Four-dimensional generalized difference matrix and some double sequence spaces

In this study, I introduce some new 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}$B(\mathcal {M}_{u})$\end{document}B(Mu), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{p})$\end{document}B(Cp), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{bp})$\end{document}B(Cbp), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal {C}_{r})$\end{document}B(Cr) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{L}_{q})$\end{document}B(Lq) as the domain of 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) 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}$\mathcal {M}_{u}$\end{document}Mu, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{C}_{p}$\end{document}Cp, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{C}_{bp}$\end{document}Cbp, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{C}_{r}$\end{document}Cr and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{L}_{q}$\end{document}Lq, respectively. I show that the 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}$B(\mathcal{M}_{u})$\end{document}B(Mu), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{bp})$\end{document}B(Cbp) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{r})$\end{document}B(Cr) are the Banach spaces under some certain conditions. I give some inclusion relations with some topological properties. Moreover, I determine the α-dual of the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(M_{u})$\end{document}B(Mu) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{bp})$\end{document}B(Cbp), the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta(\vartheta)$\end{document}β(ϑ)-duals of the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(M_{u})$\end{document}B(Mu), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{p})$\end{document}B(Cp), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{bp})$\end{document}B(Cbp), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{r})$\end{document}B(Cr) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{L}_{q})$\end{document}B(Lq), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vartheta\in\{p,bp,r\}$\end{document}ϑ∈{p,bp,r}, and the γ-dual of the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{M}_{u})$\end{document}B(Mu), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{bp})$\end{document}B(Cbp) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{L}_{q})$\end{document}B(Lq). Finally, I characterize the classes of four-dimensional matrix mappings defined on the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{M}_{u})$\end{document}B(Mu), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{p})$\end{document}B(Cp), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{bp})$\end{document}B(Cbp), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{C}_{r})$\end{document}B(Cr) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(\mathcal{L}_{q})$\end{document}B(Lq) of double sequences.


Introduction
We denote the set of all complex-valued double sequences 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 = {, , , . . .}. 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 >  there exists a natural number n  = n  ( ) and l ∈ C such that |x mn -l| < for all m, n > n  , then the double sequence x is called convergent in Pringsheim's sense to the limit point l, and we write p-lim m,n→∞ x mn = l, where C denotes the complex field. The space of all convergent double sequences in Pringsheim's sense is denoted by C p . Unlike single sequences, there are such double sequences which are convergent in Pringsheim's sense but unbounded. That is, the set C p -M u is not empty. Actually, following Boos [], p., if we define the sequence x = (x mn ) by , m ≥ , n ∈ N, then it is obvious that p-lim m,n→∞ x mn =  but x ∞ = sup m,n∈N |x mn | = ∞, so x ∈ C p -M u . Then we can consider the set C bp of double sequences which are both convergent in Pringsheim's sense and bounded, i.e., C bp = C p ∩ M u . Hardy [] showed that a sequence in the space C p is said to be regular convergent if it is a single convergent sequence with respect to each index and denoted the space of all such sequences by C r . Moreover, by C bp and C r , we denote the spaces of all double sequences converging to  contained in the sequence spaces C bp and C r , respectively. Móricz [] proved that C bp , C bp , C r and C r are Banach spaces with the norm · ∞ . By L q , we denote the space of absolutely q-summable double sequences corresponding to the space q of q-summable single sequences, that is, which is a Banach space with the norm · q defined by Başar and Sever []. Zeltser [] introduced the space L u as a special case of the space L q with q = . Let λ be a double sequence space converging with respect to some linear convergence rule ϑ-lim : λ → C.
The sum of a double series i,j x ij with respect to this rule is defined by ϑi,j x ij = ϑ-lim m,n→∞ m,n i,j= x ij . For short, throughout the text the summations without limits run from  to ∞, for instance, i,j x ij means that ∞ i,j= x ij . Here and in what follows, unless stated otherwise, we assume 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 a 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 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 sums of a double series does not imply its boundedness.
Let λ and μ be two double sequence spaces and A = (a mnkl ) be any four-dimensional complex infinite matrix. Then we say that A defines a matrix mapping from λ into μ, and we write A : λ → μ if for every sequence a mnkl x kl for each m, n ∈ N. (.) We define ϑ-summability domain λ (ϑ) A of A in a space λ of double sequences by exists and is in λ .
We say with notation (.) 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-hand side of (.) converges in the sense of ϑ for each m, n ∈ N, i.e., A mn ∈ λ β(ϑ) for all m, n ∈ N and every x ∈ λ, and we have Ax ∈ μ for all x ∈ λ, where A mn = (a mnkl ) k,l∈N for all m, n ∈ N. We say that a four- Adams [] defined that the four-dimensional infinite matrix A = (a mnkl ) is called a triangular matrix if a mnkl =  for k > m or l > n or both. We also say by [] that a triangular matrix A = (a mnkl ) is said to be a triangle if a mnmn =  for all m, n ∈ N. Moreover, by referring to Cooke [], Remark (a), p., we can say that every triangle matrix has a unique inverse which is also a triangle.
Let r, s, t, u ∈ R \ {}. Then the four-dimensional generalized difference matrix B(r, s, for all m, n, k, l ∈ N. Therefore, the B(r, s, t, u)-transform of a double sequence x = (x mn ) is given by for all m, n ∈ N. Thus, we have the inverse B - (r, s, t, u) = F(r, s, t, u) = {f mnkl (r, s, t, u)} as follows: for all m, n, k, l ∈ N. Therefore, we can obtain x = (x mn ) by applying the inverse matrix F(r, s, t, u) to (.) that Throughout the paper, we suppose that the terms of double sequence x = (x mn ) and y = (y mn ) are connected with relation (.). If p-lim{B(r, s, t, u)x} mn = l, then the sequence x = (x mn ) is said to be B(r, s, t, u) convergent to l. Note that in the case r = t =  and s = u = - for all m, n ∈ N, the four-dimensional generalized difference matrix B(r, s, t, u) is reduced to the four-dimensional difference matrix = B(, -, , -).

Some new spaces of double sequences
In this section, we define the double sequence spaces B(M u ), B(C p ), B(C bp ), B(C r ) and B(L q ) as the domain of four-dimensional generalized difference matrix B(r, s, t, u) in the double sequence spaces M u , C p , C bp , C r and L q , respectively, that is, Then we give some topological properties and inclusion relations.
This means that x = (x mn ) defined by (.) is in the space B(M u ), i.e., T is surjective and is norm-preserving. This concludes the proof of the theorem.
This means that the double sequence Proof For the first step of the proof, we show that the inclusion C p ⊂ B(C p ) holds. Let us take a sequence x = (x mn ) ∈ C p . Then there exists a complex number l such that p-lim m,n→∞ |x mn -l| = . Then we have by taking limit of the B(r, s, t, u)-transform of x as m, n → ∞ in Pringsheim's sense Since x ∈ C p , then all the subsequences of x are also convergent. Thus, To prove the fact that the inclusion C p ⊂ B(C p ) is strict, we should show that the set B(C p ) \ C p is not empty. Let us consider the double sequence x = (x mn ) defined by x mn = (mn)/(rt) for all m, n ∈ N. If we take s = -r, u = -t, then we have for all m, n ∈ N. Thus, one can easily observe that x = (x mn ) / ∈ C p . But, p-lim m,n→∞ {B(r, s, t, u)x} mn = , that is, x ∈ B(C p ). This step completes the proof.

Theorem . The inclusion C bp ⊂ B(C bp ) strictly holds.
Proof This is a natural consequence of Theorems . and .. So, we omit the details.
In order to prove the fact that the inclusion is strict, we should define a double sequence belonging to B(L q ) but not to L q . Let us define the double sequence x = (x mn ) by rt for all m, n ∈ N. If ( -s r ) >  or ( -u t ) > , or both, then it is obvious that x / ∈ L q . But, under the same restrictions, we have This says that B(r, s, t, u)x ∈ L q , i.e., x ∈ B(L q ). This completes the proof. Proof The proof of the theorem is similar to the proof of Theorem .. So, we omit the details.
Theorem . The set B(L q ) is a linear space with coordinatewise addition and scaler multiplication, and the following statements hold.
Proof (i) To show the linearity of the space B(L q ) which is a q-normed space with the given norm is a routine verification. So, we omit the details. Let us take a Cauchy sequence Then, for a given > , there exists a positive real number N( ) >  such that is satisfied for all i, j ≥ N( ). Then we conclude that {{B(r, s, t, u)x i } mn } i∈N is a Cauchy sequence for each fixed m, n ∈ N. It is known by Part (i) of Theorem . of Yeşilkayagil and Başar [] that the space L q is a complete q-normed space. Then the Cauchy sequence which means that B(r, s, t, u)x ∈ L q , that is, x ∈ B(L q ). The last conclusion says that the space B(L q ) is a complete q-normed space. Now, we should define a transform from B(L q ) to L q which is a norm-preserving bijection. Let us consider the transformation T used in the proof of the second part of Theorem . with B(L q ) and L q instead of B(M u ) and M u , respectively. It is easy to see that T is linear and bijective. Let y = (y mn ) ∈ L q and define x = (x mn ) by relation (.). Then we derive by taking summation over m, n ∈ N on the following inequality: . Thus, T is surjective. This concludes the proof of Part (i).
Since Part (ii) can be proved in a similar way, we omit the details.

The alpha-, beta-and gamma-duals of the new double sequence spaces
In  This means that (a mn ) / ∈ {B(M u )} α , which contradicts the hypothesis. Therefore, (a mn ) must belong to the space ∈ L u . Since the proof can be given for the space B(C bp ) in a similar way, we omit the details.
The α-and γ -duals of a double sequence space are unique. But β(ϑ)-dual of a double sequence space can be more than one according to the ϑ-convergence. In this part, we give the β(ϑ)-and γ -duals of the new double sequence spaces. The conditions for the characterization of the four-dimensional matrices transformed the spaces C bp , C r and C p into the space C bp are well known (see [, ] and []). Then it is derived from the last approaches that Ax ∈ M u . This completes the proof.

Lemma . ([]) Let A = (a mnkl ) be a four-dimensional matrix. Then the following statements hold:
∃β kl ∈ C ϑ-lim Theorem . The following statements hold: Proof (iii) Let us suppose that a = (a mn ) ∈ and x = (x mn ) ∈ B(C bp ). Then we have y = Bx ∈ C bp . Therefore, we have the following equality for the m, nth partial sum of k,l a kl x kl : Theorem . The following statements hold: s, t, u). s, t, u).
Proof Suppose that a = (a mn ) ∈ and x = (x mn ) ∈ B(C bp ). Then there exists a sequence y = (y mn ) ∈ C bp with Bx = y. Therefore, since (.) holds, one can conclude that ax ∈ CS ϑ whenever x = (x mn ) ∈ B(C bp ) if and only if Dy ∈ C ϑ whenever y = (y mn ) ∈ C bp . It gives us that a = (a mn ) ∈ {B(C bp )} β(ϑ) if and only if D ∈ (C bp : C ϑ ). Hence, the conditions of Lemma . are satisfied with d mnkl instead of a mnkl . That is, s, t, u). This completes the proof of Part (i). Since Parts (ii)-(vii) can be proved in a similar way by using Lemmas ., ., ., . and ., respectively, to avoid the repetition of similar statements, we omit their proofs.

Characterization of some classes of four-dimensional matrices
In this section, we characterize some four-dimensional matrix classes which are related to the double sequence spaces derived as the domain of the four-dimensional generalized difference matrix in the spaces M u , C p , C bp , C r and L q by using the concept of fourdimensional dual summability methods for double sequences introduced and studied by Başar [] and Yeşilkayagil and Başar []. Now, let us suppose that the four-dimensional matrices A = (a mnkl ) and E = (e mnkl ) transform the sequences x = (x mn ) and y = (y mn ) which are connected with relation (.) to the double sequences s = (s mn ) and z = (z mn ), respectively, that is, It is obvious that the method B is applied to the B(r, s, t, u)-transform of the sequence x, while the method A is directly applied to the elements of the sequence x. Then we can say that the methods A and E are essentially different. Let us assume that the usual matrix product EB(r, s, t, u) exists, which is a much weaker hypothesis than the conditions on the matrix E belonging to any class of matrices, in general. We can say in this case that the matrices A and E in (.) and (.) are the dual summability methods if s is reduced to z or viceversa under the application of the usual summation by parts. This leads us to the fact that EB(r, s, t, u) exists and is equal to A, and Ax = {EB(r, s, t, u)x} = E{B(r, s, t, u)x} = Ey formally holds if one side exists. This statement is equivalent to the relation between the elements of the matrices A = (a mnkl ) and E = (e mnkl ) a mnkl = sue mn,m-,n- + ste mn,m-,n + rue mnm,n- + rte mnmn or equivalently for all m, n, k, l ∈ N. It is trivial that relation (.) between the elements of the matrices A = (a mnkl ) and E = (e mnkl ) can be stated by the matrix product as follows: (r, s, t, u) or equivalently E = AF(r, s, t, u).
For the sake of brevity in notation, we may also write here and after for all m, n, k, l ∈ N that e(m, n) = That is, Av = Eu, which leads to the fact A ∈ (B(λ) : μ), as desired.
By changing the role of the spaces B(λ) and μ in Theorem ., we have the following lemma. for all m, n, k, l ∈ N. When we apply the ϑ-limit on equality (.) as m, n → ∞, we have Ax = Hy. So, Hy ∈ μ whenever y ∈ λ says that H ∈ (λ : μ). This completes the proof.

Conclusion
Zeltser [], in her PhD thesis, studied both the theory of topological double sequence spaces and the summability theory of double sequences. Altay and Başar [] have recently studied the double series spaces BS, BS(t), CS ϑ and BV whose sequences of partial sums are in the spaces M u , M u (t), C ϑ and L u , respectively, where ϑ ∈ {p, bp, r}. They studied some topological properties of those spaces and computed the α-duals of the spaces BS, CS bp and BV and the β(ϑ)-duals of the spaces CS bp and CS r of double series. Furthermore, they gave the conditions which characterize the classes of four-dimensional matrix transformations defined on the spaces CS bp , CS p and CS r .
Başar [], Chapter , p., studied the fundamental results on double sequences and related topics. Başar and Sever [] deeply studied the Banach space L q of absolutely qsummable double sequences and examined the topological properties. Moreover, they determined the α-, β(ϑ)-and γ -duals of L q , where  ≤ q < ∞ and ϑ ∈ {p, bp, r}.
The concept of matrix domain was examined by several researchers on some single sequence spaces by using some special matrices. Recently some significant studies have been done by several mathematicians for double sequence spaces and four-dimensional matrices (see [-]). In this work, I have studied the domain of four-dimensional generalized difference matrix B(r, s, t, u) on some double sequence spaces and examined some topological properties. Furthermore, I determined the α-, β(ϑ)-and γ -duals of some new double sequence spaces and characterized some classes of four-dimensional matrix transformations related to the new double sequence spaces. As a natural continuation of Yeşilkayagil and Başar [], one can obtain certain new topological properties concerning the space B(C f ) of all almost B summable double sequences.