On almost B-summable double sequence spaces

The concept of a four-dimensional generalized difference matrix and its domain on some double sequence spaces was recently introduced and studied by Tuğ and Başar (AIP Conference Proceedings, vol. 1759, 2016) and Tuğ (J. Inequal. Appl. 2017(1):149, 2017). In this present paper, as a natural continuation of (J. Inequal. Appl. 2017(1):149, 2017), we introduce new almost null and almost convergent double sequence spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{f})$\end{document}B(Cf) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{f_{0}})$\end{document}B(Cf0) as the four-dimensional generalized difference matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(r,s,t,u)$\end{document}B(r,s,t,u) domain in the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{C}_{f}$\end{document}Cf and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{C}_{f_{0}}$\end{document}Cf0, respectively. Firstly, we prove that the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{f})$\end{document}B(Cf) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{f_{0}})$\end{document}B(Cf0) of double sequences are Banach spaces under some certain conditions. Then we give an inclusion relation of these new almost convergent double sequence spaces. Moreover, we identify the α-dual, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta(bp)$\end{document}β(bp)-dual and γ-dual of the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{f})$\end{document}B(Cf). Finally, we characterize some new matrix classes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(B(\mathcal{M}_{u}):\mathcal{C}_{f})$\end{document}(B(Mu):Cf), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\mathcal{M}_{u}:B(\mathcal {C}_{f}))$\end{document}(Mu:B(Cf)), and we complete this work with some significant results.


Preliminaries, background and notation
We denote the set of all complex valued double sequence by , which is a vector space with coordinatewise addition and scalar multiplication. Any subspace of is called a double sequence space. A double sequence x = (x mn ) of complex numbers is called bounded if x ∞ = sup m,n∈N |x mn | < ∞, where N = {0, 1, 2, . . .}. The space of all bounded double sequences is denoted by M u , which is a Banach space with the norm · ∞ . Consider the double sequence x = (x mn ) ∈ . If for every > 0 there exist a natural number n 0 = n 0 ( ) and l ∈ C such that |x mn -l| < for all m, n > n 0 , then the double sequence x is said to be convergent in Pringsheim's sense to the limit point l says that p -lim m,n→∞ x mn = l; where C indicates the complex field. The space C p denotes the set of all convergent double sequences in Pringsheim's sense. Although every convergent single sequence is bounded, this is not hold for double sequences in general. That is, there are such double sequences which are convergent in Pringsheim's sense but not bounded. Actually, Boos [3, p. 16] defined the sequence x = (x mn ) by Then it is clearly seen that p -lim m,n→∞ x mn = 0 but x ∞ = sup m,n∈N |x mn | = ∞, so x ∈ C p -M u . Now, we may denote the space of all both convergent in Pringsheim's sense and bounded double sequences by the set C bp , i.e., C bp = C p ∩ M u . Hardy [4] showed that a double sequence x = (x mn ) is said to converge regularly to l if x ∈ C p and the limits x m := lim n x mn , (m ∈ N) and x n := lim m x mn , (n ∈ N) exist, and the limits lim m lim n x mn and lim n lim m x mn exist and are equal to the p-limit of x. Moreover, by C bp0 and C r0 , we may denote the spaces of all null double sequences contained in the sequence spaces C bp and C r , respectively. Móricz [5] proved that the double sequence spaces C bp , C bp0 , C r and C r0 are Banach spaces with the norm · ∞ . The space L q of all absolutely q-summable double sequences corresponding to the space q of q-summable single sequences was defined by Başar and Sever [6], that is, which is a Banach space with the norm · q . Then the space L u , which is a special case of the space L q with q = 1, was introduced by Zeltser [7]. Let λ be a double sequence space and converging with respect to some linear convergence rule is ϑ -lim : λ → C. Then the sum of a double series i,j x ij relating to this rule is defined by ϑi,j x ij = ϑ -lim m,n→∞ m,n i,j=0 x ij . Throughout, the summation from 0 to ∞ without limits, that is, i,j x ij means that ∞ i,j=0 x ij . Here and below, unless stated otherwise we consider that ϑ denotes any of the symbols p, bp or r.
The α-dual λ α , the β(ϑ)-dual λ β(ϑ) with respect to the ϑ-convergence and the γ -dual λ γ of the double sequence space λ are, respectively, defined by It is easy to see for any two spaces λ and μ of double sequences that μ α ⊂ λ α whenever λ ⊂ μ and λ α ⊂ λ γ . Additionally, it is well known that the inclusion λ α ⊂ λ β(ϑ) holds, while the inclusion λ β(ϑ) ⊂ λ γ does not hold, since the ϑ-convergence of the double sequence of partial sum of a double series does not guarantee its boundedness.
Here, we shall be concerned with a four-dimensional matrix transformation from any double sequence space λ to any double sequence space μ. Given any four-dimensional infinite matrix A = (a mnkl ), where m, n, k, l ∈ N, any double sequence x = (x kl ), we write Ax = {(Ax) mn } m,n∈N for the A-transform of x, exists for every sequence x = (x kl ) ∈ λ and it is in μ; here a mnkl x kl for each m, n ∈ N. (1.1) The four-dimensional matrix domain has fundamental importance for this article. Therefore, this concept is presented in this paragraph. The ϑ-summability domain λ (ϑ) A of A in a space λ of double sequences is described as exists and is in λ .
The notation (1.1) says that A maps the space λ into the space μ if λ ⊂ μ (ϑ) A and we denote the set of all four-dimensional matrices, transforming the space λ into the space μ, by (λ : μ). Thus, A = (a mnkl ) ∈ (λ : μ) if and only if the double series on the right side of (1.1) converges in the sense of ϑ for each m, n ∈ N, i.e., A mn ∈ λ β(ϑ) for all m, n ∈ N and we have Ax ∈ μ for all x ∈ λ; where A mn = (a mnkl ) k,l∈N for all m, n ∈ N. Moreover, the following definitions are significant in order to classify the four-dimensional matrices. A four- By using the notations of Zelster [8] we may define the double sequences e kl = (e kl mn ), e 1 , e k and e by for all k, l, m, n ∈ N and we may write the set by = span{e kl : k, l ∈ N}.
In order to establish a new sequence space, special triangular matrices were previously used. These new spaces derived by the domain of matrices are expansions or contractions of the original space, in general. Adams [9] called the four-dimensional infinite matrix A = (a mnkl ) a triangular matrix if a mnkl = 0 for k > m or l > n or both. We also see by [9] that an infinite matrix A = (a mnkl ) is said to be a triangular if a mnmn = 0 for all m, n ∈ N.
Moreover, Cooke [10] showed that every triangular matrix has a unique inverse which is also a triangular matrix.
The four-dimensional generalized difference matrix B(r, s, t, u) = {b mnkl (r, s, t, u)} and matrix domain of it on some double sequence spaces was recently defined and studied by Tuǧ and Başar [1], and Tuǧ [2]. The matrix B(r, s, t, u) = {b mnkl (r, s, t, u)} was defined by for r, s, t, u ∈ R\{0} and for all m, n, k, l ∈ N. Therefore, the B(r, s, t, u)-transform of a double sequence x = (x mn ) was defined by for all m, n ∈ N. Moreover, the matrix B -1 (r, s, t, u) = F(r, s, t, u) = {f mnkl (r, s, t, u)}, which is the inverse of the matrix B(r, s, t, u), was calculated to be In this paper, as natural continuation of [2] and [11], we introduce new almost null and almost convergent double sequence spaces B(C f ) and B(C f 0 ) as the domain of fourdimensional generalized difference matrix B(r, s, t, u) in the spaces C f and C f 0 , respectively. Throughout the paper, we suppose that the terms of the double sequence x = (x mn ) and y = (y mn ) are connected with equation (1.3) and the four-dimensional generalized difference matrix B(r, s, t, u) = (b mnkl (r, s, t, u)) will be presented with B = (b mnkl ).

2
The sequence space C f of almost convergent double sequences Lorentz [12] introduced the concept of almost convergence for a single sequence and Móricz and Rhoades [13] extended and studied this concept for a double sequence. A double sequence x = (x kl ) of complex numbers is said to be almost convergent to a generalized limit L if In this case, L is called the f 2 -limit of the double sequence x. Throughout the paper, C f denotes the space of all almost convergent double sequences, i.e., It is well known that a convergent double sequence need not be almost convergent. But it is well known that every bounded convergent double sequence is also almost convergent and every almost convergent double sequence is bounded. That is, the inclusion C bp ⊂ C f ⊂ M u holds, and each inclusion is proper. A double sequence x = (x kl ) is called almost Cauchy, which was introduced by Čunjalo [14], if for every > 0 there exists a positive integer K such that 1 (q 1 + 1)(q 1 + 1) for all q 1 , q 1 , q 2 , q 2 > K and (m 1 , n 1 ), (m 2 , n 2 ) ∈ N × N. Mursaleen and Mohiuddine [15] proved that every double sequence is almost convergent if and only if it is almost Cauchy.
Móricz and Rhoades [13] considered that four-dimensional matrices transforming every almost convergent double sequence into a bp-convergent double sequence with the same limit. Almost conservative and almost regular matrices for single sequences were characterized by King [16] and almost C ϑ -conservative and almost C ϑ -regular four-dimensional matrices for double sequences were defined and characterized by Zeltser et al. [17]. Mursaleen [18] introduced the almost strongly regularity for double sequences. A fourdimensional matrix A = (a mnkl ) is called almost strongly regular if it transforms every almost convergent double sequence into an almost convergent double sequence with the same limit.

Spaces of almost B-summable double sequences
In this present section, we define new almost convergent double sequence spaces B(C f ) and B(C f 0 ) derived by the domain of four-dimensional matrix B in the spaces of all almost convergent and almost null double sequences C f and C f 0 , respectively. Then we show that B(C f ) and B(C f 0 ) are Banach spaces with the norm x B(C f ) , and we prove an inclusion relation.

Now we may define the spaces B(C f ) and B(C f 0 ) by
Proof Since in other cases it can be similarly proved, we prove the theorem only for the space B(C f ). Let us consider a Cauchy sequence Then, for a given > 0, there exists a positive integer M( ) ∈ N such that for all i, j > M( ). Then we can read from equation [19]), it is convergent. Then we may say that there exists a double sequence x = (x kl ) ∈ C f such that as j → ∞. Now, by taking the limit as i → ∞ on the equality (3.2), for every > 0 we have for all k, l ∈ N Moreover, since {(Bx (j) ) kl } ∈ C f and every almost convergent double sequence is bounded, there exists a positive real number K such that sup m,n∈N Therefore, we are enabled to write the following inequality: Now we can say by taking the supremum over m, n ∈ N and the p-limit as q, q → ∞ from the inequality acquired above that To show this, we should prove the existence of a linear bijection between the spaces B(C f ) and C f . Let us consider the transformation T from B(C f ) to C f by x → Tx = y = Bx, with the notation of (1.2). The linearity and injectivity of T is clear. Let us take any y = (y kl ) ∈ C f and consider the sequence x = (x kl ) with respect to the sequence y by equation (1.3) for all k, l ∈ N. Then we have the following equality: for all k, l ∈ N. Thus, we arrive at the consequence that This shows that x = (x kl ) ∈ B(C f ). Then we may say that T is surjective. Moreover, one can obtain the following equality: That is, T is norm preserving. Hence, T is linear bijection and B(C f ) and C f are linearly norm isomorphic. This is what we proposed.
Proof Firstly, we should prove that the inclusions C f ⊂ B(C f ) and C f 0 ⊂ B(C f 0 ) hold. Since s = -r, t = -u, the four-dimensional matrix B = (b mnkl ) satisfy the conditions of Lemma 4.8. Then we can say that, for all In order to show that the inclusions are strict, we should show that the sets B(C f ) \ C f and B(C f 0 ) \ C f 0 are not empty, that is, there exists a double sequence x = (x mn ) which belongs to B(C f ) but not in C f . Let consider a double sequence x = (x mn ) by x mn = mn rt for all m, n ∈ N. Since it is not bounded, it is obvious that Therefore, we have the following equality with the above result: After taking the supremum over m, n ∈ N in the above equality and applying the p-limit as q, q → ∞ we see that Bx ∈ C f . It can easily be shown that the sequence x mn = m rt for all m, n ∈ N is in B(C f 0 ) \ C f 0 by the same reasoning as above. So we omit the details.

The α-, β(ϑ)and γ -duals of the sequence space B(C f )
In this section, firstly, we calculate the α-dual of the space B(C f ). Then we give some known definitions and lemmas which will be used in the proof of β(bp)-dual of the space B(C f ) and in the fourth section of this paper. Moreover, we characterize a new four-dimensional matrix class (C f : M u ) in order to calculate the γ -dual of the space B(C f ). Moreover, the inclusion C f ⊂ M u holds, and there exists a positive real number K such that sup k,l |y kl | ≤ K . Since |s/r|, |u/t| < 1, we have the following inequality: This means that (a kl ) / ∈ {B(C f )} α , which is a contradiction. Therefore, (a kl ) must belong to the space L u . So, the inclusion {B(C f )} α ⊂ L u holds. This completes the proof.
Now we have the following significant lemmas, which will be used in this present section and the fifth section of this work.

Lemma 4.2 ([17]) The following statements hold:
(a) A four-dimensional matrix A = (a mnkl ) is almost C bp -conservative, i.e., A ∈ (C bp : C f ), iff the following conditions hold: uniformly in m, n ∈ N for each i, j ∈ N, uniformly in m, n ∈ N for each j ∈ N, (4.4) where a(i, j, q, q , m, n) = m+q k=m n+q l=n a klij /[(q + 1)(q + 1)]. In this case, a = (a ij ) ∈ L u and uniformly in m, n ∈ N.
(b) A four-dimensional matrix A = (a mnkl ) is almost C bp -regular, i.e., A ∈ (C bp : C f ) reg , iff the conditions (4.1)-(4.5) hold with a ij = 0 for all i, j ∈ N and u = 1 where a(i, j, q, q , m, n) is defined as in Lemma 4.2. In this case, a = (a ij ) ∈ L u ; (u j ), (v i ) ∈ 1 and hold with a ij = u j = v i = 0 for all i, j ∈ N and u = 1.

Lemma 4.4 ([17])
The following statements hold: In this case a = (a ij ) ∈ L u , (a ij 0 ) i∈N , (a i 0 j ) j∈N ∈ ϕ where ϕ denotes the space of all finitely non-zero sequences and   for every m, n, j ∈ N, ∃K ∈ N 1 (q + 1)(q + 1) m+q k=m n+q l=n a klij = 0, for all q, q , i > K, (4.15) for every m, n, i ∈ N, ∃L ∈ N 1 (q + 1)(q + 1)  where a(i, j, q, q , m, n) is defined as in Lemma 4.2.  i, j, q, q , m, n = a i, j, q, q , m, na i + 1, j, q, q , m, n ,   01 a i, j, q, q , m, n = a i, j, q, q , m, na i, j + 1, q, q , m, Proof Suppose that a = (a mn ) ∈ and x = (x mn ) ∈ B(C f ). Then we have y = Bx ∈ C f . Therefore, we have the equality (3.16), which is in [2, Theorem 3.11, p.14] with the m, nth partial sum of k,l a kl x kl with m,n k,l=0 a kl x kl = (Dy) mn . By taking the limit as m, n → ∞ from this equality, we have the four-dimensional matrix D = (d mnkl ), which was also defined by Tuǧ for all k, l, m, n ∈ N. Then one can obtain from the above consequences ax ∈ CS bp whenever x = (x mn ) ∈ B(C f ) iff Dy ∈ C bp whenever y = (y mn ) ∈ C f . This says that a = (a mn ) ∈  This shows the fact that Ax ∈ M u , which completes the proof. This means that the γ -dual of the space B(C f ) is the set d 1 ∪ CS ϑ as mentioned.

Matrix transformations related to the sequence space B(C f )
In this section, we characterize some new four-dimensional matrix classes (B(M u ) : C f ), (M u : B(C f )). Then we complete this section with some significant results of fourdimensional matrix mapping via the dual summability methods for double sequences which have been introduced and studied by Başar [22] and Yeşilkayagil and Başar [23], and which have recently been applied in [2]. where the four-dimensional matrix E = (e mnkl ) is defined by for all m, n ∈ N. Then, by taking the ϑ-limit on (5.2) as m, n → ∞, we may say that Ax = Ey. Hence, Ey ∈ M u whenever y ∈ C f , that is, E ∈ (C f : M u ). In this instance, the conditions of Theorem 4.10 hold with E = (e mnkl ) instead of A = (a mnkl ), i.e., E mn ∈ {C f } β(ϑ) and sup m,n∈N k,l |e mnkl | < ∞. This completes the proof. Proof The proof can be shown by the same method as is followed in Theorem 5.1 by using equation (4.6) [24,Theorem 4.7] between the elements of the four-dimensional matrices A = (a mnkl ) and G = (g mnkl ). So we omit the details.
Tuǧ [2] has recently applied the dual summability methods for double sequences which has been introduced and studied by Başar [22], and Yeşilkayagil and Başar [23]. In this work, we use the relation between the four-dimensional matrices E = (e mnkl ), e(m, n), G = (g mnkl ) and H = (h mnkl ) with A = (a mnkl ), which has been proved and studied in [2,
In this work, we studied the domain of the four-dimensional generalized difference matrix B = (b mnkl ) in the spaces of almost null and almost convergent double sequences and examined some topological properties. Moreover, we determined the α-, β(bp)-and γduals of the space B(C f ) and characterized some new classes of four-dimensional matrix mappings related with the sequence space B(C f ). The characterization of the matrix classes (C f :