# Domain of the Double Sequential Band MatrixB(r~,s~) in the Sequence Space ℓ(p)∗

Authors: Havva Nergiz; Feyzi Başar
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The sequence spaceℓ(p) was introduced by Maddox (1967). Quite recently, the domain of the generalized difference matrix B(r,s) in the sequence space ℓp has been investigated by Kirişçi and Başar (2010). In the present paper, the sequence space ℓ(B~,p) of nonabsolute type has been studied which is the domain of the generalized difference matrix B(r~,s~) in the sequence space ℓ(p). Furthermore, the alpha-, beta-, and gamma-duals of the space ℓ(B~,p) have been determined, and the Schauder basis has been given. The classes of matrix transformations from the space ℓ(B~,p) to the spaces ℓ∞, c and c0 have been characterized. Additionally, the characterizations of some other matrix transformations from the space ℓ(B~,p) to the Euler, Riesz, difference, and so forth sequence spaces have been obtained by means of a given lemma. The last section of the paper has been devoted to conclusion.


---

## Body

## 1. Preliminaries, Background, and Notation

Byw, we denote the space of all real valued sequences. Any vector subspace of w is called a sequence space. We write ℓ∞, c, and c0 for the spaces of all bounded, convergent, and null sequences, respectively. Also by bs, cs, ℓ1, and ℓp, we denote the spaces of all bounded, convergent, absolutely convergent and p-absolutely convergent series, respectively, where 1<p<∞.A linear topological spaceX over the real field ℝ is said to be a paranormed space if there is a subadditive function g:X→ℝ such that g(θ)=0,g(x)=g(-x) and scalar multiplication is continuous; that is, |αn-α|→0 and g(xn-x)→0 imply g(αnxn-αx)→0 for all α’s in ℝ and all x’s in X, where θ is the zero vector in the linear space X.Assume here and after that(pk)  is a bounded sequence of strictly positive real numbers with suppk=H and M=max{1,H}. Then, the linear spaces ℓ(p) were defined by Maddox [1] (see also Simons [2] and Nakano [3]) as follows:
(1)ℓ(p)={x=(xk)∈w:∑k‍|xk|pk<∞},(0<pk≤H<∞)
which is the complete space paranormed by
(2)g(x)=(∑k‍|xk|pk)1/M.
For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. We assume throughout that pk-1+(pk′)-1=1 and denote the collection of all finite subsets of ℕ={0,1,2,…} by ℱ and use the convention that any term with negative subscript is equal to naught.Letλ, μ be any two sequence spaces and let A=(ank) be an infinite matrix of real or complex numbers ank, where n,k∈ℕ. Then, we say that A defines a matrix mapping from λ into μ, and we denote it by writing A:λ→μ; if for every sequence x=(xk)∈λ the sequence Ax={(Ax)n}, the A-transform of x, is in μ, where
(3)(Ax)n=∑kankxk,foreachn∈ℕ.
By (λ:μ), we denote the class of all matrices A such that A:λ→μ. Thus, A∈(λ:μ) if and only if the series on the right side of (3) converges for each n∈ℕ and every x∈λ, and we have Ax={(Ax)n}n∈ℕ∈μ for all x∈λ. A sequence x is said to be A-summable to α if Ax converges to α which is called the A-limit of x.The shift operatorP is defined on ω by (Px)n=xn+1 for all n∈ℕ. A Banach limit L is defined on ℓ∞, as a nonnegative linear functional, such that L(Px)=L(x) and L(e)=1, where e=(1,1,1,…). A sequence x=(xk)∈ℓ∞ is said to be almost convergent to the generalized limit l if all Banach limits of x are l and is denoted by f-limxk=l. Lorentz [4] proved that
(4)f-limxk=lifflimm→∞1m+1∑k=0mxk+n=luniformlyinn.
It is well known that a convergent sequence is almost convergent such that its ordinary and generalized limits are equal. By f, we denote the space of all almost convergent sequences; that is,
(5)f:={x=(xk)∈ω:∃l∈ℂ∋limm→∞∑k=0mxn+km+1=luniformlyinn}.Define the double sequential band matrixB(r~,s~)={bnk(rk,sk)} by
(6)bnk(rk,sk)={rk,k=n,sk,k=n-1,0,otherwise
for all k,n∈ℕ, where r~=(rk) and s~=(sk) are the convergent sequences. We should note that the double sequential band matrices were firstly used by Srivastava and Kumar [5, 6], Panigrahi and Srivastava [7], and Akhmedov and El-Shabrawy [8].The main purpose of this paper, which is a continuation of Kirişçi and Başar [9], is to introduce the sequence space ℓ(B~,p) of nonabsolute type consisting of all sequences whose B(r~,s~)-transforms are in the space ℓ(p). Furthermore, the basis is constructed and the alpha-, beta-, and gamma-duals are computed for the space ℓ(B~,p). Moreover, the matrix transformations from the space ℓ(B~,p) to some sequence spaces are characterized. Finally, we note open problems and further suggestions.It is clear thatΔ(1) can be obtained as a special case of B(r~,s~) for r~=e and s~=-e and it is also trivial that B(r~,s~) is reduced in the special case r~=re and s~=se to the generalized difference matrix B(r,s). So, the results related to the matrix domain of the matrix B(r~,s~) are more general and more comprehensive than the corresponding consequences of the matrix domains of Δ(1) and B(r,s).The rest of this paper is organized as follows. In Section2, the linear sequence space ℓ(B~,p) is defined and proved that it is a complete paranormed space with a Schauder basis. Section 3 is devoted to the determination of alpha-, beta-, and gamma-duals of the space ℓ(B~,p). In Section 4, the classes (ℓ(B~,p):ℓ∞), (ℓ(B~,p):f), (ℓ(B~,p):c), and (ℓ(B~,p):c0) of infinite matrices are characterized. Additionally, the characterizations of some other classes of matrix transformations from the space ℓ(B~,p) to the Euler, Riesz, difference, and so forth sequence spaces are obtained by means of a given lemma. In the final section of the paper, open problems and further suggestions are noted.


## 2. The Sequence Spaceℓ(B~,p) of Nonabsolute Type

In this section, we introduce the complete paranormed linear sequence spaceℓ(B~,p).Thematrix domainλA of an infinite matrix A in a sequence space λ is defined by
(7)λA={x=(xk)∈ω:Ax∈λ}.
Choudhary and Mishra [10] defined the sequence space ℓ(p)¯ which consists of all sequences such that S-transforms of them are in the space ℓ(p), where S=(snk) is defined by
(8)snk={1,0≤k≤n,0,k>n,
for all k,n∈ℕ. Başar and Altay [11] have recently examined the space bs(p) which is formerly defined by Başar in [12] as the set of all series whose sequences of partial sums are in ℓ∞(p). More recently, Aydın and Başar [13] have studied the space ar(u,p) which is the domain of the matrix Ar in the sequence space ℓ(p), where the matrix Ar={ank(r)} is defined by
(9)ank(r)={1+rkn+1uk,0≤k≤n,0,k>n,
for all k,n∈ℕ, (uk) such that uk≠0 for all k∈ℕ and 0<r<1. Altay and Başar [14] have studied the sequence space rt(p) which is derived from the sequence space ℓ(p) of Maddox by the Riesz means Rt. With the notation of (7), the spaces ℓ(p)¯,bs(p),ar(u,p), and rt(p) can be redefined by
(10)ℓ(p)¯=[ℓ(p)]S,bs(p)=[ℓ∞(p)]S,ar(u,p)=[ℓ(p)]Ar,rt(p)=[ℓ(p)]Rt.
Following Choudhary and Mishra [10], Başar and Altay [11], Altay and Başar [14–17], and Aydın and Başar [13, 18], we introduce the sequence space ℓ(B~,p) as the set of all sequences whose B(r~,s~)-transforms are in the space ℓ(p); that is
(11)ℓ(B~,p):={(xk)∈w:∑k|sk-1xk-1+rkxk|pk<∞},(0<pk≤H<∞).
It is trivial that in the case pk=p for all k∈ℕ, the sequence space ℓ(B~,p) is reduced to the sequence space ℓ~p which is introduced by Kirişçi and Başar [9]. With the notation of (7), we can redefine the space ℓ(B~,p) as follows:
(12)ℓ(B~,p):=[ℓ(p)]B(r~,s~).
Define the sequence y=(yk), which will be frequently used, as the B(r~,s~)-transform of a sequence x=(xk); that is,
(13)yk={B(r~,s~)x}k=rkxk+sk-1xk-1,∀k∈ℕ.
Since the spaces ℓ(p) and ℓ(B~,p) are linearly isomorphic by Corollary 4, one can easily observe that x=(xk)∈ℓ(B~,p) if and only if y=(yk)∈ℓ(p), where the sequences x=(xk) and y=(yk) are connected with the relation (13).Now, we may begin with the following theorem which is essential in the text.Theorem 1.
ℓ
(
B
~
,
p
) is a complete linear metric space paranormed by the paranorm
(14)h(x)=(∑k|sk-1xk-1+rkxk|pk)1/M.Proof.
It is easy to see that the spaceℓ(B~,p) is linear with respect to the coordinate-wise addition and scalar multiplication. Therefore, we first show that it is a paranormed space with the paranorm h defined by (14).
It is clear thath(θ)=0 where θ=(0,0,0,…) and h(x)=h(-x) for all x∈ℓ(B~,p).
Letx,y∈ℓ(B~,p); then by Minkowski’s inequality we have
(15)h(x+y)=[∑k|sk-1(xk-1+yk-1)+rk(xk+yk)|pk]1/M={∑k[|sk-1(xk-1+yk-1)+rk(xk+yk)|pk/M]M}1/M≤(∑k|sk-1xk-1+rkxk|pk)1/M+(∑k|sk-1yk-1+rkyk|pk)1/M=h(x)+h(y).
Let(λn) be a sequence of scalars with λn→λ, as n→∞, and let (x(n))n=0∞ be a sequence of elements x(n)∈ℓ(B~,p) with h(x(n)-x)→0, as n→∞. We observe that
(16)h(λnx(n)-λx)≤h[(λn-λ)(x(n)-x)]+h[λ(x(n)-x)]+h[(λn-λ)x].
It follows from λn→λ(n→∞) that |λn-λ|<1 for all sufficiently large n; hence
(17)limn→∞h[(λn-λ)(x(n)-x)]≤limn→∞h(x(n)-x)=0.
Furthermore, we have
(18)limn⟶∞h[λ(x(n)-x)]≤max{1,|λ|M}limn⟶∞h(x(n)-x)=0.
Also, we have(19)limn⟶∞h[(λn-λ)x]≤limn⟶∞|λn-λ|h(x)=0.
Then, we obtain from (16), (17), (18), and (19) that h(λnx(n)-λx)→0, as n→∞. This shows that h is a paranorm on ℓ(B~,p).
Furthermore, ifh(x)=0, then (∑k‍|sk-1xk-1+rkxk|pk)1/M=0. Therefore |sk-1xk-1+rkxk|pk=0 for each k∈ℕ. If we put k=0, since s-1=0 and r0≠0, we have x0=0. For k=1, since x0=0 we have x1=0. Continuing in this way, we obtain xk=0 for all k∈ℕ. That is, x=θ. This shows that h is a total paranorm.
Now, we show thatℓ(B~,p) is complete. Let {xn} be any Cauchy sequence in ℓ(B~,p) where xn={x0(n),x1(n),x2(n),…}. Here and after, for short we write B~ instead of B(r~,s~). Then for a given ε>0, there exists a positive integer n0(ε) such that h(xn-xm)<ε for all n,m>n0(ε). Since for each fixed k∈ℕ(20)|(B~xn)k-(B~xm)k|≤[∑k|(B~xn)k-(B~xm)k|pk]1/M=h(xn-xm)<ε
for every n,m>n0(ε), {(B~x0)k,(B~x1)k,(B~x2)k,…} is a Cauchy sequence of real numbers for every fixed k∈ℕ. Since ℝ is complete, it converges, say (B~xn)k→(B~x)k as n→∞. Using these infinitely many limits (B~x)0,(B~x)1,(B~x)2,… we define the sequence {(B~x)0,(B~x)1,(B~x)2,…}. For each K∈ℕ and n,m>n0(ε)(21)[∑k=0K|(B~xn)k-(B~xm)k|pk]1/M≤h(xn-xm)<ε.
By letting m,K→∞, we have for n>n0(ε) that
(22)h(xn-x)=[∑k|(B~xn)k-(B~x)k|pk]1/M<ε.
This shows us xn-x∈ℓ(B~,p). Since ℓ(B~,p) is a linear space, we conclude that x∈ℓ(B~,p); It follows that xn→x, as n→∞ in ℓ(B~,p), thus we have shown that ℓ(B~,p) is complete.Therefore, one can easily check that the absolute property does not hold on the spaceℓ(B~,p); that is, g1(x)≠g1(|x|), where |x|=(|xk|). This says that ℓ(B~,p) is the sequence space of nonabsolute type.Theorem 2.
Convergence inℓ(B~,p) is stronger than coordinate-wise convergence.Proof.
First we show thath(xn-x)→0, as n→∞ implies xkn→xk; as n→∞ for every k∈ℕ. We fix k, then we have
(23)limn→∞|sk-1xk-1(n)+rkxk(n)-sk-1xk-1-rkxk|pk≤limn→∞∑k|sk-1xk-1(n)+rkxk(n)-sk-1xk-1-rkxk|pk=limn→∞[h(xn-x)]M=0.
Hence, we have for k=0 that
(24)limn→∞|s-1x-1(n)+r0x0(n)-s-1x-1-r0x0|=0,
which gives the fact that |x0(n)-x0|→0, as n→∞. Similarly, for each k∈ℕ, we have |xk(n)-xk|→0, as n→∞.A sequence spaceλ with a linear topology is called a K-space provided each of the maps pi:λ→ℂ defined by pi(x)=xi is continuous for all i∈ℕ, where ℂ denotes the complex field. A K-space λ is called an FK-space provided λ is complete linear metric space. An FK-space whose topology is normable is called a BK-space. Given a BK-space λ⊃ϕ, we denote the nthsection of a sequencex=(xk)∈λ by x[n]:=∑k=0n‍xke(k), and we say that x=(xk) has the property AK if limn→∞∥x-x[n]∥λ=0. If AK property holds for every x∈λ, then we say that the space λ is called AK-space (cf. [19]). Now, we may give the following.Theorem 3.
(
ℓ
p
)
B
~ is the linear space under the coordinatewise addition and scalar multiplication which is the BK-space with the norm
(25)∥x∥:=(∑k|sk-1xk-1+rkxk|p)1/p,where1≤p<∞.Proof.
Because the first part of the theorem is a routine verification, we omit the detail. Sinceℓp is the BK-space with respect to its usual norm (see [20, pages 217-218]) and B(r~,s~) is a normal matrix, Theorem 4.3.2 of Wilansky [21, page 61] gives the fact that (ℓp)B~ is the BK-space, where 1≤p<∞.Let us suppose that1<pk≤sk for all k∈ℕ. Then, it is known that ℓ(p)⊂ℓ(s) which leads us to the immediate consequence that ℓ(B~,p)⊂ℓ(B~,s).With the notation of (13), define the transformation T from ℓ(B~,p) to ℓ(p) by x↦y=Tx. Since T is linear and bijection, we have the following.Corollary 4.
The sequence spaceℓ(B~,p) of nonabsolute type is linearly paranorm isomorphic to the space ℓ(p), where 0<pk≤H<∞ for all k∈ℕ.Theorem 5.
The spaceℓ(B~,p) has AK.Proof.
For eachx=(xk)∈ℓ(B~,p), we put
(26)x〈m〉=∑k=0mxke(k),∀m∈{1,2,…}.
Let ε>0 and x∈ℓ(B~,p) be given. Then, there is N=N(ε)∈ℕ such that
(27)∑k=N∞‍|sk-1xk-1+rkxk|pk<εM.
Then we have for all m≥N,
(28)h(x-x〈m〉)=h(x-∑k=1m‍xke(k))=(∑k=m+1∞‍|sk-1xk-1+rkxk|pk)1/M≤(∑k=N∞‍|sk-1xk-1+rkxk|pk)1/M<ε.
This shows that x=∑k‍xke(k).
Now we have to show that this representation is unique. We assume thatx=∑k‍λke(k). Then for each k,
(29)(|sk-1λk-1+rkλk-sk-1xk-1-rkxk|pk)1/M≤(∑k‍|sk-1λk-1+rkλk-sk-1xk-1-rkxk|pk)1/M=h(x-x)=0.
Hence, sk-1λk-1+rkλk=sk-1xk-1+rkxk for each k.
Fork=0, r0λ0=r0x0. Since r0≠0, we have λ0=x0.
Fork=1, s0λ0+r1λ1=s0x0+r1x1. Since r1≠0, we also have λ1=x1.
Continuing in this way, we obtainλk=xk for each k. Therefore, the representation is unique.We firstly define the concept of the Schauder basis for a paranormed sequence space and next give the basis of the sequence spaceℓ(B~,p).Let(X,g) be a paranormed space. A sequence (bk) of the elements of X is called a basis for X if and only if, for each x∈X, there exists a unique sequence (αk) of scalars such that
(30)limn→∞g(x-∑k=0nαkbk)=0.
The series ∑k‍αkbk which has the sum x is then called the expansion of x with respect to (bn) and written as x=∑k‍αkbk. Since it is known that the matrix domain λA of a sequence space λ has a basis if and only if λ has a basis whenever A=(ank) is a triangle (cf. [22, Remark 2.4]), we have the following.Corollary 6.
Let0<pk≤H<∞ and αk=(B~x)k for all k∈ℕ. Define the sequence b(k)={bn(k)}n∈ℕ of the elements of the space ℓ(B~,p) by
(31)bn(k):={(-1)n-krn∏‍j=kn-1sjrj,0≤k≤n,0,k>n,
for every fixed k∈ℕ. Then, the sequence {b(k)}k∈ℕ given by (31) is a basis for the space ℓ(B~,p) and any x∈ℓ(B~,p) has a unique representation of the form x:=∑k‍αkb(k).


## 3. The Alpha-, Beta-, and Gamma-Duals of the Spaceℓ(B~,p)

In this section, we state and prove the theorems determining the alpha-, beta-, and gamma-duals of the sequence spaceℓ(B~,p) of nonabsolute type.For the sequence spacesλ and μ, the set S(λ,μ) defined by
(32)S(λ,μ):={z=(zk)∈ω:xz=(xkzk)∈μ∀x=(xk)∈λ}
is called the multiplier space of the spaces λ and μ. With the notation of (32), the alpha-, beta-, and gamma-duals of a sequence space λ, which are, respectively, denoted by λα, λβ, and λγ, are defined by
(33)λα:=S(λ,ℓ1),λβ:=S(λ,cs),λγ:=S(λ,bs).Since the case0<pk≤1 may be established in similar way to the proof of the case 1<pk≤H<∞, we omit the detail of that case and give the proof only for the case 1<pk≤H<∞ in Theorems 10–12 below.We begin with quoting three lemmas which are needed in proving Theorems10–12.Lemma 7 ([23, (i) and (ii) of Theorem 1]).
LetA=(ank) be an infinite matrix. Then, the following statements hold.(i)
Let0<pk≤1 for all k∈ℕ. Then, A∈(ℓ(p):ℓ∞) if and only if
(34)supn,k∈ℕ|ank|pk<∞.(ii)
Let1<pk≤H<∞ for all k∈ℕ. Then, A∈(ℓ(p):ℓ∞) if and only if there exists an integer M>1 such that
(35)supn∈ℕ∑k‍|ankM-1|pk′<∞.Lemma 8 ([23, Corollary for Theorem 1]).
Let0<pk≤H<∞ for all k∈ℕ. Then, A=(ank)∈(ℓ(p):c) if and only if (34) and (35) hold, and
(36)limn→∞ank=βk,∀k∈ℕ.Lemma 9 ([24, Theorem 5.1.0]).
LetA=(ank) be an infinite matrix. Then, the following statements hold(i)
Let0<pk≤1 for all k∈ℕ. Then, A∈(ℓ(p):ℓ1) if and only if
(37)supN∈ℱsupk∈ℕ|∑n∈Nank|pk<∞.(ii)
Let1<pk≤H<∞ for all k∈ℕ. Then, A∈(ℓ(p):ℓ1) if and only if there exists an integer M>1 such that
(38)supN∈ℱ∑k|∑n∈NankM-1|pk′<∞.Theorem 10.
Define the setsS1(p) and S2(p) by
(39)S1(p)=⋃M>1‍{|∑n∈N(-1)n-krn∏j=kn-1‍sjrjanM-1|pk′a=(ak)∈ω:⋃B>1‍supN∈ℱ∑k|∑n∈N(-1)n-krn∏j=kn-1‍sjrjanM-1|pk′<∞|∑n∈N(-1)n-krn∏j=kn-1‍sjrjanM-1|pk′},S2(p)={a=(ak)∈ω:supN∈ℱsupk∈ℕ|∑n∈N(-1)n-krn∏j=kn-1sjrjan|pk<∞}.
Then,
(40){ℓ(B~,p)}α={S1(p),1<pk≤H<∞,∀k∈ℕ,S2(p),0<pk≤1,∀k∈ℕ.Proof.
Let us take anya=(an)∈ω. By using (13) we obtain that
(41)xn=∑k=0n(-1)n-krn∏j=kn-1sjrjyk
holds for all n∈ℕ which leads us to
(42)anxn=∑k=0n(-1)n-krn∏j=kn-1sjrjanyk=(Cy)n,(n∈ℕ),
where C=(cnk) is defined by
(43)cnk={(-1)n-krn∏‍j=kn-1sjrjan,0≤k≤n,0,k>n
for all k,n∈ℕ. Thus, we observe by combining (42) with the condition (37) of Part (i) of Lemma 9 that ax=(anxn)∈ℓ1 whenever x=(xk)∈ℓ(B~,p) if and only if Cy∈ℓ1 whenever y=(yk)∈ℓ(p). That means {ℓ(B~,p)}α=S1(p).Theorem 11.
Define the setsS3(p), S4(p), and S5(p) by
(44)S3(p)=⋃M>1‍{a=(ak)∈ω:supn∈ℕ∑k‍|∑i=kn‍(-1)i-kri∏j=ki-1‍sjrjaiM-1|pk′<∞},S4(p)={a=(ak)∈ω:∑i=k∞‍(-1)i-kri∏j=ki-1‍sjrjai<∞},S5(p)={a=(ak)∈ω:supn,k∈ℕ|∑i=kn(-1)i-kri∏j=ki-1sjrjai|pk<∞}.
Then,
(45){ℓ(B~,p)}β={S3(p)∩S4(p),1<pk≤H<∞∀k∈ℕ,S4(p)∩S5(p),0<pk≤1∀k∈ℕ.Proof.
Take anya=(ai)∈ω and consider the equation obtained with (13) that
(46)∑i=0naixi=∑i=0n[∑k=0i(-1)i-kri∏j=ki-1‍sjrjyk]ai=∑k=0n[∑i=kn(-1)i-kri∏j=ki-1‍sjrjai]yk=(Dy)n,
where D=(dnk) is defined by
(47)dnk={∑i=kn(-1)i-kri∏j=ki-1‍sjrjai,0≤k≤n,0,k>n
for all k,n∈ℕ. Thus, we deduce from Lemma 8 with (46) that ax=(aixi)∈cs whenever x=(xi)∈ℓ(B~,p) if and only if Dy∈c whenever y=(yk)∈ℓ(p). Therefore, we derive from (35) and (36) that
(48)supn∈ℕ∑k|∑i=kn(-1)i-kri∏j=ki-1‍sjrjaiM-1|pk′<∞,∑i=k∞(-1)i-kri∏j=ki-1‍sjrjai<∞.
This shows that {ℓ(B~,p)}β=S3(p)∩S4(p).Theorem 12.
(49)
{
ℓ
(
B
~
,
p
)
}
γ
=
{
S
3
(
p
)
,
1
<
p
k
≤
H
<
∞
,
∀
k
∈
ℕ
,
S
5
(
p
)
,
0
<
p
k
≤
1
,
∀
k
∈
ℕ
.Proof.
From Lemma7 and (46), we obtain that ax=(aixi)∈bs whenever x=(xi)∈ℓ(B~,p) if and only if Dy∈ℓ∞ whenever y=(yk)∈ℓ(p), where D=(dnk) is defined by (47). Therefore, we obtain from (34) and (35) that {ℓ(B~,p)}γ=S3(p) for 1<pk, {ℓ(B~,p)}γ=S5(p) for pk≤1.


## 4. Matrix Transformations on the Sequence Spaceℓ(B~,p)

In this section, we characterize some matrix transformations on the spaceℓ(B~,p). Theorem 13 gives the exact conditions of the general case 0<pk≤H<∞ by combining the cases 0<pk≤1 and 1<pk≤H<∞. We consider only the case 1<pk≤H<∞ and leave the case 0<pk≤1 to the reader because it can be proved in similar way.Theorem 13.
LetA=(ank) be an infinite matrix. Then, the following statements hold.(i)
Let1<pk≤H<∞ for all k∈ℕ. Then, A∈(ℓ(B~,p):ℓ∞) if and only if there exists an integer M>1 such that
(50)supn∈ℕ∑k|∑i=kn(-1)i-kri∏j=ki-1‍sjrjaniM-1|pk′<∞,(51)∑i=k∞(-1)i-kri∏j=ki-1sjrjani<∞.(ii)
Let0<pk≤1 for all k∈ℕ. Then, A∈(ℓ(B~,p):ℓ∞) if and only if the condition (51) holds, and
(52)supn,k∈ℕ|∑i=kn(-1)i-kri∏j=ki-1sjrjani|pk<∞.Proof.
Suppose that the conditions (50) and (51) hold, and x∈ℓ(B~,p). In this situation, since {ank}k∈ℕ∈{ℓ(B~,p)}β for every fixed n∈ℕ, the A-transform of x exists. Consider the following equality obtained by using the relation (13) that
(53)∑k=0mankxk=∑k=0m∑i=km‍(-1)i-kri∏j=ki-1sjrjaniyk
for all m,n∈ℕ. Taking into account the hypothesis we derive from (53) as m→∞ that
(54)∑kankxk=∑k∑i=k∞(-1)i-kri∏j=ki-1sjrjaniyk,foreachn∈ℕ.
Now, by combining (54) with the following inequality (see [23]) which holds for any M>0 and any a,b∈ℂ(55)|ab|≤M(|aM-1|p′+|b|p),
where p>1 and p-1+p′-1=1, one can easily see that
(56)supn∈ℕ|∑kankxk|≤supn∈ℕ∑k|∑i=k∞(-1)i-kri∏j=ki-1sjrjani||yk|≤supn∈ℕ∑kM(|∑i=k∞1ri∏j=ki-1sjrjaniM-1|pk′+|yk|pk)≤M(supn∈ℕ∑k|∑i=k∞1ri∏j=ki-1sjrjaniM-1|pk′+∑k|yk|pk)<∞.
Conversely, suppose thatA∈(ℓ(B~,p):ℓ∞) and 1<pk≤H<∞ for all k∈ℕ. Then Ax exists for every x∈ℓ(B~,p) and this implies that {ank}k∈ℕ∈{ℓ(B~,p)}β for all n∈ℕ. Now, the necessity of (51) is immediate. Besides, we have from (54) that the matrix B=(bnk) defined by bnk=∑i=k∞((-1)i-k/ri)∏j=ki-1(sj/rj)ani for all n,k∈ℕ, is in the class (ℓ(p):ℓ∞). Then, B satisfies the condition (35) which is equivalent to (50).
This completes the proof.Lemma 14 ([25, Theorem 1]).
A
=
(
a
n
k
)
∈
(
ℓ
(
p
)
:
f
) if and only if (34) and (35) hold, and
(57)∃αk∈ℂ∋f-limank=αkforeveryfixedk∈ℕ.Theorem 15.
Let the entries of the matricesE=(enk) and F=(fnk) be connected with the relation
(58)enk:=sk-1fn,k-1+rkfnkorfnk:=∑i=k∞(-1)iri∏j=ki-1sjrjeni
for all k,n∈ℕ. Then, E∈(ℓ(B~,p):f) if and only if F∈(ℓ(p):f) and
(59)Fn∈(ℓ(p):c)
for every fixed n∈ℕ, where Fn=(fmk(n)) with
(60)fmk(n):={∑‍mi=k(-1)iri∏‍j=ki-1sjrjeni,0≤k≤m,0,k>m,
for all m,k∈ℕ.Proof.
LetE=(enk)∈(ℓ(B~,p):f) and take x∈ℓ(B~,p). Then, we obtain the equality
(61)∑k=0menkxk=∑k=0menk[∑i=0k(-1)k-iri∏j=ik-1sjrjyi]=∑k=0m[∑i=km‍(-1)iri∏j=ki-1sjrj]yk=∑k=0mfmk(n)yk
for all m,n∈ℕ. Since Ex exists, Fn∈(ℓ(p):c). Letting m→∞ in the equality (61) we have Ex=Fy. Since Ex∈f, then Fy∈f. That is F∈(ℓ(p):f).
Conversely, letF∈(ℓ(p):f), and Fn∈(ℓ(p):c), and take x∈ℓ(B~,p). Then, since (fnk)k∈ℕ∈{ℓ(p)}β and F∈(ℓ(p):f) we have (enk)k∈ℕ∈{ℓ(B~,p)}β for all n∈ℕ. So, Ex exists. Therefore we obtain from equality (61) as m→∞ that Ex=Fy, that is E∈(ℓ(B~,p):f).Theorem 16.
Let0<pk≤H<∞ for all k∈ℕ. Then, A∈(ℓ(B~,p):c) if and only if (50)–(52) hold and
(62)limn→∞∑k=i∞(-1)k-irk∏j=ik-1sjrjank=αk,foreveryfixedk∈ℕ.Proof.
LetA∈(ℓ(B~,p):c) and 1<pk≤H<∞ for all k∈ℕ. Then, since the inclusion c⊂ℓ∞ holds, the necessities of (50) and (51) are immediately obtained from part (i) of Theorem 13.
To prove the necessity of (62), consider the sequence b(k) defined by (31) which is in the space ℓ(B~,p) for every fixed k∈ℕ. Because the A-transform of every x∈ℓ(B~,p) exists and is in c by the hypothesis,
(63)Ab(k)={∑k=i∞(-1)k-irk∏j=ik-1sjrjank}n∈ℕ∈c
for every fixed k∈ℕ which shows the necessity of (62).
Conversely suppose that conditions (50), (51), and (62) hold, and take any x=(xk) in the space ℓ(B~,p). Then, Ax exists. We observe for all m,n∈ℕ that
(64)∑k=0m|∑k=im(-1)k-irk∏j=ik-1sjrjankM-1|pk′≤supn∈ℕ∑km|∑k=im(-1)k-irk∏j=ik-1sjrjankM-1|pk′<∞,
which gives the fact that by letting m,n→∞ with (50) and (62) that
(65)limm,n⟶∞∑k=0m|∑k=im(-1)k-irk∏j=ik-1sjrjankM-1|pk′≤supn∈ℕ∑km|∑k=im(-1)k-irk∏j=ik-1sjrjankM-1|pk′<∞.
This shows that ∑k‍|αkM-1|pk′<∞ and so (αk)k∈ℕ∈{ℓ(B~,p)}β which implies that the series ∑k‍αkxk converges for every x∈ℓ(B~,p).
Let us now consider the equality obtained from (54) with ank-αk instead of ank(66)∑k(ank-αk)xk=∑i∑k=i(-1)k-irk∏j=ik-1sjrj(ank-αk)yi=∑kcniyi,∀n∈ℕ,
where C=(cni) defined by cni=∑k=i‍((-1)k-i/rk)∏j=ik-1(sj/rj)(ank-αk) for all n,i∈ℕ. Therefore, we have at this stage from Lemma 8 that the matrix C belongs to the class (ℓ(p):c0) of infinite matrices. Thus, we see by (66) that
(67)limn→∞∑k(ank-αk)xk=0.
Equation (67) means that Ax∈c whenever x∈ℓ(B~,p) and this is what we wished to prove.Therefore, we have the followingCorollary 17.
Let0<pk≤H<∞ for all k∈ℕ. Then, A∈(ℓ(B~,p):c0) if and only if (50)–(52) hold, and (62) also holds with αk=0 for all k∈ℕ.Now, we give the following lemma given by Başar and Altay [26] which is useful for deriving the characterizations of the certain matrix classes via Theorems 13, 15, and 16 and Corollary 17.Lemma 18 ([26, Lemma 5.3]).
Letλ, μ be any two sequence spaces, let A be an infinite matrix, and let B also be a triangle matrix. Then, A∈(λ:μB) if and only if BA∈(λ:μ).It is trivial that Lemma18 has several consequences. Indeed, combining Lemma 18 with Theorems 13, 15, and 16 and Corollary 17, one can derive the following results.Corollary 19.
LetA=(ank) be an infinite matrix and define the matrix C=(cnk) by
(68)cnk=∑j=0n(nj)(1-t)n-jtjajk,∀n,k∈ℕ.
Then, the necessary and sufficient conditions in order to A belongs to anyone of the classes (ℓ(B~,p):e∞t), (ℓ(B~,p):ect) and (ℓ(B~,p):e0t) are obtained from the respective ones in Theorems 13, 16 and Corollary 17 by replacing the entries of the matrix A by those of the matrix C; where 0<t<1, e∞t and ect, e0t, respectively, denote the spaces of all sequences whose Et-transforms are in the spaces ℓ∞ and c, c0 and are recently studied by Altay et al. [27] and Altay and Başar [28], where Et denotes the Euler mean of order t.Corollary 20.
LetA=(ank) be an infinite matrix and define the matrix C=(cnk) by
(69)cnk=san-1,k+rank,∀n,k∈ℕ.Then, the necessary and sufficient conditions in order to A belongs to the class (ℓ(B~,p):f^) is obtained from Theorem 15 by replacing the entries of the matrix A by those of the matrix C; where r,s∈ℝ∖{0} and f^ denotes the space of all sequences whose B(r,s)-transforms are in the space f and is recently studied by Başar and Kirişçi [29].Corollary 21.
LetA=(ank) be an infinite matrix and define the matrix C=(cnk) by
(70)cnk=tan-2,k+san-1,k+rank,∀n,k∈ℕ.Then, the necessary and sufficient conditions in order to A belongs to the class (ℓ(B~,p):f(B)) is obtained from Theorem 15 by replacing the entries of the matrix A by those of the matrix C; where r,s,t∈ℝ∖{0} and f(B) denotes the space of all sequences whose B(r,s,t)-transforms are in the space f and is recently studied by Sönmez [30].Corollary 22.
LetA=(ank) be an infinite matrix and define the matrix C=(cnk) by
(71)cnk=1n+1∑j=0najk,∀n,k∈ℕ.Then, the necessary and sufficient conditions in order to A belongs to the class (ℓ(B~,p):f~) is obtained from Theorem 15 by replacing the entries of the matrix A by those of the matrix C, where f~ denotes the space of all sequences whose C1-transforms are in the space f and is recently studied by Kayaduman and Şengönül [31].Corollary 23.
LetA=(ank) be an infinite matrix and let t=(tk) be a sequence of positive numbers and define the matrix C=(cnk) by
(72)cnk=1Tn∑j=0ntjajk,∀n,k∈ℕ,
where Tn=∑k=0n‍tk for all n∈ℕ. Then, the necessary and sufficient conditions in order to A belongs to anyone of the classes (ℓ(B~,p):r∞t), (ℓ(B~,p):rct) and (ℓ(B~,p):r0t) are obtained from the respective ones in Theorems 13, 16 and Corollary 17 by replacing the entries of the matrix A by those of the matrix C, where r∞t, rct, and r0t are defined by Altay and Başar in [32] as the spaces of all sequences whose Rt-transforms are, respectively, in the spaces ℓ∞, c, and c0, and are derived from the paranormed spaces r∞t(p), rct(p) and r0t(p) in the case pk=p for all k∈ℕ.Since the spacesr∞t, rct, and r0t reduce in the case t=e to the Cesàro sequence spaces X∞, c~, and c~0 of nonabsolute type, respectively, Corollary 23 also includes the characterizations of the classes (ℓ(B~,p):X∞), (ℓ(B~,p):c~), and (ℓ(B~,p):c~0), as a special case, where X∞ and c~, c~0 are the Cesàro spaces of the sequences consisting of C1-transforms are in the spaces ℓ∞ and c, c0 and studied by Ng and Lee [33] and Şengönül and Başar [34], respectively, where C1 denotes the Cesàro mean of order 1.Corollary 24.
LetA=(ank) be an infinite matrix and define the matrix C=(cnk) by cnk=ank-an+1,kfor all n,k∈ℕ. Then, the necessary and sufficient conditions in order to A belongs to anyone of the classes (ℓ(B~,p):ℓ∞(Δ)), (ℓ(B~,p):c(Δ)) and (ℓ(B~,p):c0(Δ)) are obtained from the respective ones in Theorems 13 and 16 and Corollary 17 by replacing the entries of the matrix A by those of the matrix C, where ℓ∞(Δ), c(Δ), c0(Δ) denote the difference spaces of all bounded, convergent, and null sequences and are introduced by Kızmaz [35].Corollary 25.
LetA=(ank) be an infinite matrix and define the matrix C=(cnk) by cnk=∑j=0n‍ajk for all n,k∈ℕ. Then the necessary and sufficient conditions in order to A belongs to anyone of the classes (ℓ(B~,p):bs), (ℓ(B~,p):cs) and (ℓ(B~,p):cs0) are obtained from the respective ones in Theorems 13, 16 and Corollary 17 by replacing the entries of the matrix A by those of the matrix C, where cs0 denotes the set of those series converging to zero.


## 5. Conclusion

The difference spacesℓ∞(Δ), c(Δ), and c0(Δ) were introduced by Kızmaz [35]. Since we essentially employ the infinite matrices which is more different than Kızmaz and the other authors following him, and use the technique of obtaining a new sequence space by the matrix domain of a triangle limitation method. Following this way, the domain of some triangle matrices in the sequence space ℓ(p) was recently studied and were obtained certain topological and geometric results by Altay and Başar [14, 16], Choudhary and Mishra [10], Başar et al. [36], and Aydın and Başar [13]. Although bv(e,p)=[ℓ(p)]Δ is investigated, since B(1,-1)≡Δ, our results are more general than those of Başar et al. [36]. Also in case pk=p for all k∈ℕ the results of the present study are reduced to the corresponding results of the recent paper of Kirişçi and Başar [9]. We should note that the difference spaces Δc0(p), Δc(p) and Δℓ∞(p) of Maddox’s spaces c0(p), c(p), and ℓ∞(p)  were studied by Ahmad and Mursaleen [37]. Of course, a natural continuation of the present paper is to study the sequence spaces [c0(p)]B(r~,s~), [c(p)]B(r~,s~) and [ℓ∞(p)]B(r~,s~) to generalize the main results of Ahmad and Mursaleen [37] which fills up a gap in the existing literature.It is clear thatΔ(1) can be obtained as a special case of B(r~,s~) for r~=e and s~=-e and it is also trivial that B(r~,s~) is reduced in the special case r~=re and s~=se to the generalized difference matrix B(r,s). So, the results related to the domain of the matrix B(r~,s~) are much more general and more comprehensive than the corresponding consequences of the domain of the matrix B(r,s). We should note from now that the main results of the present paper are given as an extended abstract without proof by Nergiz and Başar [38], and our next paper will be devoted to some geometric and topological properties of the space ℓ(B~,p).



---

*Source: 949282-2013-04-16.xml*