# On the Characterization of Some Classes of Four-Dimensional Matrices and AlmostB-Summable Double Sequences

Authors: Orhan Tug
Journal: Journal of Mathematics
 (2018)

---

## Abstract

We firstly summarize the related literature aboutB
r
,
s
,
t
,
u-summability of double sequence spaces and almost B
r
,
s
,
t
,
u-summable double sequence spaces. Then we characterize some new matrix classes of L
s
′
:
C
f, B
L
s
′
:
C
f, and L
s
′
:
B
C
f of four-dimensional matrices in both cases of 0
<
s
′
≤
1 and 1
<
s
′
<
∞, and we complete this work with some significant results.


---

## Body

## 1. Preliminaries, Background, and Notations

We denote the set of all complex valued double sequence byΩ which is a vector space with coordinatewise addition and scalar multiplication. Any subspace of Ω is called a double sequence space. A double sequence x
=
x
m
n of complex numbers is called bounded if x
∞
=
s
u
p
m
,
n
∈
N
x
m
n
<
∞, 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
m
n
∈
Ω. If for every ϵ
>
0 there exists a natural number n
0
=
n
0
ϵ and l
∈
C such that x
m
n
-
l
<
ϵ 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; say that p
-
l
i
m
m
,
n
→
∞
x
m
n
=
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 [1, p. 16] defined the sequence x
=
x
m
n by (1)
x
m
n
=
n
,
m
=
0
,
n
∈
N
;
0
,
m
≥
1
,
n
∈
N
.Then it is clearly seen that p
-
l
i
m
m
,
n
→
∞
x
m
n
=
0 but x
∞
=
s
u
p
m
,
n
∈
N
x
m
n
=
∞, so x
∈
C
p
-
M
u. The set C
b
p denotes the space of both bounded and convergent double sequences; that is, C
b
p
=
C
p
∩
M
u. Hardy [2] showed that a double sequence x
=
x
m
n is said to converge regularly to l if x
∈
C
p and the limits x
m
:
=
l
i
m
n
x
m
n, m
∈
N and x
n
:
=
l
i
m
m
x
m
n, n
∈
N exist, and the limits l
i
m
m
l
i
m
n
x
m
n and l
i
m
n
l
i
m
m
x
m
n exist and are equal to the p-limit of x. Moreover, by C
b
p
0 and C
r
0, we may denote the spaces of all null double sequences contained in the sequence spaces C
b
p and C
r, respectively. Móricz [3] proved that the double sequence spaces C
b
p, C
b
p
0, C
r, and C
r
0 are Banach spaces with the norm ·
∞. The space L
q of all absolutely q
-summable double sequences corresponding to the space l
q of q
-summable single sequences was defined by Başar and Sever [4]; that is, (2)
L
q
≔
x
=
x
k
l
∈
Ω
:
∑
k
,
l
x
k
l
q
<
∞
,
1
≤
q
<
∞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 is introduced by Zeltser [5].Letλ be a double sequence space and converging with respect to some linear convergence rule is ϑ
-
l
i
m
:
λ
→
C. Then, the sum of a double series ∑
i
,
j
x
i
j relating to this rule is defined by ϑ
-
∑
i
,
j
x
i
j
=
ϑ
-
l
i
m
m
,
n
→
∞
∑
i
,
j
=
0
m
,
n
x
i
j. Throughout the paper, the summations from 0 to ∞ without limits, that is, ∑
i
,
j
x
i
j, mean that ∑
i
,
j
=
0
∞
x
i
j.Here and after, unless otherwise stated, we consider thatϑ denotes any of the symbols p
,
b
p, or r.Theα
-dual λ
α, the β
(
ϑ
)
-dual λ
β
(
ϑ
) with respect to the ϑ
-convergence, and the γ
-dual λ
γ of double sequence space λ are, respectively, defined by (3)
λ
α
≔
a
=
a
k
l
∈
Ω
:
∑
k
,
l
a
k
l
x
k
l
<
∞
∀
x
=
x
k
l
∈
λ
,
λ
β
ϑ
≔
a
=
a
k
l
∈
Ω
:
ϑ
-
∑
k
,
l
a
k
l
x
k
l
exists
∀
x
=
x
k
l
∈
λ
,
λ
γ
≔
a
=
a
k
l
∈
Ω
:
s
u
p
m
,
n
∈
N
∑
k
,
l
=
0
m
,
n
a
k
l
x
k
l
<
∞
∀
x
=
x
k
l
∈
λ
.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 sum of a double series does not guarantee its boundedness.Here, we shall be concerned with four-dimensional matrix transformation from any double sequence spaceλ to any double sequence space μ. Given any four-dimensional infinite matrix A
=
a
m
n
k
l, where m
,
n
,
k
,
l
∈
N, any double sequence x
=
(
x
k
l
), we write A
x
=
{
A
x
)
m
n
m
,
n
∈
N, the A
-transform of x, exists for every sequence x
=
x
k
l
∈
λ and it is in μ, where(4)
A
x
m
n
=
ϑ
-
∑
k
,
l
a
m
n
k
l
x
k
l
for each
m
,
n
∈
N
.The four-dimensional matrix domain has fundamental importance for this article. Therefore, this concept is presented in this paragraph. The ϑ
-summability domain λ
A
(
ϑ
) of A in a space λ of double sequences is described as (5)
λ
A
ϑ
=
x
=
x
k
l
∈
Ω
:
A
x
=
ϑ
-
∑
k
,
l
a
m
n
k
l
x
k
l
m
,
n
∈
N
exists and is in
λ
.Notation (4) says that A maps the space λ into the space μ if λ
⊂
μ
A
ϑ and we denote the set of all four-dimensional matrices, transforming the space λ into the space μ, by λ
:
μ. Thus, A
=
a
m
n
k
l
∈
λ
:
μ if and only if the double series on the right side of (4) converges in the sense of ϑ for each m
,
n
∈
N; that is, A
m
n
∈
λ
β
ϑ for all m
,
n
∈
N and we have A
x
∈
μ for all x
∈
λ, where A
m
n
=
a
m
n
k
l
k
,
l
∈
N for all m
,
n
∈
N. Moreover, the following definitions are significant in order to classify the four-dimensional matrices. A four-dimensional matrix A is called C
ϑ
-
c
o
n
s
e
r
v
a
t
i
v
e if C
ϑ
⊂
(
C
ϑ
)
A and is called C
ϑ
-
r
e
g
u
l
a
r if it is C
ϑ
-
c
o
n
s
e
r
v
a
t
i
v
e and (6)
ϑ
-
l
i
m
A
x
=
ϑ
-
l
i
m
m
,
n
→
∞
A
x
m
n
=
ϑ
-
l
i
m
m
,
n
→
∞
x
m
n
,
where
x
=
x
m
n
∈
C
ϑ
.One can be obtained by using the notations of Zeltser [6] that we may define the double sequences e
k
l
=
e
m
n
k
l
,
e
1
,
e
k, and e by (7)
e
m
n
k
l
=
1
,
k
,
l
=
m
,
n
;
0
,
otherwise
.
e
1
=
∑
k
e
k
l, e
k
=
∑
l
e
k
l, and e
=
∑
k
l
e
k
l for all k
,
l
,
m
,
n
∈
N and we may write the set Φ by Φ
=
s
p
a
n
e
k
l
:
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 expansion or the contraction of the original space, in general. Adams [7] defined that the four-dimensional infinite matrix A
=
a
m
n
k
l is called a triangular matrix if a
m
n
k
l
=
0 for k
>
m or l
>
n or both. We also say by [7] that an infinite matrix A
=
a
m
n
k
l is said to be a triangular if a
m
n
m
n
≠
0 for all m
,
n
∈
N. Moreover, Cooke [8] showed that every triangular matrix has a unique inverse which is also a triangular matrix.The four-dimensional generalized difference matrixB
r
,
s
,
t
,
u
=
b
m
n
k
l
r
,
s
,
t
,
u and matrix domain of it on some double sequence spaces were recently defined by Tuğ and Başar [9] and studied by Tuğ [10–12]. The matrix B
r
,
s
,
t
,
u
=
b
m
n
k
l
r
,
s
,
t
,
u was defined by Tuğ and Başar [9] as (8)
b
m
n
k
l
r
,
s
,
t
,
u
≔
s
u
,
k
,
l
=
m
-
1
,
n
-
1
,
s
t
,
k
,
l
=
m
-
1
,
n
,
r
u
,
k
,
l
=
m
,
n
-
1
,
r
t
,
k
,
l
=
m
,
n
0
,
o
t
h
e
r
w
i
s
efor 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
m
n was defined by(9)
y
m
n
≔
B
r
,
s
,
t
,
u
x
m
n
=
∑
k
,
l
b
m
n
k
l
r
,
s
,
t
,
u
x
k
l
=
s
u
x
m
-
1
,
n
-
1
+
s
t
x
m
-
1
,
n
+
r
u
x
m
,
n
-
1
+
r
t
x
m
nfor all m
,
n
∈
N. Moreover, the matrix B
-
1
r
,
s
,
t
,
u
=
F
r
,
s
,
t
,
u
=
f
m
n
k
l
r
,
s
,
t
,
u which is the inverse of the matrix B
r
,
s
,
t
,
u was calculated as (10)
f
m
n
k
l
r
,
s
,
t
,
u
≔
-
s
/
r
m
-
k
-
u
/
t
n
-
l
r
t
,
0
≤
k
≤
m
,
0
≤
l
≤
n
,
0
,
o
t
h
e
r
w
i
s
efor all m
,
n
,
k
,
l
∈
N. Furthermore, Tuğ and Başar [9] obtained the relation between x
=
x
m
n and y
=
(
y
m
n
) by(11)
x
m
n
=
1
r
t
∑
k
,
l
=
0
m
,
n
-
s
r
m
-
k
-
u
t
n
-
l
y
k
l
∀
m
,
n
∈
N
.Tug
˘ and Başar [9] and Tuǧ [10] have introduced and studied the new double sequence spaces B
M
u, B
C
p, B
C
b
p, B
C
r, and B
L
q as the domain of four-dimensional generalized difference matrix B
r
,
s
,
t
,
u in the spaces M
u, C
p, C
b
p, C
r, and L
q, respectively. That is, (12)
B
M
u
≔
x
=
x
m
n
∈
Ω
:
s
u
p
m
,
n
∈
N
B
r
,
s
,
t
,
u
x
m
n
<
∞
,
B
C
p
≔
x
=
x
m
n
∈
Ω
:
∃
l
∈
C
∋
p
-
l
i
m
m
,
n
→
∞
B
r
,
s
,
t
,
u
x
m
n
-
l
=
0
,
B
C
b
p
≔
x
=
x
m
n
∈
Ω
:
B
r
,
s
,
t
,
u
x
∈
C
b
p
,
B
C
r
≔
x
=
x
m
n
∈
Ω
:
B
r
,
s
,
t
,
u
x
∈
C
r
,
B
L
q
≔
x
=
x
m
n
∈
Ω
:
∑
m
,
n
B
r
,
s
,
t
,
u
x
m
n
q
<
∞
,
0
<
q
<
∞
.Lorentz [13] introduced the concept of almost convergence for single sequence and Moricz and Rhoades [14] extended and studied this concept for double sequence. A double sequence x
=
x
k
l of complex numbers is said to be almost convergent to a generalized limit L if (13)
p
-
l
i
m
q
,
q
′
→
∞
s
u
p
m
,
n
>
0
1
q
+
1
q
′
+
1
∑
k
=
m
m
+
q
∑
l
=
n
n
+
q
′
x
k
l
-
L
=
0
.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; that is, (14)
C
f
≔
x
=
x
k
l
∈
Ω
:
∃
L
∈
C
∋
p
-
l
i
m
q
,
q
′
→
∞
s
u
p
m
,
n
>
0
1
q
+
1
q
′
+
1
∑
k
=
m
m
+
q
∑
l
=
n
n
+
q
′
x
k
l
-
L
=
0
,
uniformly in
m
,
n
.It is 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
b
p
⊂
C
f
⊂
M
u holds, and each inclusion is proper. A double sequence x
=
(
x
k
l
) is called almost Cauchy which was introduced by Čunjalo [15] if for every ϵ
>
0 there exists a positive integer K such that (15)
1
q
1
+
1
q
1
′
+
1
∑
k
=
m
1
m
1
+
q
1
∑
l
=
n
1
n
1
+
q
1
′
x
k
l
-
1
q
2
+
1
q
2
′
+
1
∑
k
=
m
2
m
2
+
q
2
∑
l
=
n
2
n
2
+
q
2
′
x
k
l
<
ϵfor all q
1
,
q
1
′
,
q
2
,
q
2
′
>
K and m
1
,
n
1
,
m
2
,
n
2
∈
N
×
N. Mursaleen and Mohiuddine [16] proved that every double sequence is almost convergent if and only if it is almost Cauchy.Moricz and Rhoades [14] considered that four-dimensional matrices transforming every almost convergent double sequence into a b
p
-convergent double sequence with the same limit. Almost conservative and almost regular matrices for single sequences were characterized by King [17] and almost C
ϑ
-conservative and almost C
ϑ
-regular four-dimensional matrices for double sequences were defined and characterized by Zeltser et al. [18]. Mursaleen [19] introduced the almost strongly regularity for double sequences. A four-dimensional matrix A
=
a
m
n
k
l is called almost strongly regular if it transforms every almost convergent double sequence into an almost convergent double sequence with the same limit.Definition 1 (see [18]).
A four-dimensional matrixA
=
a
m
n
k
l is said to be almost C
ϑ
-conservative matrix if it transforms every ϑ
-convergent double sequence x
=
x
k
l into an almost convergent double sequence space; that is, A
=
a
m
n
k
l
∈
C
ϑ
:
C
f.Definition 2 (see [18]).
A four-dimensional matrixA
=
(
a
m
n
k
l
) is said to be almost C
ϑ
-regular if it is C
ϑ
-conservative and f
2
-
l
i
m
A
x
=
ϑ
-
l
i
m
x for all x
∈
C
ϑ.Tuğ [11, 12] has recently studied new almost convergent double sequence spaces B
C
f and B
C
f
0 as the domain of four-dimensional generalized difference matrix B
r
,
s
,
t
,
u in the spaces C
f and C
f
0, respectively. That is, (16)
B
C
f
≔
x
=
x
k
l
∈
Ω
:
∃
L
∈
C
∋
p
-
l
i
m
q
,
q
′
→
∞
s
u
p
m
,
n
>
0
1
q
+
1
q
′
+
1
∑
k
=
m
m
+
q
∑
l
=
n
n
+
q
′
B
x
k
l
-
L
=
0
,
uniformly in
m
,
n
,
B
C
f
0
≔
x
=
x
k
l
∈
Ω
:
p
-
l
i
m
q
,
q
′
→
∞
s
u
p
m
,
n
>
0
1
q
+
1
q
′
+
1
∑
k
=
m
m
+
q
∑
l
=
n
n
+
q
′
B
x
k
l
=
0
,
uniformly in
m
,
n
.In this paper, as natural continuation of [10, 11], we characterize some new matrix classes L
s
′
:
C
f, B
L
s
′
:
C
f, and L
s
′
:
B
C
f of four-dimensional matrices in both cases 0
<
s
′
<
1 and 1
<
s
′
<
∞. Throughout the paper, we suppose that the terms of double sequence x
=
x
m
n and y
=
(
y
m
n
) are connected with relation (11) and the four-dimensional generalized difference matrix B
(
r
,
s
,
t
,
u
)
=
b
m
n
k
l
r
,
s
,
t
,
u will be presented with B
=
b
m
n
k
l.


## 2. Main Results

In this section, we characterize some new four-dimensional matrix classesL
s
′
:
C
f, B
L
s
′
:
C
f, L
s
′
:
B
C
f in both cases of 0
<
s
≤
1 and 1
<
s
<
∞, and B
C
f
:
M
u. Then we complete this section with some significant results of four-dimensional matrix mapping via the dual summability methods for double sequences which has been introduced and studied by Başar [20] and Yeşilkayagil and Başar [21] and has been recently applied in [10, 12].Theorem 3.
LetA
=
a
m
n
k
l be a four-dimensional infinite matrix. Then the following statements hold.
(a) Let0
<
s
′
≤
1. Then, A
=
a
m
n
k
l
∈
L
s
′
:
C
f if and only if(17)
s
u
p
m
,
n
,
k
,
l
∈
N
a
m
n
k
l
<
∞
,
(18)
∃
a
k
l
∈
C
such that
f
2
-
lim
m
,
n
→
∞
⁡
a
m
n
k
l
=
a
k
l
∀
k
,
l
∈
N
.
(b) Let1
<
s
′
<
∞. Then, A
=
a
m
n
k
l
∈
L
s
′
:
C
f if and only if condition (18) holds and(19)
s
u
p
m
,
n
∈
N
∑
k
,
l
a
m
n
k
l
s
′
<
∞
.Proof.
(a) Let0
<
s
′
≤
1 and A
=
a
m
n
k
l
∈
L
s
′
:
C
f. Then, A
x exists and is in C
f for all x
=
x
k
l
∈
L
s
′. Since the inclusion L
s
′
:
C
f
⊂
L
s
′
:
M
u holds, then we may say that condition (17) is necessary. Moreover, since A
x exists and is in C
f for all x
=
x
k
l
∈
L
s
′, it is also provided for x
=
e
k
l
∈
L
s
′ that A
e
k
l
=
a
m
n
k
l
m
,
n
∈
N
∈
C
f which gives us condition (18) which is also necessary.
Conversely, suppose that conditions (17) and (18) hold and x
=
(
x
k
l
) be any double sequence in L
s
′. Since A
m
n
∈
L
s
′
β
ϑ by corollary 3.2 of [22] for each m
,
n
∈
N, A
x exists. By using conditions (17) and (18) for each k
,
l
∈
N, we have the following inequality:(20)
a
k
l
=
f
2
-
l
i
m
m
,
n
→
∞
a
m
n
k
l
≤
s
u
p
m
,
n
∈
N
a
m
n
k
l
,then we may say that a
k
l
∈
M
u. Thus, the series ∑
k
,
l
a
k
l
x
k
l is convergent for all x
∈
L
s
′. Moreover, by condition (18) we may say that for every ϵ
>
0 there exists a positive n
0
∈
N such that (21)
1
q
+
1
q
′
+
1
∑
k
=
j
j
+
q
∑
l
=
i
i
+
q
′
a
m
n
k
l
-
a
k
l
<
ϵfor all m
,
n
>
n
0. Thus (22)
1
q
+
1
q
′
+
1
∑
k
=
j
j
+
q
∑
l
=
i
i
+
q
′
∑
v
,
v
′
k
,
l
a
m
n
v
,
v
′
x
v
,
v
′
-
∑
v
,
v
′
k
,
l
a
v
,
v
′
x
v
,
v
′
s
′
=
1
q
+
1
q
′
+
1
∑
k
=
j
j
+
q
∑
l
=
i
i
+
q
′
∑
v
,
v
′
k
,
l
a
m
n
v
,
v
′
-
a
v
,
v
′
x
v
,
v
′
s
′
≤
ϵ
s
′
q
+
1
q
′
+
1
∑
k
=
j
j
+
q
∑
l
=
i
i
+
q
′
∑
v
,
v
′
k
,
l
x
v
,
v
′
s
′
<
ϵ
s
′
q
+
1
q
′
+
1
∑
k
=
j
j
+
q
∑
l
=
i
i
+
q
′
∑
v
,
v
′
k
,
l
x
v
,
v
′
s
′
.It gives us that p
-
l
i
m
q
,
q
′
→
∞
1
/
q
+
1
q
′
+
1
∑
k
=
j
j
+
q
∑
l
=
i
i
+
q
′
A
x
k
l exists; that is, A
x
∈
C
f.
This completes the proof of(
a
).
(b) Let1
<
s
′
<
∞. The necessity of conditions (18) and (19) may be easily shown in the similar method which is used in (a) with the Hölder inequality. So we pass the repetition.
Conversely, suppose that conditions (18) and (19) hold and x
=
x
k
l be a double sequence in the space L
s
′. It can be written with (18) for all k
,
l
∈
N that(23)
∑
v
,
v
′
=
0
k
,
l
a
v
,
v
′
s
′
=
f
2
-
l
i
m
m
,
n
→
∞
∑
v
,
v
′
=
0
k
,
l
a
m
n
v
,
v
′
s
′
≤
s
u
p
m
,
n
∈
N
∑
v
,
v
′
=
0
k
,
l
a
m
n
v
,
v
′
s
′
<
∞
;say that a
=
a
v
,
v
′
∈
L
s
′. Therefore, the double series ∑
v
,
v
′
=
0
k
,
l
a
v
,
v
′
x
v
,
v
′ is convergent for every x
∈
L
s
′. Moreover, for any given ϵ
>
0, there exist N
0
∈
N such that (24)
1
q
+
1
q
′
+
1
∑
k
=
j
j
+
q
∑
l
=
i
i
+
q
′
∑
v
,
v
′
=
0
k
,
l
a
m
n
v
,
v
′
-
a
v
,
v
′
s
′
<
ϵfor every m
,
n
>
N
0. So, by using Hölder’s inequality with relations (23) and (24) we have power to write that (25)
1
q
+
1
q
′
+
1
∑
k
=
j
j
+
q
∑
l
=
i
i
+
q
′
∑
v
,
v
′
=
0
k
,
l
a
m
n
v
,
v
′
x
v
,
v
′
-
∑
v
,
v
′
=
0
k
,
l
a
v
,
v
′
x
v
,
v
′
=
1
q
+
1
q
′
+
1
∑
k
=
j
j
+
q
∑
l
=
i
i
+
q
′
∑
v
,
v
′
=
0
k
,
l
a
m
n
v
,
v
′
-
a
v
,
v
′
x
v
,
v
′for all m
,
n
∈
N. After passing p-limit as q
,
q
′
→
∞ we can say that A
x
∈
C
f. This last step completes the proof.Theorem 4.
LetA
=
a
m
n
k
l be a four-dimensional infinite matrix. Then the following statements hold.
(a) Let0
<
s
′
≤
1. Then, A
=
a
m
n
k
l
∈
L
s
′
:
B
C
f if and only if(26)
s
u
p
m
,
n
,
k
,
l
∈
N
∑
i
,
j
=
0
m
,
n
b
m
n
i
j
a
i
j
k
l
<
∞
,
(27)
∃
e
k
l
∈
Ω
∋
f
2
-
l
i
m
m
,
n
→
∞
∑
i
,
j
=
0
m
,
n
b
m
n
i
j
a
i
j
k
l
=
e
k
l
∀
k
,
l
∈
N
.(b) 1
<
s
′
<
∞. Then, A
=
(
a
m
n
k
l
)
∈
(
L
s
′
:
B
(
C
f
)
) if and only if condition (27) holds and(28)
s
u
p
m
,
n
∈
N
∑
k
,
l
∑
i
,
j
=
0
m
,
n
b
m
n
i
j
a
i
j
k
l
s
′
<
∞
.Proof.
Suppose that the four-dimensional matrixA
=
a
m
n
k
l
∈
L
s
′
:
B
C
f. Then, A
x exist and is in B
C
f for all x
∈
L
s
′. Since A
x
∈
B
C
f then we can say that B
A
x
∈
C
f. Moreover, we may have the following equality:(29)
∑
i
,
j
=
0
m
,
n
b
m
n
i
j
∑
v
,
v
′
=
0
k
,
l
a
i
j
v
,
v
′
x
v
,
v
′
=
∑
v
,
v
′
=
0
k
,
l
∑
i
,
j
=
0
m
,
n
b
m
n
i
j
a
i
j
v
,
v
′
x
v
,
v
′holds for all m
,
n
,
k
,
l
∈
N. Therefore, we have B
A
x
=
B
A
x by taking limit as k
,
l
→
∞ in (29). Now we can define the new four-dimensional matrix B
A
=
G
=
g
m
n
k
l as (30)
g
m
n
k
l
=
∑
i
,
j
=
0
m
,
n
b
m
n
i
j
a
i
j
v
,
v
′
.So we can write here that A
x
∈
B
C
f whenever x
∈
L
s
′ if and only if G
x
∈
C
f whenever x
∈
L
s
′ which says that G
=
g
m
n
k
l
∈
L
s
′
:
C
f. Thus conditions (17) and (18) hold for 0
<
s
′
≤
1, and conditions (18) and (19) hold for 1
<
s
′
<
∞ with g
m
n
k
l instead of a
m
n
k
l. This gives us the results as desired.Theorem 5.
LetA
=
a
m
n
k
l be a four-dimensional infinite matrix. Then the following statements hold.
(a) Let0
<
s
′
≤
1. Then, A
=
a
m
n
k
l
∈
B
L
s
′
:
C
f if and only if(31)
s
u
p
m
,
n
,
k
,
l
∈
N
∑
i
,
j
=
k
,
l
m
,
n
-
s
r
i
-
k
-
u
t
j
-
l
a
m
n
i
j
r
t
<
∞
,
(32)
∃
e
k
l
∈
Ω
∋
f
2
-
l
i
m
m
,
n
→
∞
∑
i
,
j
=
k
,
l
m
,
n
-
s
r
i
-
k
-
u
t
j
-
l
a
m
n
i
j
r
t
=
e
k
l
.
(b) Let1
<
s
′
<
∞. Then, A
=
a
m
n
k
l
∈
B
L
s
′
:
C
f if and only if condition (32) holds and(33)
s
u
p
m
,
n
∈
N
∑
k
,
l
∑
i
,
j
=
k
,
l
m
,
n
-
s
r
i
-
k
-
u
t
j
-
l
a
m
n
i
j
r
t
s
′
<
∞
.Proof.
Suppose that the four-dimensional matrixA
=
a
m
n
k
l
∈
B
L
s
′
:
C
f. Then, as the method used in the proof of Theorem 4, A
x exist and is in C
f for all x
∈
B
L
s
′. Since x
∈
B
L
s
′ then we can say that B
x
=
y
∈
L
s
′ and the equality(34)
∑
i
,
j
=
0
m
,
n
a
m
n
i
j
x
v
,
v
′
=
∑
v
,
v
′
=
0
k
,
l
∑
i
,
j
=
k
,
l
m
,
n
-
s
r
i
-
k
-
u
t
j
-
l
a
m
n
i
j
r
t
y
v
,
v
′holds for all m
,
n
,
k
,
l
∈
N. Therefore, we have A
x
=
A
B
-
1
y
=
E
y by taking limit as k
,
l
→
∞ in (34). Now we may define the following four-dimensional matrix A
B
-
1
=
E
=
e
m
n
k
l as (35)
e
m
n
k
l
=
∑
i
,
j
=
k
,
l
m
,
n
-
s
r
i
-
k
-
u
t
j
-
l
a
m
n
i
j
r
t
.So we can write here that A
x
∈
C
f whenever x
∈
B
(
L
s
′
) if and only if E
y
∈
C
f whenever y
∈
L
s
′ which says that E
=
e
m
n
k
l
∈
L
s
′
:
C
f. Thus conditions (17) and (18) hold for 0
<
s
′
≤
1, and conditions (18) and (19) hold for 1
<
s
′
<
∞ with e
m
n
k
l instead of a
m
n
k
l. This gives us the expected results and completes the proof.Corollary 6.
Suppose that the elements of four-dimensional infinite matricesA
=
(
a
m
n
k
l
) and H
=
(
h
m
n
k
l
) are connected with relation (36)
h
m
n
k
l
=
∑
i
,
j
=
k
,
l
m
,
n
b
m
n
i
j
e
i
j
k
l
∀
m
,
n
,
k
,
l
∈
N
.Then the following statements hold.
(i) Let0
<
s
′
≤
1. Then A
∈
B
L
s
′
:
B
C
f iff (17) and (18) hold with h
m
n
k
l instead of a
m
n
k
l.
(ii) Let1
<
s
′
<
∞. Then A
∈
B
L
s
′
:
B
C
f iff (18) and (19) hold with h
m
n
k
l instead of a
m
n
k
l.


## 3. Conclusion

The concept of almost convergence of single sequence was introduced by Lorentz [13]. In 2010, Mursaleen [23] investigated the certain properties of the space of almost convergent sequences denoted by f. Then many of the mathematicians have studied the matrix domain on almost null and almost convergent sequences spaces (see [24–27]).The almost convergence for double sequence was introduced by Moricz and Rhoades [14] and studied on by many researchers (see [16, 28–36]). Yeşilkayagil and Basar [37] recently studied the topological properties of the spaces of almost null and almost convergent double sequences.The domain of four-dimensional generalized difference matrixB
(
r
,
s
,
t
,
u
) on some double sequence spaces B
C
p, B
C
b
p, B
C
r, and B
L
q has been introduced and studied by Tug
˘ and Başar [9] and by Tug
˘ [10]. They examined some topological properties. Furthermore, they 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.The domain of the four-dimensional generalized difference matrixB
r
,
s
,
t
,
u in the spaces of almost null and almost convergent double sequences has been recently introduced and studied by Tug
˘ [11, 12]. He examined some topological properties and determined α
-, β
b
p
-, and γ
-duals of the space B
C
f. Moreover, he characterized some new classes of four-dimensional matrix mappings related to the sequence space B
C
f.In this work, we tried to fill one of the gaps in the four-dimensional matrix characterization of the matrix classesL
s
′
:
C
f, B
L
s
′
:
C
f, and L
s
′
:
B
C
f in both cases of 0
<
s
′
≤
1 and 1
<
s
′
<
∞. The characterization of the matrix classes C
f
:
C
p, C
f
:
C
r, B
C
f
:
C
p, andB
C
f
:
C
r and calculating the β
(
θ
) dual of the space B
(
C
f
) where θ
=
(
p
,
r
) are still open problems for readers.



---

*Source: 1826485-2018-04-19.xml*