# On Generalized Difference Hahn Sequence Spaces

Authors: Kuldip Raj; Adem Kiliçman
Journal: The Scientific World Journal
 (2014)

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## Abstract

We construct some generalized difference Hahn sequence spaces by mean of sequence of modulus functions. The topological properties and some inclusion relations of spaceshpℱ,u,Δr are investigated. Also we compute the dual of these spaces, and some matrix transformations are characterized.


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## Body

## 1. Introduction and Preliminaries

By a sequence space, we understand a linear subspace of the spacew=ℂℕ of all real or complex-valued sequences, where ℂ denotes the complex field and ℕ=0,1,2,…. For x=(xk)∈w, we write l∞, c, and c0 for the classical spaces of all bounded, convergent, and null sequences, respectively. Also by bs, cs, and lp we denote the space of all bounded, convergent, and p-absolutely convergent series, which are Banach spaces with the following norms: ∥x∥bs=∥x∥cs=sup⁡n|∑k=1n‍xk| and ∥x∥lp=(∑k‍|xk|p)1/p, respectively. Additionally, the spaces bvp and ∫λ are defined by
(1)bvp={x=(xk)∈w:∑k=1∞‍|xk-xk-1|p<∞},∫λ={x=(xk)∈w:(kxk)∈λ}.
A coordinate space (or a K-space) is a vector space of numerical sequences, where addition and scalar multiplication are defined pointwise. That is, a sequence space λ with a linear topology is called a K-space provided that each of the maps pi:λ→ℂ defined by pi(x)=xi is continuous for all i∈ℕ. A BK-space is a K-space, which is also a Banach space with continuous coordinate functionals fk(x)=xk, (k=1,2,…). A K-space λ is called an FK-space provided that λ is a complete linear metric space. An FK-space whose topology is normable is called a BK-space. If a normed sequence space λ contains a sequence (bn) with the property that for every x∈λ there is a unique sequence of scalars (αn) such that
(2)lim⁡n→∞∥x-(α0b0+α1b1+⋯+αnbn)∥=0,
then (bn) is called Schauder basis (or briefly basis) for λ. The series ∑αkbk which has the sum x is then called the expansion of x with respect to (bn) and written as x=∑αkbk. An FK-space λ is said to have AK property, if ϕ⊂λ and {ek} is a basis for λ, where ek is a sequence whose only nonzero term is 1 in kth place for each k∈ℕ and ϕ=span⁡{ek}, the set of all finitely nonzero sequences. If ϕ is dense in λ, then λ is called an AD-space, and thus AK implies AD.The notion of difference sequence spaces was introduced by Kizmaz [1], who defined the sequence spaces as follows:
(3)Z(Δ)={x=(xk)∈w:(Δxk)∈Z}mmfmmmforZ=c,c0,l∞,
where Δx=(Δxk)=(xk-xk+1). The notion was further generalized by Et and Çolak [2] by introducing the spaces. Let r be a nonnegative integer; then,
(4)Z(Δr)={x=(xk)∈w:(Δrxk)∈Z}mmmmmmmforZ=c,c0,l∞,
where Δrx=(Δrxk)=(Δr-1xk-Δr-1xk+1) and Δ0xk=xk for all k∈ℕ. The generalized difference sequence has the following binomial representation:
(5)Δrxk=∑m=0r‍(-1)m(rm)xk+m.
Later concept have been studied by Bektaş et al. [3] and Et and Esi [4]. Another type of generalization of the difference sequence spaces is due to Tripathy and Esi [5] who studied the spaces l∞(Δv), c(Δv), and c0(Δv). Recently, Esi et al. [6] and Tripathy et al. [7] have introduced a new type of generalized difference operators and unified those as follows.Letr, v be nonnegative integers; then, for Z a given sequence space, we have
(6)Z(Δvr)={x=(xk)∈w:(Δvrxk)∈Z}
for Z=c, c0 and l∞, where Δvrx=(Δvrxk)=(Δvr-1xk-Δvr-1xk+v) and Δv0xk=xk for all k∈ℕ.LetX be a linear metric space. A function p:X→ℝ is called paranorm, if(1)
p(x)≥0 for all x∈X,(2)
p(-x)=p(x) for all x∈X,(3)
p(x+y)≤p(x)+p(y) for all x, y∈X,(4)
(λn) is a sequence of scalars with λn→λ as n→∞ and (xn) is a sequence of vectors with p(xn-x)→0 as n→∞, then p(λnxn-λx)→0 as n→∞.A paranorm p for which p(x)=0 implies x=0 is called total paranorm and the pair (X,p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [8], Theorem 10.4.2, pp. 183). For more details about sequence spaces (see [9, 10]) and the references therein.A modulus function is a functionf:[0,∞)→[0,∞) such that(1)
f(x)=0 if and only if x=0,(2)
f(x+y)≤f(x)+f(y), for all x,y≥0,(3)
f is increasing,(4)
f is continuous from the right at 0.It follows that f must be continuous everywhere on [0,∞). The modulus function may be bounded or unbounded. For example, if we take f(x)=x/(x+1), then f(x) is bounded. If f(x)=xp, 0<p<1, then the modulus function f(x) is unbounded. Subsequentially, modulus function has been discussed in ([11–14]) and references therein.Letλ and μ be two sequence spaces and A=(ank) be an infinite matrix of complex numbers, where k, n∈ℕ. Then, we say that A defines a matrix mapping from λ into μ, and we denote it by writing A:λ→μ for every sequence x=(xk)∈λ. The sequence Ax={(Ax)n}, the A-transform of x, is in μ, where
(7)(Ax)n=∑k‍ankxkfor  eachn∈ℕ.
For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. 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 (7) converges for each n∈ℕ and each 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 l if Ax converges to l which is called the A-limit of x.The matrix domainλA of an infinite matrix A in a sequence space λ is defined by
(8)λA={x=(xk)∈w:Ax∈λ}
which is a sequence space (for several examples of matrix domains, see [15] p. 49–176). In [16], Başar and Altay have defined the sequence space bvp which consists of all sequences such that Δ-transforms of them are in lp, where Δ denotes the matrix Δ=(δnk) as follows:
(9)Δ=δnk={(-1)n-k,(n-1≤k≤n);0,(0≤k<n-1ork>n)
for all k, n∈ℕ. The space [l(p)]Au=bv(u,p) has been studied by Başar et al. [17], where
(10)Au=anku={(-1)n-kuk,(n-1≤k≤n);0,(0≤k<n-1ork>n).
Hahn [18] introduced the BK-space h of all sequences x=(xk) such that
(11)h={x:∑k=1∞‍k|Δxk|<∞,lim⁡k→∞xk=0},
where Δxk=xk-xk+1, for all k∈ℕ. The following norm:
(12)∥x∥h=∑k‍k|Δxk|+sup⁡k|xk|
was defined on the space h by Hahn [18] (and also [19]). Rao ([20], Proposition 2.1) defined a new norm on h as ∥x∥=∑kk|Δxk|. G. Goes and S. Goes [19] proved that the space h is a BK-space.Hahn proved the following properties of the spaceh.Lemma 1.
(i)h is a Banach space.
(ii)h⊂l1∩∫c0.
(iii)hβ=σ∞.In [19], G. Goes and S. Goes studied functional analytic properties of the BK-space bv0∩dl1. Additionally, G. Goes and S. Goes considered the arithmetic means of sequences in bv0 and bv0∩dl1 and used an important fact which the sequence of arithmetic means (n-1∑k=1n‍xk) of x∈bv0 is a quasiconvex null sequence. And also G. Goes and S. Goes proved that h=l1∩∫bv=l1∩∫bv0.Rao [20] studied some geometric properties of Hahn sequence space and gave the characterizations of some classes of matrix transformations. Balasubramanian and Pandiarani [21] defined the new sequence space h(F) called the Hahn sequence space of fuzzy numbers and proved that β and γ duals of h(F) is the Cesàro space of the set of all fuzzy bounded sequences. Kirişci [22] compiled to studies on Hahn sequence space and defined a new Hahn sequence space by Cesàro mean in [23].In [24], Kirişci introduce the sequence space hp by
(13)hp={x:∑k=1∞‍k|Δxk|p<∞,lim⁡k→∞xk=0},(1<p<∞),
where Δxk=xk-xk+1, for all k∈ℕ. If we take p=1, hp=h which are called Hahn sequence spaces. We denote the collection of all finite subsets of ℕ by F.Letℱ=(fk) be a sequence of modulus functions, p=(pk) be a bounded sequence of positive real numbers, and u=(uk) be a sequence of strictly positive real numbers. In the present paper we defined the following sequence space:
(14)hp(ℱ,u,Δr)={x:∑k=1∞‍fk[(k|ukΔrxk|)pk]<∞,lim⁡k→∞xk=0},mmmmmmmmmmmmmmmmmmmm(1<p<∞),
where Δrx=(Δrxk)=(Δr-1xk-Δr-1xk+1) and Δ0xk=xk for all k∈ℕ. Define the sequence y=(yk), which will be frequently used, by M-transform of a sequence x=(xk); that is,
(15)yk=(Mx)k=fkukk(Δr-1xk-Δr-1xk+1),
where M=(mnk) with
(16)mnk={n,(n=k);-n,(n+1=k);0,other
for all k, n∈ℕ.If we takeu=(uk)=1, r=1 and ℱ=fk(x)=x for all k∈ℕ, then we get the sequence space hp defined by [24] Kirişci. By taking u=(uk)=1, r=1,ℱ=fk(x)=x, and p=(pk)=1 for all k∈ℕ, we obtained a Hahn sequence space defined by Hanh [18].The following inequality will be used throughout the paper.Letp=(pk) be a sequence of positive real numbers with 0<pk≤sup⁡kpk=H, and let K=max⁡{1,2H-1}. Then, for the factorable sequences (ak) and (bk) in the complex plane, we have
(17)|ak+bk|pk≤K(|ak|pk+|bk|pk).
The main purpose of this paper is to study some difference Hahn sequence spaces by mean of sequence of modulus functions. We will study some topological and algebraic properties of the sequence spaces hp(ℱ,u,Δr) in Section 2. In Section 3 we will determine the α-, β-, and γ-duals of the spaces hp(ℱ,u,Δr). Finally, we also made an attempt to characterize some matrix transformations on the spaces hp(ℱ,u,Δr).


## 2. Main Results

The purpose of this section is to study the properties like linearity, paranorm, and relevant inclusion relations in the spaceshp(ℱ,u,Δr).Theorem 2.
The sequence spacehp(ℱ,u,Δr) is a linear space over the complex field ℂ.Proof.
Letx=(xk), y=(yk)∈hp(ℱ,u,Δr) and ρ, ϱ∈ℂ. Then their exist integers Mρ and Nϱ such that |ρ|≤Mρ and |ϱ|≤Nϱ. By using the inequality (17) and the properties of modulus function, we have
(18)∑k=1∞‍fk[(k|ukΔr(ρxk+ϱyk)|)pk]≤∑k=1∞‍fk[(k|uk(ρΔrxk+ϱΔryk)|)pk]≤K∑k=1∞‍fk[Mρ(k|ukΔrxk|)pk]+K∑k=1∞‍fk[Nϱ(k|ukΔryk|)pk]≤KMρH∑k=1∞‍fk[(k|ukΔrxk|)pk]+KNϱH∑k=1∞‍fk[(k|ukΔryk|)pk]<∞.
Thus, ρx+ϱy∈hp(ℱ,u,Δr). This proves that hp(ℱ,u,Δr) is a linear space over the field of complex number ℂ.Theorem 3.
Letℱ=(fk) be a sequence of modulus functions and u=(uk) be any sequence of strictly positive real numbers. Then hp(ℱ,u,Δr) is a paranormed space with the paranorm defined by
(19)g(x)=sup⁡k{fk[(k|ukΔrxk|)pk]}1/M,
where H=sup⁡kpk<∞ and M=max⁡(1,H).Proof.
Clearlyg(x)=g(-x) for all x∈hp(ℱ,u,Δr). It is trivial that ukΔrxk=0 for x=0. Since pk/M≤1 using Minkowski’s inequality, we have
(20){fk[(k|ukΔr(xk+yk)|)pk]}1/M≤{fk[(k|(ukΔrxk+ukΔryk)|)pk]}1/M≤{fk[(k|ukΔrxk|)pk]}1/M+{fk[(k|ukΔryk|)pk]}1/M.
Hence g(x+y)≤g(x)+g(y). Finally to check the continuity of scalar multiplication, let us take a complex number δ by definition, we have
(21)g(δx)=sup⁡k{fk[(k|ukΔrδxk|)pk]}1/M≤KδH/Mg(x),
where Kδ is a positive integer such that |δ|≤Kδ. Let δ→0 for any fixed x with g(x)=0. By definition for |δ|<1, we have
(22)sup⁡k{fk[(k|ukΔrxk|)pk]}1/M<ϵforn>N(ϵ).
Also for 1≤n≤N, taking δ small enough, since fk is continuous for each k, we have
(23)sup⁡k{fk[(k|ukΔrxk|)pk]}1/M<ϵ.
Equations (22) and (23) imply that g(δx)→0 as δ→0. This completes the proof.Theorem 4.
Letℱ=(fk) be a sequence of modulus functions and ϕ=lim⁡t→∞(fk(t)/t)>0. Then hp(ℱ,u,Δr)⊂hp(u,Δr).Proof.
Letϕ>0. By definition of ϕ, we have fk(t)≥ϕ·t, for all t≥0. Since ϕ>0, we have t≤(1/ϕ)fk(t) for all t≥0. Let x=(xk)∈hp(ℱ,u,Δr). Thus, we have
(24)∑k=1∞‍[(k|ukΔrxk|)pk]≤1ϕ∑k=1∞‍fk[(k|ukΔrxk|)pk]<∞.
Which implies that x=(xk)∈hp(u,Δr). This completes the proof.Theorem 5.
h
p
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ℱ
,
u
,
Δ
r
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=
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p
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Δ
r
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∩
∫
b
v
p
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∩
∫
‍
b
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0
p
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,
u
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r
).Proof.
We consider(25)fkuk(kΔrxk)≤fkuk(xk)+fkuk(Δr(kxk)).
Then, for x∈lp(ℱ,u,Δr)∩∫‍bvp(ℱ,u,Δr),
(26)∑k=1n‍fkuk(k|Δrxk|)≤∑k=1n‍fkuk|xk|+∑k=1n‍fkuk|Δr(kxk)|,
and from |a+b|pk≤2pk(|a|pk+|b|pk), (1≤pk<∞), we obtain
(27)∑k=1n‍fkuk[kpk|Δrxk|pk]≤2pkfkuk[∑k=1n‍|xk|pk+∑k=1n‍|Δr(kxk)|pk].
For each positive integer s, we get
(28)∑k=1s‍fkuk[kpkΔrxk|pk]≤2pkfkuk[∑k=1s‍|xk|pk+∑k=1s‍|Δrkxk|pk],
and as s→∞,
(29)∑k=1∞‍fkuk[kpk|Δrxk|pk]≤2pkfkuk[∑k=1∞‍|xk|pk+∑k=1∞‍|Δr(kxk)|pk]
and lim⁡k→∞xk=0. Then x∈hp(ℱ,u,Δr) and
(30)lp(ℱ,u,Δr)∩∫‍bvp(ℱ,u,Δr)⊂hp(ℱ,u,Δr).
Let x∈hp(ℱ,u,Δr), and we consider
(31)∑k=1∞‍fkuk|xk+1|pk-∑k=1∞‍fkuk|Δr(kxk)|pk≤∑k=1∞‍fkukkpk|Δr(kxk)|pk.
Then the series ∑k=1∞‍fkuk|xk+1|pk is convergent from the definition of lp. Also, ∑k=1∞‍fkuk|Δr(kxk)|pk<∞, and therefore x∈lp(ℱ,u,Δr)∩∫‍bvp(ℱ,u,Δr).
Then,(32)hp(ℱ,u,Δr)⊂lp(ℱ,u,Δr)∩∫‍bvp(ℱ,u,Δr).
From (30) and (32), we have
(33)hp(ℱ,u,Δr)=lp(ℱ,u,Δr)∩∫‍bvp(ℱ,u,Δr).Theorem 6.
The sequence spacehp(ℱ,u,Δr) is a BK-space with AK.Proof.
Ifx is any sequence, we write σn(x)=Mnx. Let ϵ>0 and x∈hp(ℱ,u,Δr). Then, their exists N such that
(34)σn(x)<ϵ2
for all n≥N. Now let m≥N be given. Then, we have for all n≥m+1 by (34)
(35)|σn(x-x[m])|≤[∑k=m+1∞‍fk(|kukΔrxk|pk)]1/pk≤|σn(x)|+|σm(x)|<ϵ2+ϵ2=ϵ,
whence ∥x-x[m]∥hp(ℱ,u,Δr)≤ϵ for all m>N. This shows that x=lim⁡m→∞x[m].Sincehp(ℱ,u,Δr) is anAK-space and everyAK-space isAD, we can give the following corollary.Corollary 7.
The sequence spacehp(ℱ,u,Δr) is AD.


## 3. Duals of Hahn Sequence Spacehp(ℱ,u,Δr)

In this section, we determining theα, β, and γ-duals of the sequence space hp(ℱ,u,Δr). Let x and y be sequences, X and Y be subsets of w, and A=(ank)n,k=0∞ be an infinite matrix of complex numbers. We write xy=(xkyk)k=0∞,x-1*Y={a∈w:ax∈Y}, and M(X,Y)=∩x∈Xx-1*Y = {a∈w:ax∈Yfor  allx∈X} for the multiplier space of X and Y. In the special cases of Y={l1,cs,bs}, we write xα=x-1*l1, xβ=x-1*cs, xγ=x-1*bs and Xα=M(X,l1), Xβ=M(X,cs), Xγ=M(X,bs) for the α-dual, β-dual, and γ-dual of X. By An=(ank)k=0∞, we denote the sequence in the nth-row of A, and we write An(x)=∑k=0∞‍ankxkn=(0,1,…) and A(x)=(An(x))n=0∞, provided that An∈xβ for all n.Given anFK-space X containing ϕ, its conjugate is denoted by X′ and its f-dual or sequential dual is denoted by Xf and is given by Xf={allsequences(f(ek)):f∈X′}. Let λ be a sequence space. Then λ is called perfect if λ=λαα, normal if y∈λ whenever |yk|≤|xk|,k≥1 for some x∈λ, and monotone if λ contains the canonical preimages of all its stepspace.Lemma 8.
(i)A∈(h:l1) if and only if
(36)∑n=1∞‍|ank|converges,(k=1,2,3,…),sup⁡k1k∑n=1∞‍|∑v=1k‍anv|<∞.
(ii)A∈(lp:l1) if and only if
(37)sup⁡K∈F∑k‍|∑n∈K‍ank|q<∞.Lemma 9.
(i)A∈(h:C) if and only if
(38)sup⁡n,k1k|∑v=1k‍anv|<∞,(39)lim⁡n→∞ankexists,(k=1,2,3,…).
(ii)A∈(lp:C) if and only if (39) holds and
(40)sup⁡n∑k‍|ank|q<∞1<p≤∞.Lemma 10.
(i)A∈(h:l∞) if and only if (38) holds.
(ii)A∈(lp:l∞) if and only if (40) holds with 1<p≤∞.Lemma 11.
(i)A∈(h:C0) if and only if (38) holds and
(41)lim⁡n→∞ank=0.Lemma 12.
(i)A∈(h:h) if and only if (41) holds and
(42)∑n=1∞‍n|ank-an+1,k|converges,(k=1,2,3,…),sup⁡k1k∑n=1∞‍n|∑v=1k‍(anv-an+1,v)|<∞.Theorem 13.
We define the set(43)d1={a=(ak)∈w:sup⁡K∈F∑k‍|∑n∈K‍1kfk(|ukΔrxk|)an|pk<∞}.
Then, [hp(ℱ,u,Δr)]α=d1.Proof.
Let us take anya=(ak)∈w. We define the matrix D=dnk by
(44)dnk={1kfk(|ukΔr|)an,k≥n;0,k<n
for all, k, n∈ℕ.
Consider the equation(45)1kfk(|ukΔr|)anxn=∑j=n∞‍1jfj(|ujΔr|)anyj=(Dy)nmmmmmmmmmmmm(n∈ℕ).
It follows from (45) with Lemma 8(ii) that ax=(anxn)∈l1 whenever x=(xk)∈hp(ℱ,u,Δr) if and only if Dy∈l1, whenever y=(yk)∈lp. This means that a=(an)∈[hp(ℱ,u,Δr)]α, whenever x=(xk)∈hp(ℱ,u,Δr) if and only if D∈(hp(ℱ,u,Δr):l1). This gives the result that [hp(ℱ,u,Δr)]α=d1.Theorem 14.
Let1<p<∞. Then [hp(ℱ,u,Δr)]β=d2, where
(46)d2={|∑j=kn‍aj|qa=(ak)∈w:sup⁡n∈ℕ(n-1)qmmm×∑k‍fk(|ukΔrxk|)|∑j=kn‍aj|q<∞}.Proof.
Consider the equation(47)∑k=1n‍1kfk(|ukΔr|)akxk=∑k=1n‍1kfk(|ukΔr|)ak(∑j=1k‍yjj)=∑k=1n‍(∑j=1k‍1kfj(|ujΔr|)aj)yk=(By)n(n∈ℕ),
where B=(bnk) are defined by
(48)bnk{∑j=1k‍1kfj(|ujΔr|)aj,k≥n;0,k<n
for all, k,n∈ℕ. Thus, we deduce from Lemma 9(ii) with (47) that ax=(akxk)∈cs whenever x=(xk)∈hp(ℱ,u,Δr) if and only if By∈c whenever y=(yk)∈lp. Thus (ak)∈cs and (ak)ind2 by (39) and (40), respectively. Nevertheless, the inclusion d2⊂cs holds, and thus, we have (ak)∈d2, whence [hp(ℱ,u,Δr)]β=d2.Lemma 15.
LetX be FK-space with X⊃φ. Then,(i)
Xβ⊂Xγ⊂Xf;(ii)
IfX has AK, Xβ=Xf;(iii)
IfX has AD, Xβ=Xγ.From Theorem6, Corollary 7, and Lemma 15, we can write the following corollary.Corollary 16.
(i)[hp(ℱ,u,Δr)]β=[hp(ℱ,u,Δr)]f;
(ii)[hp(ℱ,u,Δr)]β=[hp(ℱ,u,Δr)]γ.Lemma 17.
Letλ be a sequence space. Then, the following assertions are true:(i)
λ is perfect ⇒λ is normal ⇒λ is monotone;(ii)
λ is normal ⇒λα=λγ;(iii)
λ is monotone ⇒λα=λβ.Combining Theorem13, Theorem 14, and Lemma 17, we can give the following corollary.Corollary 18.
The spacehp(ℱ,u,Δr) is not monotone and so it is neither normal nor perfect.


## 4. Matrix Transformations

In this section we characterize some matrix transformations on the spacehp(ℱ,u,Δr).Lemma 19.
Letλ, μ be any two sequence spaces, A be an infinite matrix, and U be a triangle matrix. Then, A∈(λ:μU) if and only if UA∈(λ:U).If we definea~nk=n(ank-an+1,k), then we can give the following corollary from Lemma 19 with U=M defined by (16).Corollary 20.
(i)A∈(l1:h) if and only if
(49)sup⁡k∑n‍|a~nk|<∞.
(ii)A∈(c:h)=(c0:h)=(l∞:h) if and only if
(50)sup⁡K∈F∑n‍|sup⁡k∈Ka~nk|<∞.Theorem 21.
Suppose that the entries of the infinite matricesA=(ank) and E=(enk) are connected with the relation
(51)enk=a¯nk
for all k,n∈ℕ, where a¯nk=∑j=k∞‍(anj/j)[fj(|ujΔrxj|)] and μ is any sequence space. Then A∈(hp(ℱ,u,Δr):μ) if and only if {ank}k∈ℕ∈[hp(ℱ,u,Δr)]β, for all n∈ℕ and E∈(h:μ).Proof.
Letμ be any given sequence spaces. Suppose that (51) holds between A=(ank) and E=(enk), and take into account that the spaces hp(ℱ,u,Δr) and h are norm isomorphic. Let A∈(hp(ℱ,u,Δr):μ) and take any y=yk∈h. Then, EM exists and {ank}k∈𝕂∈[hp(ℱ,u,Δr)]β, which yields that {enk}k∈ℕ for each n∈ℕ. Hence, Ey exists, and thus,
(52)∑k‍fk(|ukΔr|)enkyk=∑k‍fk(|ukΔr|)ankxk
for all n∈ℕ. We have Ey=Ax which leads us to the consequence E∈(h:μ). Conversely, let {ank}k∈𝕂∈d1 for all n∈ℕ, and E∈(h:μ) hold, and take any x=xk∈hp(ℱ,u,Δr). Then, Ax exists. Therefore, we obtain from the equality that
(53)∑k‍fk(|ukΔr|)ankxk=[∑j=k∞‍anjj]∑k‍fk(|ukΔr|)yk
for all n∈ℕ. Thus, Ax=Ey and this shows that A∈(hp(ℱ,u,Δr):μ).If we use the Corollary20 and change the roles of the spaces hp(ℱ,u,Δr) with μ in Theorem 21, we can give the following theorem.Theorem 22.
Suppose that the entries of the infinite matricesA=(ank) and A~=(a~nk) are connected with the relation a~nk=nfk(|ukΔrxk|)(ank-an+1,k) for all k,n∈ℕ and μ is any sequence space. Then A∈(μ:hp(ℱ,u,Δr)) if and only if A~∈(μ:h).Proof.
Letz=zk∈μ and consider the following equality:
(54)∑k=0m‍a~nkzk=∑k=0m‍nfk(|ukΔr|)(ank-an+1,k)zk,aaaaaaaaaaaaaaaaaaaaaaaaa∀m,n∈ℕ
which yields that as m→∞(A~z)n=(M(Az))n for all n∈ℕ. Therefore, one can observe from here that Az∈hp(ℱ,u,Δr) whenever z∈μ if and only if A~z∈h whenever z∈μ.



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*Source: 398203-2014-05-13.xml*