# Generalized Almost Convergence and Core Theorems of Double Sequences

Authors: S. A. Mohiuddine; Abdullah Alotaibi
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The idea of[
λ
,
μ
]-almost convergence (briefly, F
[
λ
,
μ
]-convergence) has been recently introduced and studied by Mohiuddine and Alotaibi (2014). In this paper first we define a norm on F
[
λ
,
μ
] such that it is a Banach space and then we define and characterize those four-dimensional matrices which transform F
[
λ
,
μ
]-convergence of double sequences x
=
(
x
j
k
) into F
[
λ
,
μ
]-convergence. We also define a F
[
λ
,
μ
]-core of x
=
(
x
j
k
) and determine a Tauberian condition for core inclusions and core equivalence.


---

## Body

## 1. Background, Notations, and Preliminaries

We begin by recalling the definition of convergence for double sequences which was introduced by Pringsheim [1]. A double sequence x
=
(
x
j
k
) is said to be c
o
n
v
e
r
g
e
n
t to L in the Pringsheim's sense (or P-convergent to L) if for given ϵ
>
0 there exists an integer N such that |
x
j
k
-
L
|
<
ϵ whenever j
,
k
>
N. We will write this as
(1)
P
-
lim
⁡
j
,
k
→
∞
x
j
k
=
L
,
where j and k are tending to infinity independent of each other. We denote by C
P the space of P-convergent sequences.We say that a double sequencex
=
(
x
j
k
) is b
o
u
n
d
e
d if
(2)
∥
x
∥
=
sup
⁡
j
,
k
≥
0
|
x
j
k
|
<
∞
.
Denote by L
∞ the space of all bounded double sequences.If a double sequencex
=
(
x
j
k
) in L
∞ and x is also P-convergent to L, then we say that it isboundedly
P
-convergent to L (or, BP-convergent to L). We denote by C
BP the space of all boundedly P-convergent double sequences. Note that C
BP
⊂
L
∞.We remark that, in contrast to the case for single sequences, aP-convergent double sequence need not be bounded.LetA
=
(
a
p
q
j
k
:
j
,
k
=
0,1
,
2
,
…
) be a four-dimensional infinite matrix of real numbers for all p
,
q
=
0,1
,
2
,
… and S
1 a space of double sequences. Let S
2 be a double sequences space, converging with respect to a convergence rule ν
∈
{
P
,
BP
}. Define
(3)
S
1
A
,
ν
=
{
x
=
(
x
j
k
)
:
A
x
=
(
A
p
q
(
x
)
)
-
∑
j
,
k
‍
∑
k
‍
a
p
q
j
k
x
j
k
existsand
A
x
∈
S
1
h
h
h
h
h
h
h
h
h
h
h
h
h
=
ν
-

∑
j
,
k
‍
∑
k
‍

a
p
q
j
k
x
j
k
exists
,
A
x
∈
S
1
}
.
Then, we say that a four-dimensional matrix B maps the space S
2 into the space S
1 if S
2
⊂
S
1
B
,
ν and is denoted by (
S
2
,
S
1
).The idea of almost convergence of Lorentz [2] is narrowly connected with the limits of Banach (see [3]) as follows. A sequence x
=
(
x
j
) in l
∞ is almost convergent to L if all of its Banach limits are equal, where l
∞ denotes the space of all bounded sequences. Mohiuddine [4] applies this concept to established some approximation theorems for sequence of positive linear operator. Móricz and Rhoades [5] extended the notion of almost convergence from single to double sequences as follows.A double sequencex
=
(
x
j
,
k
) of real numbers is said to bealmost convergent to a number L if
(4)
lim
⁡
p
,
q
→
∞
sup
⁡
m
,
n
>
0
|
1
p
q
∑
j
=
m
m
+
p
-
1
‍
∑
k
=
n
n
+
q
-
1
‍
x
j
,
k
-
L
|
=
0
.
For more details on almost convergence for single and double sequences, one can refer to [6–13].The two-dimensional analogue of Banach limit has been defined by Mursaleen and Mohiuddine [14] as follows. A linear functional L on L
∞ is said to beBanach limit if it has the following properties:(
B
L
1
)
L
(
x
)
≥
0 if x
≥
0 (i.e., x
j
k
≥
0 for all j
,
k),
(
B
L
2
)
L
(
E
)
=
1, where E
=
(
e
j
k
) with e
j
k
=
1 for all j
,
k,
(
B
L
3
)
L
(
S
11
x
)
=
L
(
x
)
=
L
(
S
10
x
)
=
L
(
S
01
x
),where the shift operators S
01, S
10, and S
11 are defined by
(5)
S
01
x
=
(
x
j
,
k
+
1
)
,
S
10
x
=
(
x
j
+
1
,
k
)
,
S
11
x
=
(
x
j
+
1
,
k
+
1
)
.
Denote by B the set of all Banach limits on L
∞. Note that if (BL
3) holds, then we may also write L
(
S
x
)
=
L
(
x
). A double sequence x
=
(
x
j
k
) is said to bealmost convergent to a number L if L
(
x
)
=
L for all L
∈
B.Letλ
=
(
λ
m
:
m
=
0,1
,
2
,
…
) and μ
=
(
μ
n
:
n
=
0,1
,
2
,
…
) be two nondecreasing sequences of positive reals and each tending to ∞ such that λ
m
+
1
≤
λ
m
+
1, λ
1
=
0, μ
n
+
1
≤
μ
n
+
1, μ
1
=
0, and
(6)
I
m
n
(
x
)
=
1
λ
m
μ
n
∑
j
∈
J
m
‍
∑
k
∈
I
n
‍
x
j
k
,
is called thedouble generalized de la Valée-Poussin mean, where J
m
=
[
m
-
λ
m
+
1
,
m
] and I
n
=
[
n
-
μ
n
+
1
,
n
]. We denote the set of all λ and μ type sequences by using the symbol Λ.Quite recently, Mohiuddine and Alotaibi [15] presented a generalization of the notion of almost convergent double sequence with the help of de la Vallée-Poussin mean and called it [
λ
,
μ
]-almost convergent. In the same paper, they also defined and characterized some four-dimensional matrices. For more details on double sequences, four-dimensional matrices, and other related concepts, one can refer to [16–26].A double sequencex
=
(
x
j
k
) of reals is said to be [
λ
,
μ
]-almost convergent (briefly, F
[
λ
,
μ
]-convergent) [15] to some number L if x
∈
F
[
λ
,
μ
], where
(7)
F
[
λ
,
μ
]
=
{
x
=
(
x
j
k
)
:
P
-
lim
⁡
m
,
n
→
∞
Ω
m
n
s
t
(
x
)
=
L
exists
,
h
h
h
h
h
h
h
h
h
h
h
h
h
x
=
(
x
j
k
)
:
P
-
lim
⁡
m
,
n
→
∞
Ω
m
n
s
t
(
x
)
uniformly
in
s
,
t
;
L
=
F
[
λ
,
μ
]
-lim
x
}
,
Ω
m
n
s
t
(
x
)
=
1
λ
m
μ
n
∑
j
∈
J
m
‍
∑
k
∈
I
n
‍
x
j
+
s
,
k
+
t
.
Denote by F
[
λ
,
μ
] the space of all [
λ
,
μ
] almost convergent sequences (
x
j
,
k
). Note that C
BP
⊂
F
[
λ
,
μ
]
⊂
L
∞.We remark that if we takeλ
m
=
m and μ
n
=
n, then the notion of [
λ
,
μ
]-almost convergence coincides with the notion of almost convergence for double sequences due to Móricz and Rhoades [5].


## 2.(
F
[
λ
,
μ
]
,
F
[
λ
,
μ
]
)-Matrices

We will assume throughout this paper that the limit of a double sequence means limit in the Pringsheim sense. We define the following matrix classes and establish interesting results.Definition 1.
A four-dimensional matrixA
=
(
a
p
q
j
k
) is said to be [
λ
,
μ
]-almost regular if A
x
∈
F
[
λ
,
μ
] for all x
=
(
x
j
k
)
∈
C
BP with F
[
λ
,
μ
]
-lim
A
x
=
lim
⁡
x, and one denotes this by A
∈
(
C
BP
,
F
[
λ
,
μ
]
)
reg.Definition 2.
A matrixA
=
(
a
p
q
j
k
) is said to be of class (
F
[
λ
,
μ
]
,
F
[
λ
,
μ
]
) if it maps every F
[
λ
,
μ
]-convergent double sequence into F
[
λ
,
μ
]-convergent double sequence; that is, A
x
∈
F
[
λ
,
μ
] for all x
=
(
x
j
k
)
∈
F
[
λ
,
μ
]. In addition, if F
[
λ
,
μ
]
-lim
A
x
=
F
[
λ
,
μ
]
-lim
x, then A is F
[
λ
,
μ
]
-regular and, in symbol, one will write A
∈
(
F
[
λ
,
μ
]
,
F
[
λ
,
μ
]
)
reg.Now we define the norm onF
[
λ
,
μ
] as follows.Theorem 3.
F
[
λ
,
μ
] is a Banach space normed by
(8)
∥
x
∥
=
sup
⁡
m
,
n
,
s
,
t
|
Ω
m
n
s
t
(
x
)
|
.Proof.
It can be easily verified that (8) defines a norm on F
[
λ
,
μ
]. We show that F
[
λ
,
μ
] is complete. Now, let (
x
b
) be a Cauchy sequence in F
[
λ
,
μ
]. Then for each j
,
k, (
x
j
k
b
) is a Cauchy sequence in R. Therefore x
j
k
b
→
x
j
k (say). Put x
=
(
x
j
k
); given ϵ there exists an integer N
(
ϵ
)
=
N, say, such that, for each b
,
d
>
N,
(9)
∥
x
b
-
x
d
∥
<
ϵ
2
.
Hence
(10)
sup
⁡
m
,
n
,
s
,
t
|
Ω
m
n
s
t
(
x
b
-
x
d
)
|
<
ϵ
2
;
then, for each m, n, s, t and b, d
>
N, we have
(11)
|
Ω
m
n
s
t
(
x
b
-
x
d
)
|
<
ϵ
2
.
Taking limit d
→
∞, we have for b
>
N and for each of m
,
n
,
s
,
t
(12)
|
Ω
m
n
s
t
(
x
b
-
x
)
|
<
ϵ
2
.
Now for fixed b, the above inequality holds. Since for fixed b, x
b
∈
F
[
λ
,
μ
], we get
(13)
lim
⁡
m
,
n
→
∞
Ω
m
n
s
t
(
x
b
)
=
l
uniformly in s, t. For given ϵ
>
0, there exist positive integers m
0, n
0 such that
(14)
|
Ω
m
n
s
t
(
x
b
)
-
l
|
<
ϵ
2
,
for m
≥
m
0, n
≥
n
0 and for all s, t. Here m
0, n
0 are independent of s, t but depend upon ϵ. Now by using (12) and (14), we get
(15)
|
Ω
m
n
s
t
(
x
)
-
l
|
=
|
Ω
m
n
s
t
(
x
)
-
Ω
m
n
s
t
(
x
b
)
+
Ω
m
n
s
t
(
x
b
)
-
l
|
≤
|
Ω
m
n
s
t
(
x
)
-
Ω
m
n
s
t
(
x
b
)
|
+
|
Ω
m
n
s
t
(
x
b
)
-
l
|
<
ϵ
,
for m
≥
m
0, n
≥
n
0 and for all s, t. Hence x
=
(
x
j
k
)
∈
F
[
λ
,
μ
] and F
[
λ
,
μ
] is complete.Now we characterize the matrix class(
F
[
λ
,
μ
]
,
F
[
λ
,
μ
]
) as well as (
F
[
λ
,
μ
]
,
F
[
λ
,
μ
]
)
reg. Let M
[
λ
,
μ
] be the subspace of F
[
λ
,
μ
] such that P
-
lim
⁡
m
,
n
→
∞
⁡
Ω
m
n
s
t
(
x
)
=
0, uniformly in s
,
t; that is
(16)
M
[
λ
,
μ
]
=
{
x
=
(
x
j
k
)
∈
F
[
λ
,
μ
]
:
P
-
lim
⁡
m
,
n
→
∞
Ω
m
n
s
t
(
x
)
=
0
,
h
x
=
(
x
j
k
)
h
∈
F
[
λ
,
μ
]
l
F
[
λ
,
μ
]
:
P
-
lim
⁡
m
,
n
→
∞
Ω
m
n
s
t
(
x
)
uniformly
in
s
,
t
}
.
Note that every y
∈
F
[
λ
,
μ
] can be written as
(17)
y
=
x
+
l
E
,
where x
∈
M
[
λ
,
μ
]
,
l
=
P
-
lim
⁡
m
,
n
⁡
Ω
m
n
s
t
(
y
) uniformly in s
,
t, and E
=
(
e
j
k
) with e
j
k
=
1 for all j
,
k.Theorem 4.
A matrixA
=
(
a
p
q
j
k
)
∈
(
F
[
λ
,
μ
]
,
F
[
λ
,
μ
]
) if and only if (AC
1)
∥
A
∥
=
sup
⁡
p
,
q
⁡
∑
j
=
0
∞
∑
k
=
0
∞
|
a
p
q
j
k
|
<
∞,
(AC
2)
a
=
(
∑
j
=
0
∞
∑
k
=
0
∞
a
p
q
j
k
)
p
,
q
=
1
∞
∈
F
[
λ
,
μ
],
(AC
3)
A
(
S
-
I
)
∈
(
L
∞
,
F
[
λ
,
μ
]
),where S is the shift operator.Proof.
Necessity. Let A
∈
(
F
[
λ
,
μ
]
,
F
[
λ
,
μ
]
). We know that C
BP
⊂
F
[
λ
,
μ
]
⊂
L
∞, so we have A
∈
(
C
BP
,
L
∞
). Hence the necessity of (AC
1) follows. Since ∈
F
[
λ
,
μ
], then A
E
∈
F
[
λ
,
μ
]. This is equivalent to
(18)
(
∑
j
=
0
∞
∑
k
=
0
∞
a
p
q
j
k
‍
‍
)
p
,
q
=
1
∞
∈
F
[
λ
,
μ
]
;
that is, (AC
2) holds. For each x
=
(
x
j
k
)
∈
L
∞, we have S
x
-
x
∈
F
[
λ
,
μ
] because
(19)
L
(
S
x
-
x
)
=
L
(
S
x
)
-
L
(
x
)
=
0
for all Banach limit L. Hence A
(
S
x
-
x
)
∈
F
[
λ
,
μ
]; that is, (AC
3) holds.
Sufficiency. Let conditions (AC
1)–(AC
3) hold and y
=
(
y
j
k
)
∈
F
[
λ
,
μ
]. Then
(20)
y
=
x
+
l
E
,
where x
=
(
x
j
k
)
∈
M
[
λ
,
μ
], l
=
P
-
lim
⁡
m
,
n
→
∞
⁡
Ω
m
n
s
t
(
y
), uniformly in s
,
t and E
=
(
e
j
k
) with e
j
k
=
1 for all j
,
k. Taking A-transform in (20), we obtain
(21)
A
y
=
A
x
+
l
A
E
=
A
x
+
l
(
∑
j
=
0
∞
∑
k
=
0
∞
a
p
q
j
k
‍
‍
)
p
,
q
=
1
∞
.
If x
=
(
x
j
k
)
∈
L
∞, then by (AC
3) we have A
(
S
x
-
x
)
∈
F
[
λ
,
μ
]. Since by (AC
1), A is bounded linear operator on L
∞, we get A
M
[
λ
,
μ
]
⊂
F
[
λ
,
μ
]. This yields A
x
∈
F
[
λ
,
μ
]. Now from condition (AC
2) and (21), A
y
∈
F
[
λ
,
μ
]. Therefore A
∈
(
F
[
λ
,
μ
]
,
F
[
λ
,
μ
]
).Corollary 5.
A matrixA
=
(
a
p
q
j
k
)
∈
(
F
[
λ
,
μ
]
,
F
[
λ
,
μ
]
)
reg if and only if conditions(A
C
1
) and (A
C
2
) with F
[
λ
,
μ
]
-
lim
⁡
⁡
a
=
1 and (AC
3) hold.


## 3. Some Core Theorems

The core or Knopp core of a real number single sequencex is the closed interval [
liminf
⁡
x
,
limsup
⁡
x
] (see [27, 28]). In 1999, Patterson [29] extended the Knopp core from single sequences to double sequences and called it Pringsheim core (shortly, P-core) which is given by [
P
-

lim
⁡
inf
⁡
x
,
P
-

lim
⁡
sup
⁡
x
]. In the recent past, the M-core and σ-core for double sequences have been defined and studied by Mursaleen and Edely [30] and Mursaleen and Mohiuddine [31, 32], respectively, while the σ-core for single sequences is given by Mishra et al. [33]. In 2011, Kayaduman and Çakan [34] presented the concept of Cesáro core of double sequences.We define the following sublinear functional onL
∞:
(22)
Γ
(
x
)
=
limsup
⁡
m
,
n
→
∞
sup
⁡
s
,
t
1
λ
m
μ
n
∑
j
∈
J
m
‍
∑
k
∈
I
n
‍
x
j
+
s
,
k
+
t
.
Then we define the F
[
λ
,
μ
]-core of a real-valued bounded double sequence (
x
j
k
) to be the closed interval [
-
Γ
(
-
x
)
,
Γ
(
x
)
].Since everyBP-convergent double sequence is F
[
λ
,
μ
]-convergent, we have
(23)
Γ
(
x
)
≤
L
(
x
)
,
where L
(
x
)
=
P
-lim
sup
⁡
x, and hence it follows that F
[
λ
,
μ
]
-core
{
x
}
⊆
P
-core
{
x
} for all x
∈
L
∞.Theorem 6.
For everyx
=
(
x
j
k
)
∈
F
[
λ
,
μ
],
(24)
Γ
(
A
x
)
≤
Γ
(
x
)
(
o
r
F
[
λ
,
μ
]
-core
{
A
x
}
⊂
F
[
λ
,
μ
]
-core
{
x
}
)
if and only if(CR
1)
A is F
[
λ
,
μ
]-regular,
(CR
2)
lim
⁡
sup
⁡
m
,
n
→
∞
⁡
sup
⁡
s
,
t
⁡
∑
j
=
0
∞
∑
k
=
0
∞
|
α
(
m
,
n
,
s
,
t
,
j
,
k
)
|
=
1,where
(25)
α
(
m
,
n
,
s
,
t
,
j
,
k
)
=
1
λ
m
μ
n
∑
p
∈
J
m
‍
∑
q
∈
I
n
‍
a
p
+
s
,
q
+
t
,
j
,
k
.Proof.
Necessity. Suppose that (24) holds for all x
=
(
x
j
k
)
∈
F
[
λ
,
μ
]. One obtains
(26)
-
Γ
(
-
x
)
≤
-
Γ
(
-
A
x
)
≤
Γ
(
A
x
)
≤
Γ
(
x
)
;
that is,
(27)
F
[
λ
,
μ
]
-lim
inf
⁡
x
≤
-
Γ
(
-
A
x
)
≤
Γ
(
A
x
)
≤
F
[
λ
,
μ
]
-lim
sup
⁡
x
.
If x
=
(
x
j
k
)
∈
F
[
λ
,
μ
], then
(28)
-
Γ
(
-
A
x
)
=
Γ
(
A
x
)
=
F
[
λ
,
μ
]
-lim
x
;
that is,
(29)
F
[
λ
,
μ
]
-lim
(
A
x
)
=
F
[
λ
,
μ
]
-lim
x
.
Therefore A is F
[
λ
,
μ
]-regular. This yields the necessity of (CR
1).
Now, with the help of Lemma 2.1 of [35], there is a double sequence x
=
(
x
j
k
)
∈
L
∞ such that ∥
x
∥
≤
1 and
(30)
limsup
⁡
m
,
n
→
∞
sup
⁡
s
,
t
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
α
(
m
,
n
,
s
,
t
,
j
,
k
)
x
j
k
=
limsup
⁡
m
,
n
→
∞
sup
⁡
s
,
t
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
|
α
(
m
,
n
,
s
,
t
,
j
,
k
)
|
.
If a double sequence x
=
(
x
j
k
) defined by
(31)
x
j
k
=
{
1
;
if
j
=
k
,
0
;
otherwise
,
then
(32)
1
=
Γ
′
(
A
x
)
=
liminf
⁡
m
,
n
→
∞
sup
⁡
s
,
t
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
|
α
(
m
,
n
,
s
,
t
,
j
,
k
)
|
≤
Γ
(
A
x
)
≤
Γ
(
x
)
≤
∥
x
∥
≤
1
,
where
(33)
Γ
′
(
x
)
=
liminf
⁡
m
,
n
→
∞
sup
⁡
s
,
t
1
λ
m
μ
n
∑
j
=
0
p
‍
∑
k
=
0
q
‍
x
j
+
s
,
k
+
t
.
This yields the necessity of (CR
2).
Sufficiency. We know that C
BP
⊂
F
[
λ
,
μ
]. Following the lines of Theorem 2 of [31] for translation mapping, one obtains
(34)
Γ
(
A
x
)
≤
L
(
x
)
.
For any x
′
∈
M
[
λ
,
μ
], we have
(35)
Γ
(
A
x
+
A
x
′
)
≤
L
(
x
+
x
′
)
.
Taking infimum over x
′
∈
M
[
λ
,
μ
], we obtain
(36)
inf
⁡
x
′
∈
M
[
λ
,
μ
]
Γ
(
A
x
+
A
x
′
)
≤
inf
⁡
x
′
∈
M
[
λ
,
μ
]
limsup
⁡
m
,
n
→
∞
(
x
m
n
+
x
m
n
′
)
=
w
(
x
)
,
say
.
Thus
(37)
sup
⁡
s
,
t
lim
⁡
sup
⁡
m
,
n
→
∞
Ω
m
n
s
t
(
A
x
)
+
inf
⁡
x
′
∈
M
[
λ
,
μ
]
inf
⁡
s
,
t
lim
⁡
inf
⁡
m
,
n
→
∞
Ω
m
n
s
t
(
A
x
′
)
≤
w
(
x
)
.
Since A
x
′
∈
F
[
λ
,
μ
], we can write
(38)
A
x
′
=
x
¯
+
l
E
,
where x
¯
∈
M
[
λ
,
μ
]
,
l
=
F
[
λ
,
μ
]
-lim
A
x
′
(
=
F
[
λ
,
μ
]
-lim
x
′, since A is F
[
λ
,
μ
]-regular). Operating Ω
m
n
s
t to (38), one obtains
(39)
Ω
m
n
s
t
(
A
x
′
)
=
Ω
m
n
s
t
(
x
¯
)
+
Ω
m
n
s
t
(
l
E
)
.
By [
λ
,
μ
]-almost regularity, we have
(40)
liminf
⁡
m
,
n
→
∞
Ω
m
n
s
t
(
A
x
′
)
=
lim
⁡
m
,
n
→
∞
Ω
m
n
s
t
(
x
¯
)
+
l
lim
⁡
m
,
n
→
∞
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
α
(
m
,
n
,
s
,
t
,
j
,
k
)
.
From the definition of M
[
λ
,
μ
], we get
(41)
lim
⁡
m
,
n
→
∞
Ω
m
n
s
t
(
x
¯
)
=
0
uniformly in s
,
t. Also
(42)
lim
⁡
m
,
n
→
∞
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
α
(
m
,
n
,
s
,
t
,
j
,
k
)
=
1
.
Therefore we obtain from (40) that
(43)
liminf
⁡
m
,
n
→
∞
Ω
m
n
s
t
(
A
x
′
)
=
1
uniformly in s
,
t. Equations (37) and (43) give that
(44)
Γ
(
A
x
)
+
1
≤
w
(
x
)
;
that is,
(45)
Γ
(
A
x
)
≤
w
(
x
)
.
As w
(
x
)
=
Γ
(
x
), one obtains Γ
(
A
x
)
≤
Γ
(
x
).Note thatF
[
λ
,
μ
]
-core
{
x
}
⊆
P
-core
{
x
}. This motivates us to prove the following result by adding a condition to get a more general result.Theorem 7.
Forx
=
(
x
j
k
)
∈
L
∞, if
(46)
lim
⁡
s
,
t
(
x
s
t
-
x
s
+
1
,
t
+
1
)
=
0
holds, then P
-core
{
x
}
⊆
F
[
λ
,
μ
]
-core
{
x
}.Proof.
By the definition ofP-core and F
[
λ
,
μ
]-core, we have to show that L
(
x
)
≤
Γ
(
x
). Let Γ
(
x
)
=
l. Then, for given ϵ
>
0, for all j
,
k
,
s
,
t and for large m
,
n it follows from the definition of Γ that
(47)
1
λ
m
μ
n
∑
j
∈
J
m
‍
∑
k
∈
I
n
‍
x
j
+
s
,
k
+
t
<
l
+
ϵ
2
.
We can write
(48)
x
s
t
=
x
s
t
-
1
λ
m
μ
n
∑
j
∈
J
m
‍
∑
k
∈
I
n
‍
x
j
+
s
,
k
+
t
+
1
λ
m
μ
n
∑
j
∈
J
m
‍
∑
k
∈
I
n
‍
x
j
+
s
,
k
+
t
≤
|
x
s
t
-
1
λ
m
μ
n
∑
j
∈
J
m
∑
k
∈
I
n
x
j
+
s
,
k
+
t
‍
|
+
l
+
ϵ
2
.
Since (46) holds, for given ϵ
>
0, we get that
(49)
|
x
s
t
-
x
j
+
s
,
k
+
t
|
<
ϵ
2
,
for all j
,
k
≥
0. Thus we have
(50)
|
x
s
t
-
1
λ
m
μ
n
∑
j
∈
J
m
∑
k
∈
I
n
x
j
+
s
,
k
+
t
|
=
1
λ
m
μ
n
|
λ
m
μ
n
x
s
t
-
∑
j
∈
J
m
∑
k
∈
I
n
x
j
+
s
,
k
+
t
|
≤
1
λ
m
μ
n
λ
m
μ
n
|
x
s
t
-
x
j
+
s
,
k
+
t
|
,
j
,
k
≥
0
.
Equation (49) yields
(51)
|
x
s
t
-
1
λ
m
μ
n
∑
j
∈
J
m
∑
k
∈
I
n
x
j
+
s
,
k
+
t
|
<
ϵ
2
.
Taking limsup
⁡
s
,
t in (48) and using (51), one obtains L
(
x
)
≤
l
+
ϵ. Since ϵ is arbitrary, we obtain L
(
x
)
≤
Γ
(
x
).Corollary 8.
If (46) holds and x
=
(
x
j
k
) is F
[
λ
,
μ
]-convergent, then x is convergent.Finally, we define the concepts of[
λ
,
μ
]-almost uniformly positive and [
λ
,
μ
]-almost absolutely equivalent and establish a theorem related to these concepts.Definition 9.
A matrixA
=
(
a
p
q
j
k
) is said to be [
λ
,
μ
]-almost uniformly positive, denoted by F
[
λ
,
μ
]-uniformly positive, if
(52)
lim
⁡
m
,
n
→
∞
sup
⁡
s
,
t
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
1
λ
m
μ
n
|
∑
p
∈
J
m
∑
q
∈
I
n
a
p
+
s
,
q
+
t
,
j
,
k
|
=
1
.Definition 10.
LetA
=
(
a
p
q
j
k
) and B
=
(
b
p
q
j
k
) be two F
[
λ
,
μ
]-regular matrices and
(53)
y
p
q
=
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
a
p
q
j
k
x
j
k
,
y
p
q
′
=
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
b
p
q
j
k
x
j
k
.
Then A and B are said to be [
λ
,
μ
]-almost absolutely equivalent, denoted by F
[
λ
,
μ
]-absolutely equivalent, on L
∞ whenever F
[
λ
,
μ
]
-lim
(
y
p
q
-
y
p
q
′
)
=
0; that is, either (
y
p
q
) and (
y
p
q
′
) both tend to the same F
[
λ
,
μ
]-limit or neither of them tends to a F
[
λ
,
μ
]-limit, but their difference tends to F
[
λ
,
μ
]-limit zero.Before proceeding further, first we state the following lemma which we will use to our next result.Lemma 11.
Forx
,
y
∈
L
∞, if F
[
λ
,
μ
]
-lim
|
x
-
y
|
=
0, then F
[
λ
,
μ
]
-core
{
x
}
=
F
[
λ
,
μ
]
-core
{
y
}.Proof of the lemma is straightforward and thus omitted.Theorem 12.
LetA
=
(
a
p
q
j
k
) be a F
[
λ
,
μ
]-regular matrix. Then, Γ
(
A
x
)
≤
Γ
(
x
) for all x
=
(
x
j
k
)
∈
L
∞ if and only if there is a F
[
λ
,
μ
]-regular matrix B
=
(
b
p
q
j
k
) such that B is F
[
λ
,
μ
]-uniformly positive and F
[
λ
,
μ
]-absolutely equivalent with A on L
∞.Proof.
Let there be aF
[
λ
,
μ
]-regular matrix B such that B is F
[
λ
,
μ
]-uniformly positive and F
[
λ
,
μ
]-absolutely equivalent with A on L
∞. Then, by (53) and F
[
λ
,
μ
]-absolutely equivalent of A and B, we have
(54)
F
[
λ
,
μ
]
-lim
|
y
m
n
-
y
m
n
′
|
=
lim
⁡
m
,
n
→
∞
sup
⁡
s
,
t
|
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
1
λ
m
μ
n
h
h
h
h
h
h
h
h
h
h
h
h
h
×
∑
p
∈
J
m
‍
∑
q
∈
I
n
‍
[
a
p
+
s
,
q
+
t
,
j
,
k
-
b
p
+
s
,
q
+
t
,
j
,
k
]
x
j
k
|
≤
∥
x
∥
lim
⁡
m
,
n
→
∞
sup
⁡
s
,
t
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
1
λ
m
μ
n
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
|
∑
p
∈
J
m
∑
q
∈
I
n
‍
[
a
p
+
s
,
q
+
t
,
j
,
k
-
b
p
+
s
,
q
+
t
,
j
,
k
]
|
=
0
,
uniformly in s
,
t. Now, by Lemma 11, F
[
λ
,
μ
]
-core
{
A
x
}
=
F
[
λ
,
μ
]
-core
{
B
x
} for all x
∈
L
∞. By Theorem 6, we have Γ
(
A
x
)
≤
Γ
(
x
), since x is arbitrary.
Conversely, letΓ
(
A
x
)
≤
γ
(
x
) for all x
∈
L
∞. Then by Theorem 6, A is F
[
λ
,
μ
]-uniformly positive. Now we define a matrix B
=
(
b
p
q
j
k
) as
(55)
b
p
q
j
k
=
1
2
(
a
p
q
j
k
+
a
p
,
q
,
j
+
1
,
k
+
1
)
for all p, q, j, k
∈
N. Then it is easy to see that B is F
[
λ
,
μ
]-regular since A is F
[
λ
,
μ
]-regular, and
(56)
F
[
λ
,
μ
]
-lim
(
A
x
)
=
F
[
λ
,
μ
]
-lim
(
B
x
)
.
Further
(57)
lim
⁡
m
,
n
→
∞
sup
⁡
s
,
t
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
1
λ
m
μ
n
|
∑
p
∈
J
m
∑
q
∈
I
n
b
p
+
s
,
q
+
t
,
j
,
k
|
≤
1
2
[
lim
⁡
m
,
n
→
∞
sup
⁡
s
,
t
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
1
λ
m
μ
n
|
∑
p
∈
J
m
∑
q
∈
I
n
a
p
+
s
,
q
+
t
,
j
,
k
|
hhh
+
lim
⁡
m
,
n
→
∞
sup
⁡
s
,
t
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
1
λ
m
μ
n
hhhhhhhhhhhhhh
×
|
∑
p
∈
J
m
∑
q
∈
I
n
a
p
+
s
,
q
+
t
,
j
+
1
,
k
+
1
‍
|
]
.
Since B is F
[
λ
,
μ
]-regular, we have by (57) that
(58)
lim
⁡
m
,
n
→
∞
sup
⁡
s
,
t
∑
j
=
0
∞
‍
∑
k
=
0
∞
‍
1
λ
m
μ
n
|
∑
p
∈
J
m
∑
q
∈
I
n
b
p
+
s
,
q
+
t
,
j
,
k
|
=
1
.
Thus B is F
[
λ
,
μ
]-uniformly positive. Further, it follows from (56) that A and B are F
[
λ
,
μ
]-absolutely equivalent.



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*Source: 152910-2014-07-15.xml*