# Generalized Difference Spaces of Non-Absolute Type of Convergent and Null Sequences

Authors: Abdulcabbar Sönmez; Feyzi Başar
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The aim of the present paper is to introduce the spacesc0λ(B) and cλ(B) of generalized difference sequences which generalize the paper due to Mursaleen and Noman (2010). These spaces are the BK-spaces of non-absolute type and norm isomorphic to the spaces c0 and c, respectively. Furthermore, we derive some inclusion relations determine the α-, β-, and γ-duals of those spaces, and construct their Schauder bases. Finally, we characterize some matrix classes from the spaces c0λ(B), and cλ(B) to the spaces ℓp, c0, and c.


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## Body

## 1. Introduction

Byω, we denote the space of all complex valued sequences. Any vector subspace of ω is called a sequence space. A sequence space E with a linear topology is called a K-space provided each of the maps pi:E→ℂ defined by pi(x)=xi is continuous for all i∈ℕ, where ℂ denotes the complex field and ℕ={0,1,2,…}. A K-space is called an FK-space provided E is a complete linear metric space. An FK-space whose topology is normable is called a BK-space (see [1, pages 272-273]) which contains ϕ, the set of all finitely nonzero sequences. We write ℓ∞, c and c0 for the classical sequence spaces of all bounded, convergent, and null sequences, respectively, which are BK-spaces with the usual sup-norm defined by ∥x∥∞=sup|xk|, where, here and in the sequel, the supremum is taken over all k∈ℕ. Also by ℓ1 and ℓp, we denote the spaces of all absolutely and p-absolutely convergent series, respectively, which are BK-spaces with the usual norm defined by ∥x∥p=(∑k‍|xk|p)1/p, where 1≤p<∞. For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. Also by bs and cs, we denote the spaces of all bounded and convergent series, respectively.LetX and Y be two sequence spaces, and let A=(ank) be an infinite matrix of complex numbers ank, where n,k∈ℕ. Then, we say that A defines a matrix mapping from X into Y and we denote it by writing A:X→Y, if for every sequence x=(xk)∈X the sequence Ax={An(x)}, A-transform of x, exists and is in Y, where
(1.1)An(x):=∑kankxk,∀n∈ℕ.
By (X:Y), we denote the class of all infinite matrices A=(ank) such that A:X→Y. Thus A∈(X:Y) if and only if the series on the right side of (1.1) converges for each n∈ℕ and every x∈X, and Ax∈Y for all x∈X. A sequence x∈ω is said to be A-summable to l if Ax converges to l, which is called the A-limit of x.ThedomainXA of an infinite matrix A in a sequence space X is defined by
(1.2)XA:={x=(xk)∈ω:Ax∈X}.
We denote the collection of all finite subsets of ℕ by ℱ. Also, we write e(k) for the sequence whose only nonzero term is a 1 in the kth place for each k∈ℕ.The approach of constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by several authors, for example, [2–14]. They introduced the sequence spaces (ℓ∞)Nq and cNq in [14], (ℓp)C1=Xp and (ℓ∞)C1=X∞ in [10], μG=Z(u,v;μ) in [9], (ℓ∞)Rt=r∞t, cRt=rct and (c0)Rt=r0t in [8], (ℓp)Rt=rpt in [2], (c0)Et=e0r and cEr=ecr in [3], (ℓp)Er=epr and (ℓ∞)Er=e∞r in [4], (c0)Ar=a0r and cAr=acr in [5], [c0(u,p)]Ar=a0r(u,p) and [c(u,p)]Ar=acr(u,p) in [6], (ℓp)Ar=apr and (ℓ∞)Ar=a∞r in [7] and (c0)C1=c~0 and cC1=c~ in [11], νB(r,s,t)=ν(B) in [12], and fB(r,s,t)=f(B) in [13]; where, Nq, C1, Rt, and Er denote the Nörlund, Cesáro, Riesz, and Euler means, respectively, Ar, G, and B(r,s,t) are, respectively, defined in [5, 9, 12], μ∈{c0,c,ℓp}, ν∈{ℓ∞,c,c0,ℓp} and 1≤p<∞. Also c0(u,p) and c(u,p) denote the sequence spaces generated from the Maddox’s spaces c0(p) and c(p) by Başarir [15]. In the present paper, following [2–14], we introduce the difference sequence spaces c0λ(B) and cλ(B) of non-absolute type and derive some related results. We also establish some inclusion relations. Furthermore, we determine the α-, β-, and γ-duals of those spaces and construct their bases. Finally, we characterize some classes of infinite matrices concerning the spaces c0λ(B) and cλ(B).The rest of this paper is organized, as follows.In Section2, the BK-spaces c0λ(B) and cλ(B) of generalized difference sequences are introduced. Section 3 is devoted to inclusion relations concerning with the spaces c0λ(B) and cλ(B). In Sections 4 and 5, the Schauder bases of the spaces c0λ(B) and cλ(B) are given and the α-, β-, and γ-duals of the generalized difference sequence spaces c0λ(B) and cλ(B) of non-absolute type are determined, respectively. In Section 6, the classes (cλ(B):ℓp), (c0λ(B):ℓp), (cλ(B):c), (cλ(B):c0), (c0λ(B):c), and (c0λ(B):c0) of matrix transformations are characterized, where 1≤p≤∞. Also, by means of a given basic lemma, the characterizations of some other classes involving the Euler, difference, Riesz, and Cesàro sequence spaces are derived. In the final section of the paper, we note the significance of the present results in the literature related with difference sequence spaces and record some further suggestions.


## 2. The Difference Sequence Spacesc0λ(B) and cλ(B) of Non-Absolute Type

The difference sequence spaces have been studied by several authors in different ways (see e.g. [12, 16–21]). In the present section, we introduce the spaces c0λ(Δ), and cλ(Δ), and show that these spaces are BK-spaces of non-absolute type which are norm isomorphic to the spaces c0 and c, respectively.We assume throughout thatλ=(λk)k=0∞ is a strictly increasing sequence of positive reals tending to ∞, that is,
(2.1)0<λ0<λ1<⋯,limk→∞λk=∞.Recently, Mursaleen and Noman [22] studied the sequence spaces c0λ and cλ of non-absolute type, and later they introduced the difference sequence spaces c0λ(Δ) and cλ(Δ) in [21] of non-absolute type as follows:
(2.2)c0λ(Δ)={x=(xk)∈ω:limn→∞1λn∑k=0n‍(λk-λk-1)(xk-xk-1)=0},cλ(Δ)={x=(xk)∈ω:limn→∞1λn∑k=0n‍(λk-λk-1)(xk-xk-1)exists}.
Here and after, we use the convention that any term with a negative subscript is equal to zero, for example, λ-1=0 and x-1=0. With the notation of (1.2) we can redefine the spaces c0λ(Δ) and cλ(Δ) by
(2.3)c0λ(Δ)=(c0λ)Δ,cλ(Δ)=(cλ)Δ,
where Δ denotes the band matrix representing the difference operator, that is, Δx=(xk-xk-1)∈ω for x=(xk)∈ω.Letr and s be nonzero real numbers and define the generalized difference matrix B(r,s)={bnk(r,s)} by
(2.4)bnk(r,s):={r,k=n,s,k=n-1,0,otherwise,
for all k,n∈ℕ. The B(r,s)-transform of a sequence x=(xk) is
(2.5)B(r,s)k(x)=rxk+sxk-1,∀k∈ℕ.
We note that the matrix B(r,s) can be reduced to the difference matrices Δ in case r=1 and s=-1. So, the results related to the matrix domain of the matrix B(r,s) are more general and more comprehensive than the consequences of the matrices domain of Δ and include them.Now, following Başar and Altay [18] and Aydın and Başar [17], we proceed slightly differently to Kızmaz [19] and the other authors following him and employ a technique of obtaining a new sequence space by means of the matrix domain of a triangle limitation method.We thus introduce the difference sequence spacesc0λ(B) and cλ(B), which are the generalization of the spaces c0λ(Δ) and cλ(Δ) introduced by Mursaleen and Noman [21], as follows:
(2.6)c0λ(B)={x=(xk)∈ω:limn→∞1λn∑k=0n‍(λk-λk-1)(rxk+sxk-1)=0},cλ(B)={x=(xk)∈ω:limn→∞1λn∑k=0n‍(λk-λk-1)(rxk+sxk-1)exists}.
With the notation of (1.2), we can redefine the spaces c0λ(B) and cλ(B) as
(2.7)c0λ(B)=(c0λ)B,cλ(B)=(cλ)B,
where B denotes the generalized difference matrix B(r,s)={bnk(r,s)} defined by (2.4).It is immediate by (2.7) that the sets c0λ(B) and cλ(B) are linear spaces with coordinatewise addition and scalar multiplication, that is, c0λ(B) and cλ(B) are the sequence spaces of generalized differences.On the other hand, we define the triangle matrixΛ^=(λ^nk) by
(2.8)λ^nk:={r(λk-λk-1)+s(λk+1-λk)λn,k<n,r(λn-λn-1)λn,k=n,0,k>n
for all n,k∈ℕ. With a direct calculation we derive the equality
(2.9)Λ^n(x)=1λn∑k=0n‍(λk-λk-1)(rxk+sxk-1),∀n∈ℕ
and every x=(xk)∈ω which leads us together with (1.2) to the fact that
(2.10)c0λ(B)=(c0)Λ^,cλ(B)=cΛ^.Further, for any sequencex=(xk) we define the sequence y(λ)={yk(λ)} which will be frequently used as the Λ^-transform of x, that is, y(λ)=Λ^(x) and so we have
(2.11)yk(λ)=∑j=0k-1‍r(λj-λj-1)+s(λj+1-λj)λkxj+rλk-λk-1λkxk,∀k∈ℕ,
Where, here and in what follows, the summation running from 0 to k-1 is equal to zero when k=0.Moreover, it is clear by (2.9) that the relation (2.11) can be written as follows:
(2.12)yk(λ)=1λk∑j=0k‍(λj-λj-1)(rxj+sxj-1),∀k∈ℕ.
We assume throughout that the sequences x=(xk) and y=(yk) are connected by the relation (2.11).Now, we may begin with the following theorem which is essential in the text.Theorem 2.1.
The difference sequence spacesc0λ(B) and cλ(B) are BK-spaces with the norm ∥x∥c0λ(B)=∥x∥cλ(B)=∥Λ^(x)∥∞, that is,
(2.13)∥x∥c0λ(B)=∥x∥cλ(B)=supn∈ℕ|Λ^n(x)|.Proof.
Since (2.10) holds and c0 and c are BK-spaces with respect to their natural norms (see [23, pages 217-218]) and the matrix Λ^ is a triangle, Theorem 4.3.12 of Wilansky [24, page 63] gives the fact that c0λ(B) and cλ(B) are BK-spaces with the given norms. This completes the proof.Remark 2.2.
One can easily check that the absolute property does not hold on the spacesc0λ(B) and cλ(B), that is, ∥x∥c0λ(B)≠∥|x|∥c0λ(B) and ∥x∥cλ(B)≠∥|x|∥cλ(B) for at least one sequence in the spaces c0λ(B) and cλ(B), and this shows that c0λ(B) and cλ(B) are the sequence spaces of non-absolute type, where |x|=(|xk|).Now, we give the final theorem of this section.Theorem 2.3.
The sequence spacesc0λ(B) and cλ(B) of non-absolute type are norm isomorphic to the spaces c0 and c, respectively, that is, c0λ(B)≅c0 and cλ(B)≅c.Proof.
To prove this, we should show the existence of a linear bijection between the spacesc0λ(B) and c0. Consider the transformation T defined, with the notation of (2.11), from c0λ(B) to c0 by x↦y(λ). Then, Tx=y(λ)=Λ^(x)∈c0 for every x∈c0λ(B) and the linearity of T is clear. Further, it is trivial that x=θ whenever Tx=θ and hence T is injective.
Furthermore, lety=(yk)∈c0 and define the sequence x={xk(λ)} by
(2.14)xk(λ):=1r∑j=0k‍(-sr)k-j∑i=j-1j‍(-1)j-iλiλj-λj-1yi,∀k∈ℕ.
Then, we obtain
(2.15)rxk(λ)+sxk-1(λ)=∑i=k-1k‍(-1)k-iλiλk-λk-1yi,∀k∈ℕ.
Hence, for every n∈ℕ, we get by (2.9)
(2.16)Λ^n(x)=1λn∑k=0n‍(λk-λk-1)(rxk+sxk-1)=1λn∑k=0n‍∑i=k-1k‍(-1)k-iλiyi=yn.
This shows that Λ^(x)=y and since y∈c0, we conclude that Λ^(x)∈c0. Thus, we deduce that x∈c0λ(B) and Tx=y. Hence T is surjective.
Moreover, one can easily see for everyx∈c0λ(B) that
(2.17)∥Tx∥∞=∥y(λ)∥∞=∥Λ^(x)∥∞=∥x∥c0λ(B)
which means that T is norm preserving. Consequently T is a linear bijection which show that the spaces c0λ(B) and c0 are linearly isomorphic.
It is clear that if the spacesc0λ(B) and c0 are replaced by the spaces cλ(B) and c, respectively, then we obtain the fact that cλ(B)≅c. This completes the proof.


## 3. The Inclusion Relations

In the present section, we establish some inclusion relations concerning with the spacesc0λ(B) and cλ(B). We may begin with the following theorem.Theorem 3.1.
The inclusionc0λ(B)⊂cλ(B) strictly holds.Proof.
It is obvious that the inclusionc0λ(B)⊂cλ(B) holds. Further to show that this inclusion is strict, consider the sequence x=(xk) defined by xk=∑j=0k‍(-s/r)j/r for all k∈ℕ. Then, we obtain by (2.9) that
(3.1)Λ^n(x)=1λn∑k=0n‍(λk-λk-1)∀n∈ℕ
which shows that Λ^(x)=e and hence Λ^(x)∈c∖c0, where e=(1,1,1,…). Thus, the sequence x is in cλ(B) but not in c0λ(B). Hence, the inclusion c0λ(B)⊂cλ(B) is strict and this completes the proof.Theorem 3.2.
Ifs+r=0, then the inclusion c⊂c0λ(B) strictly holds.Proof.
Suppose thats+r=0 and x∈c. Then B(r,s)x=(rxk+sxk-1)∈c0 and hence B(r,s)x∈c0λ, since the inclusion c0⊂c0λ. This shows that x∈c0λ(B). Consequently, the inclusion c⊂c0λ(B) holds. Further consider the sequence y=(yk) defined by yk=k+1 for all k∈ℕ. Then, it is trivial that y∉c. On the other hand, it can easily seen that B(r,s)y∈c0. Hence, B(r,s)y∈c0λ which means that y∈c0λ(B). Thus, the sequence y is in c0λ(B) but not in c. We therefore deduce that the inclusion c⊂c0λ(B) is strict. This completes the proof.On the other hand, we recall that ifA∈(c:c) and B∈(c:c), then AB∈(c:c), namely, Λ^=(λ^nk) is stronger than the ordinary convergence, hence we have the followingCorollary 3.3.
The inclusionsc0⊂c0λ(B) and c⊂cλ(B) strictly hold.Further, it is obvious that the sequencey, defined in the proof of Theorem 3.2, is in c0λ(B) but not in ℓ∞. This leads us to the following result.Corollary 3.4.
Although the spacesℓ∞ and c0λ(B) overlap, the space ℓ∞ does not include the space c0λ(B).Now, to prove the next theorem, we need the following lemma [24, page 4].Lemma 3.5.
A
∈
(
ℓ
∞
:
c
0
) if and only if limn→∞∑k‍|ank|=0.Theorem 3.6.
The inclusionℓ∞⊂c0λ(B) strictly holds if and only if z∈c0λ, where the sequence z=(zk) is defined by
(3.2)zk=|r(λk-λk-1)+s(λk+1-λk)λk-λk-1|,∀k∈ℕ.Proof.
Suppose that the inclusionℓ∞⊂c0λ(B) holds. Then we obtain that Λ^(x)∈c0 for every x∈ℓ∞ and hence the matrix Λ^=(λ^nk) is in the class (ℓ∞:c0). Thus it follows by Lemma 3.5 that
(3.3)limn→∞∑k‍|λ^nk|=0.
Now, by taking into account the definition of the matrix Λ^=(λ^nk) given by (2.8), we have for every n∈ℕ that
(3.4)∑k‍|λ^nk|=1λn∑k=0n-1‍|r(λk-λk-1)+s(λk+1-λk)|+|r|λn-λn-1λn.
Thus, the condition (3.3) implies both
(3.5)limn→∞|r|λn-λn-1λn=0,(3.6)limn→∞1λn∑k=0n-1‍|r(λk-λk-1)+s(λk+1-λk)|=0.
Now we have for every n≥1 that
(3.7)1λn∑k=0n-1‍|r(λk-λk-1)+s(λk+1-λk)|=λn-1λn[1λn-1∑k=0n-1‍(λk-λk-1)zk]
and since limn→∞(λn-1/λn)=1 by (3.5), we obtain by (3.6) that
(3.8)limn→∞1λn-1∑k=0n-1‍(λk-λk-1)zk=0
which shows that z=(zk)∈c0λ.
Conversely, suppose thatz=(zk)∈c0λ. Then we have (3.8). Further, for every n≥1, we derive that
(3.9)1λn∑k=0n-1‍|r(λk-λk-1)+s(λk+1-λk)|=1λn∑k=0n-1‍(λk-λk-1)zk≤1λn-1∑k=0n-1‍(λk-λk-1)zk.
Then, (3.9) and (3.8) together imply that (3.6) holds. On the other hand, we have for every n≥1 that
(3.10)|rλn-1+s(λn-λ0)λn|=|1λn∑k=0n-1‍r(λk-λk-1)+s(λk+1-λk)|≤1λn∑k=0n-1‍|r(λk-λk-1)+s(λk+1-λk)|.
Therefore, it follows by (3.6) that limn→∞[rλn-1+s(λn-λ0)]/λn=0. Particularly, if we take r=1 and s=-1, then we have limn→∞[λn-λn-1-λ0]/λn=0 which shows that (3.5) holds. Thus, we deduce by the relation (3.4) that (3.3) holds. This leads us with Lemma 3.5 to the consequence that Λ^∈(ℓ∞:c0). Hence, the inclusion ℓ∞⊂c0λ(B) holds and is strict by Corollary 3.4. This completes the proof.


## 4.  The Bases for the Spacesc0λ(B) and cλ(B)

In the present section, we give two sequences of the points of the spacesc0λ(B) and cλ(B) which form the bases for those spaces.If a normed sequence spaceX contains a sequence (bn) with the property that for every x∈X there is a unique sequence of scalars (αn) such that
(4.1)limn→∞∥x-(α0b0+α1b1+⋯+αnbn)∥=0
then (bn) is called a Schauder basis (or briefly basis) for X. The series ∑k‍αkbk which has the sum x is then called the expansion of x with respect to (bn) and is written as x=∑k‍αkbk.Now, since the transformationT defined from c0λ(B) to c0 in the proof of Theorem 2.3 is an isomorphism, the inverse image of the basis {e(k)}k=0∞ of the space c0 is the basis for the new space c0λ(B). Therefore, we have the following.Theorem 4.1.
Letαk(λ)=Λ^k(x) for all k∈ℕ and l=limk→∞Λ^k(x). Define the sequence b(k)(λ)={bn(k)(λ)}k=0∞ for every fixed k∈ℕ by
(4.2)bn(k)(λ):={(-sr)n-k[λkr(λk-λk-1)+λks(λk+1-λk)],k<n,1rλk(λk-λk-1),k=n,0,k>n.
Then, the following statements hold. (a)
The sequence{b(k)(λ)}k=0∞ is a basis for the space c0λ(B) and any x∈c0λ(B) has a unique representation of the form x=∑‍kαk(λ)b(k)(λ).(b)
The sequence{b,b(0)(λ),b(1)(λ),…} is a basis for the space cλ(B) and any x∈cλ(B) has a unique representation of the form x=lb+∑‍k[αk(λ)-l]b(k)(λ), where b=(bk)={∑j=0k‍(-s/r)j/r}k=0∞.Finally, it easily follows from Theorem2.1 that c0λ(B) and cλ(B) are the Banach spaces with their natural norms. Then by Theorem 4.1 we obtain the following.Corollary 4.2.
The difference sequence spacesc0λ(B) and cλ(B) are seprable.


## 5. Theα-, β-, and γ-Duals of the Spaces c0λ(B) and cλ(B)

In this section, we state and prove the theorems determining theα-, β-, and γ-duals of the generalized difference sequence spaces c0λ(B) and cλ(B) of non-absolute type.For arbitrary sequence spacesX and Y, the set M(X,Y) defined by
(5.1)M(X,Y)={a=(ak)∈ω:ax=(akxk)∈Y∀x=(xk)∈X}
is called the multiplier space of X and Y. One can easily observe for a sequence space Z with Y⊂Z and Z⊂X that the inclusions M(X,Y)⊂M(X,Z) and M(X,Y)⊂M(Z,Y) hold, respectively.With the notation of (5.1), the α-, β-, and γ-duals of a sequence space X, which are respectively, denoted by Xα, Xβ, and Xγ, are defined by
(5.2)Xα=M(X,ℓ1),Xβ=M(X,cs),Xγ=M(X,bs).
It is clear that Xα⊂Xβ⊂Xγ. Also it can be obviously seen that the inclusions Xα⊂Yα, Xβ⊂Yβ, and Xγ⊂Yγ hold whenever Y⊂X.Now, we may begin with quoting the following lemmas (see [25]) which are needed to prove Theorems 5.5 to 5.8.Lemma 5.1.
A
=
(
a
n
k
)
∈
(
c
0
:
ℓ
1
)
=
(
c
:
ℓ
1
) if and only if
(5.3)supK∈ℱ∑n‍|∑k∈K‍ank|<∞.Lemma 5.2.
A
=
(
a
n
k
)
∈
(
c
0
:
c
) if and only if
(5.4)limn→∞ank=αkforeachfixedk∈ℕ,(5.5)supn∈ℕ∑k‍|ank|<∞.Lemma 5.3.
A
=
(
a
n
k
)
∈
(
c
:
c
) if and only if (5.4) and (5.5) hold, and
(5.6)limn→∞∑k‍ankexists.Lemma 5.4.
A
=
(
a
n
k
)
∈
(
c
:
ℓ
∞
)
=
(
c
0
:
ℓ
∞
) if and only if (5.5) holds.Now, we prove the following result.Theorem 5.5.
Theα-dual of the spaces c0λ(B) and cλ(B) is the set
(5.7)h1λ={a=(an)∈ω:supK∈ℱ∑n‍|∑k∈K‍hnkλ|<∞},
where the matrix Hλ=(hnkλ) is defined via the sequence a=(an)∈ω by
(5.8)hnk(λ)={(-sr)n-k[λkr(λk-λk-1)+λks(λk+1-λk)]an,k<n,λnr(λn-λn-1)an,k=n,0,k>n
for all n,k∈ℕ.Proof.
Leta=(an)∈ω. Then, by bearing in mind the relations (2.11) and (2.14), it is immediate that the equality
(5.9)anxn=∑k=0n‍1r(-sr)n-k∑i=k-1k‍(-1)k-iλiλk-λk-1anyi=Hnλ(y)
holds for all n∈ℕ. Thus, we observe by (5.9) that ax=(anxn)∈ℓ1 whenever x=(xk)∈c0λ(B) or cλ(B) if and only if Hλy∈ℓ1 whenever y=(yk)∈c0 or c. This means that the sequence a=(an)∈{c0λ(B)}α or a=(an)∈{cλ(B)}α if and only if Hλ∈(c0:ℓ1)=(c:ℓ1). We therefore obtain by Lemma 5.4 with Hλ instead of A that a=(an)∈{c0λ(B)}α={cλ(B)}α if and only if
(5.10)supK∈ℱ∑n‍|∑k∈K‍hnkλ|<∞
which leads us to the consequence that {c0λ(B)}α={cλ(B)}α=h1λ. This completes the proof.Theorem 5.6.
Define the setsh2λ, h3λ, h4λ, and h5λ, as follows:
(5.11)h2λ={a=(ak)∈ω:∑j=k∞‍(-sr)n-jajexistsforeachk∈ℕ},h3λ={a=(ak)∈ω:supn∈ℕ∑k=0n-1‍|a^k(n)|<∞},h4λ={a=(ak)∈ω:supn∈ℕ|1rλk(λk-λk-1)ak|<∞},h5λ={a=(ak)∈ω:∑k‍1r∑j=0k‍(-sr)jakconverges},wherea^k(n)=λk[akr(λk-λk-1)+(1r(λk-λk-1)+1s(λk+1-λk))∑j=k+1n‍(-sr)n-jaj]fork<n.
Then {c0λ(B)}β=h2λ∩h3λ∩h4λ and {cλ(B)}β=h2λ∩h3λ∩h4λ∩h5λ.Proof.
Consider the equality(5.12)∑k=0n‍akxk=∑k=0n‍{∑j=0k‍1r(-sr)k-j[∑i=j-1j‍(-1)j-iλiλj-λj-1yi]}ak=∑k=0n-1‍λk[akr(λk-λk-1)+(1r(λk-λk-1)+1s(λk+1-λk))∑j=k+1n‍(-sr)n-jaj]yk+1rλn(λn-λn-1)anyn=∑k=0n-1‍a^k(n)yk+1rλn(λn-λn-1)anyn=Tnλ(y)∀n∈ℕ,
where the matrix Tλ=(tnkλ) is defined by
(5.13)tnkλ:={a^k(n),k<n,1rλn(λn-λn-1)an,k=n,0,k>n
for all k,n∈ℕ. Then, we deduce by (5.12) that ax=(akxk)∈cs whenever x=(xk)∈c0λ(B) if and only if Tλy∈c whenever y=(yk)∈c0. This means that a=(ak)∈{c0λ(B)}β if and only if Tλ∈(c0:c). Therefore, by using Lemma 5.2, we derive from (5.4) and (5.5) that
(5.14)∑j=k+1∞‍(-sr)n-jajexistsforeachk∈ℕ,(5.15)supn∈ℕ∑k=0n-1‍|a^k(n)|<∞,supn∈ℕ|1rλn(λn-λn-1)an|<∞.
Therefore, we conclude that {c0λ(B)}β=h2λ∩h3λ∩h4λ.
Similarly, we deduce from Lemma5.3 with (5.12) that a=(ak)∈{cλ(B)}β if and only if Tλ∈(c:c). Therefore, we derive from (5.4) and (5.5) that (5.14), (5.15) hold.
Further, with a simple calculation one can easily see that the equality(5.16)∑k=0n‍1r∑j=0k‍(-sr)jak=∑k=0n-1‍a^k(n)+1rλn(λn-λn-1)an=∑k‍tnkλ
holds for all n∈ℕ. Consequently, from (5.6) we obtain that
(5.17){1r∑j=0k‍(-sr)jak}∈cs.
Hence, we deduce that {cλ(B)}β=h2λ∩h3λ∩h4λ∩h5λ. This completes the proof.Remark 5.7.
We may note by combining (5.17) with the conditions (5.15) that {∑j=0k‍(-s/r)jak/r}∈bs for every sequence a=(ak)∈{c0λ(B)}β.Finally, we close this section with the following theorem which determines theγ-dual of the spaces c0λ(B) and cλ(B):Theorem 5.8.
Theγ-duals of the spaces c0λ(B) and cλ(B) are the set h3λ∩h4λ.Proof.
The proof of this result follows the same lines that in the proof of Theorem5.6 using Lemma 5.4 instead of Lemma 5.2.


## 6. Certain Matrix Mappings Related to the Spacesc0λ(B) and cλ(B)

In this final section, we characterize the matrix classes(cλ(B):ℓp), (c0λ(B):ℓp), (cλ(B):c), (cλ(B):c0), (c0λ(B):c), and (c0λ(B):c0), where 1≤p≤∞. Also, by means of a given basic lemma, we derive the characterizations of some other classes involving the Euler, difference, Riesz, and Cesàro sequence spaces.For an infinite matrixA=(ank), we write for brevity that
(6.1)a^nk(m)=λk[ankr(λk-λk-1)+(1r(λk-λk-1)+1s(λk+1-λk))∑j=k+1m‍(-sr)n-janj]ifk<m,a^nk=λk[ankr(λk-λk-1)+(1r(λk-λk-1)+1s(λk+1-λk))∑j=k+1∞‍(-sr)n-janj]
for all k,m,n∈ℕ provided the convergence of the series.The following lemmas will be needed in proving our main results.Lemma 6.1 (see [24, page 57]).
The matrix mappings between the BK-spaces are continuous.Lemma 6.2 (see [25, pages 7-8]).
A
=
(
a
n
k
)
∈
(
c
:
ℓ
p
) if and only if
(6.2)supK∈ℱ∑n‍|∑k∈K‍ank|p<∞,(1≤p<∞).Lemma 6.3 (see [25, page 5]).
A
=
(
a
n
k
)
∈
(
c
:
c
0
) if and only if (5.5) holds and
(6.3)limn→∞ank=0∀k∈ℕ,limn→∞∑k‍ank=0.Lemma 6.4 (see [25, page 5]).
A
=
(
a
n
k
)
∈
(
c
0
:
c
0
) if and only if (5.5) and (6.3) hold.Now, we give the following results on the matrix transformations.Theorem 6.5.
LetA=(ank) be an infinite matrix over the complex field. Then, the following statements hold. (i) Let(1≤p<∞). Then, A∈(cλ(B):ℓp) if and only if
(6.4)∑j=k+1∞‍(-sr)n-janjexistsforeachfixedk∈ℕ,(6.5)supK∈ℱ∑n‍|∑k∈K‍a^nk|p<∞,(6.6)supm∈ℕ∑k=0m-1‍|a^nk(m)|<∞,∀n∈ℕ,(6.7){1r∑j=0k‍(-sr)jank}k=0∞∈csforeachfixedn∈ℕ,(6.8)limk→∞λkr(λk-λk-1)ank=anforeachfixedn∈ℕ,(6.9)(an)∈ℓp.(ii)A∈(cλ(B):ℓ∞) if and only if (6.7) and (6.8) hold, and
(6.10)supn∈ℕ∑k‍|a^nk|<∞,(6.11)(an)∈ℓ∞.Proof.
Suppose that the conditions (6.4)–(6.9) hold and take any x=(xk)∈cλ(B). Then, we have by Theorem 5.6 that {ank}k∈ℕ∈{cλ(B)}β for all n∈ℕ and this implies that the A-transform of x exists. Also, it is clear that the associated sequence y=(yk) is in the space c and hence yk→l as k→∞ for some suitable l. Further, it follows by combining Lemma 6.2 with (6.5) that the matrix A^=(a^nk) is in the class (c:ℓp), where 1≤p<∞.
Let us now consider the following equality derived by using the relation (2.11) from the mth partial sum of the series ∑k‍ankxk:
(6.12)∑k=0m‍ankxk=∑k=0m-1‍a^nk(m)yk+λmr(λm-λm-1)anmym,∀n,m∈ℕ.
Then, since y∈c and A^∈(c:ℓp), the series ∑k‍a^nkyk converges for every n∈ℕ. Furthermore, it follows by (6.4) that the series ∑j=k∞‍(-s/r)n-janj converges for all n,k∈ℕ and hence a^nk(m)→a^nk as m→∞. Therefore, if we pass to limit in (6.12) as m→∞ then we obtain by (6.8) that
(6.13)∑k‍ankxk=∑k‍a^nkyk+lan,∀n∈ℕ
which can be written as follows:
(6.14)An(x)=A^n(y)+lan,∀n∈ℕ.
This yields by taking p-norm that
(6.15)∥Ax∥p≤∥A^y∥p+|l|∥(an)∥p<∞
which leads us to the consequence that Ax∈ℓp. Hence, A∈(cλ(B):ℓp).
Conversely, suppose thatA∈(cλ(B):ℓp), where 1≤p<∞. Then {ank}k∈ℕ∈{cλ(B)}β for all n∈ℕ which implies with Theorem 5.6 that the conditions (6.6) and (6.7) are necessary.
On the other hand, sincecλ(B) and ℓp are BK-spaces, we have by Lemma 6.1 that there is a constant M>0 such that
(6.16)∥Ax∥p≤M∥x∥cλ(B)
holds for all x∈cλ(B). Now, K∈ℱ. Then, the sequence z=∑k∈K‍b(k)(λ) is in cλ(B), where the sequence b(k)(λ)={bn(k)(λ)}n∈ℕ is defined by (4.2) for every fixed k∈ℕ.
SinceΛ^(b(k)(λ))=e(k) for each fixed k∈ℕ, we have
(6.17)∥z∥cλ(B)=∥Λ^(z)∥∞=∥∑k∈K‍Λ^(b(k)(λ))∥∞=∥∑k∈K‍e(k)∥∞=1.
Furthermore, for everyn∈ℕ, we obtain by (4.2) that
(6.18)An(z)=∑k∈K‍An(b(k)(λ))=∑k∈K‍∑j‍anjbj(k)(λ)=∑k∈K‍a^nk
Hence, since the inequality (6.16) is satisfied for the sequence z∈cλ(B), we have for any K∈ℱ that
(6.19)(∑n‍|∑k∈K‍a^nk|p)1/p≤M
which shows the necessity of (6.5). Thus, it follows by Lemma 6.2 that A^=(a^nk)∈(c:ℓp).
Now, lety=(yk)∈c∖c0 and consider the sequence x=(xk) defined by (2.14) for every k∈ℕ. Then, x∈cλ(B) such that y=Λ^(x), that is, the sequences x and y are connected by the relation (2.11). Therefore, Ax and A^y exist. This leads us to the convergence of the series ∑k‍ankxk and ∑k‍a^nkyk for every n∈ℕ. We thus deduce that
(6.20)limm→∞∑k=0m-1‍a^nk(m)yk=∑k‍a^nkyk,∀n∈ℕ.
Consequently, we obtain from (6.12) as m→∞ that
(6.21)limm→∞λmr(λm-λm-1)anmymexistsforeachfixedn∈ℕ
and since y=(yk)∈c∖c0, we conclude that
(6.22)limm→∞λmr(λm-λm-1)anmexistsforeachfixedn∈ℕ
which shows the necessity of (6.8). Then relation (6.14) holds.
Finally, sinceAx∈ℓp and A^y∈ℓp, the necessity of (6.9) is immediate by (6.14). This completes the proof of Part (i) of the theorem.
Since Part (ii) can be proved by using the similar way that used in the proof of Part (i) with Lemma5.4 instead of Lemma 6.2, we leave the details to the reader.Remark 6.6.
It is clear by (6.10) that the limit
(6.23)limm→∞∑k=0m-1‍|a^nk(m)|=∑k‍|a^nk|
exists for each n∈ℕ. This just tells us that condition (6.10) implies condition (6.6).Now, we may note that(c0:ℓp)=(c:ℓp) for 1≤p≤∞, (see [25, pages 7-8]). Thus, by means of Theorem 5.6 and Lemmas 6.2 and 5.4, we immediately conclude the following theorem.Theorem 6.7.
LetA=(ank) be an infinite matrix over the complex field. Then, the following statements hold. (i) Let1≤p<∞. Then, A∈(c0λ(B):ℓp) if and only if (6.5) and (6.6) hold, and
(6.24)∑j=k∞‍(-sr)n-janjexistsforalln,k∈ℕ,{λkr(λk-λk-1)ank}k=0∞∈ℓ∞,∀n∈ℕ.(ii)A∈(c0λ(B):ℓ∞) if and only if (6.10) and (6.24) hold.Proof.
It is natural that the present theorem can be proved by the same technique used in the proof of Theorem6.5, above, and so we omit the proof.Theorem 6.8.
A
=
(
a
n
k
)
∈
(
c
λ
(
B
)
:
c
) if and only if (6.7), (6.8), and (6.10) hold, and
(6.25)limn→∞an=a,(6.26)limn→∞a^nk=αk,foreachk∈ℕ,(6.27)limn→∞∑k‍a^nk=α.Proof.
Suppose thatA satisfies the conditions (6.7), (6.8), (6.10), (6.25), (6.26), and (6.27) and take any x∈cλ(B). Then, since (6.10) implies (6.6), we have by Theorem 5.6 that {ank}k∈ℕ∈{cλ(B)}β for all n∈ℕ and hence Ax exists. We also observe from (6.10) and (6.26) that
(6.28)∑j=0k‍|αj|≤supm∈ℕ∑j‍|a^nk|<∞
holds for every k∈ℕ. This implies that (αk)∈ℓ1 and hence the series ∑k‍αk(yk-l) converges, where y=(yk)∈c is the sequence connected with x=(xk) by the relation (2.11) such that yk→l as k→∞. Further it is obvious by combining Lemma 5.3 with the condition (6.10), (6.26), and (6.27) that the matrix A^=(a^nk) is in the class (c:c).
Now reasoning as in the proof of Theorem6.5, we obtain that the relation (6.13) holds which can be written as follows:
(6.29)∑k‍ankxk=∑k‍a^nk(yk-l)+l∑k‍a^nk+lan,∀n∈ℕ.
In this situation, we see by passing to the limit in (6.29) as n→∞ that the first term on the right tends to ∑k‍αk(yk-l) by (6.10) and (6.26). The second term tends to lα by (6.27) and the last term to la by (6.25). Consequently, we have
(6.30)limn→∞An(x)=∑k‍αk(yk-l)+l(α+a),
which shows that Ax∈c, that is to say that A∈(cλ(B):c).
Conversely, Suppose thatA∈(cλ(B):c). Then, since the inclusion c⊂ℓ∞ holds, it is trivial that A∈(cλ(B):ℓ∞). Therefore, the necessity of the conditions (6.7), (6.8), and (6.10) are obvious from Theorem 6.5. Further, consider the sequence b(k)(λ)={bn(k)(λ)}n∈ℕ∈cλ(B) defined by (4.2) for every fixed k∈ℕ. Then, it is easily seen that Ab(k)(λ)={a^nk}n∈ℕ and hence {a^nk}n∈ℕ∈c for every k∈ℕ which shows the necessity of (6.26). Let z=∑k‍b(k)(λ). Then, since the linear transformation T:cλ(B)→c, defined as in the proof of Theorem 2.3 by analogy, is continuous and Λ^(b(k)(λ))=e(k) for each fixed k∈ℕ, we obtain that
(6.31)Λ^n(z)=∑k‍Λ^n(b(k)(λ))=∑k‍δnk=1foreachn∈ℕ
which shows that Λ^(z)=e∈c and hence z∈cλ(B). On the other hand, since cλ(B) and c are the BK-spaces, Lemma 6.1 implies the continuity of the matrix mapping A:cλ(B)→c. Thus, we have for every n∈ℕ that
(6.32)An(z)=∑k‍An(b(k)(λ))=∑k‍a^nk.
This shows the necessity of (6.27).
Now, it follows by (6.10), (6.26), and (6.27) with Lemma 5.3 that A^=(a^nk)∈(c:c). So by (6.7), and (6.8), relation (6.14) holds for all x∈cλ(B) and y∈c, and x and y are connected by relation (2.11), where yk→l (k→∞).
Lastly, sinceAx∈c and A^x∈c; the necessity of (6.25) is immediate by (6.14). This step concludes the proof.Theorem 6.9.
A
=
(
a
n
k
)
∈
(
c
λ
(
B
)
:
c
0
) if and only if (6.7), (6.8), and (6.10) hold, and
(6.33)limn→∞an=0,limn→∞a^nk=0foreachk∈ℕ,limn→∞∑k‍a^nk=0.Proof.
Since the present theorem can be proved by the similar way used in the proof of Theorem6.8 with Lemma 6.3 instead of Lemma 5.3, we omit the detailed proof.Theorem 6.10.
A
=
(
a
n
k
)
∈
(
c
0
λ
(
B
)
:
c
) if and only if (6.10), (6.24), and (6.26) hold.Proof.
This is similarly obtained by using Lemma5.2, Theorem 5.6, and Part (ii) of Theorem 6.7.Theorem 6.11.
A
=
(
a
n
k
)
∈
(
c
0
λ
(
B
)
:
c
0
) if and only if (6.10) and (6.24) hold, and (6.26) also holds with αk=0 for all k∈ℕ.Proof.
This is immediate by Lemma6.4, Theorems 5.6 and 6.10.


## 7. Conclusion

Letν denotes any of the classical sequence spaces ℓ∞, c, or c0. Then, ν(Δ) consisting of the sequences x=(xk) such that Δx=(xk-xk+1)∈ν is called as the difference sequence space which was introduced by Kızmaz [19]. Kızmaz [19] proved that ν(Δ) is a Banach space with the norm ∥x∥Δ=|x1|+∥Δx∥∞, where x=(xk)∈ν(Δ) and the inclusion relation ν⊂ν(Δ) strictly holds. He also determined the α-, β-, and γ-duals of the difference spaces and characterized the classes (ν(Δ):μ) and (μ:ν(Δ)) of infinite matrices, where ν,μ∈{ℓ∞,c}. Following Kızmaz [19], Sarıgöl [26] extended the difference space ν(Δ) to the space ν(Δr) defined by
(7.1)ν(Δr):={x=(xk)∈ω:Δrx={kr(xk-xk+1)}∈νforr<1}
and computed the α-, β-, and γ-duals of the space ν(Δr), where ν∈{ℓ∞,c,c0}. It is easily seen that ν(Δr)⊂ν(Δ), if 0<r<1 and ν(Δ)⊂ν(Δr), if r<0. Recently, the difference spaces bvp consisting of the sequences x=(xk) such that (xk-xk-1)∈ℓp have been studied in the case 0<p<1 by Altay and Başar [27] and in the case 1≤p≤∞ by Başar and Altay [18], Çolak et al. [28], and Malkowsky et al. [29]. Quite recently, Mursaleen and Noman have introduced the spaces cλ and c0λ of λ-convergent and λ-null sequences and nextly studied the difference spaces cλ(Δ) and c0λ(Δ) in [21, 22], respectively. Of course, there is a wide literature concerning the difference sequence spaces. By the domain of the generalized difference matrix B(r,s) in the spaces of λ-convergent and λ-null sequences we have generalized the difference spaces cλ(Δ) and c0λ(Δ) defined by Mursaleen and Noman [21]. Since the generalized difference matrix B(r,s) reduces, in the special case r=1, s=-1, to the usual difference matrix Δ; our results are more general and more comprehensive than the corresponding results of Mursaleen and Noman [21].Since the difference spaces ofλ-bounded and absolutely λp-summable sequences are not studied, the domain of both the difference matrix Δ, and the generalized difference matrix B(r,s) in those spaces are still open. So, it is meaningful to fill this gap.



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*Source: 435076-2012-11-08.xml*