On the Domain of the Four-Dimensional Sequential Band Matrix in Some Double Sequence Spaces

: Let E represent any of the spaces M u , C ϑ ( ϑ = { b , bp , r } ) , and L q ( 0 < q < ∞ ) of bounded, ϑ -convergent, and q -absolutely summable double sequences, respectively, and (cid:101) E be the domain of the four-dimensional (4D) inﬁnite sequential band matrix B ( (cid:101) r , (cid:101) s , (cid:101) t , (cid:101) u ) in the double sequence space E , where (cid:101) r = ( r m ) ∞ m = 0 , (cid:101) s = ( s m ) ∞ m = 0 , (cid:101) t = ( t n ) ∞ n = 0 , and (cid:101) u = ( u n ) ∞ n = 0 are given sequences of real numbers in the set c \ c 0 . In this paper, we investigate the double sequence spaces (cid:101) E . First, we determine some topological properties and prove several inclusion relations under some strict conditions. Then, we examine α -, β ( ϑ ) -, and γ -duals of (cid:101) E . Finally, we characterize some new classes of 4D matrix mappings related to our new double sequence spaces and conclude the paper with some signiﬁcant consequences.

a double sequence such that it converges in the Pringsheim sense, but is unbounded. For instance, the sequence x = (x mn ) is defined as: x mn = n , m = 0, n ∈ N; 0 , m ≥ 1, n ∈ N.
which is obviously in x ∈ C p \ M u . A matrix transformation between double sequence spaces E 1 and E 2 is given by any 4D infinite matrix A = (a mnkl ), where m, n, k, l ∈ N. This means that A transforms any double sequence x = (x kl ) ∈ E 1 to Ax = {(Ax) mn } m,n∈N ∈ E 2 , where: (Ax) mn = ϑ − ∑ k,l a mnkl x kl for each m, n ∈ N. (2) The set E

(ϑ)
A , which is the ϑ-summability domain of a 4D infinite matrix A in a double sequence space E, is defined by: a mnkl x kl m,n∈N exists and is in E which is a sequence space. The above transformation (2) indicates that A maps E 1 into E 2 provided A , and the set of all 4D matrices, transforming the space E 1 into the space E 2 , is denoted by (E 1 : E 2 ). Thus, A = (a mnkl ) ∈ (E 1 : E 2 ) if and only if the double series ∑ k,l a mnkl x kl m,n∈N ϑ-converges for each m, n ∈ N, i.e., A mn ∈ E β(ϑ) 1 for all m, n ∈ N, and we have Ax ∈ E 2 for all x ∈ E 1 ; where A mn = (a mnkl ) k,l∈N for all m, n ∈ N.
It is a common method to create a new sequence space from a given sequence space E as the matrix domains E A of matrices A in E. Following Adams [1], a 4D infinite matrix A = (amnkl) is said to be a triangular matrix if a mnkl = 0 for k > m or l > n or both. From [1], we can say that an infinite matrix A = (a mnkl ) refers to a triangle if a mnmn = 0 for all m, n ∈ N. It was shown in [2] that every triangle has unique and equal left and right inverses. One can easily observe that, if A is a triangle, then E (ϑ) A and E are linearly isomorphic. The adventure of double sequences and their topological properties started by distinguished works of several mathematicians, and those core ideas helped to describe some new double sequence spaces by the summability theory. Zeltser [3] studied the theoretical and topological properties of double sequence spaces by applying the summability theory. Then, other mathematicians (see [4][5][6][7][8][9][10]) studied the topological properties, inclusion relations, dual spaces, and matrix transformations of some new spaces of double sequences. Moreover, many characterizations of 4D matrix classes have been achieved by several mathematicians (see [11][12][13][14]). We list the domains E A of some 4D triangle matrices A in a certain double sequence space E in the following Table 1. C M u , C 0p , C p , C r , C bp , L qMu ,C 0p ,C p ,C r ,C bp ,L q [15] ∆(1, −1, 1, −1) M u , C 0p , C p , C r , L q M u (∆), C 0p (∆), C p (∆), C r (∆), L q (∆) [16] CM u ,C 0p ,C p ,C r ,C bp ,L qM u (t),C 0p (t),C p (t),C r (t),C bp (t),L q (t) [17] [24] where the matrices C, E(r, s), ∆(1, −1, 1, −1), R qt , and B(r, s, t, u) denote the Cesàro mean, the double Euler mean, the 4D difference matrix, the Riesz mean, and the 4D generalized difference matrix, respectively.
More recently, the 4D generalized difference matrix B(r, s, t, u) was defined and studied by Tuǧ [11,21,22,[24][25][26][27]. The results were much more general than the consequences derived by the 4D difference matrix ∆(1, −1, 1, −1) domains in some double sequence spaces. After carefully searching in the existing literature, one can see clearly that another generalization of the 4D infinite matrix B(r, s, t, u) is possible by defining this matrix sequentially. In this work, we introduce some new double sequence spaces B(M u ), B(C p ), B(C bp ), B(C r ), and B(L q ),(0 < q < ∞) whose 4D sequential band matrix B-transforms are in the spaces M u , C p , C bp , C r , andL q , respectively. Moreover, the goal of the following sections of this paper is to provide extensive proofs of some topological properties, inclusion relations, dual spaces of these new double sequence spaces, and the characterization of some 4D infinite matrix classes.
Let r = (r m ) ∞ m=0 , s = (s m ) ∞ m=0 , t = (t n ) ∞ n=0 , and u = (u n ) ∞ n=0 be given sequences of real numbers in the set c \ c 0 . We define the 4D sequential band matrix B( r, s, t, u) = {b mnkl ( r, s, t, u)} by: r m t n , (k, l) = (m, n), r m u n , (k, l) = (m, n − 1), s m t n , (k, l) = (m − 1, n), s m u n , (k, l) = (m − 1, n − 1), 0 , elsewhere, for all m, n, k, l ∈ N. Thus, the matrix B( r, s, t, u)-transforms a double sequence x = (x mn ) into the double sequence y = (y mn ) as: = s m−1 u n−1 x m−1,n−1 + s m−1 t n x m−1,n + r m u n−1 x m,n−1 + r m t n x mn for all m, n ∈ N. By means of the above equation in (4), the inverse B −1 ( r, s, t, u) = F( r, s, t, u) = { f mnkl ( r, s, t, u)} of B( r, s, t, u) is defined by: for all m, n, k, l ∈ N. Therefore, by using the inverse matrix F( r, s, t, u) from Equation (5) and the relation between x = (x mn ) and y = (y mn ), we obtain: −u j t j y m−k,n−l , for all m, n ∈ N. (6) Throughout this work, we will consider crucial relations between Equations (4) and (6) in terms of the sequences x = (x mn ) and y = (y mn ). Moreover, the sequence x = (x mn ) is called B( r, s, t, u) convergent to L provided p − lim{B( r, s, t, u)x} mn = L in the Pringsheim sense.

Remark 1.
In the case of s m = s, r m = r for all m ∈ N and t n = t, u n = u for all n ∈ N, then the matrix B( r, s, t, u) is decomposed into the matrix B(r, s, t, u). However, if we consider s m = −r m = 1 for all m ∈ N and t n = −u n = −1 for all n ∈ N, then the matrix B( r, s, t, u) reduces to the matrix ∆(1, −1, 1, −1). Therefore, the results obtained by the domain of the matrix B( r, s, t, u) are more general than the corresponding consequences achieved by the domains of matrices ∆(1, −1, 1, −1) and B(r, s, t, u).

New Double Sequence Spaces
In this section, the matrix domains of the matrix B in the spaces E are introduced and denoted by From definition (3), we can define the spaces B(E), where E = {C p , M u , C bp , C r , L q }, and 1 ≤ q < ∞ as: Now, we investigate some topological properties and inclusion relations subject to some strict conditions. Proof. To prove this claim, we must show the existence of a linear transformation, which is one-to-one and between the spaces B(C ϑ ) and C ϑ . Assume the transformation T from B(C ϑ ) to C ϑ is given by x → Tx = y = Bx. It is obvious that T is linear for each assumption of B(C ϑ ) and C ϑ . Moreover, it can be seen that x = θ whenever Tx = θ, which says that T is injective.
By using (4), taking an arbitrary y = (y kl ) ∈ C ϑ , and defining x = (x mn ) through the sequence y = (y kl ) and the relation (6) for all m, n ∈ N, we obtain the following: { Bx} mn = s m−1 u n−1 x m−1,n−1 + s m−1 t n x m−1,n + r m u n−1 x m,n−1 + r m t n x mn for all m, n ∈ N. Therefore, we can derive the fact that: This shows that x = (x mn ) ∈ B(C ϑ ) whenever y = (y mn ) ∈ C ϑ , where ϑ = {p, bp, r}. Thus, T is surjective, and finally, T is a linear bijection between the spaces B(C ϑ ) and C ϑ .
Proof. We can analogously use a similar pattern in Theorem 2 to prove this by considering the following equality: |{ Bx} mn | = |s m−1 u n−1 x m−1,n−1 + s m−1 t n x m−1,n + r m u n−1 x m,n−1 + r m t n x mn | = |y mn |.
since y = (y mn ) ∈ M u . Hence, the proof is completed.
Proof. The technique of the proof is similar to that used to prove Theorem 2. We consider the following equality: It shows that ∑ m,n |{ Bx} mn | q = ∑ m,n |y mn | q < ∞ since y = (y mn ) ∈ L q . This is what we claimed.
Theorem 5. The following statements hold.
(a) If sup m s m inf m r m < 1 and sup n u n inf n t n < 1 for all m, n ∈ N, then: sup n u n inf n t n ≥ 1 for all m, n ∈ N, then the following inclusions strictly hold.
Let sup m s m inf m r m < 1, and sup n u n inf n t n < 1 for all m, n ∈ N. Since the matrix F = ( f mnkl ) defined in (5) is the inverse of B = ( b mnkl ), it should satisfy the following conditions: This completes the proof of (a)(i). Parts (a)(ii) and (a)(iii) can be proven in a similar way by applying [11], Lemma 3.4, p. 11, and [11], Lemma 3.2, p. 11, respectively. Therefore, we omit the details.
Proof of (b): Suppose that sup m s m inf m r m = 1, and sup n u n inf n t n = 1 and (s m ) = (r m ) = s for all m ∈ N and (u n ) = (t n ) = u for all n ∈ N. Let us consider the sequence This concludes the proof of the theorem. Proof. The proof of (a) can clearly be achieved by referring to [14], Theorem 2.2, since the matrix B satisfies the condition (7) and the inverse matrix F satisfies the condition (8) with the assumptions sup m s m inf m r m < 1 and sup n u n inf n t n < 1 for all m, n ∈ N. Therefore, we omit the details. For the proof of (b), suppose that sup m s m inf m r m = 1 and sup n u n inf n t n = 1. We consider the sequence x = (x mn ) with: Thus, the sequence x ∈ B(M u ) \ M u , since the B-transform of x is: This proves the statement.

Theorem 7. If
− sup m s m inf m r m ≥ 1 and − sup n u n inf n t n ≥ 1 for all m, n ∈ N, then the inclusion L q ⊂ B(L q ) strictly holds, where 1 ≤ q < ∞.
Proof. Suppose that an arbitrary double sequence x = (x mn ) is in L q , where 1 ≤ q < ∞. Thus, ∑ m,n |x mn | q < ∞. We have to show Bx ∈ L q . It follows from: that Bx ∈ L q . Thus, x ∈ B(L q ), and the inclusion L q ⊂ B(L q ) holds. Now, it is our main goal to prove the existence of a double sequence x = (x mn ), which is contained in the set B(L q ) \ L q . We define the double sequence x = (x mn ) by: Hence, Bx ∈ L q , i.e., x ∈ B(L q ). This completes the proof.

Dual Spaces
In this section, we start by determining the α-dual of the spaces B(M u ) and B(C bp ). Then, we summarize the literature for the characterizations of the 4D matrix classes in Table 2 and the necessary and sufficient conditions for the members in these classes in Table 3. Finally, we state some results derived from Table 2 and Lemma 1 as a corollary. For the first inclusion, let a = (a mn ) ∈ L u and x = (x mn ) ∈ B(M u ). Therefore, there exists a double sequence y = (y mn ) ∈ M u with the relations (4) and (6) such that sup m,n∈N |y mn | ≤ K, where K ∈ R + . Then, we have: This means that (a mn ) / ∈ B(M u ) α , which contradicts our assumption. Therefore, (a mn ) must belong to the space L u .
We summarize the characterizations of some 4D matrix classes as in the following useful results, which have been established in [11][12][13][14].
We list the necessary and sufficient conditions for each class in the following table. Note that * shows the unknown characterization of respective 4D matrix class. We recall that the α-and γ-duals of spaces of double sequences are uniquely determined. To determine the β(ϑ)-dual of the spaces of double sequences, we need to consider ϑ-convergence, where ϑ = {p, bp, r}. Therefore, there are different β(ϑ)-duals of a double sequence space. Now, we state the following basic lemma, which will enable us to determine the γ-and β(ϑ)-duals of some new spaces of double sequences.

Lemma 1.
Suppose that E is any double sequence space, and let the 4D matrix D = ( d mnkl ) be defined with the double sequence a = (a kl ) ∈ Ω and the inverse matrix F = ( f mnkl ) of the matrix B = ( b mnkl ) as: , elsewhere for all m, n, k, l ∈ N. Then, we have: where ϑ = {p, bp, r}.
Proof. We take a = (a mn ) ∈ Ω and x = (x mn ) ∈ B(C ϑ ). Then, clearly, y = Bx ∈ C ϑ . Therefore, the (m, n) th -partial sum of ∑ k,l a kl x kl can be stated by the following equality: where the 4D matrix D = ( d mnkl ) is defined by: , elsewhere for all m, n, k, l ∈ N. Therefore, we can easily obtain the result by the definition of the γ-dual of a double sequence space (see (1) The following theorems are direct consequences of Table 2 by applying the above Lemma 1. We consider the following sets d i with i = {1, 2, . . . , 15} by: −u ρ t ρ a ij = 0 ∀l > l 0 and ∀m, n ∈ N , −u ρ t ρ a ij = 0 ∀k > k 0 and ∀m, n ∈ N , Theorem 9. The following characterizations of γ-duals hold: Proof. The proof of (i), (ii), (iii) and (iv) easily follows from Lemma 1 with matrix D = ( d mnkl ) instead of A = (a mnkl ) in the matrix classes 1, 5, 8, and 11 in Table 2, respectively.
Theorem 10. The following characterizations of β(ϑ)-duals hold: Proof. The proof is similar to that of Theorem 9, and the proof of (i), (ii) and (iii) easily follows from Lemma 1 with matrix D = ( d mnkl ) instead of A = (a mnkl ) in the matrix classes 4, 6 and 7 in Table 2, respectively.
Theorem 11. The following characterizations of β(bp)-duals hold: Proof. The proof is similar to that of Theorem 9, and the proof of (i), (ii) and (iii) easily follows from Lemma 1 with matrix D = ( d mnkl ) instead of A = (a mnkl ) in the matrix classes 3, 9 and 12 in Table 2, respectively.
Proof. The proof is similar to that of Theorem 9, and the proof of (i) easily follows from Lemma 1 with matrix D = ( d mnkl ) instead of A = (a mnkl ) in the matrix class 2 in Table 2, respectively. This section is organized as follows. First, we state two significant theorems including dual summability matrices A = (a mnkl ) and C = ( c mnkl ), and A = (a mnkl ) and G = ( g mnkl ), respectively. Then, we establish necessary and sufficient conditions of some new matrix classes including the spaces of the double sequence whose B-transforms are in the spaces M u , C p , C bp , C r , and L q , respectively, where 0 < q < ∞. Finally, we conclude this section with some significant consequences.

Matrix Transformations
We note here that since B(E) ∼ = E for any arbitrary double sequence space E, it clearly holds that x ∈ B(E) if and only if y = Bx ∈ E holds. In this context, we now prove the following basic theorems: Theorem 13. We assume that the 4D infinite matrices A = (a mnkl ) and C = ( c mnkl ) are connected with the relation: for any m, n, k, l ∈ N and E 2 is any given double sequence space. Then, for any m, n ∈ N and C ∈ (E 1 : Proof. To prove the necessity, we assume A ∈ ( B(E 1 ) : E 2 ). Then, Ax exists and Ax ∈ E 2 for every x ∈ B(E 1 ). Therefore, A mn ∈ [ B(E 1 )] β(ϑ) is a necessary condition for all m, n ∈ N. Thus, by the relation in Equation (27) and the connection in (6) between the terms of the sequences y = (y kl ) and x = (x kl ), we obtain the (m, n) th -partial sum of the series ∑ k,l a mnkl x kl as: c mnkl y kl for all m, n ∈ N. Thus, taking the ϑ-limit in (28) as m, n → ∞, we obtain Ax = Cy. Hence, Cy ∈ E 2 whenever y ∈ E 1 , that is, C ∈ (E 1 : E 2 ). This concludes the proof of the necessity. To prove the sufficiency, suppose that A mn ∈ [ B(E 1 )] β(ϑ) for all m, n ∈ N and C ∈ (E 1 : E 2 ), where the matrix C = ( c mnkl ) is defined as in (27). Now, we assume that an arbitrary double sequence satisfies v = (v kl ) ∈ B(E 1 ) if and only if u = Bv ∈ E 1 . Then, for this sequence v = (v kl ), Av exists, and it is enough to show that Av ∈ E 2 . Clearly, v = B −1 u = Fu holds. We obtain the (m, n) th -partial sum of the series ∑ k,l a mnkl v kl for all m, n, k, l ∈ N as: −u ρ t ρ a mnij u kl .
Taking the ϑ-limit as m, n → ∞, we obtain: ∑ k,l a mnkl v kl = ∑ k,l c mnkl u kl for all m, n ∈ N.
Therefore, Av = Cu for arbitrary u = (u kl ) ∈ E 1 , and since C ∈ (E 1 ; E 2 ) by assumption, then clearly, Av ∈ E 2 . This concludes the proof of the sufficiency of A ∈ ( B(E 1 ) : E 2 ). Theorem 14. Suppose that the following connection between the 4D infinite matrices A = (a mnkl ) and G = ( g mnkl ) holds: = s m−1 u n−1 a m−1,n−1,k,l + s m−1 t n a m−1,n,k,l + r m u n−1 a m,n−1,k,l + r m t n a mnkl for all m, n, k, l ∈ N. Then, A ∈ (E 2 : B(E 1 )) if and only if G ∈ (E 2 : Proof. Suppose that A ∈ (E 2 : B(E 1 )). Then, Ax exists and is in B(E 1 ) for each sequence x = (x kl ) ∈ E 2 , and clearly, B(Ax) ∈ E 1 . We obtain the (m, n) th -partial sum by Relation (29)  for all m, n ∈ N. Thus, by taking the ϑ-limit as m, n → ∞, we conclude Ax ∈ B(E 1 ) whenever x = (x kl ) ∈ E 2 if and only if Gx ∈ E 1 whenever x = (x kl ) ∈ E 2 . Therefore, G ∈ (E 2 ; E 1 ) as desired.
Here, we organize the following new Tables 4-7 by referring to Tables 2 and 3. The aim is to summarize all the consequences derived by Theorems 13 and 14 by avoiding the repetition in the existing literature.

Corollary 1.
Let A = (a mnkl ) be a 4D infinite matrix and connected with the 4D matrix C = ( c mnkl ) by (27). Then, the following characterizations of the 4D matrix classes ( B(E 1 ); E 2 ), where E 1 , E 2 ∈ {M u , C p , C bp , C r , L q } given by: Table 4. 4D matrix classes ( B(E 1 ); E 2 ), where E 1 , E 2 ∈ {M u , C p , C bp , C r , L q }. Hold, and the necessary and sufficient conditions for each characterization of the new classes are given by: Table 5. Characterization of the classes ( B(E 1 ); E 2 ), where E 1 , E 2 ∈ {M u , C p , C bp , C r , L q }.
hold(s) with c mnkl instead of a mnkl .

Conclusions
In this work, we first defined the sequentially defined 4D infinite band matrix B = ( b mnkl ) and the new double sequence spaces B(M u ), B(C p ), B(C bp ), B(C r ), and B(L q ),(0 < q < ∞) whose B-transforms are in the spaces M u , C p , C bp , C r , L q , respectively. Then, we proved the isomorphism between the new double sequence spaces and some double sequence spaces. We also established some inclusion relations subject to some strict conditions. Moreover, we determined the dual spaces of our new double sequence spaces, and finally, we characterized some new 4D matrix classes related to the spaces B(M u ), B(C p ), B(C bp ), B(C r ), and B(L q ),(0 < q < ∞). The results stated in this paper were much more inclusive than the consequences in the works [11,25,27,28]. As a natural continuation of this paper and [21,22,24,26], calculating the B = ( b mnkl ) domain on almost convergent and strongly almost convergent double sequence spaces B(C f ) and B[C f ] is still an open problem.