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

. 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,

. 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

(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,

(2)

The alpha-, beta-, and gamma-duals μ^α, μ^β, and μ^γ of a sequence space μ are, respectively, defined by

(3)

If A is an infinite matrix with complex entries a_nk, where k, n ∈ ℕ, then we write A = (a_nk) instead of

. Also, we write A_n for the sequence in the nth row of the matrix A, that is,

for every n ∈ ℕ. Further, if x = (x_k) ∈ ω then we define the A-transform of x as the sequence

, where

(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

(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

(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 and

(7)

Let us define the sequence y = (y_n) by the A(λ)-transform of a sequence x = (x_k), that is,

(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:

(9)

Now, we define the infinite matrix

(10)

for all n, k ∈ ℕ. Then, A(λ)-transform of a sequence x ∈ ω is the sequence

, 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,

(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

(12)

is finite for every x = (x_k) ∈ A_λ(X), and

is a norm on A_λ(X).

Denote by ∥·∥_bv the usual bv-norm, that is, to say that

(13)

With the notation of (5), we can redefine the spaces A_λ(c₀), A_λ(c), and A_λ(ℓ_∞) as follows:

(14)

Theorem 4. The sequence spaces A_λ(c₀), A_λ(c), and A_λ(ℓ_∞) are BK-spaces with the norm given by

(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 by

(16)

where k ∈ ℕ. Then, one has the following.

(1)
The sequence is a Schauder basis for the space A_λ(c₀), and every x ∈ A_λ(c₀) has a unique representation: .
(2)
The sequence is a Schauder basis for the space A_λ(c), and every x ∈ A_λ(c) has a unique representation: , 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

(17)

for all k ∈ ℕ. Then, it follows that

(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)
if and only if liminf _n→∞((Δλ_n+1 − Δλ_n)/Δλ_n) = 0;
(ii)
if and only if liminf _n→∞((Δλ_n+1 − Δλ_n)/Δλ_n) > 0.

Proof. (i) Let us start with the expression

(19)

After some calculations, we get

(20)

On the other hand, from the definition of the sequence (λ_n) we have

(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 is bounded. Part (b) is satisfied if and only if 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 .

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 if

(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 . 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

(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

(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

(25)

for all n ∈ ℕ. Then, we get the following estimation:

(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

(27)

Now, let us define the sequence x = (x_n) by

(28)

for all n ∈ ℕ. It follows from (28) that x ∉ c. On the other hand,

(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 if

(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 if

(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 set

(32)

Proof. Define the matrix B = (b_nk) with the aid of a sequence a = (a_n) as follows:

(33)

Then, x = (x_n) ∈ A_λ (c₀), we have from Theorem 5

(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

(35)

Now, from definition of the sets K, N and the matrix B = (b_nk), it follows that (35) holds if and only if

(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:

(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

(38)

where

(39)

and the matrix B = (b_nk) is defined by

(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

(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 if

(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 ∑_k a_nkx_k:

(44)

for all m, n ∈ ℕ. Therefore, we obtain from (44) with (42), as m → ∞, that

(45)

Now, by taking the sup-norm in (45), we derive that

(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

(47)

Since A_λ(ℓ_∞)≅ℓ_∞, A_λ(c)≅c and A_λ(c₀)≅c₀, it should follow with (45) that

. 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

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 − lim x_k = α. Let Pⁱ be the composition of P with itself i times and write for a sequence x = (x_k)

(48)

Lorentz [8] proved that f − lim x_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 if

(49)

(50)

(51)

Theorem 22. A = (a_nk)∈(A_λ(ℓ_∞) : f) if and only if the conditions (42) and (43) hold, and

(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 , 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 for all k ∈ ℕ, that is, the condition (52) is necessary.

Define the matrix B = (b_nk) by 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 . 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 for all k ∈ ℕ.

Theorem 24. A = (a_nk)∈(A_λ(ℓ_∞) : c) if and only if the condition (42) holds, and the conditions

(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

(56)

Theorem 27. A = (a_nk)∈(A_λ(c) : f) if and only if the conditions (42), (43), and (52) hold, and the condition

(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 , 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

(58)

which is contrary to the condition (57) with

. 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)
if and only if (42) and (43) hold with instead of a_nk, where denotes the space of all sequences x = (x_k) such that 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 and was introduced by Ng and Lee [13].
(iv)
if and only if (42) and (43) hold with instead of a_nk, where denotes the space of all sequences x = (x_k) such that 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)
if and only if (42), (54), and (55) hold with instead of a_nk, where denotes the space of all sequences x = (x_k) such that E^rx ∈ c and was introduced by Altay and Başar [16].
(iii)
if and only if (42), (54), and (55) hold with e(n, k)/(n + 1) instead of a_nk, where denotes the space of all sequences x = (x_k) such that C₁x ∈ c and was introduced by Şengönül and Başar [17].
(iv)
if and only if (42), (54), and (55) hold with instead of a_nk, where 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)
if and only if (42), (43), (52), and (53) hold with d_nk instead of a_nk, where 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)
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 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)
if and only if (42), (43), (52), and (53) hold with d_nk instead of a_nk.
(iv)
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)
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)
if and only if (42), (43), (52), and (53) hold with c_nk instead of a_nk, where denotes the space of all sequences x = (x_k) such that C₁x ∈ f and was introduced by Kayaduman and Şengönül [20].
(ii)
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 denotes the space of all sequences x = (x_k) such that C₁x ∈ f₀ and was introduced by Kayaduman and Şengönül [20].
(iii)
if and only if (42), (43), (52), and (53) hold with c_nk instead of a_nk.
(iv)
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)
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 , c^λ, , and 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 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 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.

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

Abstract

1. Introduction

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

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

4. Some Inclusion Relations Related to the New Spaces

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

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

7. Conclusion

Acknowledgment

References

References

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

Abstract

1. Introduction

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

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

4. Some Inclusion Relations Related to the New Spaces

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

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

7. Conclusion

Acknowledgment

References

References

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

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