On Some Properties of New Paranormed Sequence Space of Nonabsolute Type

Abstract

We introduce some new generalized sequence space related to the space ℓ(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.

1. Introduction

In studying the sequence spaces, especially, to obtain new sequence spaces, in general, the matrix domain 𝜇_𝐴 of an infinite matrix 𝐴 defined by 𝜇_𝐴={𝑥=(𝑥_𝑘)∈𝑤∶𝐴𝑥∈𝜇} is used. In most cases, the new sequence space 𝜇_𝐴 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 𝜇_𝑆⊂𝜇 strictly holds for 𝜇∈{ℓ_∞,𝑐,𝑐₀}. Similarly one can deduce that the inclusion 𝜇⊂𝜇_Δ 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),𝑟^𝑡_𝑐(𝑝),𝑟^𝑡₀(𝑝). In [6], Karakaya and Polat defined and examined the spaces 𝑒^𝑟₀(Δ;𝑝),𝑒^𝑟(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 𝑅^𝑡 and 𝐸^𝑟 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].

By 𝑤, 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 space 𝑋 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, |𝛼_𝑛−𝛼|→0 and ℎ(𝑥_𝑛−𝑥)→0 imply ℎ(𝛼_𝑛𝑥_𝑛−𝛼𝑥)→0 for all 𝛼 in ℝ and 𝑥 in 𝑋, where 𝜃 is the zero in the linear space 𝑋.

Let 𝜇,𝜈 be any two sequence spaces, and let 𝐴=(𝑎_𝑛𝑘) be any infinite matrix of real number 𝑎_𝑛𝑘, 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=(𝑥_𝑘)∈𝜇, the sequence 𝐴𝑥=(𝐴_𝑛(𝑥)), the 𝐴-transform of 𝑥, is in 𝜈, where 𝐴_𝑛(𝑥)=_𝑘𝑎_𝑛𝑘𝑥_𝑘(𝑛∈ℕ). (1.1) 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 (𝑝_𝑘), (𝑞_𝑘) are bounded sequences of strictly positive real numbers with sup𝑝_𝑘=𝐻 and 𝑀=max(1,𝐻), and also let 𝑝^‵_𝑘=𝑝_𝑘/(𝑝_𝑘−1) for 1<𝑝_𝑘<∞ and for all 𝑘∈ℕ. The linear space ℓ(𝑝) was defined by Maddox [14] as follows: ℓ𝑝=𝑥=𝑥_𝑛∈𝑤∶^∞_𝑛=0𝑥_𝑛^𝑝𝑛<∞ (1.2) which are the complete space paranormed by ℎ(𝑥)=^∞_𝑛=0𝑥_𝑛^{𝑝𝑛1/𝑀}. (1.3) Throughout this work, by Ϝ and 𝑁_𝑘, 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 Space ℓ(𝜆,𝑝)

In this section, we define the sequence space ℓ(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=(𝜆_𝑛𝑘)^∞_𝑛,𝑘=0 by 𝜆_𝑛𝑘=𝜆_𝑘−𝜆_𝑘−1𝜆_𝑛()0(),0≤𝑘≤𝑛𝑘>𝑛 (2.1) where 𝜆=(𝜆_𝑘)^∞_𝑘=0 is a strictly increasing sequence of positive reals tending to ∞, that is, 0<𝜆₀<𝜆₁<⋯ and 𝜆_𝑘→∞ as 𝑘→∞. Now, by using (2.1) we define new sequence space as follows: ℓ𝜆,𝑝=𝑥=𝑥_𝑘∈𝑤∶^∞_𝑛=01𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑥_𝑘^𝑝𝑛<∞. (2.2) For any 𝑥=(𝑥_𝑛)∈𝑤, we define the sequence 𝑦=(𝑦_𝑛), which will frequently be used, as the Λ-transform of 𝑥, that is, 𝑦=Λ(𝑥), and hence 𝑦_𝑛=^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝜆_𝑛𝑥_𝑘(𝑛∈𝑁). (2.3) We now may begin with the following theorem.

Theorem 2.1. The sequence space ℓ(𝜆,𝑝) is the complete linear metric space with respect to paranorm defined by ℎ(𝑥)=^∞_𝑛=01𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑥_𝑘^{𝑝𝑛1/𝑀}. (2.4)

Proof. The linearity of ℓ(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]):^∞_𝑛=01𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑥_𝑘+𝑡_𝑘^{𝑝𝑛1/𝑀}≤^∞_𝑛=01𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑥_𝑘^{𝑝𝑛1/𝑀}+^∞_𝑛=01𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑡_𝑘^{𝑝𝑛1/𝑀} (2.5) and for any 𝛼∈ℝ (see, [16]) |𝛼|^𝑝𝑘≤max1,|𝛼|^𝑀. (2.6) 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 {𝑥^𝑚} be any sequence of points 𝑥^𝑚∈ℓ(𝜆,𝑝) such that ℎ(𝑥^𝑚−𝑥)→0 and (𝛼_𝑚) also any sequence of scalars such that 𝛼_𝑚→𝛼. Then, since the inequality ℎ𝑥^𝑚≤ℎ(𝑥)+ℎ𝑥^𝑚−𝑥 (2.7) holds by subadditivity of ℎ, we can write that {ℎ(𝑥^𝑚)} is bounded and we thus have ℎ𝛼_𝑚𝑥^𝑚−𝛼𝑥=^∞_𝑛=01𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝛼_𝑚𝑥^𝑚_𝑘−𝛼𝑥_𝑘^{𝑝𝑛1/𝑀}≤𝛼_𝑚⟶𝛼^1/𝑀ℎ𝑥^𝑚+|𝛼|^1/𝑀ℎ𝑥^𝑚−𝑥 (2.8) 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 {𝑥^𝑗} be any Cauchy sequence in the space ℓ(𝜆,𝑝), where 𝑥^𝑗={𝑥₀^(𝑗),𝑥₁^(𝑗),𝑥₂^(𝑗),…}. Then, for a given 𝜀>0, there exists a positive integer 𝑚₀(𝜀) such that ℎ(𝑥^𝑗−𝑥^𝑖)<𝜀/2 for all 𝑖,𝑗>𝑚₀(𝜀). Using definition of ℎ, we obtain for each fixed 𝑛∈ℕ that Λ_𝑛𝑥^𝑗−Λ_𝑛𝑥^𝑖≤^∞_𝑛=0Λ_𝑛𝑥^𝑗−Λ_𝑛𝑥^{𝑖𝑝𝑛1/𝑀}<𝜀2 (2.9) for every 𝑖,𝑗>𝑚₀(𝜀) which leads us to the fact that {Λ_𝑛(𝑥⁰),Λ_𝑛(𝑥¹),Λ_𝑛(𝑥²),…} is a Cauchy sequence of real numbers for every fixed 𝑛∈ℕ. Since ℝ is complete, it converges, say Λ_𝑛(𝑥^𝑖)−Λ_𝑛(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 Λ_𝑛𝑥^𝑗−Λ_𝑛(𝑥<𝜀)2,𝑗≥𝑚₀(𝜀) (2.10) for every fixed 𝑛∈ℕ. By using (2.9) and boundedness of Cauchy sequence, we have ^∞_𝑛=0Λ_𝑛(𝑥)^{𝑝𝑛1/𝑀}≤^∞_𝑛=0Λ_𝑛𝑥^𝑗−Λ_𝑛𝑥^{𝑖𝑝𝑛1/𝑀}+^∞_𝑛=0Λ_𝑛𝑥^{𝑗𝑝𝑛1/𝑀}<∞. (2.11) Hence, we get 𝑥∈ℓ(𝜆,𝑝). So, the space ℓ(𝜆,𝑝) is complete.

Theorem 2.2. The sequence space ℓ(𝜆,𝑝) of nonabsolute type is linearly isomorphic to the space ℓ(𝑝), where 0<𝑝_𝑘≤𝐻<∞.

Proof. To prove the theorem, we would show the existence of linear bijection between the spaces ℓ(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.
Let 𝑦∈ℓ(𝑝) and define the sequence 𝑥={𝑥_𝑛}𝑥_𝑛(𝜆)=^𝑛_{𝑘=𝑛−1}(−1)^𝑛−𝑘𝜆_𝑘𝜆_𝑛−𝜆_𝑛−1𝑦_𝑘(𝑛,𝑘∈ℕ). (2.12) Then, we have ℎ_{ℓ(𝜆,𝑝)}(𝑥)=^∞_𝑛=01𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑥_𝑘^{𝑝𝑛1/𝑀}=^∞_𝑛=0𝑦_𝑛^{𝑝𝑛1/𝑀}=ℎ_ℓ(𝑝)𝑦. (2.13) 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 space ℓ(𝜆,𝑝). Before giving the theorems about the section, we give a lemma given in [1].

Lemma 3.1. For any sequence 𝑥=(𝑥_𝑘)∈𝑤, the equalites 𝑆_𝑛(𝑥)=𝑥_𝑛−Λ_𝑛(𝑥),𝑆_𝑛(𝑥)=𝜆_𝑛−1𝜆_𝑛−𝜆_𝑛−1Λ_𝑛(𝑥)−Λ_𝑛−1(𝑥) (3.1)(3.2) hold, where the sequence 𝑆(𝑥)={𝑆_𝑛(𝑥)} is defined by 𝑆₀(𝑥)=0,𝑆_𝑛(𝑥)=1𝜆_𝑛^𝑛_𝑘=1𝜆_𝑘−1𝑥_𝑘−𝑥_𝑘−1(𝑛≥1). (3.3)

Theorem 3.2. The inclusion ℓ(𝜆,𝑝)⊂𝑐₀(𝜆,𝑝) holds.

Proof. Let 𝑥∈ℓ(𝜆,𝑝). It can be written Λ𝑥∈ℓ(𝑝). By the definition of the space ℓ(𝑝),  Λ_𝑛𝑥→∞ as 𝑛→∞, we obtain Λ𝑥∈𝑐₀. Hence we get 𝑥∈𝑐₀(𝜆,𝑝).

Theorem 3.3. The inclusion ℓ(𝜆,𝑝)⊂ℓ(𝑝) if and only if 𝑆(𝑥)∈ℓ(𝑝) for every sequence 𝑥∈ℓ(𝜆,𝑝), where 1≤𝑝_𝑘≤𝐻.

Proof. We suppose that ℓ(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 [ℎ(𝑆(𝑥))]_ℓ(𝑝)≤[ℎ(𝑥)]_ℓ(𝑝)+[ℎ(Λ𝑥)]_ℓ(𝑝)=[ℎ(𝑥)]_ℓ(𝑝)+[ℎ(𝑥)]_{ℓ(𝜆,𝑝)} (3.4) which yields that 𝑆(𝑥)∈ℓ(𝑝).
Conversely, let 𝑥∈ℓ(𝜆,𝑝) be given. Then we have by the hypothesis that 𝑆(𝑥)∈ℓ(𝑝). Again by using (3.1) [ℎ(𝑥)]_ℓ(𝑝)≤[ℎ(𝑆(𝑥))]_ℓ(𝑝)+[ℎ(Λ𝑥)]_ℓ(𝑝)=[ℎ(𝑆(𝑥))]_ℓ(𝑝)+[ℎ(𝑥)]_{ℓ(𝜆,𝑝)} (3.5) which shows that 𝑥∈ℓ(𝑝). Hence the inclusion ℓ(𝜆,𝑝)⊂ℓ(𝑝) holds. This completes the proof.

Theorem 3.4. (i) If 𝑝_𝑛>1 for all 𝑛∈𝑁, then the inclusion ℓ^𝜆_𝑝⊂ℓ(𝜆,𝑝) holds.
(ii) If 𝑝_𝑛<1 for all 𝑛∈𝑁, then the inclusion ℓ(𝜆,𝑝)⊂ℓ^𝜆_𝑝 holds.

Proof. (i) If 𝑝=(𝑝_𝑛) for all 𝑛∈𝑁, then we write ℓ^𝜆_𝑝 in place ofℓ(𝜆,𝑝). Let 𝑥∈ℓ^𝜆_𝑝. It is clear that Λ(𝑥)∈ℓ_𝑝. One can find 𝑚∈𝑁 such that |Λ_𝑛(𝑥)|<1 for all 𝑛≥𝑚. Under condition (i), we have |Λ_𝑛(𝑥)|^𝑝𝑛<|Λ_𝑛(𝑥)| for all 𝑛≥𝑚. Hence we get 𝑥∈ℓ(𝜆,𝑝).
(ii) We suppose that 𝑥∈ℓ(𝜆,𝑝). Then Λ(𝑥)∈ℓ(𝑝) and there exists 𝑚∈𝑁 such that |Λ_𝑛(𝑥)|^𝑝𝑛<1 for all 𝑛≥𝑚. To obtain the result, we consider the following inequality: Λ_𝑛(𝑥)=Λ_𝑛(𝑥)^{𝑝𝑛1/𝑝𝑛}<Λ_𝑛(𝑥)^𝑝𝑛 (3.6) for all 𝑛≥𝑚. So, we get 𝑥∈ℓ^𝜆_𝑝.

4. Some Matrix Transformations and Duals of the Space ℓ(𝜆,𝑝)

In this section, we give the theorems determining the 𝛼-,  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 space 𝜇 and 𝜈, the set 𝑆(𝜇,𝜈) defined by 𝑆𝜇,𝜈=𝑎=𝑎_𝑘∈𝑤∶𝑎𝑥∈𝜈∀𝑥∈𝜇 (4.1) 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 𝜇^𝛼, 𝜇^𝛽, and 𝜇^𝛾, are defined by 𝜇^𝛼=𝑆𝜇,ℓ₁,𝜇^𝛽=𝑆𝜇,𝑐𝑠,𝜇^𝛾=𝑆𝜇,𝑏𝑠. (4.2) We may begin with the following theorem which computes the 𝛼-dual of the space ℓ(𝜆,𝑝).

Theorem 4.1. Let 𝐾₁={𝑘∈ℕ∶𝑝_𝑘≤1} and 𝐾₂={𝑘∈ℕ∶𝑝_𝑘>1}. Define the matrix 𝐷^𝑎=(𝑑^𝑎_𝑛𝑘) by 𝑑^𝑎_𝑛𝑘=(−1)^𝑛−𝑘𝜆_𝑘𝜆_𝑛−𝜆_𝑛−1𝑎_𝑛,()()().𝑛−1≤𝑘≤𝑛0,0≤𝑘≤𝑛−1or𝑘>𝑛 (4.3) Then ℓ^𝛼_𝐾1𝜆,𝑝=𝑎=𝑎_𝑛∈𝑤∶𝐷^𝑎∈ℓ𝑝;ℓ_∞ℓ^𝛼_𝐾2𝜆,𝑝=𝑎=𝑎_𝑛∈𝑤∶𝐷^𝑎∈ℓ𝑝;ℓ₁. (4.4)

Proof. We consider the following equality: 𝑎_𝑛𝑥_𝑛=^𝑛_{𝑘=𝑛−1}𝑑^𝑎_𝑛𝑘𝑦_𝑘=𝐷^𝑎𝑦_𝑛(𝑛∈ℕ), (4.5) where 𝐷^𝑎=(𝑑^𝑎_𝑛𝑘) is defined by (4.3).
From (4.5), it can be obtained that 𝑎𝑥=(𝑎_𝑛𝑥_𝑛)∈ℓ₁ or 𝑎𝑥=(𝑎_𝑛𝑥_𝑛)∈ℓ_∞ whenever 𝑥∈ℓ(𝜆,𝑝) if and only if 𝐷^𝑎𝑦∈ℓ₁ or 𝐷^𝑎𝑦∈ℓ_∞ whenever 𝑦∈ℓ(𝑝).This means 𝑎∈ℓ^𝛼_𝐾1(𝜆,𝑝) or 𝑎∈ℓ^𝛼_𝐾2(𝜆,𝑝) if and only if 𝐷^𝑎∈(ℓ(𝑝);ℓ₁) or 𝐷^𝑎∈(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 Theorem 4.1, we have the following.

Corollary 4.2. Let 𝐾^∗={𝑘∈ℕ∶𝑛−1≤𝑘≤𝑛}∩𝐾 for 𝐾∈ϝ. Then(i)ℓ^𝛼_𝐾1(𝜆,𝑝)={𝑎=(𝑎_𝑛)∈𝑤∶sup_𝑁sup_𝑘∈ℕ|∑_{𝑛∈𝐾∗}𝑑^𝑎_𝑛𝑘|^𝑝𝑘<∞}, (ii)ℓ^𝛼_𝐾2(𝜆,𝑝)=⋃_𝑀>1{𝑎=(𝑎_𝑛)∈𝑤∶sup_𝐾∈ϝ∑_𝑘|∑_{𝑛∈𝐾∗}𝑑^𝑎_𝑛𝑘𝑀⁻¹|^𝑝!𝑘<∞}

In the following theorem, we characterize the 𝛽- and 𝛾-duals of the space ℓ(𝜆,𝑝).

Theorem 4.3. Let 𝐾₁={𝑘∈ℕ∶𝑝_𝑘≤1}, 𝐾₂={𝑘∈ℕ∶𝑝_𝑘>1}, and let Δ𝑥_𝑘=𝑥_𝑘−𝑥_𝑘+1. Define the sequence 𝑠¹=(𝑠¹_𝑘),  𝑠²=(𝑠²_𝑘) and the matrix 𝐵^𝑎=(𝑏^𝑎_𝑛𝑘) by 𝑠¹_𝑘𝑎=Δ^𝑘𝜆_𝑘−𝜆_𝑘−1𝜆_𝑘,𝑠²_𝑘=𝑎_𝑘𝜆_𝑘𝜆_𝑘−𝜆_𝑘−1𝑏^𝑎_𝑛𝑘=𝑠¹_𝑘,()𝑠0≤𝑘≤𝑛−1_2𝑘,()()𝑘=𝑛0,𝑘>𝑛 (4.6) for all 𝑛,𝑘∈ℕ. Then ℓ^𝛽_𝐾1𝜆,𝑝=ℓ^𝛾_𝐾1𝜆,𝑝=𝑎=𝑎_𝑛∈𝑤∶𝐵^𝑎∈ℓ𝑝;ℓ_∞,ℓ^𝛽_𝐾2𝜆,𝑝=ℓ^𝛾_𝐾2𝜆,𝑝=𝑎=𝑎_𝑛∈𝑤∶𝐵^𝑎∈ℓ𝑝;𝑐. (4.7)(4.8)

Proof. Consider the equality ^𝑛_𝑘=0𝑎_𝑘𝑥_𝑘=^𝑛−1_𝑘=0𝑠¹_𝑘𝑦_𝑘+𝑠²_𝑛𝑦𝑛=𝐵^𝑎𝑦_𝑛. (4.9) From (4.9), it can be obtained that 𝑎𝑥=(𝑎_𝑛𝑥_𝑛)∈𝑐𝑠 or 𝑏𝑠 whenever 𝑥=(𝑥_𝑛)∈ℓ(𝜆,𝑝) if and only if 𝐵^𝑎𝑦∈𝑐 or ℓ_∞ whenever 𝑦=(𝑦_𝑘)∈ℓ(𝑝). This means that 𝑎=(𝑎_𝑛)∈{ℓ^𝛽_𝐾1(𝜆,𝑝)orℓ^𝛽_𝐾2(𝜆,𝑝)} or 𝑎=(𝑎_𝑛)∈{ℓ^𝛾_𝐾1(𝜆,𝑝)orℓ^𝛾_𝐾2(𝜆,𝑝)} if and only if 𝐵^𝑎∈(ℓ(𝑝);𝑐) or 𝐵^𝑎∈(ℓ(𝑝);ℓ_∞). Hence this completes the proof.

We can write the following corollary from Theorem 4.3.

Corollary 4.4. Let 𝑝^‵_𝑘=𝑝_𝑘/(𝑝_𝑘−1) for 1<𝑝_𝑘<∞ and for all 𝑘∈ℕ. Then(i)ℓ^𝛽_𝐾1(𝜆,𝑝)=ℓ^𝛾_𝐾1(𝜆,𝑝)={𝑎=(𝑎_𝑛)∈𝑤∶𝑠¹,𝑠²∈ℓ_∞(𝑝)}, (ii)ℓ^𝛽_𝐾2(𝜆,𝑝)=ℓ^𝛾_𝐾2(𝜆,𝑝)=⋃_𝑀>1{𝑎=(𝑎_𝑛)∈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 space ℓ(𝜆,𝑝).

Let 𝑥,𝑦∈𝑤 be connected by the relation 𝑦=Λ(𝑥). For an infinite matrix 𝐴=(𝑎_𝑛𝑘), we have by using (4.9) of Theorem 4.3 that ^𝑚_𝑘=0𝑎_𝑛𝑘𝑥_𝑘=^𝑚−1_𝑘=0𝑎_𝑛𝑘𝑦_𝑘+𝜆_𝑚𝜆_𝑚−𝜆_𝑚−1𝑎_𝑛𝑚𝑦_𝑚(𝑚,𝑛∈ℕ), (4.10) where 𝑎_𝑛𝑘=𝑎_𝑛𝑘𝜆_𝑘−𝜆_𝑘−1−𝑎_𝑛,𝑘+1𝜆_𝑘+1−𝜆_𝑘𝜆_𝑘(𝑛,𝑘∈ℕ). (4.11) 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∶𝑝_𝑘≤1} and 𝐾₂={𝑘∈ℕ∶𝑝_𝑘>1} and also let us put 𝑝^‵_𝑘=𝑝_𝑘/(𝑝_𝑘−1) for 1<𝑝_𝑘<∞ and for all 𝑘∈ℕ. Before giving the theorems, let us suppose that (𝑞_𝑛) is a nondecreasing bounded sequence of positive real numbers and consider the following conditions: sup_𝑁sup_{𝑘∈𝐾1𝑛∈𝑁}𝑎_𝑛𝑘^𝑞𝑛<∞,∃𝑀sup_{𝑁𝑘∈𝐾2𝑛∈𝑁}𝑎_𝑛𝑘𝑀^{−1‵𝑝𝑘}<∞,∃𝑀sup_𝑘𝑛𝑎_𝑘𝑀^{−1/𝑝𝑘𝑞𝑛}<∞,lim_𝑛𝑎_𝑛𝑘^𝑞𝑛=0(∀𝑘∈ℕ),∀𝐿,sup_𝑛sup_𝑘∈𝐾1𝑎_𝑛𝑘𝐿^{1/𝑞𝑛𝑝𝑘}<∞,∀𝐿,∃𝑀sup_{𝑛𝑘∈𝐾2}𝑎_𝑘𝐿^1/𝑞𝑛𝑀^{−1‵𝑝𝑘}<∞,sup_𝑛sup_𝑘∈𝐾1𝑎_𝑛𝑘^𝑝𝑘<∞,∃𝑀sup_{𝑛𝑘∈𝐾2}𝑎_𝑘𝑀^{−1‵𝑝𝑘}<∞,∀𝐿,sup_𝑛sup_𝑘∈𝐾1𝑎_𝑛𝑘−𝑎_𝑘𝐿^{1/𝑞𝑛𝑝𝑘}<∞,lim_𝑛𝑎_𝑛𝑘−𝑎_𝑘^𝑞𝑛=0,∀𝑘.∀𝐿,∃𝑀sup_{𝑛𝑘∈𝐾2}𝑎_𝑛𝑘−𝑎_𝑘𝐿^1/𝑞𝑛𝑀^{−1‵𝑝𝑘},∃𝐿,sup_𝑛sup_𝑘∈𝐾1𝑎_𝑛𝑘𝐿^{−1/𝑞𝑛𝑝𝑘}<∞,∃𝐿,sup_{𝑛𝑘∈𝐾2}𝑎_𝑛𝑘𝐿^{−1/𝑞𝑛‵𝑝𝑘}𝜆<∞,_𝑘𝜆_𝑘−𝜆_𝑘−1𝑎_𝑛𝑘^∞_𝑘=0∈𝑐₀𝑞)𝜆(∀𝑛∈ℕ_𝑘𝜆_𝑘−𝜆_𝑘−1𝑎_𝑛𝑘^∞_𝑘=0𝑞()𝜆∈𝑐∀𝑛∈ℕ_𝑘𝜆_𝑘−𝜆_𝑘−1𝑎_𝑛𝑘^∞_𝑘=0∈ℓ_∞𝑞(∀𝑛∈ℕ). (4.12)(4.13)(4.14)(4.15)(4.16)(4.17)(4.18)(4.19)(4.20)(4.21)(4.22)(4.23)(4.24)(4.25)(4.26)(4.27)

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.