# On Some Properties of New Paranormed Sequence Space of Nonabsolute Type

Authors: Vatan Karakaya; Necip Şimşek
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce some new generalized sequence space
related to the spaceℓ(p). Furthermore we investigate some topological properties
as the completeness, the isomorphism, and also we give some inclusion
relations between this sequence space and some of the other sequence spaces.
In addition, we compute α-, β-, and γ-duals of this space and characterize
certain matrix transformations on this sequence space.


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## Body

## 1. Introduction

In studying the sequence spaces, especially, to obtain new sequence spaces, in general, the matrix domainμA of an infinite matrix A defined by μA={x=(xk)∈w:Ax∈μ} is used. In most cases, the new sequence space μA generated by a sequence space μ is the expansion or the contraction of the original space μ. In some cases, these spaces could be overlapped. Indeed, one can easily see that the inclusion μS⊂μ strictly holds for μ∈{ℓ∞,c,c0}. Similarly one can deduce that the inclusion μ⊂μΔ also strictly holds for μ∈{ℓ∞,c,c0}, where S and Δ are matrix operators.Recently, in [1], Mursaleen and Noman constructed new sequence spaces by using matrix domain over a normed space. They also studied some topological properties and inclusion relations of these spaces.It is well known that paranormed spaces have more general properties than the normed spaces. In this work, we generalize the normed sequence spaces defined by Mursaleen and Noman [1] to the paranormed spaces. Furthermore we introduce new sequence space over the paranormed space. Next we investigate behaviors of this sequence space according to topological properties and inclusion relations. Finally we give certain matrix transformation on this sequence space and its duals.In the literature, by using the matrix domain over the paranormed spaces, many authors have defined new sequence spaces. Some of them are as follows. For example, Choudhary and Mishra [2] have defined the sequence space ℓ(p)¯ where the S-transform is in ℓ(p), Başar and Altay [3, 4] defined the spaces λ(u,v;p)={λ(p)}G for λ∈{ℓ∞,c,c0} and ℓ(u,v;p)={ℓ(p)}G, respectively, and Altay and Başar [5] have defined the spaces r∞t(p),rct(p),r0t(p). In [6], Karakaya and Polat defined and examined the spaces e0r(Δ;p),er(Δ;p),e∞r(Δ;p), and Karakaya et al. [7] have recently introduced and studied the spaces ℓ∞(λ,p),  c(λ,p), c0(λ,p), where Rt and Er denote the Riesz and the Euler means, respectively, Δ denotes the band matrix of the difference operators, and Λ, G are defined in [1, 8], respectively. Also, the information on matrix domain of sequence spaces can be found in [9–13].Byw, we denote the space of all real valued sequences. Any vector subspace of w is called a sequence space. By the spaces ℓ1,cs, and bs, we denote the spaces of all absolutely convergent series, convergent series, and bounded series, respectively.A linear topological spaceX over the real field ℝ is said to be a paranormed space if there is a subadditivity function h:X→ℝ such that h(θ)=0, h(x)=h(-x), and scalar multiplication is continuous, that is, |αn-α|→0 and h(xn-x)→0 imply h(αnxn-αx)→0 for all α in ℝ and x in X, where θ is the zero in the linear space X.Letμ,ν be any two sequence spaces, and let A=(ank) be any infinite matrix of real number ank, where n,k∈ℕ with ℕ={0,1,2,…}. Then we say that A defines a matrix mapping from μ into ν by writingA:μ→ν, if for every sequence x=(xk)∈μ, the sequence Ax=(An(x)), the A-transform of x, is in ν, where
(1.1)An(x)=∑kankxk(n∈N).
By (μ,ν), we denote the class of all matrices A such that A:μ→ν. Thus, A∈(μ,ν) if and only if the series on the right hand side of (1.1) converges for each n∈ℕ and every x∈μ, and we have Ax∈ν for all x∈μ. A sequence x is said to be A-summable to a if Ax converges to a which is called as the A-limit of x.Assume here and after that(pk), (qk) are bounded sequences of strictly positive real numbers with suppk=H and M=max(1,H), and also let p`k=pk/(pk-1) for 1<pk<∞ and for all k∈ℕ. The linear space ℓ(p) was defined by Maddox [14] as follows:
(1.2)l(p)={x=(xn)∈w:∑n=0∞|xn|pn<∞}
which are the complete space paranormed by
(1.3)h(x)=(∑n=0∞|xn|pn)1/M.
Throughout this work, by Ϝ and Nk, respectively, we will denote the collection of all subsets of ℕ and the set of all n∈ℕ such that n≥k and e=(1,1,1,…).


## 2. The Sequence Spaceℓ(λ,p)

In this section, we define the sequence spaceℓ(λ,p) and prove that this sequence space according to its paranorm is complete paranormed linear space. In [1], Mursaleen and Noman defined the matrix Λ=(λnk)n,k=0∞ by
(2.1)λnk={λk-λk-1λn(0≤k≤n)0(k>n),
where λ=(λk)k=0∞ is a strictly increasing sequence of positive reals tending to ∞, that is, 0<λ0<λ1<⋯ and λk→∞ as k→∞. Now, by using (2.1) we define new sequence space as follows:
(2.2)l(λ,p)={x=(xk)∈w:∑n=0∞‍|1λn∑k=0n(λk-λk-1)xk|pn<∞}.
For any x=(xn)∈w, we define the sequence y=(yn), which will frequently be used, as the Λ-transform of x, that is, y=Λ(x), and hence
(2.3)yn=∑k=0n‍(λk-λk-1λn)xk(n∈N).
We now may begin with the following theorem.Theorem 2.1.
The sequence spaceℓ(λ,p) is the complete linear metric space with respect to paranorm defined by
(2.4)h(x)=(∑n=0∞‍|1λn∑k=0n(λk-λk-1)xk|pn)1/M.Proof.
The linearity ofℓ(λ,p) with respect to the coordinatewise addition and scalar multiplication follows from the following inequalities which are satisfied for x,t∈ℓ(λ,p) (see, [15]):(2.5)(∑n=0∞|1λn∑k=0n(λk-λk-1)(xk+tk)|pn)1/M≤(∑n=0∞‍|1λn∑k=0n(λk-λk-1)xk|pn)1/M+(∑n=0∞‍|1λn∑k=0n(λk-λk-1)tk|pn)1/M
and for any α∈ℝ (see, [16])
(2.6)|α|pk≤max⁡{1,|α|M}.
It is clear that h(θ)=0, h(x)=h(-x) for all x∈ℓ(λ,p). Again inequalities (2.5) and (2.6) yield the subadditivity of h and hence h(αx)≤max{1,|α|M}h(x). Let {xm} be any sequence of points xm∈ℓ(λ,p) such that h(xm-x)→0 and (αm) also any sequence of scalars such that αm→α. Then, since the inequality
(2.7)h(xm)≤h(x)+h(xm-x)
holds by subadditivity of h, we can write that {h(xm)} is bounded and we thus have
(2.8)h(αmxm-αx)=(∑n=0∞‍|1λn∑k=0n(λk-λk-1)(αmxkm-αxk)|pn)1/M≤|αm⟶α|1/Mh(xm)+|α|1/Mh(xm-x)
which tends to zero as n→∞. Therefore, the scalar multiplication is continuous. Hence h is a paranorm on the space ℓ(λ,p). It remains to prove the completeness of the space ℓ(λ,p). Let {xj} be any Cauchy sequence in the space ℓ(λ,p), where xj={x0(j),x1(j),x2(j),…}. Then, for a given ɛ>0, there exists a positive integer m0(ɛ) such that h(xj-xi)<ɛ/2 for all i,j>m0(ɛ). Using definition of h, we obtain for each fixed n∈ℕ that
(2.9)|Λn(xj)-Λn(xi)|≤(∑n=0∞‍|Λn(xj)-Λn(xi)|pn)1/M<ɛ2
for every i,j>m0(ɛ) which leads us to the fact that {Λn(x0),Λn(x1),Λn(x2),…} is a Cauchy sequence of real numbers for every fixed n∈ℕ. Since ℝ is complete, it converges, say Λn(xi)-Λn(x) as i→∞. Using these infinitely many limits, we may write the sequence {Λ0(x),Λ1(x),Λ2(x),…}. From (2.9) as i→∞, we have
(2.10)|Λn(xj)-Λn(x)|<ɛ2,(j≥m0(ɛ))
for every fixed n∈ℕ. By using (2.9) and boundedness of Cauchy sequence, we have
(2.11)(∑n=0∞‍|Λn(x)|pn)1/M≤(∑n=0∞‍|Λn(xj)-Λn(xi)|pn)1/M+(∑n=0∞‍|Λn(xj)|pn)1/M<∞.
Hence, we get x∈ℓ(λ,p). So, the space ℓ(λ,p) is complete.Theorem 2.2.
The sequence spaceℓ(λ,p) of nonabsolute type is linearly isomorphic to the space ℓ(p), where 0<pk≤H<∞.Proof.
To prove the theorem, we would show the existence of linear bijection between the spacesℓ(λ,p) and ℓ(p). With the notation of (2.3), we define transformation T from ℓ(λ,p) to ℓ(p) by x→y=Tx. The linearity of T is trivial. Furthermore, it is obvious that x=θ whenever Tx=θ and hence T is injective.
Lety∈ℓ(p) and define the sequence x={xn}(2.12)xn(λ)=∑k=n-1n((-1)n-kλkλn-λn-1)yk(n,k∈N).
Then, we have
(2.13)hl(λ,p)(x)=(∑n=0∞‍|1λn∑k=0n(λk-λk-1)xk|pn)1/M=(∑n=0∞‍|yn|pn)1/M=hl(p)(y).
Thus, we have that x∈ℓ(λ,p) and consequently T is surjective. Hence, T is a linear bijection and this tells us that the spaces ℓ(λ,p) and ℓ(p) are linearly isomorphic. This completes the proof.


## 3. Some Inclusion Relations

In this section, we give some inclusion relations concerning the spaceℓ(λ,p). Before giving the theorems about the section, we give a lemma given in [1].Lemma 3.1.
For any sequencex=(xk)∈w, the equalites
(3.1)Sn(x)=xn-Λn(x),(3.2)Sn(x)=λn-1λn-λn-1[Λn(x)-Λn-1(x)]
hold, where the sequence S(x)={Sn(x)} is defined by
(3.3)S0(x)=0,Sn(x)=1λn∑k=1nλk-1(xk-xk-1)(n≥1).Theorem 3.2.
The inclusionℓ(λ,p)⊂c0(λ,p) holds.Proof.
Letx∈ℓ(λ,p). It can be written Λx∈ℓ(p). By the definition of the space ℓ(p),  Λnx→∞ as n→∞, we obtain Λx∈c0. Hence we get x∈c0(λ,p).Theorem 3.3.
The inclusionℓ(λ,p)⊂ℓ(p) if and only if S(x)∈ℓ(p) for every sequence x∈ℓ(λ,p), where 1≤pk≤H.Proof.
We suppose thatℓ(λ,p)⊂ℓ(p) holds and take any x∈ℓ(λ,p). Then x∈ℓ(p) by hypothesis. Thus we obtain from (3.1) that
(3.4)[h(S(x))]l(p)≤[h(x)]l(p)+[h(Λx)]l(p)=[h(x)]l(p)+[h(x)]l(λ,p)
which yields that S(x)∈ℓ(p).
Conversely, letx∈ℓ(λ,p) be given. Then we have by the hypothesis that S(x)∈ℓ(p). Again by using (3.1)
(3.5)[h(x)]l(p)≤[h(S(x))]l(p)+[h(Λx)]l(p)=[h(S(x))]l(p)+[h(x)]l(λ,p)
which shows that x∈ℓ(p). Hence the inclusion ℓ(λ,p)⊂ℓ(p) holds. This completes the proof.Theorem 3.4.
(i) Ifpn>1 for all n∈N, then the inclusion ℓpλ⊂ℓ(λ,p) holds.
(ii) Ifpn<1 for all n∈N, then the inclusion ℓ(λ,p)⊂ℓpλ holds.Proof.
(i) Ifp=(pn) for all n∈N, then we write ℓpλ in place ofℓ(λ,p). Let x∈ℓpλ. It is clear that Λ(x)∈ℓp. One can find m∈N such that |Λn(x)|<1 for all n≥m. Under condition (i), we have |Λn(x)|pn<|Λn(x)| for all n≥m. Hence we get x∈ℓ(λ,p).
(ii) We suppose thatx∈ℓ(λ,p). Then Λ(x)∈ℓ(p) and there exists m∈N such that |Λn(x)|pn<1 for all n≥m. To obtain the result, we consider the following inequality:
(3.6)|Λn(x)|=(|Λn(x)|pn)1/pn<|Λn(x)|pn
for all n≥m. So, we get x∈ℓpλ.


## 4. Some Matrix Transformations and Duals of the Spaceℓ(λ,p)

In this section, we give the theorems determining theα-,  β-, and γ-duals of the space ℓ(λ,p). In proving the theorem, we apply the technique used in [3]. Also we give some matrix transformations from the space ℓ(λ,p) into paranormed spaces ℓ(q) by using the matrix given in [1].For the sequence spaceμ and ν, the set S(μ,ν) defined by
(4.1)S(μ,ν)={a=(ak)∈w:ax∈ν∀x∈μ}
is called the multiplier space of μ and ν. The α-,β-, and γ-duals of a sequence space μ, which are, respectively, denoted by μα, μβ, and μγ, are defined by
(4.2)μα=S(μ,l1),μβ=S(μ,cs),μγ=S(μ,bs).
We may begin with the following theorem which computes the α-dual of the space ℓ(λ,p).Theorem 4.1.
LetK1={k∈ℕ:pk≤1} and K2={k∈ℕ:pk>1}. Define the matrix Da=(dnka) by
(4.3)dnka={(-1)n-kλkλn-λn-1an,(n-1≤k≤n)0,(0≤k≤n-1)or(k>n).
Then
(4.4)lK1α(λ,p)={a=(an)∈w:Da∈(l(p);l∞)}lK2α(λ,p)={a=(an)∈w:Da∈(l(p);l1)}.Proof.
We consider the following equality:(4.5)anxn=∑k=n-1ndnkayk=(Day)n(n∈N),
where Da=(dnka) is defined by (4.3).
From (4.5), it can be obtained that ax=(anxn)∈ℓ1 or ax=(anxn)∈ℓ∞ whenever x∈ℓ(λ,p) if and only if Day∈ℓ1 or Day∈ℓ∞ whenever y∈ℓ(p).This means a∈ℓK1α(λ,p) or a∈ℓK2α(λ,p) if and only if Da∈(ℓ(p);ℓ1) or Da∈(ℓ(p);ℓ∞). Hence this completes the proof.The result of the Theorem above corresponds the Theorem 5.1(0, 8, 12) given in [17].As a direct consequence of Theorem4.1, we have the following.Corollary 4.2.
LetK*={k∈ℕ:n-1≤k≤n}∩K for K∈Ϝ. Then(i)
ℓK1α(λ,p)={a=(an)∈w:supNsupk∈ℕ|∑n∈K*dnka|pk<∞},(ii)
ℓK2α(λ,p)=⋃M>1{a=(an)∈w:supK∈Ϝ∑k|∑n∈K*dnkaM-1|pk!<∞}In the following theorem, we characterizethe β- and γ-duals of the space ℓ(λ,p).Theorem 4.3.
LetK1={k∈ℕ:pk≤1}, K2={k∈ℕ:pk>1}, and let Δxk=xk-xk+1. Define the sequence s1=(sk1),  s2=(sk2) and the matrix Ba=(bnka) by
(4.6)sk1=Δ(akλk-λk-1)λk,sk2=akλkλk-λk-1bnka={sk1,(0≤k≤n-1)sk2,(k=n)0,(k>n)
for all n,k∈ℕ. Then
(4.7)lK1β(λ,p)=lK1γ(λ,p)={a=(an)∈w:Ba∈(l(p);l∞)},(4.8)lK2β(λ,p)=lK2γ(λ,p)={a=(an)∈w:Ba∈(l(p);c)}.Proof.
Consider the equality(4.9)∑k=0nakxk=∑k=0n-1sk1yk+sn2yn=(Bay)n.
From (4.9), it can be obtained that ax=(anxn)∈cs or bs whenever x=(xn)∈ℓ(λ,p) if and only if Bay∈c or ℓ∞ whenever y=(yk)∈ℓ(p). This means that a=(an)∈{ℓK1β(λ,p)orℓK2β(λ,p)} or a=(an)∈{ℓK1γ(λ,p)orℓK2γ(λ,p)} if and only if Ba∈(ℓ(p);c) or Ba∈(ℓ(p);ℓ∞). Hence this completes the proof.We can write the following corollary from Theorem4.3.Corollary 4.4.
Letp`k=pk/(pk-1) for 1<pk<∞ and for all k∈ℕ. Then(i)
ℓK1β(λ,p)=ℓK1γ(λ,p)={a=(an)∈w:s1,s2∈ℓ∞(p)},(ii)
ℓK2β(λ,p)=ℓK2γ(λ,p)=⋃M>1{a=(an)∈w:s1M-1,s2M-1∈ℓ(p′)∩ℓ∞(p′)}.After this step, we can give our theorems on the characterization of some matrix classes concerning the sequence spaceℓ(λ,p).Letx,y∈w be connected by the relation y=Λ(x). For an infinite matrix A=(ank), we have by using (4.9) of Theorem 4.3 that
(4.10)∑k=0m‍ankxk=∑k=0m-1‍a~nkyk+λmλm-λm-1anmym(m,n∈N),
where
(4.11)a~nk=(ankλk-λk-1-an,k+1λk+1-λk)λk(n,k∈N).
The necessary and sufficient conditions characterizing the matrix mapping of the sequence space ℓ(p) of Maddox have been determined by Grosse-Erdmann [17]. Let L and M be the natural numbers and define the sets by K1={k∈ℕ:pk≤1} and K2={k∈ℕ:pk>1} and also let us put p`k=pk/(pk-1) for 1<pk<∞ and for all k∈ℕ. Before giving the theorems, let us suppose that (qn) is a nondecreasing bounded sequence of positive real numbers and consider the following conditions:
(4.12)sup⁡N⁡sup⁡k∈K1⁡|∑n∈Na~nk|qn<∞,(4.13)∃Msup⁡N⁡∑k∈K2|n∈Na~nkM-1|p`k<∞,(4.14)∃Msup⁡k⁡∑n|a~kM-1/pk|qn<∞,(4.15)lim⁡n⁡|a~nk|qn=0(∀k∈N),(4.16)∀L,sup⁡n⁡sup⁡k∈K1⁡|a~nkL1/qn|pk<∞,(4.17)∀L,∃Msup⁡n⁡∑k∈K2|a~kL1/qnM-1|p`k<∞,(4.18)sup⁡n⁡sup⁡k∈K1|a~nk|pk<∞,(4.19)∃Msup⁡n⁡∑k∈K2|a~kM-1|p`k<∞,(4.20)∀L,sup⁡n⁡supk∈K1(|a~nk-a~k|L1/qn)pk<∞,(4.21)lim⁡n⁡|a~nk-a~k|qn=0,∀k.(4.22)∀L,∃Msup⁡n∑k∈K2(|a~nk-a~k|L1/qnM-1)p`k,(4.23)∃L,sup⁡nsup⁡k∈K1|a~nkL-1/qn|pk<∞,(4.24)∃L,sup⁡n∑k∈K2|a~nkL-1/qn|p`k<∞,(4.25)(λkλk-λk-1ank)k=0∞∈c0(q)(∀n∈N)(4.26)(λkλk-λk-1ank)k=0∞∈c(q)(∀n∈N)(4.27)(λkλk-λk-1ank)k=0∞∈l∞(q)(∀n∈N).By using (4.7), (4.10), and Corollary 4.4, we have the following results.Theorem 4.5.
One has the following:(i)
A∈(ℓ(λ,p):ℓ(q)) if and only if (4.12), (4.13), (4.14), and (4.25) hold,(ii)
A∈(ℓ(λ,p):c0(q)) if and only if (4.15), (4.16), (4.17), and (4.25) hold,(iii)
A∈(ℓ(λ,p):c(q)) if and only if (4.18), (4.19), (4.20), (4.21), (4.22), and (4.26) hold,(iv)
A∈(ℓ(λ,p):ℓ∞(q)) if and only if (4.23), (4.24), and (4.27) hold.



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