On the Domain of the Triangle 𝐴(𝜆) on the Spaces of Null, Convergent, and Bounded Sequences

Abstract

We introduce the spaces of 𝐴(𝜆)-null, 𝐴(𝜆)-convergent, and 𝐴(𝜆)-bounded sequences. We examine some topological properties of the spaces and give some inclusion relations concerning these sequence spaces. Furthermore, we compute 𝛼-, 𝛽-, and 𝛾-duals of these spaces. Finally, we characterize some classes of matrix transformations from the spaces of 𝐴(𝜆)-bounded and 𝐴(𝜆)-convergent sequences to the spaces of bounded, almost convergent, almost null, and convergent sequences and present a Steinhaus type theorem.

1. Introduction

By 𝜔, we denote the space of all complex sequences. If 𝑥∈𝜔, then we simply write 𝑥=(𝑥_𝑘) instead of 𝑥=(𝑥_𝑘)^∞_𝑘=0. Also, we will use the conventions that 𝑒=(1,1,…), and 𝑒^(𝑛) is the sequence whose only nonzero term is 1 in the 𝑛th place for each 𝑛∈ℕ, where ℕ={0,1,2,…}. Any vector subspace of 𝜔 is called a sequence space. We will write ℓ_∞, 𝑐 and 𝑐₀ for the sequence spaces of all bounded, convergent, and null sequences, respectively. Further, by ℓ_𝑝 with 1≤𝑝<∞, we denote the sequence space of all 𝑝-absolutely convergent series, that is, ℓ_𝑝={𝑥=(𝑥_𝑘)∈𝜔∶∑_𝑘|𝑥_𝑘|^𝑝<∞}. For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. Moreover, we write 𝑏𝑠 and 𝑐𝑠 for the spaces of all bounded and convergent series, respectively. A sequence space 𝜇 is called an FK-space if it is a complete linear metric space with continuous coordinates 𝑝_𝑛∶𝜇→ℂ, where ℂ denotes the complex field and 𝑝_𝑛(𝑥)=𝑥_𝑛 for all 𝑥=(𝑥_𝑛)∈𝜇 and every 𝑛∈ℕ. A normed FK-space is called a BK-space, that is, a BK-space is a Banach space with continuous coordinates. The sequence spaces 𝑐₀ and 𝑐 are BK-spaces with the usual sup-norm given by ‖𝑥‖_∞=sup_𝑛∈ℕ|𝑥_𝑛|. Also, the space ℓ_𝑝 is a BK-space with the usual norm ‖⋅‖_𝑝 defined by ‖𝑥‖_𝑝=_𝑛𝑥_𝑛^𝑝1/𝑝, (1) where 1≤𝑝<∞. A sequence (𝑦_𝑛) in a normed space 𝑋 is called a Schauder basis for 𝑋 if for every 𝑥∈𝑋 there is a unique sequence (𝛼_𝑛) of scalars such that 𝑥=∑_𝑛𝛼_𝑛𝑦_𝑛, that is, lim_𝑛→∞𝑥−𝛼₀𝑦₀+𝛼₁𝑦₁+𝛼₂𝑦₂+⋯+𝛼_𝑛𝑦_𝑛=0. (2) The alpha-, beta-, and gamma-duals 𝜇^𝛼, 𝜇^𝛽, and 𝜇^𝛾 of a sequence space 𝜇 are, respectively, defined by 𝜇^𝛼=𝑎=𝑎_𝑘∈𝜔∶𝑎𝑥=𝑎_𝑘𝑥_𝑘∈ℓ₁∀𝑥=𝑥_𝑘,𝜇∈𝜇_𝛽=𝑎=𝑎_𝑘∈𝜔∶𝑎𝑥=𝑎_𝑘𝑥_𝑘∈𝑐𝑠∀𝑥=𝑥_𝑘,𝜇∈𝜇_𝛾=𝑎=𝑎_𝑘∈𝜔∶𝑎𝑥=𝑎_𝑘𝑥_𝑘∈𝑏𝑠∀𝑥=𝑥_𝑘∈𝜇. (3)

If 𝐴 is an infinite matrix with complex entries 𝑎_𝑛𝑘, where 𝑘,𝑛∈ℕ, then we write 𝐴=(𝑎_𝑛𝑘) instead of 𝐴=(𝑎_𝑛𝑘)^∞_𝑛,𝑘=0. Also, we write 𝐴_𝑛 for the sequence in the 𝑛th row of the matrix 𝐴, that is, 𝐴_𝑛=(𝑎_𝑛𝑘)^∞_𝑘=0 for every 𝑛∈ℕ. Further, if 𝑥=(𝑥_𝑘)∈𝜔 then we define the 𝐴-transform of 𝑥 as the sequence 𝐴𝑥={(𝐴𝑥)_𝑛}^∞_𝑛=0, where (𝐴𝑥)_𝑛=_𝑘𝑎_𝑛𝑘𝑥_𝑘 (4) provided the series on the right hand side of (4) convergent for each 𝑛∈ℕ.

Furthermore, the sequence 𝑥 is said to be 𝐴-summable to 𝑙∈ℂ if 𝐴𝑥 converges to 𝑙 which is called the 𝐴-limit of 𝑥. In addition, let 𝜇 and 𝜈 be sequence spaces. Then, we say that 𝐴 defines a matrix mapping from 𝜇 into 𝜈 if for every sequence 𝑥∈𝜇 the 𝐴-transform of 𝑥 exists and is in 𝜈. Moreover, we write (𝜇∶𝜈) for the class of all infinite matrices that map 𝜇 into 𝜈. Thus, 𝐴∈(𝜇∶𝜈) if and only if 𝐴_𝑛∈𝜇^𝛽 for all 𝑛∈ℕ and 𝐴𝑥∈𝜈 for all 𝑥∈𝜇. The matrix domain 𝜇_𝐴 of an infinite matrix 𝐴 in a sequence space 𝜇 is defined by 𝜇_𝐴=𝑥∈𝜔∶𝐴𝑥∈𝜇 (5) which is a sequence space. The approach constructing a new sequence space by means of the matrix domain of a triangle matrix was employed by several authors, see for instance [1–4]. In this paper, we introduce the spaces of 𝐴(𝜆)-null, 𝐴(𝜆)-convergent, and 𝐴(𝜆)-bounded sequences which generalize the results given in [2]. Further, we define some related BK-spaces and construct their bases. Moreover, we establish some inclusion relations concerning those spaces and determine their alpha-, beta-, and gamma-duals. Finally, we characterize some classes of matrix transformations on these sequence spaces.

2. Notion of 𝐴(𝜆)-Null, 𝐴(𝜆)-Convergent, and 𝐴(𝜆)-Bounded Sequences

Let 𝜆=(𝜆_𝑘) be a strictly increasing sequence of positive real numbers tending to infinity, as 𝑘→∞ and 𝜆_𝑛+1≥2𝜆_𝑛 for each 𝑛∈ℕ. From this last relation, it follows that Δ²𝜆_𝑛≥0. The first and second differences are defined as follows: Δ𝜆_𝑘=𝜆_𝑘−𝜆_𝑘−1 and Δ²𝜆_𝑘=Δ(Δ𝜆_𝑘)=𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2 for all 𝑘∈ℕ, where 𝜆₋₁=𝜆₋₂=0.

Let 𝑥=(𝑥_𝑘) be a sequence of complex numbers, such that 𝑥₋₁=𝑥₋₂=0. We say that the sequence 𝑥=(𝑥_𝑘) is 𝐴(𝜆)-strongly convergent to a number 𝑙 if lim_𝑛→∞1𝜆_𝑛−𝜆^𝑛_{𝑛−1𝑘=0}𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2𝑥_𝑘−𝑙=0. (6) This generalizes the concept of Λ-strong convergence (see [5]).

Lemma 1 (see [5]). A sequence 𝑥=(𝑥_𝑛) of complex numbers 𝜆-strongly converges to a number 𝑙 if and only if 𝑥=(𝑥_𝑛) converges to 𝑙 in the ordinary sense andlim_𝑛→∞1𝜆_𝑛−𝜆^𝑛_{𝑛−1𝑘=0}𝜆_𝑘−1𝑥_𝑘−𝑥_𝑘−1=0. (7)

Let us define the sequence 𝑦=(𝑦_𝑛) by the 𝐴(𝜆)-transform of a sequence 𝑥=(𝑥_𝑘), that is, 𝑦_𝑛=𝐴_𝜆𝑥_𝑛=1𝜆_𝑛−𝜆^𝑛_{𝑛−1𝑘=0}𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2𝑥_𝑘 (8) for all 𝑛∈ℕ. Throughout the text, we suppose that the terms of the sequences 𝑥=(𝑥_𝑘) and 𝑦=(𝑦_𝑘)   are connected with the relation (8).

Lemma 2 (see [5]). If a sequence (𝑦_𝑛) converges to 𝑙 in the ordinary sense and condition (7) of Lemma 1 holds, then the sequence 𝑥=(𝑥_𝑛) of complex numbers 𝐴(𝜆)-strongly converges to 𝑙.

Remark 3 (see [5]). From above results, we can conclude the following. The sequence 𝑥=(𝑥_𝑛) of complex numbers 𝐴(𝜆)-strongly converges to 𝑙 if and only if the following relation holds: lim_𝑛→∞1𝜆_𝑛−𝜆^𝑛_{𝑛−1𝑘=0}𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2𝑥_𝑘−𝑙=0. (9)

Now, we define the infinite matrix 𝐴(𝜆)={𝑎_𝑛𝑘(𝜆)}^∞_𝑛,𝑘=0 by 𝑎_𝑛𝑘(𝜆)=𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2𝜆_𝑛−𝜆_𝑛−1,0≤𝑘≤𝑛;0,𝑘>𝑛 (10) for all 𝑛,𝑘∈ℕ. Then, 𝐴(𝜆)-transform of a sequence 𝑥∈𝜔 is the sequence 𝐴_𝜆𝑥={(𝐴_𝜆𝑥)_𝑛}^∞_𝑛=0, where (𝐴_𝜆𝑥)_𝑛 is given by the relation (8) for every 𝑛∈ℕ. Thus, the sequence 𝑥 is 𝐴(𝜆)-convergent if and only if 𝑥 is 𝐴(𝜆)-summable. Further, if 𝑥 is 𝐴(𝜆)-convergent then the 𝐴(𝜆)-limit of 𝑥 exists and coincides with the ordinary limit of 𝑥, that is, to say that the method 𝐴(𝜆) is regular.

3. The Spaces of 𝐴(𝜆)-Null, 𝐴(𝜆)-Convergent, and 𝐴(𝜆)-Bounded Sequences

We introduce the classes 𝐴_𝜆(𝑐₀), 𝐴_𝜆(𝑐), and 𝐴_𝜆(ℓ_∞) of all 𝐴(𝜆)-null, 𝐴(𝜆)-convergent, and 𝐴(𝜆)-bounded sequences of complex numbers, that is, 𝐴_𝜆𝑐₀=𝑥=𝑥_𝑘∈𝜔∶lim_𝑛→∞𝐴_𝜆𝑥_𝑛,𝐴=0_𝜆(𝑐)=𝑥=𝑥_𝑘∈𝜔∶∃𝑙∈ℂ∋lim_𝑛→∞𝐴_𝜆𝑥_𝑛,𝐴=𝑙_𝜆ℓ_∞=𝑥=𝑥_𝑘∈𝜔∶sup_𝑛∈ℕ𝐴_𝜆𝑥_𝑛<∞. (11) Obviously, 𝐴_𝜆(𝑐₀), 𝐴_𝜆(𝑐), and 𝐴_𝜆(ℓ_∞) are the linear spaces with respect to the usual operations coordinatewise addition and scalar multiplication of sequences. Here and after, by 𝑋 we denote any of the spaces 𝑐₀, 𝑐, and ℓ_∞. It is not hard to see that the quantity ‖𝑥‖_{𝐴𝜆(𝑋)}∶=sup_𝑛∈ℕ1𝜆_𝑛−𝜆^𝑛_{𝑛−1𝑘=0}𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2𝑥_𝑘 (12) is finite for every 𝑥=(𝑥_𝑘)∈𝐴_𝜆(𝑋), and ‖⋅‖_{𝐴𝜆(𝑋)} is a norm on 𝐴_𝜆(𝑋).

Denote by ‖⋅‖_𝑏𝑣 the usual 𝑏𝑣-norm, that is, to say that ‖𝑥‖_𝑏𝑣∶=_𝑘𝑥_𝑘−𝑥_𝑘−1. (13) With the notation of (5), we can redefine the spaces 𝐴_𝜆(𝑐₀), 𝐴_𝜆(𝑐), and 𝐴_𝜆(ℓ_∞) as follows: 𝐴_𝜆𝑐₀=𝑐_0𝐴(𝜆),𝐴_𝜆(𝑐)=(𝑐)_𝐴(𝜆),𝐴_𝜆ℓ_∞=(ℓ_∞)_𝐴(𝜆). (14)

Theorem 4. The sequence spaces 𝐴_𝜆(𝑐₀), 𝐴_𝜆(𝑐), and 𝐴_𝜆(ℓ_∞) are 𝐵𝐾-spaces with the norm given by‖𝑥‖_{𝐴𝜆(𝑋)}=𝐴_𝜆𝑥_∞=sup_𝑛∈ℕ𝐴_𝜆𝑥_𝑛. (15)

Proof. This follows from Theorem  4.3.12 given in [6] and the relations (14).

Theorem 5. The sequence spaces 𝐴_𝜆(𝑐₀), 𝐴_𝜆(𝑐), and 𝐴_𝜆(ℓ_∞) are norm isomorphic to the spaces 𝑐₀, 𝑐, and ℓ_∞, respectively.

Proof. Since the matrix 𝐴(𝜆) is triangle, it has unique inverse which is also triangle matrix (see [6, 1.4.8]). Therefore, the linear operator, defined by 𝑇∶𝐴_𝜆(𝑋)→𝑋, 𝑇𝑥=𝐴_𝜆𝑥 for all 𝑥∈𝐴_𝜆(𝑋), is bijective and norm preserving by relation (15).

As a consequence of Theorems 4 and 5, we get the following result.

Corollary 6. Define the sequence 𝑒^(𝑛)(𝜆)∈𝐴_𝜆(𝑐₀) for every fixed 𝑛∈ℕ by𝑒_𝑘^(𝑛)(𝜆)=(−1)^𝑘−𝑛𝜆_𝑛−𝜆_𝑛−1𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2,𝑛≤𝑘≤𝑛+1,0,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, (16)where 𝑘∈ℕ. Then, one has the following.(1)The sequence {𝑒_𝑘⁽⁰⁾(𝜆),𝑒_𝑘⁽¹⁾(𝜆),𝑒_𝑘⁽²⁾(𝜆),…} is a Schauder basis for the space 𝐴_𝜆(𝑐₀), and every 𝑥∈𝐴_𝜆(𝑐₀) has a unique representation: 𝑥=∑_𝑛(𝐴_𝜆𝑥)_𝑛𝑒_𝑘^(𝑛)(𝜆). (2)The sequence {𝑒,𝑒_𝑘⁽⁰⁾(𝜆),𝑒_𝑘⁽¹⁾(𝜆),𝑒_𝑘⁽²⁾(𝜆),…}  is a Schauder basis for the space 𝐴_𝜆(𝑐), and every 𝑥∈𝐴_𝜆(𝑐) has a unique representation: 𝑥=𝑙𝑒+∑_𝑛[(𝐴_𝜆𝑥)_𝑛−𝑙]𝑒_𝑘^(𝑛)(𝜆), where 𝑙=lim_𝑛→∞(𝐴_𝜆𝑥)_𝑛.

4. Some Inclusion Relations Related to the New Spaces

In this section, we give some inclusion relations concerning the spaces 𝐴_𝜆(𝑐₀), 𝐴_𝜆(𝑐), and 𝐴_𝜆(ℓ_∞).

Theorem 7. The inclusions 𝐴_𝜆(𝑐₀)⊂𝐴_𝜆(𝑐)⊂𝐴_𝜆(ℓ_∞) strictly hold.

Proof. Let us suppose that 𝑥=(𝑥_𝑛)∈𝐴_𝜆(𝑐₀), then it follows that 𝑥=(𝑥_𝑛)∈𝐴_𝜆(𝑐) and 𝑥=(𝑥_𝑛)∈𝐴_𝜆(ℓ_∞). In what follows we show that these inclusions are strict. The first inclusion follows from the fact that every sequence, which converges in ordinary sense, converges in 𝐴(𝜆)-sense to the same limit. To prove the strictness of the inclusion 𝐴_𝜆(𝑐)⊂𝐴_𝜆(ℓ_∞), define the sequence 𝑥=(𝑥_𝑘) by 𝑥_𝑘=(−1)^𝑘𝜆_𝑘−𝜆_𝑘−2𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2 (17) for all 𝑘∈ℕ. Then, it follows that 𝐴_𝜆𝑥_𝑛=1𝜆_𝑛−𝜆^𝑛_{𝑛−1𝑘=0}𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2𝑥_𝑘=1𝜆_𝑛−𝜆^𝑛_{𝑛−1𝑘=0}(−1)^𝑘𝜆_𝑘−𝜆_𝑘−2=(−1)^𝑛. (18) Therefore, it is trivial that 𝑥=(𝑥_𝑘)∈𝐴_𝜆(ℓ_∞)⧵𝐴_𝜆(𝑐).

Theorem 8. The equality 𝐴_𝜆(𝑐₀)∩𝑐=𝑐₀ holds.

Proof. First, we prove that 𝐴_𝜆(𝑐₀)∩𝑐⊂𝑐₀. If a sequence 𝑥=(𝑥_𝑛) converges in the ordinary sense to 𝑙 then it follows that 𝑥_𝑛→𝑙 converges in 𝐴(𝜆)-sense, too. This gives the first inclusion. The converse inclusion follows from Lemma 1, in [5].

In what follows we describe some properties of the sequence (𝜆_𝑛) in the space ℓ_∞.

Theorem 9. For the sequence (𝜆_𝑛) which is given in Section 2, the following relations are satisfied:(i)(Δ𝜆_𝑛−1/(Δ𝜆_𝑛−Δ𝜆_𝑛−1))^∞_𝑛=1∉ℓ_∞  if and only if liminf_𝑛→∞((Δ𝜆_𝑛+1−Δ𝜆_𝑛)/Δ𝜆_𝑛)=0;(ii)(Δ𝜆_𝑛−1/(Δ𝜆_𝑛−Δ𝜆_𝑛−1))^∞_𝑛=1∈ℓ_∞  if and only if liminf_𝑛→∞((Δ𝜆_𝑛+1−Δ𝜆_𝑛)/Δ𝜆_𝑛)>0.

Proof. (i) Let us start with the expression 𝑠_𝑛(𝑥)=𝑥_𝑛−𝐴_𝜆𝑥_𝑛=1𝜆_𝑛−𝜆^𝑛_{𝑛−1𝑘=0}𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2𝑥_𝑛−𝑥_𝑘. (19) After some calculations, we get 𝑠_𝑛(𝑥)=Δ𝜆_𝑛−1Δ𝜆_𝑛𝑠_𝑛(𝑥)+𝐴_𝜆𝑥_𝑛+𝐴_𝜆𝑥_𝑛−1=Δ𝜆_𝑛−1Δ𝜆_𝑛−Δ𝜆_𝑛−1𝐴_𝜆𝑥_𝑛−𝐴_𝜆𝑥_𝑛−1. (20)
On the other hand, from the definition of the sequence (𝜆_𝑛) we have 𝜆_𝑛≥2𝜆_𝑛−1⟹Δ²𝜆_𝑛≥0⟹Δ𝜆_𝑛≥Δ𝜆_𝑛−1⟹Δ𝜆_𝑛Δ𝜆_𝑛−1≥1. (21)
From the last relation, we have following two possibilities:(a)liminf_𝑛→∞((Δ𝜆_𝑛−Δ𝜆_𝑛−1)/Δ𝜆_𝑛−1)>0 or (b)liminf_𝑛→∞((Δ𝜆_𝑛−Δ𝜆_𝑛−1)/Δ𝜆_𝑛−1)=0.
Part (a) is satisfied if and only if (Δ𝜆_𝑛−1/(Δ𝜆_𝑛−Δ𝜆_𝑛−1))^∞_𝑛=1 is bounded. Part (b) is satisfied if and only if (Δ𝜆_𝑛−1/(Δ𝜆_𝑛−Δ𝜆_𝑛−1))^∞_𝑛=1 is unbounded.

Lemma 10. The inclusions 𝑐₀⊂𝐴_𝜆(𝑐₀) and 𝑐⊂𝐴_𝜆(𝑐) hold. Those spaces coincide if and only if 𝑠(𝑥)∈𝑐₀ for every 𝑥∈𝐴_𝜆(𝑐₀), respectively, 𝐴_𝜆(𝑐), where 𝑠(𝑥)={𝑠_𝑛(𝑥)}^∞_𝑛=0.

Lemma 11. The inclusion ℓ_∞⊂𝐴_𝜆(ℓ_∞) holds. Those spaces coincide if and only if 𝑠(𝑥)∈ℓ_∞  for every 𝑥∈𝐴_𝜆(ℓ_∞).

Theorem 12. The inclusions 𝑐₀⊂𝐴_𝜆(𝑐₀), 𝑐⊂𝐴_𝜆(𝑐) and ℓ_∞⊂𝐴_𝜆(ℓ_∞) strictly hold if and only ifliminf_𝑛→∞Δ𝜆_𝑛+1−Δ𝜆_𝑛Δ𝜆_𝑛>0. (22)

Proof. Let us suppose that ℓ_∞⊂𝐴_𝜆(ℓ_∞) is strict. Then, from Lemma 11, it follows that there exists a sequence 𝑥=(𝑥_𝑛)∈𝐴_𝜆(ℓ_∞) such that 𝑠(𝑥)={𝑠_𝑛(𝑥)}^∞_𝑛=0∉ℓ_∞. Since 𝑥=(𝑥_𝑛)∈𝐴_𝜆(ℓ_∞), we have 𝐴_𝜆𝑥∈ℓ_∞ which leads us to the fact that {𝑥_𝑛−𝑠_𝑛(𝑥)}∈ℓ_∞. On the other hand, from relation (20), it follows that (Δ𝜆_𝑛−1/(Δ𝜆_𝑛−Δ𝜆_𝑛−1))∉ℓ_∞. The last relation is equivalent to liminf_𝑛→∞Δ𝜆_𝑛+1−Δ𝜆_𝑛Δ𝜆_𝑛=0 (23) by part (i) of Theorem 9. In a similar way we can conclude that the inclusions 𝑐₀⊂𝐴_𝜆(𝑐₀), 𝑐⊂𝐴_𝜆(𝑐) are strict. In what follows we prove the sufficiency. Let liminf_𝑛→∞Δ𝜆_𝑛+1−Δ𝜆_𝑛Δ𝜆_𝑛=0. (24) Then, from, part (i) of Theorem 9, it follows that (Δ𝜆_𝑛−1/(Δ𝜆_𝑛−Δ𝜆_𝑛−1))∉ℓ_∞ and ((Δ𝜆_𝑛−1+Δ𝜆_𝑛)/(Δ𝜆_𝑛−Δ𝜆_𝑛−1))∉ℓ_∞. Let us define the sequence 𝑥=(𝑥_𝑛) by 𝑥_𝑛=(−1)^𝑛Δ𝜆_𝑛−1+Δ𝜆_𝑛Δ𝜆_𝑛−Δ𝜆_𝑛−1 (25) for all 𝑛∈ℕ. Then, we get the following estimation: 𝐴_𝜆𝑥_𝑛=1𝜆_𝑛−𝜆_𝑛−1^𝑛_𝑘=0(−1)^𝑘Δ𝜆_𝑘−1+Δ𝜆_𝑘Δ𝜆_𝑘−Δ𝜆_𝑘−1×𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2=1𝜆_𝑛−𝜆_𝑛−1^𝑛_𝑘=0(−1)^𝑘Δ𝜆_𝑘−1+Δ𝜆_𝑘=1𝜆_𝑛−𝜆_𝑛−1(−1)^𝑛Δ𝜆_𝑛=1. (26) Hence, 𝐴_𝜆𝑥∈ℓ_∞ which means that 𝑥∈𝐴_𝜆(ℓ_∞)⧵ℓ_∞. If liminf_𝑛→∞((Δ𝜆_𝑛+1−Δ𝜆_𝑛)/Δ𝜆_𝑛)=0. Then, there exists a subsequence (𝑛_𝑟) such that lim_𝑟→∞Δ𝜆_𝑛𝑟+1−Δ𝜆_𝑛𝑟Δ𝜆_𝑛𝑟=0. (27) Now, let us define the sequence 𝑥=(𝑥_𝑛) by 𝑥_𝑛=1,𝑛=𝑛_𝑟,−Δ𝜆_𝑘−1−Δ𝜆_𝑘−2Δ𝜆_𝑘−Δ𝜆_𝑘−1,𝑛=𝑛_𝑟,0,otherwise, (28) for all 𝑛∈ℕ. It follows from (28) that 𝑥∉𝑐. On the other hand, 𝐴_𝜆𝑥_𝑛=Δ𝜆_𝑘−Δ𝜆_𝑘−1Δ𝜆_𝑘,𝑛=𝑛_𝑟,0,otherwise. (29) Now, from the relations (27) and (29), we derive that 𝑥=(𝑥_𝑛)∈𝐴_𝜆(𝑐₀)⊂𝐴_𝜆(𝑐). This completes the proof.

As an immediate result of Theorem 12, we have the following.

Corollary 13. The equalities 𝑐₀=𝐴_𝜆(𝑐₀), 𝑐=𝐴_𝜆(𝑐), and ℓ_∞=𝐴_𝜆(ℓ_∞) are satisfied if and only ifliminf_𝑛→∞Δ𝜆_𝑛+1−Δ𝜆_𝑛Δ𝜆_𝑛>0. (30)

Proposition 14. The following statements hold.(i)Although 𝑐 and 𝐴_𝜆(𝑐₀) overlap, the space 𝐴_𝜆(𝑐₀) does not include the space 𝑐. (ii) Although ℓ_∞ and 𝐴_𝜆(𝑐) overlap, the space 𝐴_𝜆(𝑐) does not include the space ℓ_∞.

Proposition 15. If  liminf_𝑛→∞((Δ𝜆_𝑛+1−Δ𝜆_𝑛)/Δ𝜆_𝑛)=0, then the following statements hold.(i)Neither of the spaces 𝑐 and 𝐴_𝜆(𝑐₀) includes the other.(ii)Neither of the spaces 𝐴_𝜆(𝑐₀) and ℓ_∞ includes the other.(iii)Neither of the spaces 𝐴_𝜆(𝑐) and ℓ_∞ includes the other.

5. The 𝛼-, 𝛽-, and 𝛾-Duals of the Spaces 𝐴_𝜆(𝑐₀), 𝐴_𝜆(𝑐), and 𝐴_𝜆(ℓ_∞)

In this section, we determine the alpha-, beta-, and gamma-duals of the spaces 𝐴_𝜆(𝑐₀), 𝐴_𝜆(𝑐), and 𝐴_𝜆(ℓ_∞).

We need the following lemma due to Stieglitz and Tietz [3] in proving Theorem 17.

Lemma 16. 𝐴=(𝑎_𝑛𝑘)∈(𝑐₀∶ℓ₁)=(𝑐∶ℓ₁) if and only ifsup_{𝐾,𝑁∈ℱ𝑛∈𝑁𝑘∈𝐾}𝑎_𝑛𝑘<∞. (31) Here and after, by ℱ one denotes the collection of all finite subsets of ℕ.

Theorem 17. The 𝛼-dual of the spaces 𝐴_𝜆(𝑐₀), 𝐴_𝜆(𝑐), and 𝐴_𝜆(ℓ_∞) is the set𝑎₁(𝜆)=𝑎=𝑎_𝑛∈𝜔∶_𝑛𝜆_𝑛−𝜆_𝑛−1𝜆_𝑛−2𝜆_𝑛−1+𝜆_𝑛−2𝑎_𝑛<∞. (32)

Proof. Define the matrix 𝐵=(𝑏_𝑛𝑘) with the aid of a sequence 𝑎=(𝑎_𝑛) as follows: 𝑏_𝑛𝑘=(−1)^𝑛−𝑘𝜆_𝑘−𝜆_𝑘−1𝜆_𝑛−2𝜆_𝑛−1+𝜆_𝑛−2𝑎_𝑛,𝑛−1≤𝑘≤𝑛,0,0≤𝑘≤𝑛−1or𝑘>𝑛. (33) Then, 𝑥=(𝑥_𝑛)∈𝐴_𝜆(𝑐₀), we have from Theorem 5𝑎_𝑛𝑥_𝑛=𝑎_𝑛^𝑛_{𝑘=𝑛−1}(−1)^𝑛−𝑘𝜆_𝑘−𝜆_𝑘−1𝜆_𝑛−2𝜆_𝑛−1+𝜆_𝑛−2𝑦_𝑘=𝐵𝑦_𝑛 (34) for all 𝑛∈ℕ. From the relation (34), it follows that 𝑎𝑥=(𝑎_𝑛𝑥_𝑛)∈ℓ₁ whenever 𝑥∈𝐴_𝜆(𝑐₀) if and only if 𝐵𝑦∈ℓ₁ whenever 𝑦=(𝑦_𝑘)∈𝑐₀, that is, 𝑎∈{𝐴_𝜆(𝑐₀)}^𝛼 if and only if 𝐵∈(𝑐₀∶ℓ₁). By Lemma 16, this is possible if and only if sup_{𝐾,𝑁∈ℱ}∑_𝑛∈𝑁∑_𝑘∈𝐾𝑏_𝑛𝑘<∞. (35) Now, from definition of the sets 𝐾,𝑁 and the matrix 𝐵=(𝑏_𝑛𝑘), it follows that (35) holds if and only if _𝑛𝜆_𝑛−𝜆_𝑛−1𝜆_𝑛−2𝜆_𝑛−1+𝜆_𝑛−2𝑎_𝑛<∞ (36) which gives that {𝐴_𝜆(𝑐₀)}^𝛼=𝑎₁(𝜆).
In a similar way, one can show that 𝑎₁(𝜆) is the 𝛼-dual of the spaces 𝐴_𝜆(𝑐) and 𝐴_𝜆(ℓ_∞). So, we omit the details.

Theorem 18. Define the sets 𝐴, 𝐵, 𝐶, and 𝐷 as follows:𝐴=𝑎=𝑎_𝑘∈𝜔∶_𝑘Δ𝑎_𝑘𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2𝜆_𝑘−𝜆_𝑘−1,𝐵=𝑎=𝑎_𝑘∈𝜔∶sup_𝑘∈ℕ𝑎_𝑘𝜆_𝑘−𝜆_𝑘−1𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2,𝐶=𝑎=𝑎_𝑘∈𝜔∶lim_𝑘→∞𝑎_𝑘𝜆_𝑘−𝜆_𝑘−1𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2,𝑎𝑒𝑥𝑖𝑠𝑡𝑠𝐷=𝑎=_𝑘∈𝜔∶lim_𝑘→∞𝑎_𝑘𝜆_𝑘−𝜆_𝑘−1𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2=0. (37) Then, one has {𝐴_𝜆(𝑐₀)}^𝛽=𝐴∩𝐵, {𝐴_𝜆(𝑐)}^𝛽=𝐴∩𝐶 and {𝐴_𝜆(ℓ_∞)}^𝛽=𝐴∩𝐷.

Proof. Since the proof is similar for the spaces 𝐴_𝜆(𝑐₀) and 𝐴_𝜆(ℓ_∞), we consider only the space 𝐴_𝜆(𝑐). Let 𝑢=(𝑢_𝑘)∈𝜔. Then, taking into account the relation (8) between the sequences 𝑥=(𝑥_𝑘) and 𝑦=(𝑦_𝑘), we obtain that ^𝑛_𝑘=0𝑢_𝑘𝑥_𝑘=^𝑛_𝑘=0𝑢_𝑘^𝑘_{𝑗=𝑘−1}(−1)^𝑘−𝑗𝜆_𝑗−𝜆_𝑗−1𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2𝑦_𝑗=^𝑛−1_𝑘=0𝜆_𝑘−𝜆_𝑘−1Δ𝑢_𝑘𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2𝑦_𝑘+𝜆_𝑛−𝜆_𝑛−1𝑢_𝑛𝑦_𝑛𝜆_𝑛−2𝜆_𝑛−1+𝜆_𝑛−2,=𝐵𝑦_𝑛,∀𝑛∈ℕ, (38) where Δ𝑢_𝑘𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2=𝑢_𝑘𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2−𝑢_𝑘+1𝜆_𝑘+1−2𝜆_𝑘+𝜆_𝑘−1, (39) and the matrix 𝐵=(𝑏_𝑛𝑘) is defined by 𝑏_𝑛𝑘=𝜆_𝑘−𝜆_𝑘−1𝑢×Δ^𝑘𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2𝜆,0≤𝑘≤𝑛−1,_𝑛−𝜆_𝑛−1𝑢_𝑛𝜆_𝑛−2𝜆_𝑛−1+𝜆_𝑛−2,𝑘=𝑛,0,𝑘>𝑛 (40) for all 𝑘,𝑛∈ℕ. Therefore, one can easily see from (38) that 𝑢𝑥=(𝑢_𝑘𝑥_𝑘)∈𝑐𝑠 with 𝑥=(𝑥_𝑘)∈𝐴_𝜆(𝑐) if and only if 𝐵𝑦∈𝑐 with 𝑦=(𝑦_𝑘)∈𝑐, where 𝐵=(𝑏_𝑛𝑘) is defined by (40). That is, to say that 𝑢=(𝑢_𝑘)∈{𝐴_𝜆(𝑐)}^𝛽 if and only if 𝐵 is a matrix satisfying the conditions of Kojima-Schur's theorem (cf. Başar [7, Theorem 3.3.3, page 35]). This leads to the fact that {𝐴_𝜆(𝑐)}^𝛽=𝐴∩𝐶.

Theorem 19. The 𝛾-dual of the spaces 𝐴_𝜆(𝑐₀), 𝐴_𝜆(𝑐), and 𝐴_𝜆(ℓ_∞) is the set 𝐴∩𝐵.

Proof. This is similar to the proof of Theorem 18. So, we omit the details.

6. Some Matrix Transformation Related to Sequence Spaces 𝐴_𝜆(𝑐₀), 𝐴_𝜆(𝑐), and 𝐴_𝜆(ℓ_∞)

In this section, we characterize the matrix transformations from the spaces 𝐴_𝜆(ℓ_∞) and 𝐴_𝜆(𝑐) into the spaces ℓ_∞, 𝑓, 𝑓₀, 𝑐, and 𝑐₀ of bounded, almost convergent, almost null, convergent, and null sequences, respectively. We write throughout for brevity that Δ𝑎_𝑛𝑘=𝑎_𝑛𝑘−𝑎_𝑛,𝑘+1,𝑎_𝑛𝑘=𝜆_𝑘−𝜆_𝑘−1Δ𝑎_𝑛𝑘𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2,𝑎(𝑛,𝑘)=^𝑛_𝑗=0𝑎_𝑗𝑘,𝑐_𝑛𝑘=^𝑛_𝑗=01𝑎𝑗+1^𝑗𝑘,𝑑_𝑛𝑘=𝑠𝑎_{𝑛−1,𝑘}+𝑟𝑎_𝑛𝑘,𝑒_𝑛𝑘=𝑡𝑎_{𝑛−2,𝑘}+𝑠𝑎_{𝑛−1,𝑘}+𝑟𝑎_𝑛𝑘,𝑎()=1𝑛,𝑘,𝑚𝑚+1^𝑚_𝑗=0𝑎_{𝑛+𝑗,𝑘} (41) for all 𝑘,𝑚,𝑛∈ℕ, and we use these abbreviations with other letters, where 𝑟,𝑠,𝑡∈ℝ⧵{0}.

Theorem 20. 𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑋)∶ℓ_∞) if and only if𝜆_𝑘−𝜆_𝑘−1𝜆_𝑘−2𝜆_𝑘−1+𝜆_𝑘−2𝑎_{𝑛𝑘𝑘∈ℕ}∈𝑐₀𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑓𝑖𝑥𝑒𝑑𝑛∈ℕsup_{𝑛∈ℕ𝑘}𝑎_𝑛𝑘<∞. (42)(43)

Proof. Suppose that the conditions (42) and (43) hold, and take any 𝑥=(𝑥_𝑘)∈𝐴_𝜆(𝑋). Then, the sequence (𝑎_𝑛𝑘)_𝑘∈ℕ∈{𝐴_𝜆(𝑋)}^𝛽 for all 𝑛∈ℕ, and this implies the existence of the 𝐴-transform of 𝑥.
Let us now consider the following equality derived by using the relation (8) from the 𝑚th partial sum of the series ∑_𝑘𝑎_𝑛𝑘𝑥_𝑘: ^𝑚_𝑘=0𝑎_𝑛𝑘𝑥_𝑘=^𝑚−1_𝑘=0𝑎_𝑛𝑘𝑦_𝑘+𝜆_𝑚−𝜆_𝑚−1𝜆_𝑚−2𝜆_𝑚−1+𝜆_𝑚−2𝑎_𝑛𝑚𝑦_𝑚 (44) for all 𝑚,𝑛∈ℕ. Therefore, we obtain from (44) with (42), as 𝑚→∞, that _𝑘𝑎_𝑛𝑘𝑥_𝑘=_𝑘𝑎_𝑛𝑘𝑦_𝑘,∀𝑛∈ℕ. (45) Now, by taking the sup-norm in (45), we derive that sup_𝑛∈ℕ(𝐴𝑥)_𝑛≤𝑦_∞sup_{𝑛∈ℕ𝑘}𝑎_𝑛𝑘<∞ (46) which shows the sufficiency of the conditions (42) and (43).
Conversely, suppose that 𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑋)∶ℓ_∞). Then, since (𝑎_𝑛𝑘)_𝑘∈ℕ∈{𝐴_𝜆(𝑋)}^𝛽 for all 𝑛∈ℕ by the hypothesis, the necessity of (42) is trivial and (45) holds. Consider the continuous linear functionals 𝑓_𝑛 defined on 𝐴_𝜆(𝑋) by the sequences 𝑎_𝑛=(𝑎_𝑛𝑘)_𝑘∈ℕ as 𝑓_𝑛(𝑥)=_𝑘𝑎_𝑛𝑘𝑥_𝑘,∀𝑛∈ℕ. (47) Since 𝐴_𝜆(ℓ_∞)≅ℓ_∞, 𝐴_𝜆(𝑐)≅𝑐 and 𝐴_𝜆(𝑐₀)≅𝑐₀, it should follow with (45) that ‖𝑓_𝑛‖=‖𝑎_𝑛‖_∞. This just says that the functionals defined by the rows of 𝐴 on 𝐴_𝜆(𝑋) are pointwise bounded. Hence, by Banach-Steinhaus theorem, 𝑓_𝑛's are uniformly bounded which gives that there exists a constant 𝐾>0 such that ‖𝑓_𝑛‖≤𝐾 for all 𝑛∈ℕ. It therefore follows that ∑_𝑘|𝑎_𝑛𝑘|=‖𝑓_𝑛‖≤𝐾 holds for all 𝑛∈ℕ which shows the necessity of the condition (43).
This step completes the proof.

Prior to characterizing the class of infinite matrices from the space 𝐴_𝜆(ℓ_∞) into the space of almost convergent sequences, we give a short knowledge on the concept of almost convergence. The shift operator 𝑃 is defined on 𝜔 by 𝑃_𝑛(𝑥)=𝑥_𝑛+1 for all 𝑛∈ℕ. 𝐴 Banach limit 𝐿 is defined on ℓ_∞, as a non-negative linear functional, such that 𝐿(𝑃𝑥)=𝐿(𝑥) and 𝐿(𝑒)=1. 𝐴 sequence 𝑥=(𝑥_𝑘)∈ℓ_∞ is said to be almost convergent to the generalized limit 𝛼 if all Banach limits of 𝑥 coincide and are equal to 𝛼 [8] and is denoted by 𝑓−lim𝑥_𝑘=𝛼. Let 𝑃^𝑖 be the composition of 𝑃 with itself 𝑖 times and write for a sequence 𝑥=(𝑥_𝑘)𝑡_𝑚𝑛(𝑥)=1𝑚+1^𝑚_𝑖=0𝑃^𝑖_𝑛(𝑥),∀𝑚,𝑛∈ℕ. (48) Lorentz [8] proved that 𝑓−lim𝑥_𝑘=𝛼 if and only if lim_𝑚→∞𝑡_𝑚𝑛(𝑥)=𝛼, uniformly in 𝑛. It is well known that a convergent sequence is almost convergent such that its ordinary and generalized limits are equal. By 𝑓₀ and 𝑓, we denote the spaces of almost null and almost convergent sequences, respectively. Now, we can give the lemma characterizing the almost coercive matrices.

Lemma 21 (see [9, Theorem 1]). 𝐴=(𝑎_𝑛𝑘)∈(ℓ_∞∶𝑓) if and only ifsup_{𝑛∈ℕ𝑘}𝑎_𝑛𝑘<∞;∃𝛼_𝑘∈ℂ∋𝑓−lim𝑎_𝑛𝑘=𝛼_𝑘𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑓𝑖𝑥𝑒𝑑𝑘∈ℕ;lim_{𝑚→∞𝑘}𝑎(𝑛,𝑘,𝑚)−𝛼_𝑘=0𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑛. (49)(50)(51)

Theorem 22. 𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑓) if and only if the conditions (42) and (43) hold, and ∃𝛼_𝑘∈ℂ∋𝑓−lim𝑎_𝑛𝑘=𝑎_𝑘𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑓𝑖𝑥𝑒𝑑𝑘∈ℕ;lim_{𝑚→∞𝑘}𝑎(𝑛,𝑘,𝑚)−𝑎_𝑘=0𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑛. (52)(53)

Proof. Let 𝐴∈(𝐴_𝜆(ℓ_∞)∶𝑓). Then, since 𝑓⊂ℓ_∞, the necessity of (42) and (43) is immediately obtained from Theorem 20. To prove the necessity of (52), consider the sequence 𝑒^(𝑛)(𝜆)={𝑒_𝑘^(𝑛)(𝜆)}_𝑛∈ℕ∈𝐴_𝜆(ℓ_∞), defined by (16) for every fixed 𝑘∈ℕ. Since 𝐴𝑥 exists and is in 𝑓 for every 𝑥∈𝐴_𝜆(ℓ_∞), one can easily see that 𝐴𝑒^(𝑛)(𝜆)=(𝑎_𝑛𝑘)_𝑛∈ℕ∈𝑓 for all 𝑘∈ℕ, that is, the condition (52) is necessary.
Define the matrix 𝐵=(𝑏_𝑛𝑘) by 𝑏_𝑛𝑘=𝑎_𝑛𝑘 for all 𝑘,𝑛∈ℕ. Then, we derive from the equality (45) that 𝐴𝑥=𝐵𝑦. Since 𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑓) by the hypothesis, we have 𝐵∈(ℓ_∞∶𝑓). Therefore, the matrix 𝐵 satisfies the condition (51) of Lemma 21 which is equivalent to the condition (53).
Conversely, suppose that the matrix 𝐴 satisfies the conditions (42), (43), (52), and (53), and 𝑥∈𝐴_𝜆(ℓ_∞). Reconsider the equality 𝐴𝑥=𝐵𝑦 obtained from (45) with 𝑏_𝑛𝑘 instead of 𝑎_𝑛𝑘. Then, the conditions (49), (50), and (51) are satisfied by the matrix 𝐵. Hence, 𝐵 is almost coercive by Lemma 21 and this gives by passing to 𝑓-limit in (45) that 𝐴𝑥∈𝑓, that is, 𝐴∈(𝐴_𝜆(ℓ_∞)∶𝑓), as desired.
This concludes the proof.

As a direct consequence of Theorem 22, we have the following.

Corollary 23. 𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑓₀) if and only if the conditions (42) and (43) hold, and (52) and (53) hold with 𝑎_𝑘=0 for all 𝑘∈ℕ.

Theorem 24. 𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑐) if and only if the condition (42) holds, and the conditions_𝑘𝑎_𝑛𝑘𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑛∈ℕ;∃𝛼_𝑘∈ℂ∋lim_𝑛→∞𝑎_𝑛𝑘=𝛼_𝑘𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑓𝑖𝑥𝑒𝑑𝑘∈ℕ. (54)(55)

Corollary 25. 𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑐₀)  if and only if the conditions (42) and (54) hold, and (55) also holds with 𝛼_𝑘=0 for all 𝑘∈ℕ.

Now, we give the following lemma due to King [10] characterizing the class of almost conservative matrices.

Lemma 26. 𝐴=(𝑎_𝑛𝑘)∈(𝑐∶𝑓) if and only if (49) and (50) hold, and∃𝛼∈ℂ∋𝑓−lim∑_𝑘𝑎_𝑛𝑘=𝛼. (56)

Theorem 27. 𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑐)∶𝑓) if and only if the conditions (42), (43), and (52) hold, and the condition∃𝛼∈ℂ∋𝑓−lim_𝑘𝑎_𝑛𝑘=𝑎. (57)also holds.

Proof. This is obtained by a similar way used in proving Theorem 22 with Lemma 26 instead of Lemma 21. So, to avoid the repetition of the similar statements we omit the details.

Corollary 28. 𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑐)∶𝑓)_𝜌 if and only if the conditions (42) and (43) hold, and the conditions (52) and (57) also hold with 𝛼_𝑘=0 for all 𝑘∈ℕ and 𝑎=𝜌, respectively; where by (𝐴_𝜆(𝑐)∶𝑓)_𝜌, we denote the class of infinite matrices 𝐴 such that 𝑓−lim𝐴𝑥=𝜌[𝐴(𝜆)−lim𝑥] for all 𝑥∈𝐴_𝜆(𝑐).

Now, we give the following Steinhaus type theorem.

Theorem 29. The classes (𝐴_𝜆(ℓ_∞)∶𝑓) and (𝐴_𝜆(𝑐)∶𝑓)_𝜌 are disjoint, where 𝜌∈ℝ⧵{0}.

Proof. Suppose that the classes (𝐴_𝜆(ℓ_∞)∶𝑓) and (𝐴_𝜆(𝑐)∶𝑓)_𝜌 are not disjoint. Then, there is at least one 𝐴=(𝑎_𝑛𝑘) in the set (𝐴_𝜆(ℓ_∞)∶𝑓)∩(𝐴_𝜆(𝑐)∶𝑓)_𝜌. Therefore, one can derive by combining (53) and (52) with 𝛼_𝑘=0 for all 𝑘∈ℕ that lim_{𝑚→∞𝑘}|𝑎(𝑛,𝑘,𝑚)|=0uniformlyin𝑛 (58) which is contrary to the condition (57) with 𝑎=𝜌≠0. This completes the proof.

Lemma 30 (see [11, Lemma 5.3]). Let 𝜇, 𝜈 be any two sequence spaces, 𝐴 an infinite matrix, and 𝐵 a triangle matrix. Then, 𝐴∈(𝜇∶𝜈_𝐵) if and only if 𝐵𝐴∈(𝜇∶𝜈).

It is trivial that Lemma 30 has several consequences. Indeed, combining Lemma 30 with Theorems 20, 22, 24, and 27 and Corollaries 23, 25, and 28 by choosing 𝐵 as one of the special matrices 𝐶₁, 𝐸^𝑟, 𝑅^𝑡, Δ, Δ⁽¹⁾, 𝐴^𝑟, or 𝑆, one can easily obtain the following results.

Corollary 31. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix over the complex field. Then, the following statements hold.(i)𝐸=(𝑒_𝑛𝑘)∈(𝐴_𝜆(𝑋)∶𝑏𝑣_∞) if and only if (42) and (43) hold with 𝑒_𝑛𝑘−𝑒_{𝑛−1,𝑘} instead of 𝑎_𝑛𝑘, where 𝑏𝑣_∞ denotes the space of all sequences 𝑥=(𝑥_𝑘) such that (𝑥_𝑘−𝑥_𝑘−1)∈ℓ_∞ and was introduced by Başar and Altay [11]. (ii)𝐸=(𝑒_𝑛𝑘)∈(𝐴_𝜆(𝑋)∶𝑒^𝑟_∞) if and only if (42) and (43) hold with ∑^𝑛_𝑗=0^𝑛_𝑗(1−𝑟)^𝑛−𝑗𝑟^𝑗𝑒_𝑗𝑘 instead of 𝑎_𝑛𝑘, where 𝑒^𝑟_∞ denotes the space of all sequences 𝑥=(𝑥_𝑘) such that (∑^𝑛_𝑗=0^𝑛_𝑗(1−𝑟)^𝑛−𝑗𝑟^𝑗𝑒_𝑘𝑗𝑥_𝑗)∈ℓ_∞ and was introduced by Altay et al. [12]. (iii)𝐸=(𝑒_𝑛𝑘)∈(𝐴_𝜆(𝑋)∶𝑋_∞) if and only if (42) and (43) hold with 𝑒(𝑛,𝑘)/(𝑛+1) instead of 𝑎_𝑛𝑘, where 𝑋_∞ denotes the space of all sequences 𝑥=(𝑥_𝑘) such that (∑^𝑛_𝑗=0𝑥_𝑗/(𝑛+1))∈ℓ_∞ and was introduced by Ng and Lee [13].(iv)𝐸=(𝑒_𝑛𝑘)∈(𝐴_𝜆(𝑋)∶𝑟^𝑡_∞) if and only if (42) and (43) hold with ∑^𝑛_𝑗=0𝑡_𝑗𝑒_𝑗𝑘/𝑇_𝑛 instead of 𝑎_𝑛𝑘, where 𝑟^𝑡_∞ denotes the space of all sequences 𝑥=(𝑥_𝑘) such that (∑^𝑛_𝑗=0𝑡_𝑗𝑥_𝑗/𝑇_𝑛)∈ℓ_∞ and was introduced by Altay and Başar [14].(v)𝐸=(𝑒_𝑛𝑘)∈(𝐴_𝜆(𝑋)∶𝑏𝑠) if and only if (42) and (43) hold with 𝑒(𝑛,𝑘) instead of 𝑎_𝑛𝑘.

Corollary 32. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix over the complex field. Then, the following statements hold.(i)𝐸=(𝑒_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑐(Δ)) if and only if (42), (54), and (55) hold with 𝑒_𝑛𝑘−𝑒_𝑛+1,𝑘 instead of 𝑎_𝑛𝑘, where 𝑐(Δ) denotes the space of all sequences 𝑥=(𝑥_𝑘) such that (𝑥_𝑘−𝑥_𝑘+1)∈𝑐 and was introduced by Kızmaz [15]. (ii)𝐸=(𝑒_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑒^𝑟_𝑐) if and only if (42), (54), and (55) hold with ∑^𝑛_𝑗=0^𝑛_𝑗(1−𝑟)^𝑛−𝑗𝑟^𝑗𝑒_𝑗𝑘 instead of 𝑎_𝑛𝑘, where 𝑒^𝑟_𝑐 denotes the space of all sequences 𝑥=(𝑥_𝑘) such that 𝐸^𝑟𝑥∈𝑐 and was introduced by Altay and Başar [16].(iii)𝐸=(𝑒_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑐)̃ if and only if (42), (54), and (55) hold with 𝑒(𝑛,𝑘)/(𝑛+1) instead of 𝑎_𝑛𝑘, where 𝑐̃ denotes the space of all sequences 𝑥=(𝑥_𝑘) such that 𝐶₁𝑥∈𝑐  and was introduced by Şengönül and Başar [17].(iv)𝐸=(𝑒_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑟^𝑡_𝑐) if and only if (42), (54), and (55) hold with ∑^𝑛_𝑗=0𝑡_𝑗𝑒_𝑗𝑘/𝑇_𝑛 instead of 𝑎_𝑛𝑘, where 𝑟^𝑡_𝑐 denotes the space of all sequences 𝑥=(𝑥_𝑘) such that 𝑅^𝑡𝑥∈𝑐 and was introduced by Altay and Başar [18].(v)𝐸=(𝑒_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑐𝑠) if and only if (42), (54) and (55) hold with 𝑒(𝑛,𝑘) instead of 𝑎_𝑛𝑘.

Corollary 33. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix over the complex field. Then, the following statements hold.(i)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑓) if and only if (42), (43), (52), and (53) hold with 𝑑_𝑛𝑘 instead of 𝑎_𝑛𝑘, where 𝑓 denotes the space of all sequences 𝑥=(𝑥_𝑘) such that 𝐵(𝑟,𝑠)𝑥∈𝑓 and was introduced by Başar and Kirişçi [19].(ii)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑓₀) if and only if (42) and (43) hold, and (52) and (53) hold with 𝛼_𝑘=0 for all 𝑘∈ℕ and 𝑑_𝑛𝑘 instead of 𝑎_𝑛𝑘, where 𝑓₀ denotes the space of all sequences 𝑥=(𝑥_𝑘) such that 𝐵(𝑟,𝑠)𝑥∈𝑓₀ and was introduced by Başar and Kirişçi [19].(iii)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑐)∶𝑓) if and only if (42), (43), (52), and (53) hold with 𝑑_𝑛𝑘 instead of 𝑎_𝑛𝑘.(iv)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑐)∶𝑓₀) if and only if (42) and (43) hold, and (52) and (53) also hold with 𝛼_𝑘=0 for all 𝑘∈ℕ and 𝛼=1, respectively, hold with 𝛼_𝑘=0 for all 𝑘∈ℕ and 𝑑_𝑛𝑘 instead of 𝑎_𝑛𝑘.(v)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑐)∶𝑓)_𝜌 if and only if the conditions of Corollary 28 hold with 𝑑_𝑛𝑘 instead of 𝑎_𝑛𝑘.

Corollary 34. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix over the complex field. Then, the following statements hold. (i)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑓) if and only if (42), (43), (52), and (53) hold with 𝑐_𝑛𝑘 instead of 𝑎_𝑛𝑘, where 𝑓 denotes the space of all sequences 𝑥=(𝑥_𝑘) such that 𝐶₁𝑥∈𝑓 and was introduced by Kayaduman and Şengönül [20]. (ii)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑓₀) if and only if (42) and (43) hold, and (52) and (53) hold with 𝛼_𝑘=0 for all 𝑘∈ℕ and 𝑐_𝑛𝑘 instead of 𝑎_𝑛𝑘, where 𝑓₀ denotes the space of all sequences 𝑥=(𝑥_𝑘) such that 𝐶₁𝑥∈𝑓₀ and was introduced by Kayaduman and Şengönül [20]. (iii)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑐)∶𝑓) if and only if (42), (43), (52), and (53) hold with 𝑐_𝑛𝑘 instead of 𝑎_𝑛𝑘. (iv)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑐)∶𝑓₀) if and only if (42) and (43) hold, and (52) and (53) also hold with 𝛼_𝑘=0 for all 𝑘∈ℕ and 𝛼=1, respectively, with 𝑐_𝑛𝑘 instead of 𝑎_𝑛𝑘. (v)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑐)∶𝑓)_𝜌 if and only if the conditions of Corollary 28 hold with 𝑐_𝑛𝑘 instead of 𝑎_𝑛𝑘.

Corollary 35. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix over the complex field. Then, the following statements hold.(i)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑓(𝐵)) if and only if (42), (43), (52), and (53) hold with 𝑒_𝑛𝑘 instead of 𝑎_𝑛𝑘, where 𝑓(𝐵) denotes the space of all sequences 𝑥=(𝑥_𝑘) such that 𝐵(𝑟,𝑠,𝑡)𝑥∈𝑓 and was introduced by Sönmez [21]. (ii)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(ℓ_∞)∶𝑓₀(𝐵)) if and only if (42) and (43) hold, and (52) and (53) hold with 𝛼_𝑘=0 for all 𝑘∈ℕ and 𝑒_𝑛𝑘 instead of 𝑎_𝑛𝑘, where 𝑓₀(𝐵) denotes the space of all sequences 𝑥=(𝑥_𝑘) such that 𝐵(𝑟,𝑠,𝑡)𝑥∈𝑓₀ and was introduced by Sönmez [21]. (iii)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑐)∶𝑓(𝐵)) if and only if the conditions (42), (43), (52), and (53) hold with 𝑒_𝑛𝑘 instead of 𝑎_𝑛𝑘. (iv)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑐)∶𝑓₀(𝐵)) if and only if the conditions (42) and (43) hold, and the conditions (52) and (53) also hold with 𝛼_𝑘=0 for all 𝑘∈ℕ and 𝛼=1, respectively, hold with 𝛼_𝑘=0 for all 𝑘∈ℕ and 𝑒_𝑛𝑘 instead of 𝑎_𝑛𝑘. (v)𝐴=(𝑎_𝑛𝑘)∈(𝐴_𝜆(𝑐)∶𝑓(𝐵))_𝜌 if and only if the conditions of Corollary 28 hold with 𝑒_𝑛𝑘 instead of 𝑎_𝑛𝑘.

7. Conclusion

Mursaleen and Noman [2, 22, 23] have studied the domains ℓ^𝜆_∞, 𝑐^𝜆, 𝑐^𝜆₀, and ℓ^𝜆_𝑝 of the matrix Λ in the classical sequence spaces ℓ_∞, 𝑐, 𝑐₀, and ℓ_𝑝, respectively. Malkowsky and Rakočević [24] characterized some classes of matrix transformations and investigated related compact operators involving the spaces of Λ-null, Λ-convergent, and Λ-bounded sequences. Quite recently, Sönmez and Başar [25] have introduced the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) of generalized difference sequences which generalize the paper due to Mursaleen and Noman [26]. Mursaleen and Noman [26] have derived some inclusion relations and determined the alpha-, beta-, and gamma-duals of those spaces and constructed their Schauder bases. Finally, Sönmez and Başar [25] have characterized some matrix classes from the spaces 𝑐^𝜆₀(𝐵) and 𝑐^𝜆(𝐵) to the spaces ℓ_𝑝, 𝑐₀, and 𝑐. In the present paper, we emphasize the domains 𝐴_𝜆(𝑐₀), 𝐴_𝜆(𝑐), and 𝐴_𝜆(ℓ_∞) of the matrix 𝐴(𝜆) in the classical sequence spaces 𝑐₀, 𝑐, and ℓ_∞. Our results are more general and comprehensive than the corresponding results of Mursaleen and Noman [2, 22, 23] derived with the matrix Λ. We should note that, as a natural continuation of the present paper, one can study the domains 𝐴_𝜆(ℓ_𝑝) and 𝐴_𝜆(𝑏𝑣_𝑝) of the matrix 𝐴(𝜆) in the classical sequence space ℓ_𝑝 and in the sequence space 𝑏𝑣_𝑝 with 0<𝑝<1 and 1≤𝑝<∞, where 𝑏𝑣_𝑝 denotes the space of all sequences 𝑥=(𝑥_𝑘) such that (𝑥_𝑘−𝑥_𝑘−1)∈ℓ_𝑝 and introduced in the case 1≤𝑝<∞ by Başar and Altay [11] and in the case 0<𝑝<1 by Altay and Başar [27].

Acknowledgment

The authors would like to express their pleasure to the anonymous referees for constructive criticism of an earlier version of this paper which improved its readability.