# On Some New Generalized Difference Sequence Spaces of Nonabsolute Type

Authors: Osman Duyar; Serkan Demiriz; Osman Özdemir
Journal: Journal of Mathematics
 (2014)

---

## Abstract

We define a new triangle matrixW^=(wnkλ) by the composition of the matrices Λ=(λnk) and B(r,s,t). Also, we introduce the sequence spaces c0λ(B^),cλ(B^),l∞λ(B^), and lpλ(B^) by using matrix domain of the matrix W^ on the classical sequence spaces c0,c,l∞, and lp, respectively, where 1≤p<∞. Moreover, we show that the space μλ(B^) is norm isomorphic to μ for μ∈{c0,c,l∞,lp}. Furthermore, we establish some inclusion relations concerning those spaces and determine   α-, β-, and γ-duals of those spaces and construct the Schauder bases c0λ(B^),cλ(B^), and lpλ(B^). Finally, we characterize the classes (μ1λ(B^):μ2) of infinite matrices where μ1∈{c,c0,lp} and μ2∈{l∞,c,c0,lp}.


---

## Body

## 1. Preliminaries, Background, and Notation

By a sequence space, we understand a linear subspace of the spaceω of all complex sequences which contains ϕ, the set of all finitely nonzero sequences. We write l∞,c,c0, and lp for the classical sequence spaces of all bounded, convergent, null, and absolutely p-summable sequences, respectively, where 1≤p<∞. Also by bs and cs, we denote the spaces of all bounded and convergent series, respectively. We assume throughout unless stated otherwise that p,q>1 with p-1+q-1=1 and use the convention that any term with negative subscript is equal to zero. We denote throughout that the collection of all finite subsets of N by F.LetA=(ank) be an infinite matrix and X,Y two sequence spaces. Then, A defines a matrix mapping from X to Y and is denoted by A:X→Y if for every sequence x=(xk)∈X the sequence Ax={(Ax)n}n∈N, the A-transform of x, is in Y, where
(1)(Ax)n=∑k‍ankxk,(n∈N).
By (X:Y), denote the class of all matrices A such that A:X→Y. Thus, A∈(X:Y) if and only if the series on the right hand side of (1) converges for each n∈N and x∈X, and we have Ax={(Ax)n}n∈N∈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.A matrixA=(ank) is called a triangle if ank=0 for k>n and ann≠0 for all n,k∈N. It is trivial that A(Bx)=(AB)x holds for the triangle matrices A,B and a sequence x. Further, a triangle matrix U has a unique inverse U-1=V which is also a triangle matrix. Then, x=U(Vx)=V(Ux) holds for all x∈ω.Let us give the definition of some triangle limitation matrices which are needed in the text. Letq=(qk) be a sequence of positive reals and write
(2)Qn=∑k=0n‍qk,(n∈N).
Then the Cesàro mean of order one, Riesz mean with respect to the sequence q=(qk), and Euler mean of order r with 0<r<1 are, respectively, defined by the matrices C=(cnk),  Rq=(rnkq), and Er=(enkr), where
(3)cnk={1n+1,(0≤k≤n),0,(k>n),rnkq={qkQn,(0≤k≤n),0,(k>n),enkr={(nk)(1-r)n-krk,(0≤k≤n),0,(k>n)
for all k,n∈N. We write U for the set of all sequences u=(uk) such that uk≠0 for all k∈N. For u∈U, let 1/u=(1/uk). Let z,u,v∈U, and define the summation matrix S=(snk), the difference matrix Δ=(Δnk(1)), and the generalized weighted mean or factorable matrix G(u,v)=(gnk),  Aur={ankr(u)},  Δ(m)=(Δnk(m)) by
(4)snk={1,(0≤k≤n),0,(k>n),Δnk(1)={(-1)n-k,(n-1≤k≤n),0,(0≤k<n-1ork>n),gnk={unvk,(0≤k≤n),0,(k>n),ankr(u)={1+rkn+1uk,(0≤k≤n),0,(k>n),Δnk(m)={(-1)n-k(mn-k),(max⁡⁡{0,n-m}≤k≤n),0,(0≤k<n-1ork>n)
for all k,n∈N, where un and vk depend only on n and k, respectively. Let r and s be nonzero real numbers, and define the generalized difference matrix B(r,s)={bnk(r,s)} by
(5)bnk(r,s)={r,(k=n),s,(k=n-1),0,(0≤k<n-1ork>n),
for all k,n∈N. We note that if we choose r=1 and s=-1 then the matrix B(r,s) is reduced to the backward difference.For a sequence spaceX, the matrix domain XA of an infinite matrix A in the space X is defined by
(6)XA={x=(xk)∈ω:Ax∈X},
which is a sequence space. If A is triangle, then one can easily observe that the sequence spaces XA and X are linearly isomorphic; that is, XA≅X.Following Başar [1, page 51], we note that although in the most cases the new sequence space XA generated by the limitation matrix A from a sequence space X is the expansion or the contraction of the original space X, it may be observed in some cases that those spaces overlap. Indeed, one can easily see that the inclusion, XS⊂X, strictly holds for X∈{l∞,c,c0}. As this, one can deduce that the inclusion X⊂XΔ(1) also strictly holds for X∈{l∞,c,c0,lp}. However, if we define X=c0⊕span⁡{z} with z=((-1))k, that is, x∈X if and only if x∶=β+αz for some β∈c0 and some α∈C, and consider the matrix A with the rows An defined by An=(-1)ne(n) for all n∈N, we have Ae=z∈X but Az=e∉X which lead us to the consequences that z∈X∖XA and e∈XA∖X, where e=(1,1,1,…) and e(n) is a sequence whose only nonzero term is a 1 in nth place for each n∈N. That is to say, the sequence spaces XA and X overlap but neither contains the other.The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by Wang [2], Ng and Lee [3], Malkowsky [4], Altay and Başar [5], Malkowsky and Savaş [6], Başarır [7], Aydın and Başar [8], Başar et al. [9], Şengönül and Başar [10], Altay [11], Polat and Başar [12], and Malkowsky et al. [13]. Δ,Δ2 and Δm are the transposes of the matrices Δ(1),Δ(2), and Δ(m), respectively, and c0(u,p) are the spaces consisting of the sequences x=(xk) such that ux=(ukxk) in the spaces c0(p) and c(p) for u∈U, respectively, studied by Başarır [7]. More recently, the generalized difference matrix B(r,s)={bnk(r,s)} has been used by Kirişçi and Başar [14] for generalizing the difference spaces l∞(Δ),c(Δ),c0(Δ), and bvp. Finally, the new technique for deducing certain topological properties, for example, AB-,  KB-, and  AD-properties, and determining the β- and γ-duals of the domain of a triangle matrix in a sequence space has been given by Altay and Başar [15].Letr,s, and t be nonzero real numbers, and define the generalized difference matrix B^=B(r,s,t)={bnk(r,s,t)} by
(7)bnk(r,s,t)={r,(k=n),s,(k=n-1),t,(k=n-2),0,(0≤k<n-1ork>n)
for all n,k∈N. The inverse B-1(r,s,t)={dnk(r,s,t)} of B(r,s,t)={bnk(r,s,t)} is given by
(8)dnk(r,s,t)={1r∑v=0n-k‍(-s+s2-4tr2r)n-k-vk×(-s-s2-4tr2r)v,(0≤k≤n),0,(k>n).
We should record here that B(r,s,0)=B(r,s), B(1,-2,1)=Δ(2), and B(1,-1,0)=Δ(1). So, the results related to the matrix domain of the triple band matrix B(r,s,t) are more general and more comprehensive than the consequences on the matrix domain of B(r,s),  Δ(2) and Δ(1) and include them. We assume throughout that λ=(λk)k=0∞ is a strictly increasing sequence of positive reals tending to ∞; that is
(9)0<λ0<λ1<⋯,lim⁡k→∞λk=∞.The main purpose of the present paper is to introduce the sequence spacesμλ(B^) and to determine the α-,  β-, and γ-duals of this space, where μ denotes throughout any of the classical spaces l∞,c,  c0, or lp, and B^ is the triple band matrix B(r,s,t) and the sequence λ=(λk) is defined in (9). Furthermore, the Schauder bases for the spaces c0λ(B^),  cλ(B^) and lpλ(B^) are given, and some topological properties of the spaces c0λ(B^),  cλ(B^), and lpλ(B^) are examined. Finally, some classes of matrix mappings on the spaces μλ(B^) are characterized.The paper is organized as follows. In Section2, the BK-spaces c0λ(B^),  cλ(B^),  l∞λ(B^), and lpλ(B^) of generalized difference sequence spaces are introduced and the Schauder bases of the spaces c0λ(B^),  cλ(B^), and lpλ(B^) are given. In Section 3, some inclusion relations concerning these spaces are examined. In Section 4, the α-,  β-, and γ-duals of the generalized difference sequence spaces μλ(B^) of nonabsolute type are determined. In Section 5, the classes (μ1λ(B^):μ2) of infinite matrices are characterized, where μ1∈{c,c0,lp} and μ2∈{l∞,c,c0,lp}.


## 2. The Difference Sequence Spacesμλ(B^) of Nonabsolute Type

The difference sequence spaces have been studied by several authors in different ways [16–34]. In the present section, we introduce the spaces c0λ(B^),  cλ(B^),  l∞λ(B^), and lpλ(B^) and show that these spaces are BK-spaces of nonabsolute type which are norm isomorphic to the spaces c0,  c,  l∞, and lp, respectively. Furthermore, we give the bases of the spaces c0λ(B^),  cλ(B^), and lpλ(B^).We say that a sequencex=(xk)∈ω is λ-convergent to the number l∈C, called the λ-limit of x, if (Λx)n→l as n→∞ where
(10)(Λx)n=1λn∑k=0n‍(λk-λk-1)xk;(n∈N).
In particular, we say that x is a λ-null sequence if (Λx)n→0 as n→∞. Further, we say that x is λ-bounded if sup⁡n∈N|(Λx)n|<∞, [35]. Recently, Mursaleen and Noman [35, 36] studied the sequence spaces c0λ,  cλ,  l∞λ, and lpλ of nonabsolute type as follows:
(11)c0λ={x=(xk)∈ω:lim⁡n→∞1λn∑k=0n‍(λk-λk-1)xk=0},cλ={x=(xk)∈ω:lim⁡n→∞1λn∑k=0n‍(λk-λk-1)xkexists},l∞λ={x=(xk)∈ω:sup⁡n∈N|1λn∑k=0n‍(λk-λk-1)xk|<∞},lpλ={x=(xk)∈ω:∑n=0∞‍|1λn∑k=0n‍(λk-λk-1)xk|p<∞}.
On the other hand, we define the matrix Λ=(λnk) for all n,k∈N by
(12)λnk={λk-λk-1λn,(0≤k≤n)0,(k>n).
Then, the space μλ can be restated with the notation of (6) that
(13)c0λ={c0}Λ,cλ=cΛ,l∞λ={l∞}Λ,lpλ={lp}Λ.More recently, Sönmez [37] has defined the sequence spaces l∞(B^),c(B^),c0(B^), and lp(B^) as follows:
(14)l∞(B^)={x=(xk)∈ω:sup⁡k∈N|rxk+sxk-1+txk-2|<∞},c(B^)={x=(xk)∈ω:lim⁡k→∞|rxk+sxk-1+txk-2-l|k∃l∈C∋lim⁡k→∞|rxk+sxk-1+txk-2-l|=0},c0(B^)={x=(xk)∈ω:lim⁡k→∞|rxk+sxk-1+txk-2|=0},lp(B^)={x=(xk)∈ω:∑k‍|rxk+sxk-1+txk-2|p<∞}.
In fact, the sequence spaces l∞(B^),c(B^),c0(B^), and lp(B^) can be considered as the set of all sequences whose B(r,s,t)-transforms are in the spaces l∞,c,c0, and lp, respectively. That is,
(15)l∞(B^)={l∞}B(r,s,t),c(B^)=cB(r,s,t),c0(B^)={c0}B(r,s,t),lp(B^)={lp}B(r,s,t).Now, we introduce the difference sequence spacesl∞λ(B^),cλ(B^),c0λ(B^), and lpλ(B^) as follows:
(16)l∞λ(B^)={x=(xn)∈ω:∑k=0n‍ksup⁡n∈N|1λn∑k=0n‍(λk-λk-1)×∑k=0n‍(rxk+sxk-1+txk-2)|<∞},cλ(B^)={x=(xn)∈ω:∑k=0n‍klim⁡n→∞1λn∑k=0n‍(λk-λk-1)×∑k=0n‍(rxk+sxk-1+txk-2)exists},c0λ(B^)={x=(xn)∈ω:∑k=0n‍klim⁡n→∞1λn∑k=0n‍(λk-λk-1)k×∑k=0n‍(rxk+sxk-1+txk-2)=0},lpλ(B^)={x=(xn)∈ω:1λn∑k=0n‍∑n‍|1λn∑k=0n‍(λk-λk-1)k×1λn∑k=0n‍(rxk+sxk-1+txk-2)|p<∞}.
Now, we define the triangle matrix W^=(wnkλ)=ΛB^; that is,
(17)wnkλ={(r(λk-λk-1)+s(λk+1-λk)+t(λk+2-λk+1))×(λn)-1,(k<n-1),r(λn-1-λn-2)+s(λn-λn-1)λn,(k=n-1),r(λn-λn-1)λn,(k=n),0,(otherwise)
for all k,n∈N. Further, for any sequence x=(xk) we define the sequence y={yk(λ)} which will be used, as the W^-transform of x; that is,
(18)yk(λ)=∑j=0k-2‍r(λj-λj-1)+s(λj+1-λj)+t(λj+2-λj+1)λkxj+r(λk-1-λk-2)+s(λk-λk-1)λkxk-1+r(λk-λk-1)λkxk;(k∈N).
In fact, the sequence spaces c0λ(B^),cλ(B^),l∞λ(B^), and lpλ(B^) can be considered as the set of all sequences whose W^-transforms are in the spaces c0,c,l∞, and lp, respectively. That is,
(19)c0λ(B^)={c0}W^,cλ(B^)=cW^,l∞λ(B^)={l∞}W^,lpλ(B^)={lp}W^.Since the proof may also be obtained in a similar way as the other spaces, to avoid the repetition of the similar statements, we give the proof only for one of those spaces. Now, we may begin with the following theorem which is essential in the study.Theorem 1.
(i) The difference sequence spacesc0λ(B^), cλ(B^), and l∞λ(B^) are BK-spaces with the norm ∥x∥c0λ(B^)=∥x∥cλ(B^)=∥x∥l∞λ(B^)=∥W^x∥∞; that is,
(20)∥x∥c0λ(B^)=∥x∥cλ(B^)=∥x∥l∞λ(B^)=sup⁡n∈N|(W^x)n|.
(ii) Let 1≤p<∞. Then lpλ(B^) is a BK-space with the norm ∥x∥lpλ(B^)=∥W^x∥lp; that is,
(21)∥x∥lpλ(B^)=(∑n‍|(W^x)n|p)1/p.Proof.
Since (19) holds and c0,c,l∞, and lp are BK-spaces with respect to their natural norms (see [1, pages 16-17]) and the matrix W^ is a triangle, Theorem 4.3.12 of Wilansky [38, page 63] gives the fact that c0λ(B^), cλ(B^),  l∞λ(B^), and lpλ(B^) are BK-spaces with the given norms. This completes the proof.Remark 2.
Let|x|=(|xk|). Then the absolute property does not hold on the space μλ(B^); that is, ∥x∥μλ(B^)≠∥|x|∥μλ(B^). This can be shown for at least one sequence in those spaces. Hence μλ(B^) is the sequence space of nonabsolute type.Theorem 3.
The sequence spacesc0λ(B^), cλ(B^), l∞λ(B^), and lpλ(B^) of nonabsolute type are norm isomorphic to the spaces c0, c, l∞, and lp, respectively; that is, c0λ(B^)≅c0, cλ(B^)≅c, l∞λ(B^)≅l∞, and lpλ(B^)≅lp.Proof.
We prove the theorem for the spacec0λ(B^). To prove our assertion we should show the existence of a linear bijection between the spaces c0λ(B^) and c0. Let T:c0λ(B^)→c0 be defined by (18). Then, Tx=y=W^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.
Moreover, lety=(yk)∈c0 and we define the sequence x={xk(λ)} by
(22)xk(λ)=∑j=0k‍1r∑v=0k-j‍(-s+s2-4tr2r)k-j-v(-s-s2-4tr2r)v×∑i=j-1j‍(-1)j-iλiλj-λj-1yi.
Then we obtain
(23)rxk(λ)+sxk-1(λ)+txk-2(λ)=∑j=k-1k‍(-1)k-jλjλk-λk-1yj,∀k∈N.
Hence, we get for every n∈N that
(24)(W^x)n=1λn∑k=0n‍(λk-λk-1)(rxk+sxk-1+txk-2)=1λn∑k=0n‍∑j=k-1k‍(-1)k-jλjyj=yn.
This shows that W^x=y and since y∈c0, we conclude that W^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
(25)∥Tx∥∞=∥y∥∞=∥W^x∥∞=∥x∥c0λ(B^)
which means that T is norm preserving. Consequently T is a linear bijection which shows that the spaces c0λ(B^) and c0 are linearly isomorphic, as desired.Let(X,∥·∥) be a normed space. A sequence (bk) of points of X is called a Schauder basis for X if and only if, for each x∈X, there exists a unique sequence (αk) of scalars such that x=∑k‍αkbk; that is,
(26)lim⁡n→∞∥x-∑k=0n‍αkbk∥=0.Because of the isomorphismT, which is defined in the proof of Theorem 3, the inverse image of the basis of the spaces c0,c, and lp are the basis of new spaces c0λ(B^),cλ(B^), and lpλ(B^), respectively. Therefore, we have the following.Theorem 4.
Letαk(λ)=(W^x)k for all k∈N and lim⁡k→∞(W^x)k=l. Define the sequence b(k)(λ)={bn(k)(λ)}n∈N for every fixed k∈N by
(27)bn(k)(λ)={dnk(r,s,t)λkλk-λk-1-dn,k+1(r,s,t)λkλk+1-λk,(n>k)1rλk(λk-λk-1),(n=k)0,(n<k).
Then, the following statements hold.(i)
The sequence{b(k)(λ)}k∈N is a basis for the spaces c0λ(B^) and lpλ(B^); any x∈c0λ(B^) or x∈lpλ(B^) has a unique representation of the form x=∑k‍αk(λ)b(k)(λ).(ii)
The sequence{b,b(0)(λ),b(1)(λ),b(2)(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=0k‍dkj).


## 3. The Inclusion Relations

In the present section, we prove some inclusion relations concerning the spacesc0λ(B^),cλ(B^),l∞λ(B^), and lpλ(B^).Theorem 5.
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
(28)xk=∑j=0k‍dkj
for all k∈N. Then
(29)rxk+sxk-1+txk-2=∑j=k-1k‍(-1)k-jλjλk-λk-1.
Then one can easily see that W^x=e∈c∖c0, where e=(1,1,1,…). Thus, this 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 of the theorem.Theorem 6.
Ifr+s+t=0 then the inclusion c⊂c0λ(B^) strictly holds.Proof.
Suppose thatr+s+t=0 and x∈c. Then B^x=(rxk+sxk-1+txk-2)∈c0 and hence B^x∈c0λ since the inclusion c0⊂c0λ holds [35]. This shows that x∈c0λ(B^); that is, c⊂c0λ(B^). Further consider the sequence y=(yk) defined by yk=ln⁡(k+3) for all k∈N. Then it is trivial that y∉c. On the other hand, it can easily be seen that B^y∈c0. Hence, y∈c0λ(B^). Thus the sequence y∈c0λ(B^)∖c. Hence, the inclusion c⊂c0λ(B^) is strict. This completes the proof.Lemma 7.
A
∈
(
l
∞
:
c
0
) if and only if  lim⁡n→∞∑k‍|ank|=0.Theorem 8.
Letz=(zk) be defined by
(30)zk=|r(λk-λk-1)+s(λk+1-λk)+t(λk+2-λk+1)λk-λk-1|
for all k∈N. Then the inclusion l∞⊂c0λ(B^) strictly holds if and only if z∈c0λ.Proof.
Letl∞ be a subset of c0λ(B^). Then we obtain that W^x∈c0 for every x∈l∞ and the matrix W^=(wnkλ) is in the class (l∞:c0). By using Lemma 7 it follows that
(31)lim⁡n→∞∑k‍|wnkλ|=0.
Now, by taking into account the definition of matrix W^=(wnkλ) given by (17), we have for every n∈N that
(32)∑k‍|wnkλ|=1λn∑k=0n-2‍|r(λk-λk-1)+s(λk+1-λk)+t(λk+2-λk+1)|+|r(λn-1-λn-2)+s(λn-λn-1)λn|+|r|(λn-λn-1)λn.
By Lemma 7, we have that
(33)lim⁡n→∞|r|(λn-λn-1)λn=0,(34)lim⁡n→∞|r(λn-1-λn-2)+s(λn-λn-1)λn|=0,(35)lim⁡n→∞1λn×∑k=0n-2‍|r(λk-λk-1)+s(λk+1-λk)+t(λk+2-λk+1)|=0.
Now, we have for every n≥1 that
(36)1λn∑k=0n-2‍|r(λk-λk-1)+s(λk+1-λk)+t(λk+2-λk+1)|=λn-2λn[1λn-2∑k=0n-2‍(λk-λk-1)zk]
and since lim⁡n→∞(λn-2/λn)=1 by (33) and (34), we obtain that by (35)
(37)lim⁡n→∞1λn-2∑k=0n-2‍(λk-λk-1)zk=0
which shows that z=(zk)∈c0λ where the sequence z=(zk) is defined by
(38)zk=|r(λk-λk-1)+s(λk+1-λk)+t(λk+2-λk+1)λk-λk-1|
for all k∈N.
Conversely, we suppose thatz=(zk)∈c0λ. Then we have (37). Further, for every n≥2, we derive that(39)1λn∑k=0n-2‍|r(λk-λk-1)+s(λk+1-λk)+t(λk+2-λk+1)|=1λn∑k=0n-2‍(λk-λk-1)zk≤1λn-2∑k=0n-2‍(λk-λk-1)zk.
Then, (37) and (39) together imply that (35) holds. On the other hand, we have for every n≥2 that
(40)|rλn-2+s(λn-1-λ0)+t(λn-λ1)λn|=|1λn∑k=0n-2‍r(λk-λk-1)+s(λk+1-λk)+t(λk+2-λk+1)|≤1λn∑k=0n-2‍|r(λk-λk-1)+s(λk+1-λk)+t(λk+2-λk+1)|.
Therefore, it follows by (35)
(41)lim⁡n→∞rλn-2+s(λn-1-λ0)+t(λn-λ1)λn=0.
Particularly if (i) r=0, s=-1, t=1, then we obtain lim⁡n→∞[(λn-λn-1)/λn]=0, which shows that (33) holds. (ii) r=-1, s=1, t=0, and then we obtain lim⁡n→∞[(λn-1-λn-2)/λn]=0, which shows that (34) holds. Thus, we deduce by relation (32) that (31) holds. This leads us with Lemma 7 to the consequence that W^∈(l∞:c0). Hence, the inclusion l∞⊂c0λ(B^) holds. Finally, it is obvious that the sequence y, defined in the proof of Theorem 6, is in c0λ(B^) but not in l∞, so the inclusion l∞⊂c0λ(B^) is strict.Theorem 9.
The inclusionl∞⊂l∞λ(B^) strictly holds.Proof.
To prove the validity of the inclusionl∞⊂l∞λ(B^), it suffices to show that, for every x∈l∞λ(B^), there exists a positive real number K such that ∥x∥l∞λ(B^)≤K∥x∥l∞. Let x∈l∞. Then, we have
(42)∥x∥l∞λ(B^)=sup⁡n∈N|1λn∑k=0n‍(λk-λk-1)(rxk+sxk-1+txk-2)|≤sup⁡0≤k≤n|rxk+sxk-1+txk-2|sup⁡n∈N|1λn∑k=0n‍(λk-λk-1)|≤max⁡{|r|+|s|+|t|}∥x∥l∞
so that x∈l∞λ(B^) and hence l∞⊂l∞λ(B^). Furthermore, we consider x=(xk) defined by xk=∑j=0k‍dkj for all k∈N. Then we have W^x=e. Thus, we deduce that ∥x∥l∞λ(B^)=∥W^x∥∞=1. Hence x=(xk)∈l∞λ(B^).
On the other hand, we know from Theorem8 that the inclusion l∞⊂c0λ(B^) is strict. Since c0λ(B^)⊂l∞λ(B^), the inclusion l∞⊂l∞λ(B^) strictly holds.


## 4. Theα-, β-, and γ-Duals of the Spaces μλ(B^) of Nonabsolute Type

In this section, we determine theα-, β-, and γ-duals of the generalized difference sequence spaces c0λ(B^),cλ(B^),l∞λ(B^), and lpλ(B^) of nonabsolute type.We will firstly give the definition ofα-, β-, and γ-duals of sequence spaces and secondly we give the lemmas due to [39] which are needed in proving the theorems given in Sections 4 and 5.For the sequence spacesλ and μ, define the set S(λ,μ) by
(43)S(λ,μ)={z=(zk)∈w:xz=(xkzk)∈μ∀x∈λ}.With the notation of (43), the α-, β-, and γ-duals λα,λβ, and λγ of a sequence space λ are defined by
(44)λα=S(λ,l1),λβ=S(λ,cs),λγ=S(λ,bs).Lemma 10.
(i)A∈(c0:l1)=(c:l1)=(l∞:l1) if and only if
(45)sup⁡K∈F∑n‍|∑k∈K‍ank|<∞.
(ii) A=(ank)∈(lp:l1) if and only if
(46)sup⁡N∈F∑k‍|∑n∈N‍ank|q<∞.Lemma 11.
A
∈
(
c
0
:
c
) if and only if
(47)lim⁡n→∞ank=αk,foreachfixedk∈N,(48)sup⁡n∈N⁡∑k|ank|<∞.Lemma 12.
A
∈
(
c
:
c
) if and only if (47) and (48) hold, and
(49)lim⁡n→∞∑k‍ankexists.Lemma 13.
A
∈
(
l
∞
:
c
) if an only if (47) holds and
(50)lim⁡n→∞∑k‍|ank|=∑k‍|αk|.Lemma 14.
A
=
(
a
n
k
)
∈
(
l
p
:
c
) if and only if (47) holds and
(51)sup⁡n∈N∑k‍|ank|q<∞.Lemma 15.
A
∈
(
c
:
l
∞
)
=
(
c
0
:
l
∞
)
=
(
l
∞
:
l
∞
) if and only if (48) holds.Lemma 16.
A
=
(
a
n
k
)
∈
(
l
p
:
l
∞
) if and only if (51) holds.Now we consider the following sets:(52)f1λ={a=(an)∈w:sup⁡K∈F∑n‍|∑k∈K‍fnkλ|<∞},f2λ={a=(an)∈w:∑j=k∞‍djkajexistsforeachk∈N},f3λ={a=(an)∈w:sup⁡n∈N∑k=0n-1‍|g^k(n)|q<∞},f4λ={a=(an)∈w:sup⁡n∈N|1rλn(λn-λn-1)an|<∞},f5λ={a=(an)∈w:lim⁡n→∞∑k=0n‍[∑j=0k‍dkj]akexists},f6λ={a=(an)∈w:sup⁡N∈F∑k=0∞‍|∑n∈N‍fnkλ|q<∞},f7λ={a=(an)∈w:lim⁡n→∞∑k‍|vnkλ|=∑k‍|lim⁡n→∞vnkλ|},
where the matrices Fλ=(fnkλ) and Vλ=(vnkλ) are defined as follows:
(53)fnkλ={(dnkλkλk-λk-1-dn,k+1λkλk+1-λk)an,(k<n),1rλn(λn-λn-1)an,(k=n),0,(k>n),vnkλ={g^k(n),(k<n),1rλn(λn-λn-1)an,(k=n),0,(k>n)
for all k,n∈N and the g^k(n) is defined by
(54)g^k(n)=λk(1λk-λk-1∑j=kn‍djkaj-1λk+1-λk∑j=k+1n‍dj,k+1aj),fork<n.Theorem 17.
(i){c0λ(B^)}α={cλ(B^)}α={l∞λ(B^)}α=f1λ.
(ii){lpλ(B^)}α=f6λ.Proof.
We prove the theorem for the spacec0λ(B^). Let a=(an)∈w. Then, we obtain the equality
(55)anxn=∑k=0n‍dnk∑j=k-1k‍(-1)k-jλjλk-λk-1anyi=(Fλy)n,∀n∈N,
by relation (22). Thus we observe by (55) that ax=(anxn)∈l1 whenever x=(xk)∈c0λ(B^) if and only if Fλy∈l1 whenever y=(yk)∈c0. This means that the sequence a=(an)∈{c0λ(B^)}α if and only if Fλ∈(c0:l1). Therefore we obtain by Lemma 10 with Fλ instead of A that a=(an)∈{c0λ(B^)}α if and only if
(56)sup⁡K∈F∑n‍|∑k∈K‍fnkλ|<∞
which leads us to the consequence that {c0λ(B^)}α=f1λ. This completes the proof.Theorem 18.
(i){c0λ(B^)}β=f2λ∩f3λ∩f4λ  (withq=1).
(ii){cλ(B^)}β=f2λ∩f3λ∩f4λ∩f5λ  (withq=1).
(iii){l∞λ(B^)}β=f2λ∩f4λ∩f7λ.
(iv){lpλ(B^)}β=f2λ∩f3λ∩f4λ.Proof.
Consider the equality(57)∑k=0n‍akxk=∑k=0n‍[∑j=0k‍dkj∑i=j-1i‍(-1)j-iλiλj-λj-1yi]ak=∑k=0n‍(λkyk-λk-1yk-1)[1λk-λk-1∑j=kn‍djkaj]=∑k=0n-1‍λk[∑j=kndjkajλk-λk-1-∑j=k+1ndj,k+1ajλk+1-λk]yk+1ranλn(λn-λn-1)yn=∑k=0n-1‍g^k(n)yk+1ranλn(λn-λn-1)yn=(Vλy)n;(n∈N).
Then we deduce by (57) that ax=(akxk)∈cs whenever x=(xk)∈c0λ(B^) if and only if Vλy∈c whenever y=(yk)∈c0. This means that a=(ak)∈{c0λ(B^)}β if and only if Vλ∈(c0:c). Therefore, by using Lemma 11, we obtain
(58)∑j=k∞‍djkajexistsforeachk∈N,sup⁡n∈N∑k=0n-1‍|g^k(n)|<∞,sup⁡k∈N|1rλk(λk-λk-1)ak|<∞.
Hence, we conclude that {c0λ(B^)}β=f2λ∩f3λ∩f4λ.Finally, we ended this section with the following theorem which determines theγ-duals of sequence spaces c0λ(B^),cλ(B^),l∞λ(B^), and lpλ(B^).Theorem 19.
(i){c0λ(B^)}γ={cλ(B^)}γ={l∞λ(B^)}γ=f3λ∩f4λ  (withq=1).
(ii){lpλ(B^)}γ=f3λ∩f4λ.


## 5. Some Matrix Transformations Related to the Spacesμλ(B^) of Nonabsolute Type for μ∈{c0,c,l∞,lp}

In this final section, we state some results on the characterization of several classes of matrix mappings on thec0λ(B^),cλ(B^),l∞λ(B^), and lpλ(B^). We will write throughout for brevity that
(59)g^nk(m)=λk(1λk-λk-1∑j=km‍djkanj-1λk+1-λk∑j=k+1m‍dj,k+1anj)fork<m,g^nk=λk(1λk-λk-1∑j=k∞‍djkanj-1λk+1-λk∑j=k+1∞‍dj,k+1anj)
for all k,m,n∈N. We assume that the series are convergent in the last equation.We will begin with lemmas which are needed in the proof of our theorems.Lemma 20.
The matrix mappings between theBK-spaces are continuous.Lemma 21.
A
=
(
a
n
k
)
∈
(
c
:
l
p
) if and only if
(60)sup⁡F∈F∑n‍|∑k∈F‍ank|p<∞.Lemma 22.
A
=
(
a
n
k
)
∈
(
c
:
c
0
) if and only if (48) holds and
(61)lim⁡n→∞ank=0,foreachfixedk∈N,lim⁡n→∞∑k‍ank=0.Lemma 23.
A
∈
(
c
0
:
c
0
) if and only if (48) and (61) hold.Now, we may give our matrix transformations.Theorem 24.
LetA=(ank) be an infinite matrix and 1<p<∞. Then, A∈(cλ(B^):lp) if and only if
(62)sup⁡F∈F∑n‍|∑k∈F‍g^nk|p<∞,(63)∑j=k∞‍djkanjexistsforallfixedk,n∈N,(64)sup⁡m∈N∑k=0m-1‍|g^nk(m)|<∞,(65)lim⁡k→∞1rλk(λk-λk-1)ank=anexistsforeachfixedn∈N,(66){∑j=k∞‍djkanj}∈cs;(n∈N),(67)(an)∈lp.Proof.
If conditions (62)–(67) hold and x=(xk) is any sequence in the space cλ(B^) then by using Theorem 18 we have that {ank}k∈N∈{cλ(B^)}β for all n∈N. Hence, A-transform of x exists. Furthermore, since the associated sequence y=(yk) is in the space c, we may write lim⁡kyk=l for some suitable l. Also, the matrix A^=(g^nk) is in the class (c:lp) by Lemma 21 and condition (62), where 1<p<∞.
Now, we may consider themth partial sum of the series ∑k‍ankxk which is derived by using relation (22):
(68)∑k=0m‍ankxk=∑k=0m-1‍g^nk(m)yk+1rλm(λm-λm-1)anmym;∀n,m∈N.
Then, since y∈c and A^∈(c:lp), A^y exists and so the series ∑k‍g^nkyk converges for every n∈N. Also, it follows by (63) that we have lim⁡m→∞g^nk(m)=g^nk. Therefore, if we pass to limit in (68) as m→∞, then we obtain by (65) that
(69)∑k‍ankxk=∑k‍g^nkyk+lan,∀n∈N
which can be written as follows:
(70)(Ax)n=(A^y)n+lan,∀n∈N.
This yields by taking lp-norm that
(71)∥Ax∥lp≤∥A^y∥lp+|l|∥(an)∥lp<∞.
Consequently, we have that Ax∈lp; that is, A∈(cλ(B^):lp).
Conversely, ifA∈(cλ(B^):lp), we have {ank}k∈N∈{cλ(B^)}β for all n∈N which implies with Theorem 18 that conditions (63), (64), and (66) are necessary.
On the other hand, sincecλ(B^) and lp are BK-spaces, we have by Lemma 20 that there is a constant M>0 such that
(72)∥Ax∥p≤M∥x∥cλ(B^)
holds for all x∈cλ(B^). Let F∈F. Then, the sequence z=∑k∈F‍b(k)(λ) is in cλ(B^), where the sequences b(k)(λ)={bn(k)(λ)}n∈N are defined by (27) for every fixed k∈N.
SinceW^b(k)(λ)=e(k) for each fixed k∈N, we have
(73)∥z∥cλ(B^)=∥W^(z)∥l∞=∥∑k∈F‍W^b(k)(λ)∥l∞=∥∑k∈F‍e(k)∥l∞=1.
Furthermore, for every n∈N, we obtain by (27) that
(74)An(z)=∑k∈F‍An(b(k)(λ))=∑k∈F‍∑j‍anjbj(k)(λ)=∑k∈F‍g^nk.
Hence, since inequality (72) is satisfied for the sequence z∈cλ(B^), we have for any F∈F that
(75)(∑n‍|∑k∈F‍g^nk|p)1/p≤M
which shows the necessity of (62). Thus, it follows by Lemma 21 that A^=(g^nk)∈(c:lp).
Moreover, we consider the sequencex=(xk) defined by (22) for every k∈N and suppose that y=(yk)∈c∖c0. Then, since x∈cλ(B^) such that y=W^x by (18), the transforms Ax and A^y exist. Hence, the series ∑k‍ankxk and ∑k‍g^nkyk are convergent for every n∈N. So we infer that
(76)lim⁡m→∞∑k=0m-1‍g^nk(m)yk=∑‍g^nkyk;(n∈N).
Consequently, we obtain from (68) that
(77)lim⁡m→∞1rλm(λm-λm-1)anmymexistsforeachfixedn∈N,
Hence, we deduce that
(78)lim⁡m→∞1rλm(λm-λm-1)anmexistsforeachfixedn∈N.
which leads us to the necessity of (65) and so the relation (68) holds, where l=lim⁡kyk.
Finally, sinceAx∈lp and A^y∈lp, the necessity of (67) is immediate by (68). This completes the proof.Theorem 25.
A
=
(
a
n
k
)
∈
(
c
λ
(
B
^
)
:
l
∞
) if and only if (65) and (66) hold, and
(79)sup⁡n∈N∑k‍|g^nk|<∞,(80)(an)∈l∞.Proof.
This is an immediate consequence of Lemma15 and Theorem 24.Theorem 26.
(i)A=(ank)∈(c0λ(B^):lp) if and only if (62),(63), and (64) hold and
(81)sup⁡k|1rλk(λk-λk-1)ank|<∞,(n∈N).
(ii) A∈(c0λ(B^):l∞) if and only if (64), (79), and (81) hold.Proof.
This may be obtained by proceedings as in Theorem24, above. So, we omit the detail.Theorem 27.
A
=
(
a
n
k
)
∈
(
c
λ
(
B
^
)
:
c
) if and only if (65), (66), and (79) hold and
(82)lim⁡nan=a,(83)lim⁡ng^nk=αk,foreachfixedk∈N,(84)lim⁡n∑k‍g^nk=α.Proof.
Suppose thatA satisfies conditions (65), (66), (79), (82), (83), and (84), and let x=(xk) be a sequence in the space cλ(B^). Since (79) implies (64), we have by Theorem 18 that {ank}k∈N∈{cλ(B^)}β for all n∈N. Hence, Ax exists. We also observe from (79) and (83) that
(85)∑j=0k‍|αj|≤sup⁡n∈N∑j‍|g^nk|<∞
holds for every k∈N. So (αk)∈l1 and hence the series ∑k‍αk(yk-l) converges, where y=(yk)∈c is the sequence connected with x=(xk) by relation (18) such that lim⁡kyk=l. Also, it is obvious by combining Lemma 12 with conditions (79), (83), and (84) that the matrix A^=(g^nk) is in the class (c:c).
Now, by following the similar way used in the proof of Theorem24, we obtain that relation (69) holds, which can be written as follows:
(86)∑k‍ankxk=∑k‍g^nk(yk-l)+l∑k‍g^nk+lan,∀n∈N.
If we pass the limit in (86) as n→∞ we have that
(87)lim⁡n(Ax)n=∑k‍αk(yk-l)+l(α+a),
which shows that Ax∈c; that is, A∈(cλ(B^):c).
Conversely, suppose thatA∈(cλ(B^):c). Since the inclusion c⊂l∞ holds, A∈(cλ(B^):l∞). Therefore, the necessity of conditions (65), (66), and (79) is obvious from Theorem 25. Furthermore, consider the sequence b(k)(λ)={bn(k)(λ)}n∈N∈cλ(B^) defined by (27) for every fixed k∈N. Then, one can see that Ab(k)(λ)={g^nk}n∈N and hence {g^nk}n∈N∈c for every k∈N which shows the necessity of (83). Let z=∑k‍b(k)(λ). Then, since the linear transformation T:cλ(B^)→c, defined as in the proof of Theorem 3 by analogy, is continuous and W^b(k)(λ)=e(k) for each fixed k∈N, we obtain that
(88)(W^z)n=∑k‍(W^b(k)(λ))n=∑k‍δnk=1,foreachn∈N
which shows that (W^z)n=e∈c and hence z∈cλ(B^). On the other hand, since cλ(B^) and c are the BK-spaces, Lemma 20 implies the continuity of the matrix mapping A:cλ(B^)→c. Thus, we have for every n∈N that
(89)(Az)n=∑k‍(Ab(k)(λ))n=∑k‍g^nk.
This shows the necessity of (84).
Now, it follows by (79), (83), and (84) with Lemma 12 that A^=(g^nk)∈(c:c). So by (65), (66) and relation (70) hold for all x∈cλ(B^) and y∈c.
Finally, the necessity of (82) is immediate by (70) since Ax∈c and A^x∈c. This completes the proof.Theorem 28.
A
=
(
a
n
k
)
∈
(
c
λ
(
B
^
)
:
c
0
) if and only if (65), (66), and (79) hold and
(90)lim⁡nan=0,lim⁡ng^nk=0,foreachfixedk∈N,lim⁡n∑k‍g^nk=0.Proof.
This is obtained in the similar way used in the proof of Theorem27 with Lemma 22 instead of Lemma 12 and so we omit the detail.Theorem 29.
A
=
(
a
n
k
)
∈
(
c
0
λ
(
B
^
)
:
c
) if and only if (63), (79), (81), and (83) hold.Proof.
This is an immediate consequence of Lemma11 and Theorems 18 and 26 (ii).Theorem 30.
A
=
(
a
n
k
)
∈
(
c
0
λ
(
B
^
)
:
c
0
) if and only if (63), (79), and (83) hold.Proof.
This is an immediate consequence of Lemma23 and Theorems 18 and 29.Theorem 31.
A
=
(
a
n
k
)
∈
(
l
p
λ
(
B
^
)
:
l
∞
) if and only if (64) and (81) hold and
(91)sup⁡n∈N∑k=0∞‍|g^nk|q<∞,(n∈N).Proof.
Suppose thatA=(ank) satisfies conditions (64), (81), and (91) and take any x=(xk)∈lpλ(B^). Then, it is obvious that {ank}k∈N∈{lpλ(B^)}β for each n∈N. Hence, (Ax)n=∑k=0∞‍ankxk is convergent for all x∈lpλ(B^); that is, Ax exists. Also, it is obvious by combining Lemma 16 with condition (91) that A^=(g^nk)∈(lp:l∞).
Now, we will show thatAx∈l∞ for all x∈lpλ(B^). By using relation (68) we have that
(92)∑k=0∞‍ankxk=∑k=0∞‍g^nkyk.
By applying Hölder’s inequality to (92) we obtain that
(93)|(Ax)n|=|∑k=0∞‍ankxk|≤∑k=0∞‍|g^nk||yk|≤∥y∥lp(∑k=0∞‍|g^nk|q)1/q
which gives us by taking supremum over n∈N in this last inequality that
(94)sup⁡n∈N|(Ax)n|≤∥y∥lpsup⁡n∈N(∑k=0∞‍|g^nk|q)1/q<∞
which implies that Ax∈l∞; that is, A∈(lpλ(B^):l∞).
Conversely, suppose thatA=(ank)∈(lpλ(B^):l∞). Thus, Ax exists and bounded for all x∈lpλ(B^). Also, {ank}k∈N∈{lpλ(B^)}β for all n∈N which implies the necessity of (64) and (81). If we define the sequences g^n such that g^n={g^nk}k∈N then we have that
(95)∥g^n∥lq<∞.
So, bearing in mind (81) one can easily obtain relation (92) by using relation (68). On the other hand, the sequences an={ank}k∈N define the continuous linear functionals on the space lpλ(B^) as follows:
(96)fn(x)=∑k=0∞‍ankxk;(n∈N).
Since the spaces lpλ(B^) and lpλ are linear isomorphic, we have that
(97)∥fn∥=∥g^n∥lq.
Also, since {fn(x)}n∈N∈l∞, it is obvious that
(98)sup⁡∥fn(x)∥<∞.
Therefore, by using the Banach-Steinhaus Theorem [38, see pages: 1-2] we obtain that
(99)sup⁡n∈N∥fn∥=sup⁡n∈N∥g^n∥lq=sup⁡n∈N(∑k=0∞‍|g^nk|q)1/q<∞
which shows the necessity of (91). This step completes the proof.Theorem 32.
A
=
(
a
n
k
)
∈
(
l
p
λ
(
B
^
)
:
c
) if and only if (64), (81), (83), and (91) hold.Proof.
Suppose that conditions (64), (81), (83), and (91) hold. Then, {ank}k∈N∈{lpλ(B^)}β for each n∈N. Hence, the series ∑k=0∞‍ankxk is convergent for each n∈N and for all x∈lpλ(B^) which implies that Ax exists. By using (83) we obtain that
(100)|g^nk|q⟶|αk|q;(n⟶∞).
Since
(101)(∑k=0m‍|g^nk|q)1/q≤sup⁡n∈N(∑k=0∞‍|g^nk|q)1/q=M
for all m>0, in this situation, we see by passing to the limit in the last inequality as n→∞ that
(102)(∑k=0m‍|αk|q)1/q≤sup⁡n∈N(∑k=0∞‍|g^nk|q)1/q.
Because (102) holds for all positive integers m, we have that
(103)(∑k=0∞‍|αk|q)1/q<∞.
We remember thatx=(xk) and y=(yk) are associated sequences by relation (22) where y=(yk)∈lp for x=(xk)∈lpλ(B^).
Letɛ be any positive number. Then, there exists a number N such that
(104)(∑k=N∞‍|yk|q)1/q<ɛ4M.
On the other hand, there is an integer N1 such that
(105)|∑k=0N‍{g^nk-αk}yk|≤ɛ2
whenever n≥N1. Therefore, we obtain
(106)|∑k=0∞‍{g^nk-αk}yk|≤|∑k=0N‍{g^nk-αk}yk|+|∑k=N+1∞‍{g^nk-αk}yk|≤ɛ2+(∑k=N+1∞‍[|g^nk|+|αk|]q)1/q(∑k=N+1∞‍|yk|p)1/p≤ɛ2+2Mɛ4M=ɛ,
which shows that
(107)lim⁡n→∞∑k=0∞‍g^nkyk=∑k=0∞‍αkyk.
Since
(108)lim⁡n→∞(Ax)n=lim⁡n→∞∑k=0∞‍g^nkyk=∑k=0∞‍αkyk
we have that Ax∈c which implies that A∈(lpλ(B^):c).
The necessity part can be proved by using the similar way in the proof of Theorem27, so we omit the detail.Theorem 33.
A
=
(
a
n
k
)
∈
(
l
p
λ
(
B
^
)
:
c
0
) if and only if (64), (81), and (91) hold and
(109)lim⁡ng^nk=0,foreachfixedk∈N.Proof.
This result can be proved similarly as the proof of Theorem32.



---

*Source: 876813-2014-09-09.xml*