∈`_∞(∆^m−1_λ ). Then we have

Λ˜_n(x)=

1 λ_n

n

X

k=0

(λ_k−λk−1) ∆^m−1x_k

≤K (12)

for K > 0. We obtain the following equality that x ∈ `_∞(∆^m_λ), hence the inclusion `∞(∆^m−1_λ )⊂`∞(∆^m_λ) holds.

1 λ_n

n

X

k=0

(λk−λk−1)∆^mxk

≤

1 λ_n

n

X

k=0

(λk−λk−1)∆^m−1xk

+

1 λ_n

n

X

k=0

(λk−λk−1)∆^m−1xk−1

. (13)

To show strict, we consider x = (x_k) defined by x = (k^m), then we obtain x∈`∞(∆^m_λ)−`∞(∆^m−1_λ ).

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4 The Bases for the Spaces c (∆

^m_λ

) and c

₀

(∆

^m_λ

)

If a normed sequence spaceX contains a sequence (b_n) with the property that for every x∈X there is a unique sequence (α_n) of scalars such that

limn kx−(α₀b₀+α₁b₁ +...+α_nb_n)k= 0. (14) Then (bn) is called a Schauder basis (or briefly basis) forX.The series^Pαkbk

which has the sum x is then called the expansion of x with respect to (b_n) , and written as x=^P

k

α_kb_k.

Theorem 4.1 Define the sequence b^(k)(λ, m) = ⁿb^(k)_n (λ, m)^o^∞

k=0 for every fixed k, m∈N and by

b^(k)_n (λ, m) =











m+n−k−1 n−k

!

λk

λ_k−λk−1 − ^m+n−k−2

n−k−1

!

λk

λ_k+1−λ_k, n > k

λ_k

λk−λ_k−1, n=k

0, n < k

. (15)

Then, the sequenceⁿb^(k)_n (λ, m)^o^∞

k=0 is a basis for the space c₀(∆^m_λ) and every x∈c₀(∆^m_λ) has a unique representation of the form

x=^X

k

α_k(λ)b^(k)(λ, m) (16)

whereα_k(λ) = ˜Λ_k(x) for all k ∈N.

Theorem 4.2 The sequenceⁿb, b⁽⁰⁾(λ, m), b⁽¹⁾(λ, m), ...^ois a basis for the space c(∆^m_λ) and every x∈c(∆^m_λ) has a unique representation of the form

x=lb+^X

k

[αk(λ)−l]b^(k)(λ, m) ; (17) where α_k(λ) = ˜Λ_k(x) for all k ∈N, the sequence b = (b_k) is defined by

b_k=

k

X

j=0

m+k−j−1 k−j

!

. (18)

Corollary 4.3 The difference sequence spacesc(∆^m_λ)andc₀(∆^m_λ)are seper- able.

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5 The α−, β− and γ −Duals of the Spaces c (∆

^m_λ

) and c

₀

(∆

^m_λ

)

In this section, we introduce and prove the theorems determining theα−, β−

and γ− duals of the difference sequence spaces c(∆^m_λ) and c0(∆^m_λ) of non- absolute type.

For arbitrary sequence spaces X and Y ,the set M(X, Y) defined by M(X, Y) ={a = (a_k)∈w:ax= (a_kx_k)∈Y f or all x= (x_k)∈X} (19) is called the multipier space of X and Y.

With the notation of (19); the α−, β− and γ−duals of a sequence space X,which are respectively denoted by X^α, X^β and X^γ are defined by

X^α =M(X, `₁), X^β =M(X, cs) and X^γ =M(X, bs). (20) Now, we may begin with lemmas which are needed in proving theorems.

Lemma 5.1 A∈(c₀ :`₁) = (c:`₁) if and only if sup

K∈F

X

n

X

k∈K

a_nk

<∞. (21)

Lemma 5.2 A∈(c0 :c) if and only if

limn a_nk exists f or each k ∈N, (22) sup

n

X

k

|a_nk|<∞. (23) Lemma 5.3 A∈(c:c) if and only if (22) and (23) hold, and

limn

X

k

a_nk exists. (24)

Lemma 5.4 A∈(c0 :`∞) = (c:`∞) if and only if (23) holds.

Lemma 5.5 A∈(`∞:c) if and only if (22) holds and

n→∞lim

X

k

|a_nk|=^X

k

|α_k|. (25) Theorem 5.6 The α−dual of the space c₀(∆^m_λ) and c(∆^m_λ) is the set

b^λ₁ =







a= (a_k)∈w: sup

K∈F

X

n

X

k∈K

b_nk(λ, m)

<∞







; (26)

where the matrixB^λ =b^λm_nk is defined via the sequence a= (a_k) by

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b^(k)_n (λ, m) =











"

m+n−k−1 n−k

!

λk

λ_k−λk−1 − ^m+n−k−2

n−k−1

!

λk

λ_k+1−λ_k

#

a_n, n > k

λn

λn−λn−1a_n, n =k

0, n < k

.

(27) Proof: Let a= (a_k)∈w. Then, we obtain the equality

a_kx_k =

n

X

k=0

m+n−k−1 n−k

! _k X

j=k−1

(−1)^k−j λ_j

λ_k−λ_k−1y_j =B_n^λ(y), (n ∈N). (28) Thus, we observe by (28) that ax= (a_kx_k)∈`₁ whenever x= (x_k)∈c₀(∆^m_λ) orc(∆^m_λ) if and only ifB^λy∈`₁ whenever y= (y_k)∈c₀ orc.This means that the the sequencea = (a_k) is in the α−dual of the spaces c₀(∆^m_λ) or c(∆^m_λ) if and only if B^λ ∈ (c₀ :`₁) = (c:`₁). We therefore obtain by Lemma 5.1 with B^λ instead ofA that a ∈ {c₀(∆^m_λ)}^α ={c(∆^m_λ)}^α if and only if

sup

K∈F

X

n

X

k∈K

b_nk(λ, m)

<∞. (29)

Which leads us to the consequence that {c₀(∆^m_λ)}^α = {c(∆^m_λ)}^α = b^λ₁. This concludes proof.

Theorem 5.7 Define the sets b^λ₂ =







a= (a_k)∈w:

∞

X

j=k

m+n−j−1 n−j

!

a_j exists f or each k ∈N.







(30)

b^λ₃ =

(

a= (a_k)∈w: sup

n∈N n−1

X

k=0

|g_k(n)|<∞.

)

(31) b^λ₄ =

(

a= (a_k)∈w: sup

n∈N

λ_n λ_n−λn−1

a_n

<∞.

)

(32)

b^λ₅ =







a= (ak)∈w: lim

n→∞

n

X

k=0 k

X

j=0

m+k−j−1 k−j

!

ak exists.







(33) b^λ₆ =

(

a= (a_k)∈w: lim

n→∞

X

k

t^λ_nk=^X

k

lim

n→∞t^λ_nk

)

(34) where the matriceT^λ =t^λ_nk is defined as follow:

t^λ_nk =







a_k(n), k < n

λn

λn−λn−1a_n, k =n 0, k > n

(35)

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for all k, n∈Nand the a_k(n) is defined by a_k(n) =λ_k





1 λ_k−λk−1

n

X

j=k

m+j−k−1 j−k

!

a_j − 1 λ_k+1−λ_k

n

X

j=k

m+j−k−2 j−k−1

!

a_j



y_k (36) for k < n.Then {c₀(∆^m_λ)}^β =b^λ₂ ∩b^λ₃ ∩b^λ₄, {c(∆^m_λ)}^β =b^λ₂ ∩b^λ₃ ∩b^λ₄ ∩b^λ₅ and {`∞(∆^m_λ)}^β =b^λ₂ ∩b^λ₄ ∩b^λ₆.

Proof: We have from (5)

n

X

k=0

a_kx_k =

n

X

k=0





k

X

j=0

m+k−j−1 k−j

! _j X

i=j−1

(−1)^j−i λi

λ_j −λj−1

y_i



a_k

=

n−1

X

k=0

λk







n

P

j=k

m+j−k−1 j−k

!

a_j λ_k−λk−1

−

n

P

j=k+1

m+j−k−2 j−k−1

!

a_j λ_k+1−λ_k







yk+ a_nλ_n λ_n−λn−1

yn

=

n−1

X

k=0

a_k(n)y_k+ a_nλ_n λ_n−λn−1

y_n =T^λy

n; (n ∈N).

Then we derive thatax= (a_kx_k)∈cswheneverx= (x_k)∈c₀(∆^m_λ) if and only if T^λy ∈c whenever y = (y_k)∈ c₀. This means that a = (a_k)∈ {c₀(∆^m_λ)}^β if and only ifT^λ ∈(c₀ :c). Therefore, by using Lemma 5.2, we obtain

∞

X

j=k

m+k−j−1 k−j

!

a_j exists f or each k∈N, (37)

sup

n∈N n−1

X

k=0

|a_k(n)|<∞ (38) and

sup

k∈N n−1

X

k=0

λ_k λ_k−λk−1

a_k

<∞. (39)

Hence we conclude that {c₀(∆^m_λ)}^β =b^λ₂ ∩b^λ₃ ∩b^λ₄.

Theorem 5.8 {c₀(∆^m_λ)}^γ ={c(∆^m_λ)}^γ ={`_∞(∆^m_λ)}^γ =b^λ₃ ∩b^λ₄.

Proof: It can be proved similalry as the proof of the Theorem 5.7 with Lemma 5.4 instead of Lemma 5.2.

Acknowledgements: We thank the anonymous referees for their com- ments and suggestions that improved the presentation of this paper.

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