Generalized Difference Spaces of Non-Absolute Type of Convergent and Null Sequences

Abstract

The aim of the present paper is to introduce the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) of generalized difference sequences which generalize the paper due to Mursaleen and Noman (2010). These spaces are the BK-spaces of non-absolute type and norm isomorphic to the spaces 𝑐₀ and 𝑐, respectively. Furthermore, we derive some inclusion relations determine the 𝛼-, 𝛽-, and 𝛾-duals of those spaces, and construct their Schauder bases. Finally, we characterize some matrix classes from the spaces 𝑐^𝜆₀(𝐵), and 𝑐^𝜆(𝐵) to the spaces ℓ_𝑝, 𝑐₀, and c.

1. Introduction

By 𝜔, we denote the space of all complex valued sequences. Any vector subspace of 𝜔 is called a sequence space. A sequence space 𝐸 with a linear topology is called a 𝐾-space provided each of the maps 𝑝_𝑖∶𝐸→ℂ defined by 𝑝_𝑖(𝑥)=𝑥_𝑖 is continuous for all 𝑖∈ℕ, where ℂ denotes the complex field and ℕ={0,1,2,…}. A K-space is called an 𝐹𝐾-space provided 𝐸 is a complete linear metric space. An 𝐹𝐾-space whose topology is normable is called a 𝐵𝐾-space (see [1, pages 272-273]) which contains 𝜙, the set of all finitely nonzero sequences. We write ℓ_∞, 𝑐 and 𝑐₀ for the classical sequence spaces of all bounded, convergent, and null sequences, respectively, which are 𝐵𝐾-spaces with the usual sup-norm defined by ‖𝑥‖_∞=sup|𝑥_𝑘|, where, here and in the sequel, the supremum is taken over all 𝑘∈ℕ. Also by ℓ₁ and ℓ_𝑝, we denote the spaces of all absolutely and 𝑝-absolutely convergent series, respectively, which are 𝐵𝐾-spaces with the usual norm defined by ‖𝑥‖_𝑝=(∑_𝑘|𝑥_𝑘|^𝑝)^1/𝑝, where 1≤𝑝<∞. For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. Also by 𝑏𝑠 and 𝑐𝑠, we denote the spaces of all bounded and convergent series, respectively.

Let 𝑋 and 𝑌 be two sequence spaces, and let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix of complex numbers 𝑎_𝑛𝑘, where 𝑛,𝑘∈ℕ. Then, we say that 𝐴 defines a matrix mapping from 𝑋 into 𝑌 and we denote it by writing 𝐴∶𝑋→𝑌, if for every sequence 𝑥=(𝑥_𝑘)∈𝑋 the sequence 𝐴𝑥={𝐴_𝑛(𝑥)}, 𝐴-transform of 𝑥, exists and is in 𝑌, where 𝐴_𝑛(𝑥)∶=_𝑘𝑎_𝑛𝑘𝑥_𝑘,∀𝑛∈ℕ. (1.1) By (𝑋∶𝑌), we denote the class of all infinite matrices 𝐴=(𝑎_𝑛𝑘) such that 𝐴∶𝑋→𝑌. Thus 𝐴∈(𝑋∶𝑌) if and only if the series on the right side of (1.1) converges for each 𝑛∈ℕ and every 𝑥∈𝑋, and 𝐴𝑥∈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.

The domain 𝑋_𝐴 of an infinite matrix 𝐴 in a sequence space 𝑋 is defined by 𝑋_𝐴∶=𝑥=𝑥_𝑘∈𝜔∶𝐴𝑥∈𝑋. (1.2) We denote the collection of all finite subsets of ℕ by ℱ. Also, we write 𝑒^(𝑘) for the sequence whose only nonzero term is a 1 in the 𝑘th place for each 𝑘∈ℕ.

The approach of constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by several authors, for example, [2–14]. They introduced the sequence spaces (ℓ_∞)_𝑁𝑞 and c_𝑁𝑞 in [14], (ℓ_𝑝)_𝐶1=𝑋_𝑝 and (ℓ_∞)_𝐶1=𝑋_∞ in [10], 𝜇_𝐺=𝑍(𝑢,𝑣;𝜇) in [9], (ℓ_∞)_𝑅^𝑡=𝑟^𝑡_∞, 𝑐_𝑅^𝑡=𝑟^𝑡_𝑐 and (𝑐₀)_𝑅𝑡=𝑟^𝑡₀ in [8], (ℓ_𝑝)_𝑅^𝑡=𝑟^𝑡_𝑝 in [2], (𝑐₀)_𝐸^𝑡=𝑒^𝑟₀ and 𝑐_𝐸^𝑟=𝑒^𝑟_𝑐 in [3], (ℓ_𝑝)_𝐸^𝑟=𝑒^𝑟_𝑝 and (ℓ_∞)_𝐸^𝑟=𝑒^𝑟_∞ in [4], (𝑐₀)_𝐴^𝑟=𝑎^𝑟₀ and 𝑐_𝐴^𝑟=𝑎^𝑟_𝑐 in [5], [𝑐₀(𝑢,𝑝)]_𝐴^𝑟=𝑎^𝑟₀(𝑢,𝑝) and [𝑐(𝑢,𝑝)]_𝐴^𝑟=𝑎^𝑟_𝑐(𝑢,𝑝) in [6], (ℓ_𝑝)_𝐴^𝑟=𝑎^𝑟_𝑝 and (ℓ_∞)_𝐴^𝑟=𝑎^𝑟_∞ in [7] and (𝑐₀)_𝐶1=𝑐₀̃ and 𝑐_𝐶1=𝑐̃ in [11], 𝜈_{𝐵(𝑟,𝑠,𝑡)}=𝜈(𝐵) in [12], and 𝑓_{𝐵(𝑟,𝑠,𝑡)}=𝑓(𝐵) in [13]; where, 𝑁_𝑞, 𝐶₁, 𝑅^𝑡, and 𝐸^𝑟 denote the Nörlund, Cesáro, Riesz, and Euler means, respectively, 𝐴^𝑟, 𝐺, and 𝐵(𝑟,𝑠,𝑡) are, respectively, defined in [5, 9, 12], 𝜇∈{𝑐₀,𝑐,ℓ_𝑝}, 𝜈∈{ℓ_∞,𝑐,𝑐₀,ℓ_𝑝} and 1≤𝑝<∞. Also 𝑐₀(𝑢,𝑝) and 𝑐(𝑢,𝑝) denote the sequence spaces generated from the Maddox’s spaces 𝑐₀(𝑝) and 𝑐(𝑝) by Başarir [15]. In the present paper, following [2–14], we introduce the difference sequence spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) of non-absolute type and derive some related results. We also establish some inclusion relations. Furthermore, we determine the 𝛼-, 𝛽-, and 𝛾-duals of those spaces and construct their bases. Finally, we characterize some classes of infinite matrices concerning the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵).

The rest of this paper is organized, as follows.

In Section 2, the 𝐵𝐾-spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) of generalized difference sequences are introduced. Section 3 is devoted to inclusion relations concerning with the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵). In Sections 4 and 5, the Schauder bases of the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) are given and the 𝛼-, 𝛽-, and 𝛾-duals of the generalized difference sequence spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) of non-absolute type are determined, respectively. In Section 6, the classes (𝑐^𝜆(𝐵)∶ℓ_𝑝), (𝑐^𝜆₀(𝐵)∶ℓ_𝑝), (𝑐^𝜆(𝐵)∶𝑐), (𝑐^𝜆(𝐵)∶𝑐₀), (𝑐^𝜆₀(𝐵)∶𝑐), and (𝑐^𝜆₀(𝐵)∶𝑐₀) of matrix transformations are characterized, where 1≤𝑝≤∞. Also, by means of a given basic lemma, the characterizations of some other classes involving the Euler, difference, Riesz, and Cesàro sequence spaces are derived. In the final section of the paper, we note the significance of the present results in the literature related with difference sequence spaces and record some further suggestions.

2. The Difference Sequence Spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) of Non-Absolute Type

The difference sequence spaces have been studied by several authors in different ways (see e.g. [12, 16–21]). In the present section, we introduce the spaces 𝑐^𝜆₀(Δ), and 𝑐^𝜆(Δ), and show that these spaces are 𝐵𝐾-spaces of non-absolute type which are norm isomorphic to the spaces 𝑐₀ and 𝑐, respectively.

We assume throughout that 𝜆=(𝜆_𝑘)^∞_𝑘=0 is a strictly increasing sequence of positive reals tending to ∞, that is, 0<𝜆₀<𝜆₁<⋯,lim_𝑘→∞𝜆_𝑘=∞. (2.1)

Recently, Mursaleen and Noman [22] studied the sequence spaces 𝑐^𝜆₀ and 𝑐^𝜆 of non-absolute type, and later they introduced the difference sequence spaces 𝑐^𝜆₀(Δ) and 𝑐^𝜆(Δ) in [21] of non-absolute type as follows: 𝑐^𝜆₀(Δ)=𝑥=𝑥_𝑘∈𝜔∶lim_𝑛→∞1𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑥_𝑘−𝑥_𝑘−1,𝑐=0_𝜆(Δ)=𝑥=𝑥_𝑘∈𝜔∶lim_𝑛→∞1𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑥_𝑘−𝑥_𝑘−1exists. (2.2) Here and after, we use the convention that any term with a negative subscript is equal to zero, for example, 𝜆₋₁=0 and 𝑥₋₁=0. With the notation of (1.2) we can redefine the spaces 𝑐^𝜆₀(Δ) and 𝑐^𝜆(Δ) by 𝑐^𝜆₀(Δ)=𝑐^𝜆_0Δ,𝑐^𝜆(Δ)=𝑐^𝜆_Δ, (2.3) where Δ denotes the band matrix representing the difference operator, that is, Δ𝑥=(𝑥_𝑘−𝑥_𝑘−1)∈𝜔 for 𝑥=(𝑥_𝑘)∈𝜔.

Let 𝑟 and 𝑠 be nonzero real numbers and define the generalized difference matrix 𝐵(𝑟,𝑠)={𝑏_𝑛𝑘(𝑟,𝑠)} by 𝑏_𝑛𝑘(𝑟,𝑠)∶=𝑟,𝑘=𝑛,𝑠,𝑘=𝑛−1,0,otherwise, (2.4) for all 𝑘,𝑛∈ℕ. The 𝐵(𝑟,𝑠)-transform of a sequence 𝑥=(𝑥_𝑘) is 𝐵(𝑟,𝑠)_𝑘(𝑥)=𝑟𝑥_𝑘+𝑠𝑥_𝑘−1,∀𝑘∈ℕ. (2.5) We note that the matrix 𝐵(𝑟,𝑠) can be reduced to the difference matrices Δ in case 𝑟=1 and 𝑠=−1. So, the results related to the matrix domain of the matrix 𝐵(𝑟,𝑠) are more general and more comprehensive than the consequences of the matrices domain of Δ and include them.

Now, following Başar and Altay [18] and Aydın and Başar [17], we proceed slightly differently to Kızmaz [19] and the other authors following him and employ a technique of obtaining a new sequence space by means of the matrix domain of a triangle limitation method.

We thus introduce the difference sequence spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵), which are the generalization of the spaces 𝑐^𝜆₀(Δ) and 𝑐^𝜆(Δ) introduced by Mursaleen and Noman [21], as follows: 𝑐^𝜆₀(𝐵)=𝑥=𝑥_𝑘∈𝜔∶lim_𝑛→∞1𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑟𝑥_𝑘+𝑠𝑥_𝑘−1,𝑐=0_𝜆(𝐵)=𝑥=𝑥_𝑘∈𝜔∶lim_𝑛→∞1𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑟𝑥_𝑘+𝑠𝑥_𝑘−1exists. (2.6) With the notation of (1.2), we can redefine the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) as 𝑐^𝜆₀(𝐵)=𝑐^𝜆_0𝐵,𝑐^𝜆(𝐵)=𝑐^𝜆_𝐵, (2.7) where 𝐵 denotes the generalized difference matrix 𝐵(𝑟,𝑠)={𝑏_𝑛𝑘(𝑟,𝑠)} defined by (2.4).

It is immediate by (2.7) that the sets 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) are linear spaces with coordinatewise addition and scalar multiplication, that is, 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) are the sequence spaces of generalized differences.

On the other hand, we define the triangle matrix Λ=(𝜆_𝑛𝑘) by 𝜆_𝑛𝑘𝑟𝜆∶=^𝑘−𝜆_𝑘−1+𝑠𝜆_𝑘+1−𝜆_𝑘𝜆_𝑛𝑟𝜆,𝑘<𝑛,_𝑛−𝜆_𝑛−1𝜆_𝑛,𝑘=𝑛,0,𝑘>𝑛 (2.8) for all 𝑛,𝑘∈ℕ. With a direct calculation we derive the equality Λ_𝑛(𝑥)=1𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑟𝑥_𝑘+𝑠𝑥_𝑘−1,∀𝑛∈ℕ (2.9) and every 𝑥=(𝑥_𝑘)∈𝜔 which leads us together with (1.2) to the fact that 𝑐^𝜆₀(𝐵)=𝑐₀Λ,𝑐^𝜆(𝐵)Λ=𝑐. (2.10)

Further, for any sequence 𝑥=(𝑥_𝑘) we define the sequence 𝑦(𝜆)={𝑦_𝑘(𝜆)} which will be frequently used as the Λ-transform of 𝑥, that is, 𝑦(𝜆)=Λ(𝑥) and so we have 𝑦_𝑘(𝜆)=^𝑘−1_𝑗=0𝑟𝜆_𝑗−𝜆_𝑗−1+𝑠𝜆_𝑗+1−𝜆𝑗𝜆_𝑘𝑥_𝑗𝜆+𝑟^𝑘−𝜆_𝑘−1𝜆_𝑘𝑥_𝑘,∀𝑘∈ℕ, (2.11) Where, here and in what follows, the summation running from 0 to 𝑘−1 is equal to zero when 𝑘=0.

Moreover, it is clear by (2.9) that the relation (2.11) can be written as follows: 𝑦_𝑘(𝜆)=1𝜆_𝑘^𝑘_𝑗=0𝜆_𝑗−𝜆_𝑗−1𝑟𝑥_𝑗+𝑠𝑥_𝑗−1,∀𝑘∈ℕ. (2.12) We assume throughout that the sequences 𝑥=(𝑥_𝑘) and 𝑦=(𝑦_𝑘) are connected by the relation (2.11).

Now, we may begin with the following theorem which is essential in the text.

Theorem 2.1. The difference sequence spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) are BK-spaces with the norm ‖𝑥‖_𝑐^𝜆_0(𝐵)=‖𝑥‖_{𝑐𝜆(𝐵)}=‖Λ(𝑥)‖_∞, that is, ‖𝑥‖_𝑐^𝜆_0(𝐵)=‖𝑥‖_{𝑐𝜆(𝐵)}=sup_𝑛∈ℕΛ_𝑛(𝑥). (2.13)

Proof. Since (2.10) holds and 𝑐₀ and 𝑐 are 𝐵𝐾-spaces with respect to their natural norms (see [23, pages 217-218]) and the matrix Λ is a triangle, Theorem 4.3.12 of Wilansky [24, page 63] gives the fact that 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) are 𝐵𝐾-spaces with the given norms. This completes the proof.

Remark 2.2. One can easily check that the absolute property does not hold on the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵), that is, ‖𝑥‖_𝑐^𝜆_0(𝐵)≠‖|𝑥|‖_𝑐^𝜆_0(𝐵) and ‖𝑥‖_{𝑐𝜆(𝐵)}≠‖|𝑥|‖_{𝑐𝜆(𝐵)} for at least one sequence in the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵), and this shows that 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) are the sequence spaces of non-absolute type, where |𝑥|=(|𝑥_𝑘|).

Now, we give the final theorem of this section.

Theorem 2.3. The sequence spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) of non-absolute type are norm isomorphic to the spaces 𝑐₀ and 𝑐, respectively, that is, 𝑐^𝜆₀(𝐵)≅𝑐₀ and 𝑐^𝜆(𝐵)≅𝑐.

Proof. To prove this, we should show the existence of a linear bijection between the spaces 𝑐^𝜆₀(𝐵) and 𝑐₀. Consider the transformation 𝑇 defined, with the notation of (2.11), from 𝑐^𝜆₀(𝐵) to 𝑐₀ by 𝑥↦𝑦(𝜆). Then, 𝑇𝑥=𝑦(𝜆)=Λ(𝑥)∈𝑐₀ for every 𝑥∈𝑐^𝜆₀(𝐵) and the linearity of 𝑇 is clear. Further, it is trivial that 𝑥=𝜃 whenever 𝑇𝑥=𝜃 and hence 𝑇 is injective.
Furthermore, let 𝑦=(𝑦_𝑘)∈𝑐₀ and define the sequence 𝑥={𝑥_𝑘(𝜆)} by 𝑥_𝑘(𝜆)1∶=𝑟^𝑘_𝑗=0−𝑠𝑟^{𝑗𝑘−𝑗}_{𝑖=𝑗−1}(−1)^𝑗−𝑖𝜆_𝑖𝜆_𝑗−𝜆_𝑗−1𝑦_𝑖,∀𝑘∈ℕ. (2.14) Then, we obtain 𝑟𝑥_𝑘(𝜆)+𝑠𝑥_𝑘−1(𝜆)=^𝑘_{𝑖=𝑘−1}(−1)^𝑘−𝑖𝜆_𝑖𝜆_𝑘−𝜆_𝑘−1𝑦_𝑖,∀𝑘∈ℕ. (2.15) Hence, for every 𝑛∈ℕ, we get by (2.9) Λ_𝑛(𝑥)=1𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑟𝑥_𝑘+𝑠𝑥_𝑘−1=1𝜆_𝑛^𝑛𝑘_{𝑘=0𝑖=𝑘−1}(−1)^𝑘−𝑖𝜆_𝑖𝑦_𝑖=𝑦_𝑛. (2.16) This shows that Λ(𝑥)=𝑦 and since 𝑦∈𝑐₀, we conclude that Λ(𝑥)∈𝑐₀. Thus, we deduce that 𝑥∈𝑐^𝜆₀(𝐵) and 𝑇𝑥=𝑦. Hence 𝑇 is surjective.
Moreover, one can easily see for every 𝑥∈𝑐^𝜆₀(𝐵) that ‖𝑇𝑥‖_∞=𝑦(𝜆)_∞=Λ(𝑥)_∞=‖𝑥‖_𝑐^𝜆_0(𝐵) (2.17) which means that 𝑇 is norm preserving. Consequently 𝑇 is a linear bijection which show that the spaces 𝑐^𝜆₀(𝐵) and 𝑐₀ are linearly isomorphic.
It is clear that if the spaces 𝑐^𝜆₀(𝐵) and 𝑐₀ are replaced by the spaces 𝑐^𝜆(𝐵) and 𝑐, respectively, then we obtain the fact that 𝑐^𝜆(𝐵)≅𝑐. This completes the proof.

3. The Inclusion Relations

In the present section, we establish some inclusion relations concerning with the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵). We may begin with the following theorem.

Theorem 3.1. The inclusion 𝑐^𝜆₀(𝐵)⊂𝑐^𝜆(𝐵) strictly holds.

Proof. It is obvious that the inclusion 𝑐^𝜆₀(𝐵)⊂𝑐^𝜆(𝐵) holds. Further to show that this inclusion is strict, consider the sequence 𝑥=(𝑥_𝑘) defined by 𝑥_𝑘=∑^𝑘_𝑗=0(−𝑠/𝑟)^𝑗/𝑟 for all 𝑘∈ℕ. Then, we obtain by (2.9) that Λ_𝑛(𝑥)=1𝜆_𝑛^𝑛_𝑘=0𝜆_𝑘−𝜆_𝑘−1∀𝑛∈ℕ (3.1) which shows that Λ(𝑥)=𝑒 and hence Λ(𝑥)∈𝑐⧵𝑐₀, where 𝑒=(1,1,1,…). Thus, the sequence 𝑥 is in 𝑐^𝜆(𝐵) but not in 𝑐^𝜆₀(𝐵). Hence, the inclusion 𝑐^𝜆₀(𝐵)⊂𝑐^𝜆(𝐵) is strict and this completes the proof.

Theorem 3.2. If 𝑠+𝑟=0, then the inclusion 𝑐⊂𝑐^𝜆₀(𝐵) strictly holds.

Proof. Suppose that 𝑠+𝑟=0 and 𝑥∈𝑐. Then 𝐵(𝑟,𝑠)𝑥=(𝑟𝑥_𝑘+𝑠𝑥_𝑘−1)∈𝑐₀ and hence 𝐵(𝑟,𝑠)𝑥∈𝑐^𝜆₀, since the inclusion 𝑐₀⊂𝑐^𝜆₀. This shows that 𝑥∈𝑐^𝜆₀(𝐵). Consequently, the inclusion 𝑐⊂𝑐^𝜆₀(𝐵) holds. Further consider the sequence 𝑦=(𝑦_𝑘) defined by 𝑦_𝑘=√𝑘+1 for all 𝑘∈ℕ. Then, it is trivial that 𝑦∉𝑐. On the other hand, it can easily seen that 𝐵(𝑟,𝑠)𝑦∈𝑐₀. Hence, 𝐵(𝑟,𝑠)𝑦∈𝑐^𝜆₀ which means that 𝑦∈𝑐^𝜆₀(𝐵). Thus, the sequence 𝑦 is in 𝑐^𝜆₀(𝐵) but not in 𝑐. We therefore deduce that the inclusion 𝑐⊂𝑐^𝜆₀(𝐵) is strict. This completes the proof.

On the other hand, we recall that if 𝐴∈(𝑐∶𝑐) and 𝐵∈(𝑐∶𝑐), then 𝐴𝐵∈(𝑐∶𝑐), namely, Λ=(𝜆_𝑛𝑘) is stronger than the ordinary convergence, hence we have the following

Corollary 3.3. The inclusions 𝑐₀⊂𝑐^𝜆₀(𝐵) and 𝑐⊂𝑐^𝜆(𝐵) strictly hold.

Further, it is obvious that the sequence 𝑦, defined in the proof of Theorem 3.2, is in 𝑐^𝜆₀(𝐵) but not in ℓ_∞. This leads us to the following result.

Corollary 3.4. Although the spaces ℓ_∞ and 𝑐^𝜆₀(𝐵) overlap, the space ℓ_∞ does not include the space 𝑐^𝜆₀(𝐵).

Now, to prove the next theorem, we need the following lemma [24, page 4].

Lemma 3.5. 𝐴∈(ℓ_∞∶𝑐₀) if and only if lim_𝑛→∞∑_𝑘|𝑎_𝑛𝑘|=0.

Theorem 3.6. The inclusion ℓ_∞⊂𝑐^𝜆₀(𝐵) strictly holds if and only if 𝑧∈𝑐^𝜆₀, where the sequence 𝑧=(𝑧_𝑘) is defined by 𝑧_𝑘=𝑟𝜆_𝑘−𝜆_𝑘−1+𝑠𝜆_𝑘+1−𝜆_𝑘𝜆_𝑘−𝜆_𝑘−1,∀𝑘∈ℕ. (3.2)

Proof. Suppose that the inclusion ℓ_∞⊂𝑐^𝜆₀(𝐵) holds. Then we obtain that Λ(𝑥)∈𝑐₀ for every 𝑥∈ℓ_∞ and hence the matrix Λ=(𝜆_𝑛𝑘) is in the class (ℓ_∞∶𝑐₀). Thus it follows by Lemma 3.5 that lim_{𝑛→∞𝑘}𝜆_𝑛𝑘=0. (3.3) Now, by taking into account the definition of the matrix Λ=(𝜆_𝑛𝑘) given by (2.8), we have for every 𝑛∈ℕ that _𝑘𝜆_𝑛𝑘=1𝜆_𝑛^𝑛−1_𝑘=0𝑟𝜆_𝑘−𝜆_𝑘−1+𝑠𝜆_𝑘+1−𝜆_𝑘+|𝑟|𝜆_𝑛−𝜆_𝑛−1𝜆_𝑛. (3.4) Thus, the condition (3.3) implies both lim_𝑛→∞|𝑟|𝜆_𝑛−𝜆_𝑛−1𝜆_𝑛=0,lim_𝑛→∞1𝜆_𝑛^𝑛−1_𝑘=0𝑟𝜆_𝑘−𝜆_𝑘−1+𝑠𝜆_𝑘+1−𝜆_𝑘=0. (3.5)(3.6) Now we have for every 𝑛≥1 that 1𝜆_𝑛^𝑛−1_𝑘=0𝑟𝜆_𝑘−𝜆_𝑘−1+𝑠𝜆_𝑘+1−𝜆_𝑘=𝜆_𝑛−1𝜆_𝑛1𝜆_𝑛−1^𝑛−1_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑧_𝑘 (3.7) and since lim_𝑛→∞(𝜆_𝑛−1/𝜆_𝑛)=1 by (3.5), we obtain by (3.6) that lim_𝑛→∞1𝜆_𝑛−1^𝑛−1_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑧_𝑘=0 (3.8) which shows that 𝑧=(𝑧_𝑘)∈𝑐^𝜆₀.
Conversely, suppose that 𝑧=(𝑧_𝑘)∈𝑐^𝜆₀. Then we have (3.8). Further, for every 𝑛≥1, we derive that 1𝜆_𝑛^𝑛−1_𝑘=0𝑟𝜆_𝑘−𝜆_𝑘−1+𝑠𝜆_𝑘+1−𝜆_𝑘=1𝜆_𝑛^𝑛−1_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑧_𝑘≤1𝜆_𝑛−1^𝑛−1_𝑘=0𝜆_𝑘−𝜆_𝑘−1𝑧_𝑘. (3.9) Then, (3.9) and (3.8) together imply that (3.6) holds. On the other hand, we have for every 𝑛≥1 that 𝑟𝜆_𝑛−1+𝑠𝜆_𝑛−𝜆₀𝜆_𝑛=1𝜆_𝑛^𝑛−1_𝑘=0𝑟𝜆_𝑘−𝜆_𝑘−1+𝑠𝜆_𝑘+1−𝜆_𝑘≤1𝜆_𝑛^𝑛−1_𝑘=0𝑟𝜆_𝑘−𝜆_𝑘−1+𝑠𝜆_𝑘+1−𝜆_𝑘. (3.10) Therefore, it follows by (3.6) that lim_𝑛→∞[𝑟𝜆_𝑛−1+𝑠(𝜆_𝑛−𝜆₀)]/𝜆_𝑛=0. Particularly, if we take 𝑟=1 and 𝑠=−1, then we have lim_𝑛→∞[𝜆_𝑛−𝜆_𝑛−1−𝜆₀]/𝜆_𝑛=0 which shows that (3.5) holds. Thus, we deduce by the relation (3.4) that (3.3) holds. This leads us with Lemma 3.5 to the consequence that Λ∈(ℓ_∞∶𝑐₀). Hence, the inclusion ℓ_∞⊂𝑐^𝜆₀(𝐵) holds and is strict by Corollary 3.4. This completes the proof.

4. The Bases for the Spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵)

In the present section, we give two sequences of the points of the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) which form the bases for those spaces.

If a normed sequence space 𝑋 contains a sequence (𝑏_𝑛) with the property that for every 𝑥∈𝑋 there is a unique sequence of scalars (𝛼_𝑛) such that lim_𝑛→∞𝑥−𝛼₀𝑏₀+𝛼₁𝑏₁+⋯+𝛼_𝑛𝑏_𝑛=0 (4.1) then (𝑏_𝑛) is called a Schauder basis (or briefly basis) for 𝑋. The series ∑_𝑘𝛼_𝑘𝑏_𝑘 which has the sum 𝑥 is then called the expansion of 𝑥 with respect to (𝑏_𝑛) and is written as 𝑥=∑_𝑘𝛼_𝑘𝑏_𝑘.

Now, since the transformation 𝑇 defined from 𝑐^𝜆₀(𝐵) to 𝑐₀ in the proof of Theorem 2.3 is an isomorphism, the inverse image of the basis {𝑒^(𝑘)}^∞_𝑘=0 of the space 𝑐₀ is the basis for the new space 𝑐^𝜆₀(𝐵). Therefore, we have the following.

Theorem 4.1. Let 𝛼_𝑘(𝜆)=Λ_𝑘(𝑥) for all 𝑘∈ℕ and 𝑙=lim_𝑘→∞Λ_𝑘(𝑥). Define the sequence 𝑏^(𝑘)(𝜆)={𝑏_𝑛^(𝑘)(𝜆)}^∞_𝑘=0 for every fixed 𝑘∈ℕ by 𝑏_𝑛^(𝑘)(𝜆)∶=−𝑠𝑟^𝑛−𝑘𝜆_𝑘𝑟𝜆_𝑘−𝜆_𝑘−1+𝜆_𝑘𝑠𝜆_𝑘+1−𝜆_𝑘1,𝑘<𝑛,𝑟𝜆_𝑘𝜆_𝑘−𝜆_𝑘−1,𝑘=𝑛,0,𝑘>𝑛. (4.2) Then, the following statements hold. (a)The sequence {𝑏^(𝑘)(𝜆)}^∞_𝑘=0 is a basis for the space 𝑐^𝜆₀(𝐵) and any 𝑥∈𝑐^𝜆₀(𝐵) has a unique representation of the form 𝑥=∑_𝑘𝛼_𝑘(𝜆)𝑏^(𝑘)(𝜆). (b)The sequence {𝑏,𝑏⁽⁰⁾(𝜆),𝑏⁽¹⁾(𝜆),…} is a basis for the space 𝑐^𝜆(𝐵) and any 𝑥∈𝑐^𝜆(𝐵) has a unique representation of the form 𝑥=𝑙𝑏+∑_𝑘[𝛼_𝑘(𝜆)−𝑙]𝑏^(𝑘)(𝜆), where 𝑏=(𝑏_𝑘)={∑^𝑘_𝑗=0(−𝑠/𝑟)^𝑗/𝑟}^∞_𝑘=0.

Finally, it easily follows from Theorem 2.1 that 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) are the Banach spaces with their natural norms. Then by Theorem 4.1 we obtain the following.

Corollary 4.2. The difference sequence spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) are seprable.

5. The 𝛼-, 𝛽-, and 𝛾-Duals of the Spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵)

In this section, we state and prove the theorems determining the 𝛼-, 𝛽-, and 𝛾-duals of the generalized difference sequence spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) of non-absolute type.

For arbitrary sequence spaces 𝑋 and 𝑌, the set 𝑀(𝑋,𝑌) defined by 𝑀(𝑋,𝑌)=𝑎=𝑎_𝑘∈𝜔∶𝑎𝑥=𝑎_𝑘𝑥_𝑘∈𝑌∀𝑥=𝑥_𝑘∈𝑋 (5.1) is called the multiplier space of 𝑋 and 𝑌. One can easily observe for a sequence space 𝑍 with 𝑌⊂𝑍 and 𝑍⊂𝑋 that the inclusions 𝑀(𝑋,𝑌)⊂𝑀(𝑋,𝑍) and 𝑀(𝑋,𝑌)⊂𝑀(𝑍,𝑌) hold, respectively.

With the notation of (5.1), the 𝛼-, 𝛽-, and 𝛾-duals of a sequence space 𝑋, which are respectively, denoted by 𝑋^𝛼, 𝑋^𝛽, and 𝑋^𝛾, are defined by 𝑋^𝛼=𝑀𝑋,ℓ₁,𝑋^𝛽=𝑀(𝑋,𝑐𝑠),𝑋^𝛾=𝑀(𝑋,𝑏𝑠). (5.2) It is clear that 𝑋^𝛼⊂𝑋^𝛽⊂𝑋^𝛾. Also it can be obviously seen that the inclusions 𝑋^𝛼⊂𝑌^𝛼, 𝑋^𝛽⊂𝑌^𝛽, and 𝑋^𝛾⊂𝑌^𝛾 hold whenever 𝑌⊂𝑋.

Now, we may begin with quoting the following lemmas (see [25]) which are needed to prove Theorems 5.5 to 5.8.

Lemma 5.1. 𝐴=(𝑎_𝑛𝑘)∈(𝑐₀∶ℓ₁)=(𝑐∶ℓ₁) if and only if sup_{𝐾∈ℱ𝑛𝑘∈𝐾}𝑎_𝑛𝑘<∞. (5.3)

Lemma 5.2. 𝐴=(𝑎_𝑛𝑘)∈(𝑐₀∶𝑐) if and only if lim_𝑛→∞𝑎_𝑛𝑘=𝛼_𝑘foreachfixed𝑘∈ℕ,sup_{𝑛∈ℕ𝑘}𝑎_𝑛𝑘<∞. (5.4)(5.5)

Lemma 5.3. 𝐴=(𝑎_𝑛𝑘)∈(𝑐∶𝑐) if and only if (5.4) and (5.5) hold, and lim_{𝑛→∞𝑘}𝑎_𝑛𝑘exists. (5.6)

Lemma 5.4. 𝐴=(𝑎_𝑛𝑘)∈(𝑐∶ℓ_∞)=(𝑐₀∶ℓ_∞) if and only if (5.5) holds.

Now, we prove the following result.

Theorem 5.5. The 𝛼-dual of the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) is the set ℎ^𝜆₁=𝑎=𝑎_𝑛∈𝜔∶sup_{𝐾∈ℱ𝑛𝑘∈𝐾}ℎ^𝜆_𝑛𝑘<∞, (5.7) where the matrix 𝐻^𝜆=(ℎ^𝜆_𝑛𝑘) is defined via the sequence 𝑎=(𝑎_𝑛)∈𝜔 by ℎ^(𝜆)_𝑛𝑘=−s𝑟^𝑛−𝑘𝜆_𝑘𝑟𝜆_𝑘−𝜆_𝑘−1+𝜆_𝑘s𝜆_𝑘+1−𝜆_𝑘a_𝑛𝜆,𝑘<𝑛,_𝑛𝑟𝜆_𝑛−𝜆_𝑛−1𝑎_𝑛,𝑘=𝑛,0,𝑘>𝑛 (5.8) for all 𝑛,𝑘∈ℕ.

Proof. Let 𝑎=(𝑎_𝑛)∈𝜔. Then, by bearing in mind the relations (2.11) and (2.14), it is immediate that the equality 𝑎_𝑛𝑥_𝑛=^𝑛_𝑘=01𝑟−𝑠𝑟^{𝑘𝑛−𝑘}_{𝑖=𝑘−1}(−1)^𝑘−𝑖𝜆_𝑖𝜆_𝑘−𝜆_𝑘−1𝑎_𝑛𝑦_𝑖=𝐻^𝜆_𝑛𝑦 (5.9) holds for all 𝑛∈ℕ. Thus, we observe by (5.9) that 𝑎𝑥=(𝑎_𝑛𝑥_𝑛)∈ℓ₁ whenever 𝑥=(𝑥_𝑘)∈𝑐^𝜆₀(𝐵) or 𝑐^𝜆(𝐵) if and only if 𝐻^𝜆𝑦∈ℓ₁ whenever 𝑦=(𝑦_𝑘)∈𝑐₀ or 𝑐. This means that the sequence 𝑎=(𝑎_𝑛)∈{𝑐^𝜆₀(𝐵)}^𝛼 or 𝑎=(𝑎_𝑛)∈{𝑐^𝜆(𝐵)}^𝛼 if and only if 𝐻^𝜆∈(𝑐₀∶ℓ₁)=(𝑐∶ℓ₁). We therefore obtain by Lemma 5.4 with 𝐻^𝜆 instead of 𝐴 that 𝑎=(𝑎_𝑛)∈{𝑐^𝜆₀(𝐵)}^𝛼={𝑐^𝜆(𝐵)}^𝛼 if and only if sup_{𝐾∈ℱ𝑛𝑘∈𝐾}ℎ^𝜆_𝑛𝑘<∞ (5.10) which leads us to the consequence that {𝑐^𝜆₀(𝐵)}^𝛼={𝑐^𝜆(𝐵)}^𝛼=ℎ^𝜆₁. This completes the proof.

Theorem 5.6. Define the sets ℎ^𝜆₂, ℎ^𝜆₃, ℎ^𝜆₄, and ℎ^𝜆₅, as follows: ℎ^𝜆₂=𝑎=𝑎_𝑘∈𝜔∶^∞_𝑗=𝑘−𝑠𝑟^𝑛−𝑗𝑎_𝑗,ℎexistsforeach𝑘∈ℕ_𝜆3=𝑎=𝑎_𝑘∈𝜔∶sup_𝑛∈ℕ^𝑛−1_𝑘=0𝑎_𝑘(𝑛,ℎ)<∞_𝜆4=𝑎=𝑎_𝑘∈𝜔∶sup_𝑛∈ℕ1𝑟𝜆_𝑘𝜆_𝑘−𝜆_𝑘−1𝑎_𝑘,ℎ<∞_𝜆5=𝑎=𝑎_𝑘∈𝜔∶_𝑘1𝑟^𝑘_𝑗=0−𝑠𝑟^𝑗𝑎_𝑘𝑎converges,where_𝑘(𝑛)=𝜆_𝑘𝑎_𝑘𝑟𝜆_𝑘−𝜆_𝑘−1+1𝑟𝜆_𝑘−𝜆_𝑘−1+1𝑠𝜆_𝑘+1−𝜆_𝑘^𝑛_𝑗=𝑘+1−𝑠𝑟^𝑛−𝑗𝑎_𝑗for𝑘<𝑛. (5.11) Then {𝑐^𝜆₀(𝐵)}^𝛽=ℎ^𝜆₂∩ℎ^𝜆₃∩ℎ^𝜆₄ and {𝑐^𝜆(𝐵)}^𝛽=ℎ^𝜆₂∩ℎ^𝜆₃∩ℎ^𝜆₄∩ℎ^𝜆₅.

Proof. Consider the equality ^𝑛_𝑘=0𝑎_𝑘𝑥_𝑘=^𝑛_𝑘=0^𝑘_𝑗=01𝑟−𝑠𝑟^{𝑘−𝑗𝑗}_{𝑖=𝑗−1}(−1)^𝑗−𝑖𝜆_𝑖𝜆_𝑗−𝜆_𝑗−1𝑦_𝑖𝑎_𝑘=^𝑛−1_𝑘=0𝜆_𝑘𝑎_𝑘𝑟𝜆_𝑘−𝜆_𝑘−1+1𝑟𝜆_𝑘−𝜆_𝑘−1+1𝑠𝜆_𝑘+1−𝜆_𝑘^𝑛_𝑗=𝑘+1−𝑠𝑟^𝑛−𝑗𝑎_𝑗𝑦_𝑘+1𝑟𝜆_𝑛𝜆_𝑛−𝜆_𝑛−1𝑎_𝑛𝑦_𝑛=^𝑛−1_𝑘=0𝑎_𝑘(𝑛)𝑦_𝑘+1𝑟𝜆_𝑛𝜆_𝑛−𝜆_𝑛−1𝑎_𝑛𝑦_𝑛=𝑇^𝜆_𝑛𝑦∀𝑛∈ℕ, (5.12) where the matrix 𝑇^𝜆=(𝑡^𝜆_𝑛𝑘) is defined by 𝑡^𝜆_𝑛𝑘𝑎∶=^𝑘(𝑛)1,𝑘<𝑛,𝑟𝜆_𝑛𝜆_𝑛−𝜆_𝑛−1𝑎_𝑛,𝑘=𝑛,0,𝑘>𝑛 (5.13) for all 𝑘,𝑛∈ℕ. Then, we deduce by (5.12) that 𝑎𝑥=(𝑎_𝑘𝑥_𝑘)∈𝑐𝑠 whenever 𝑥=(𝑥_𝑘)∈𝑐^𝜆₀(𝐵) if and only if 𝑇^𝜆𝑦∈𝑐 whenever 𝑦=(𝑦_𝑘)∈𝑐₀. This means that 𝑎=(𝑎_𝑘)∈{𝑐^𝜆₀(𝐵)}^𝛽 if and only if 𝑇^𝜆∈(𝑐₀∶𝑐). Therefore, by using Lemma 5.2, we derive from (5.4) and (5.5) that ^∞_𝑗=𝑘+1−𝑠𝑟^𝑛−𝑗𝑎_𝑗existsforeach𝑘∈ℕ,sup_𝑛∈ℕ^𝑛−1_𝑘=0𝑎_𝑘(𝑛)<∞,sup_𝑛∈ℕ1𝑟𝜆_𝑛𝜆_𝑛−𝜆_𝑛−1𝑎_𝑛<∞. (5.14)(5.15) Therefore, we conclude that {𝑐^𝜆₀(𝐵)}^𝛽=ℎ^𝜆₂∩ℎ^𝜆₃∩ℎ^𝜆₄.
Similarly, we deduce from Lemma 5.3 with (5.12) that 𝑎=(𝑎_𝑘)∈{𝑐^𝜆(𝐵)}^𝛽 if and only if 𝑇^𝜆∈(𝑐∶𝑐). Therefore, we derive from (5.4) and (5.5) that (5.14), (5.15) hold.
Further, with a simple calculation one can easily see that the equality ^𝑛_𝑘=01𝑟^𝑘_𝑗=0−𝑠𝑟^𝑗𝑎_𝑘=^𝑛−1_𝑘=0𝑎_𝑘(𝑛)+1𝑟𝜆_𝑛𝜆_𝑛−𝜆_𝑛−1𝑎_𝑛=_𝑘𝑡^𝜆_𝑛𝑘 (5.16) holds for all 𝑛∈ℕ. Consequently, from (5.6) we obtain that 1𝑟^𝑘_𝑗=0−𝑠𝑟^𝑗𝑎_𝑘∈𝑐𝑠. (5.17) Hence, we deduce that {𝑐^𝜆(𝐵)}^𝛽=ℎ^𝜆₂∩ℎ^𝜆₃∩ℎ^𝜆₄∩ℎ^𝜆₅. This completes the proof.

Remark 5.7. We may note by combining (5.17) with the conditions (5.15) that {∑^𝑘_𝑗=0(−𝑠/𝑟)^𝑗𝑎_𝑘/𝑟}∈𝑏𝑠 for every sequence 𝑎=(𝑎_𝑘)∈{𝑐^𝜆₀(𝐵)}^𝛽.

Finally, we close this section with the following theorem which determines the 𝛾-dual of the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵):

Theorem 5.8. The 𝛾-duals of the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) are the set ℎ^𝜆₃∩ℎ^𝜆₄.

Proof. The proof of this result follows the same lines that in the proof of Theorem 5.6 using Lemma 5.4 instead of Lemma 5.2.

6. Certain Matrix Mappings Related to the Spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵)

In this final section, we characterize the matrix classes (𝑐^𝜆(𝐵)∶ℓ_𝑝), (𝑐^𝜆₀(𝐵)∶ℓ_𝑝), (𝑐^𝜆(𝐵)∶𝑐), (𝑐^𝜆(𝐵)∶𝑐₀), (𝑐^𝜆₀(𝐵)∶𝑐), and (𝑐^𝜆₀(𝐵)∶𝑐₀), where 1≤𝑝≤∞. Also, by means of a given basic lemma, we derive the characterizations of some other classes involving the Euler, difference, Riesz, and Cesàro sequence spaces.

For an infinite matrix 𝐴=(𝑎_𝑛𝑘), we write for brevity that 𝑎_𝑛𝑘(𝑚)=𝜆_𝑘𝑎_𝑛𝑘𝑟𝜆_𝑘−𝜆_𝑘−1+1𝑟𝜆_𝑘−𝜆_𝑘−1+1𝑠𝜆_𝑘+1−𝜆_𝑘^𝑚_𝑗=𝑘+1−𝑠𝑟^𝑛−𝑗𝑎_𝑛𝑗𝑎if𝑘<𝑚,_𝑛𝑘=𝜆_𝑘𝑎_𝑛𝑘𝑟𝜆_𝑘−𝜆_𝑘−1+1𝑟𝜆_𝑘−𝜆_𝑘−1+1𝑠𝜆_𝑘+1−𝜆_𝑘^∞_𝑗=𝑘+1−𝑠𝑟^𝑛−𝑗𝑎_𝑛𝑗 (6.1) for all 𝑘,𝑚,𝑛∈ℕ provided the convergence of the series.

The following lemmas will be needed in proving our main results.

Lemma 6.1 (see [24, page 57]). The matrix mappings between the BK-spaces are continuous.

Lemma 6.2 (see [25, pages 7-8]). 𝐴=(𝑎_𝑛𝑘)∈(𝑐∶ℓ_𝑝) if and only if sup_{𝐾∈ℱ𝑛𝑘∈𝐾}𝑎_𝑛𝑘^𝑝<∞,1≤𝑝<∞. (6.2)

Lemma 6.3 (see [25, page 5]). 𝐴=(𝑎_𝑛𝑘)∈(𝑐∶𝑐₀) if and only if (5.5) holds and lim_𝑛→∞𝑎_𝑛𝑘=0∀𝑘∈ℕ,lim_{𝑛→∞𝑘}𝑎_𝑛𝑘=0. (6.3)

Lemma 6.4 (see [25, page 5]). 𝐴=(𝑎_𝑛𝑘)∈(𝑐₀∶𝑐₀) if and only if (5.5) and (6.3) hold.

Now, we give the following results on the matrix transformations.

Theorem 6.5. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix over the complex field. Then, the following statements hold.  (i) Let (1≤𝑝<∞). Then, 𝐴∈(𝑐^𝜆(𝐵)∶ℓ_𝑝) if and only if ^∞_𝑗=𝑘+1−𝑠𝑟^𝑛−𝑗𝑎_𝑛𝑗existsforeachfixed𝑘∈ℕ,sup_{𝐾∈ℱ𝑛𝑘∈𝐾}𝑎_𝑛𝑘^𝑝<∞,sup_𝑚∈ℕ^𝑚−1_𝑘=0𝑎_𝑛𝑘(𝑚1)<∞,∀𝑛∈ℕ,𝑟^𝑘_𝑗=0−𝑠𝑟^𝑗𝑎_𝑛𝑘^∞_𝑘=0∈𝑐𝑠foreachfixed𝑛∈ℕ,lim_𝑘→∞𝜆_𝑘𝑟𝜆_𝑘−𝜆_𝑘−1𝑎_𝑛𝑘=𝑎_𝑛𝑎foreachfixed𝑛∈ℕ,_𝑛∈ℓ_𝑝. (6.4)(6.5)(6.6)(6.7)(6.8)(6.9) (ii) 𝐴∈(𝑐^𝜆(𝐵)∶ℓ_∞) if and only if (6.7) and (6.8) hold, and sup_{𝑛∈ℕ𝑘}𝑎_𝑛𝑘𝑎<∞,_𝑛∈ℓ_∞. (6.10)(6.11)

Proof. Suppose that the conditions (6.4)–(6.9) hold and take any 𝑥=(𝑥_𝑘)∈𝑐^𝜆(𝐵). Then, we have by Theorem 5.6 that {𝑎_𝑛𝑘}_𝑘∈ℕ∈{𝑐^𝜆(𝐵)}^𝛽 for all 𝑛∈ℕ and this implies that the 𝐴-transform of 𝑥 exists. Also, it is clear that the associated sequence 𝑦=(𝑦_𝑘) is in the space 𝑐 and hence 𝑦_𝑘→𝑙 as 𝑘→∞ for some suitable 𝑙. Further, it follows by combining Lemma 6.2 with (6.5) that the matrix 𝐴=(𝑎_𝑛𝑘) is in the class (𝑐∶ℓ_𝑝), where 1≤𝑝<∞.
Let us now consider the following equality derived by using the relation (2.11) from the 𝑚th partial sum of the series ∑_𝑘𝑎_𝑛𝑘𝑥_𝑘: ^𝑚_𝑘=0𝑎_𝑛𝑘𝑥_𝑘=^𝑚−1_𝑘=0𝑎_𝑛𝑘(𝑚)𝑦_𝑘+𝜆_𝑚𝑟𝜆_𝑚−𝜆_𝑚−1𝑎_𝑛𝑚𝑦_𝑚,∀𝑛,𝑚∈ℕ. (6.12) Then, since 𝑦∈𝑐 and 𝐴∈(𝑐∶ℓ_𝑝), the series ∑_𝑘𝑎_𝑛𝑘𝑦_𝑘 converges for every 𝑛∈ℕ. Furthermore, it follows by (6.4) that the series ∑^∞_𝑗=𝑘(−𝑠/𝑟)^𝑛−𝑗𝑎_𝑛𝑗 converges for all 𝑛,𝑘∈ℕ and hence 𝑎_𝑛𝑘(𝑚)→𝑎_𝑛𝑘 as 𝑚→∞. Therefore, if we pass to limit in (6.12) as 𝑚→∞ then we obtain by (6.8) that _𝑘𝑎_𝑛𝑘𝑥_𝑘=_𝑘𝑎_𝑛𝑘𝑦_𝑘+𝑙𝑎_𝑛,∀𝑛∈ℕ (6.13) which can be written as follows: 𝐴_𝑛(𝑥)=𝐴_𝑛𝑦+𝑙𝑎_𝑛,∀𝑛∈ℕ. (6.14) This yields by taking 𝑝-norm that ‖𝐴𝑥‖_𝑝≤𝐴𝑦_𝑝+|𝑙|(𝑎_𝑛)_𝑝<∞ (6.15) which leads us to the consequence that 𝐴𝑥∈ℓ_𝑝. Hence, 𝐴∈(𝑐^𝜆(𝐵)∶ℓ_𝑝).
Conversely, suppose that 𝐴∈(𝑐^𝜆(𝐵)∶ℓ_𝑝), where 1≤𝑝<∞. Then {𝑎_𝑛𝑘}_𝑘∈ℕ∈{𝑐^𝜆(𝐵)}^𝛽 for all 𝑛∈ℕ which implies with Theorem 5.6 that the conditions (6.6) and (6.7) are necessary.
On the other hand, since 𝑐^𝜆(𝐵) and ℓ_𝑝 are 𝐵𝐾-spaces, we have by Lemma 6.1 that there is a constant 𝑀>0 such that ‖𝐴𝑥‖_𝑝≤𝑀‖𝑥‖_{𝑐𝜆(𝐵)} (6.16) holds for all 𝑥∈𝑐^𝜆(𝐵). Now, 𝐾∈ℱ. Then, the sequence 𝑧=∑_𝑘∈𝐾𝑏^(𝑘)(𝜆) is in 𝑐^𝜆(𝐵), where the sequence 𝑏^(𝑘)(𝜆)={𝑏_𝑛^(𝑘)(𝜆)}_𝑛∈ℕ is defined by (4.2) for every fixed 𝑘∈ℕ.
Since Λ(𝑏^(𝑘)(𝜆))=𝑒^(𝑘) for each fixed 𝑘∈ℕ, we have ‖𝑧‖_{𝑐𝜆(𝐵)}=Λ(𝑧)_∞=_𝑘∈𝐾Λ𝑏^(𝑘)(𝜆)_∞=_𝑘∈𝐾𝑒^(𝑘)_∞=1. (6.17)
Furthermore, for every 𝑛∈ℕ, we obtain by (4.2) that 𝐴_𝑛(𝑧)=_𝑘∈𝐾𝐴_𝑛𝑏^(𝑘)(𝜆)=_{𝑘∈𝐾𝑗}𝑎_𝑛𝑗𝑏_𝑗^(𝑘)(𝜆)=_𝑘∈𝐾𝑎_𝑛𝑘 (6.18) Hence, since the inequality (6.16) is satisfied for the sequence 𝑧∈𝑐^𝜆(𝐵), we have for any 𝐾∈ℱ that _{𝑛𝑘∈𝐾}𝑎_𝑛𝑘^𝑝1/𝑝≤𝑀 (6.19) which shows the necessity of (6.5). Thus, it follows by Lemma 6.2 that 𝐴=(𝑎_𝑛𝑘)∈(𝑐∶ℓ_𝑝).
Now, let 𝑦=(𝑦_𝑘)∈𝑐⧵𝑐₀ and consider the sequence 𝑥=(𝑥_𝑘) defined by (2.14) for every 𝑘∈ℕ. Then, 𝑥∈𝑐^𝜆(𝐵) such that 𝑦=Λ(𝑥), that is, the sequences 𝑥 and 𝑦 are connected by the relation (2.11). Therefore, 𝐴𝑥 and 𝐴𝑦 exist. This leads us to the convergence of the series ∑_𝑘𝑎_𝑛𝑘𝑥_𝑘 and ∑_𝑘𝑎_𝑛𝑘𝑦_𝑘 for every 𝑛∈ℕ. We thus deduce that lim_𝑚→∞^𝑚−1_𝑘=0𝑎_𝑛𝑘(𝑚)𝑦_𝑘=_𝑘𝑎_𝑛𝑘𝑦_𝑘,∀𝑛∈ℕ. (6.20) Consequently, we obtain from (6.12) as 𝑚→∞ that lim_𝑚→∞𝜆_𝑚𝑟𝜆_𝑚−𝜆_𝑚−1𝑎_𝑛𝑚𝑦_𝑚existsforeachfixed𝑛∈ℕ (6.21) and since 𝑦=(𝑦_𝑘)∈𝑐⧵𝑐₀, we conclude that lim_𝑚→∞𝜆_𝑚𝑟𝜆_𝑚−𝜆_𝑚−1𝑎_𝑛𝑚existsforeachfixed𝑛∈ℕ (6.22) which shows the necessity of (6.8). Then relation (6.14) holds.
Finally, since 𝐴𝑥∈ℓ_𝑝 and 𝐴𝑦∈ℓ_𝑝, the necessity of (6.9) is immediate by (6.14). This completes the proof of Part (i) of the theorem.
Since Part (ii) can be proved by using the similar way that used in the proof of Part (i) with Lemma 5.4 instead of Lemma 6.2, we leave the details to the reader.

Remark 6.6. It is clear by (6.10) that the limit lim_𝑚→∞^𝑚−1_𝑘=0𝑎_𝑛𝑘(𝑚)=_𝑘𝑎_𝑛𝑘 (6.23) exists for each 𝑛∈ℕ. This just tells us that condition (6.10) implies condition (6.6).

Now, we may note that (𝑐₀∶ℓ_𝑝)=(𝑐∶ℓ_𝑝) for 1≤𝑝≤∞, (see [25, pages 7-8]). Thus, by means of Theorem 5.6 and Lemmas 6.2 and 5.4, we immediately conclude the following theorem.

Theorem 6.7. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix over the complex field. Then, the following statements hold.  (i) Let 1≤𝑝<∞. Then, 𝐴∈(𝑐^𝜆₀(𝐵)∶ℓ_𝑝) if and only if (6.5) and (6.6) hold, and ^∞_𝑗=𝑘−𝑠𝑟^𝑛−𝑗𝑎_𝑛𝑗𝜆existsforall𝑛,𝑘∈ℕ,_𝑘𝑟(𝜆_𝑘−𝜆_𝑘−1)𝑎_𝑛𝑘^∞_𝑘=0∈ℓ_∞,∀𝑛∈ℕ. (6.24) (ii) 𝐴∈(𝑐^𝜆₀(𝐵)∶ℓ_∞) if and only if (6.10) and (6.24) hold.

Proof. It is natural that the present theorem can be proved by the same technique used in the proof of Theorem 6.5, above, and so we omit the proof.

Theorem 6.8. 𝐴=(𝑎_𝑛𝑘)∈(𝑐^𝜆(𝐵)∶𝑐) if and only if (6.7), (6.8), and (6.10) hold, and lim_𝑛→∞𝑎_𝑛=𝑎,lim_𝑛→∞𝑎_𝑛𝑘=𝛼_𝑘,foreach𝑘∈ℕ,lim_{𝑛→∞𝑘}𝑎_𝑛𝑘=𝛼. (6.25)(6.26)(6.27)

Proof. Suppose that 𝐴 satisfies the conditions (6.7), (6.8), (6.10), (6.25), (6.26), and (6.27) and take any 𝑥∈𝑐^𝜆(𝐵). Then, since (6.10) implies (6.6), we have by Theorem 5.6 that {𝑎_𝑛𝑘}_𝑘∈ℕ∈{𝑐^𝜆(𝐵)}^𝛽 for all 𝑛∈ℕ and hence 𝐴𝑥 exists. We also observe from (6.10) and (6.26) that ^𝑘_𝑗=0𝛼_𝑗≤sup_{𝑚∈ℕ𝑗}𝑎_𝑛𝑘<∞ (6.28) holds for every 𝑘∈ℕ. This implies that (𝛼_𝑘)∈ℓ₁ and hence the series ∑_𝑘𝛼_𝑘(𝑦_𝑘−𝑙) converges, where 𝑦=(𝑦_𝑘)∈𝑐 is the sequence connected with 𝑥=(𝑥_𝑘) by the relation (2.11) such that 𝑦_𝑘→𝑙 as 𝑘→∞. Further it is obvious by combining Lemma 5.3 with the condition (6.10), (6.26), and (6.27) that the matrix 𝐴=(𝑎_𝑛𝑘) is in the class (𝑐∶𝑐).
Now reasoning as in the proof of Theorem 6.5, we obtain that the relation (6.13) holds which can be written as follows: _𝑘𝑎_𝑛𝑘𝑥_𝑘=_𝑘𝑎_𝑛𝑘𝑦_𝑘−𝑙+𝑙_𝑘𝑎_𝑛𝑘+𝑙𝑎_𝑛,∀𝑛∈ℕ. (6.29) In this situation, we see by passing to the limit in (6.29) as 𝑛→∞ that the first term on the right tends to ∑_𝑘𝛼_𝑘(𝑦_𝑘−𝑙) by (6.10) and (6.26). The second term tends to 𝑙𝛼 by (6.27) and the last term to 𝑙𝑎 by (6.25). Consequently, we have lim_𝑛→∞𝐴_𝑛(𝑥)=_𝑘𝛼_𝑘𝑦_𝑘−𝑙+𝑙(𝛼+𝑎), (6.30) which shows that 𝐴𝑥∈𝑐, that is to say that 𝐴∈(𝑐^𝜆(𝐵)∶𝑐).
Conversely, Suppose that 𝐴∈(𝑐^𝜆(𝐵)∶𝑐). Then, since the inclusion 𝑐⊂ℓ_∞ holds, it is trivial that 𝐴∈(𝑐^𝜆(𝐵)∶ℓ_∞). Therefore, the necessity of the conditions (6.7), (6.8), and (6.10) are obvious from Theorem 6.5. Further, consider the sequence 𝑏^(𝑘)(𝜆)={𝑏_𝑛^(𝑘)(𝜆)}_𝑛∈ℕ∈𝑐^𝜆(𝐵) defined by (4.2) for every fixed 𝑘∈ℕ. Then, it is easily seen that 𝐴𝑏^(𝑘)(𝜆)={𝑎_𝑛𝑘}_𝑛∈ℕ and hence {𝑎_𝑛𝑘}_𝑛∈ℕ∈𝑐 for every 𝑘∈ℕ which shows the necessity of (6.26). Let 𝑧=∑_𝑘𝑏^(𝑘)(𝜆). Then, since the linear transformation 𝑇∶𝑐^𝜆(𝐵)→𝑐, defined as in the proof of Theorem 2.3 by analogy, is continuous and Λ(𝑏^(𝑘)(𝜆))=𝑒^(𝑘) for each fixed 𝑘∈ℕ, we obtain that Λ_𝑛(𝑧)=_𝑘Λ_𝑛𝑏^(𝑘)(𝜆)=_𝑘𝛿_𝑛𝑘=1foreach𝑛∈ℕ (6.31) which shows that Λ(𝑧)=𝑒∈𝑐 and hence 𝑧∈𝑐^𝜆(𝐵). On the other hand, since 𝑐^𝜆(𝐵) and 𝑐 are the BK-spaces, Lemma 6.1 implies the continuity of the matrix mapping 𝐴∶𝑐^𝜆(𝐵)→𝑐. Thus, we have for every 𝑛∈ℕ that 𝐴_𝑛(𝑧)=_𝑘𝐴_𝑛𝑏^(𝑘)(𝜆)=_𝑘𝑎_𝑛𝑘. (6.32) This shows the necessity of (6.27).
Now, it follows by (6.10), (6.26), and (6.27) with Lemma 5.3 that 𝐴=(𝑎_𝑛𝑘)∈(𝑐∶𝑐). So by (6.7), and (6.8), relation (6.14) holds for all 𝑥∈𝑐^𝜆(𝐵) and 𝑦∈𝑐, and 𝑥 and 𝑦 are connected by relation (2.11), where 𝑦_𝑘→𝑙 (𝑘→∞).
Lastly, since 𝐴𝑥∈𝑐 and 𝐴𝑥∈𝑐; the necessity of (6.25) is immediate by (6.14). This step concludes the proof.

Theorem 6.9. 𝐴=(𝑎_𝑛𝑘)∈(𝑐^𝜆(𝐵)∶𝑐₀) if and only if (6.7), (6.8), and (6.10) hold, and lim_𝑛→∞𝑎_𝑛=0,lim_𝑛→∞𝑎_𝑛𝑘=0foreach𝑘∈ℕ,lim_{𝑛→∞𝑘}𝑎_𝑛𝑘=0. (6.33)

Proof. Since the present theorem can be proved by the similar way used in the proof of Theorem 6.8 with Lemma 6.3 instead of Lemma 5.3, we omit the detailed proof.

Theorem 6.10. 𝐴=(𝑎_𝑛𝑘)∈(𝑐^𝜆₀(𝐵)∶𝑐) if and only if (6.10), (6.24), and (6.26) hold.

Proof. This is similarly obtained by using Lemma 5.2, Theorem 5.6, and Part (ii) of Theorem 6.7.

Theorem 6.11. 𝐴=(𝑎_𝑛𝑘)∈(𝑐^𝜆₀(𝐵)∶𝑐₀) if and only if (6.10) and (6.24) hold, and (6.26) also holds with 𝛼_𝑘=0 for all 𝑘∈ℕ.

Proof. This is immediate by Lemma 6.4, Theorems 5.6 and 6.10.

7. Conclusion

Let 𝜈 denotes any of the classical sequence spaces ℓ_∞, 𝑐, or 𝑐₀. Then, 𝜈(Δ) consisting of the sequences 𝑥=(𝑥_𝑘) such that Δ𝑥=(𝑥_𝑘−𝑥_𝑘+1)∈𝜈 is called as the difference sequence space which was introduced by Kızmaz [19]. Kızmaz [19] proved that 𝜈(Δ) is a Banach space with the norm ‖𝑥‖_Δ=|𝑥₁|+‖Δ𝑥‖_∞, where 𝑥=(𝑥_𝑘)∈𝜈(Δ) and the inclusion relation 𝜈⊂𝜈(Δ) strictly holds. He also determined the 𝛼-, 𝛽-, and 𝛾-duals of the difference spaces and characterized the classes (𝜈(Δ)∶𝜇) and (𝜇∶𝜈(Δ)) of infinite matrices, where 𝜈,𝜇∈{ℓ_∞,𝑐}. Following Kızmaz [19], Sarıgöl [26] extended the difference space 𝜈(Δ) to the space 𝜈(Δ_𝑟) defined by 𝜈Δ_𝑟∶=𝑥=𝑥_𝑘∈𝜔∶Δ_𝑟𝑥=𝑘^𝑟𝑥_𝑘−𝑥_𝑘+1∈𝜈for𝑟<1 (7.1) and computed the 𝛼-, 𝛽-, and 𝛾-duals of the space 𝜈(Δ_𝑟), where 𝜈∈{ℓ_∞,𝑐,𝑐₀}. It is easily seen that 𝜈(Δ_𝑟)⊂𝜈(Δ), if 0<𝑟<1 and 𝜈(Δ)⊂𝜈(Δ_𝑟), if 𝑟<0. Recently, the difference spaces 𝑏𝑣_𝑝 consisting of the sequences 𝑥=(𝑥_𝑘) such that (𝑥_𝑘−𝑥_𝑘−1)∈ℓ_𝑝 have been studied in the case 0<𝑝<1 by Altay and Başar [27] and in the case 1≤𝑝≤∞ by Başar and Altay [18], Çolak et al. [28], and Malkowsky et al. [29]. Quite recently, Mursaleen and Noman have introduced the spaces 𝑐^𝜆 and 𝑐^𝜆₀ of 𝜆-convergent and 𝜆-null sequences and nextly studied the difference spaces 𝑐^𝜆(Δ) and 𝑐^𝜆₀(Δ) in [21, 22], respectively. Of course, there is a wide literature concerning the difference sequence spaces. By the domain of the generalized difference matrix 𝐵(𝑟,𝑠) in the spaces of 𝜆-convergent and 𝜆-null sequences we have generalized the difference spaces 𝑐^𝜆(Δ) and 𝑐^𝜆₀(Δ) defined by Mursaleen and Noman [21]. Since the generalized difference matrix 𝐵(𝑟,𝑠) reduces, in the special case 𝑟=1, 𝑠=−1, to the usual difference matrix Δ; our results are more general and more comprehensive than the corresponding results of Mursaleen and Noman [21].

Since the difference spaces of 𝜆-bounded and absolutely 𝜆𝑝-summable sequences are not studied, the domain of both the difference matrix Δ, and the generalized difference matrix 𝐵(𝑟,𝑠) in those spaces are still open. So, it is meaningful to fill this gap.

Acknowledgments

The authors have benefited a lot from discussion on the main results of the earlier version of the paper with Professor Bilâl Altay, Department of Mathematical Education, Faculty of Education, İnönü University, Turkey. So, they wish to express their gratitude for his many helpful suggestions and interesting comments.