# On Vector-Valued Generalized Lorentz Difference Sequence Space

Authors: Birsen Sağır; Oğuz Oğur
Journal: Journal of Operators
 (2014)

---

## Abstract

We introduce generalized Lorentz difference sequence spacesd
(
v
,
Δ
,
p
). Also we study some topologic properties of this space and obtain some inclusion relations.


---

## Body

## 1. Introduction

Throughout this work,N, R, and C denote the set of positive integers, real numbers, and complex numbers, respectively.The notion of difference sequence space was introduced by Kızmaz in [1] in 1981 as follows:
(1)
X
(
Δ
)
=
{
x
=
{
x
k
}
∈
w
:
(
Δ
x
k
)
∈
X
}
for X
=
l
∞
,
c, c
0, where Δ
x
k
=
x
k
-
x
k
+
1 for all k
∈
N. Et and Çolak in [2] defined the sequence space
(2)
X
(
Δ
m
)
=
{
x
=
{
x
k
}
∈
w
:
(
Δ
m
x
k
)
∈
X
}
for X
=
l
∞
,
c, c
0, where m
∈
N
,
Δ
0
x
k
=
{
x
k
}, Δ
x
k
=
x
k
-
x
k
+
1, Δ
m
x
k
=
Δ
m
-
1
x
k
-
Δ
m
-
1
x
k
+
1
=
∑
v
=
1
m
(
-
1
)
v
(
m
v
)
x
k
+
v for all k
∈
N, and showed that this space is a Banach space with norm
(3)
∥
x
∥
Δ
=
∑
i
=
1
m
‍
|
x
i
|
+
∥
Δ
m
x
∥
∞
.
Subsequently difference sequence spaces has been discussed in Ahmad and Mursaleen [3], Malkowsky and Parashar [4], Et and Basarir [5], and others.Let(
E
,
∥
·
∥
) be a Banach space. The Lorentz sequence space l
(
p
,
q
,
E
)
(
or
l
p
,
q
(
E
)
) for 1
≤
p, q
≤
∞ is the collection of all sequences {
a
i
}
∈
c
0
(
E
) such that
(4)
∥
{
a
i
}
∥
p
,
q
=
{
(
∑
i
=
1
∞
‍
i
q
/
p
-
1
∥
a
ϕ
(
i
)
∥
q
)
1
/
q
for
1
≤
p
≤
∞
,
1
≤
q
<
∞
sup
⁡
i
⁡
i
1
/
p
∥
a
ϕ
(
i
)
∥
for
1
≤
p
<
∞
,
q
=
∞
is finite, where {
∥
a
ϕ
(
i
)
∥
} is nonincreasing rearrangement of {
∥
a
i
∥
} (we can interpret that the decreasing rearrangement {
∥
a
ϕ
(
i
)
∥
} is obtained by rearranging {
∥
a
i
∥
} in decreasing order). This space was introduced by Miyazaki in [6] and examined comprehensively by Kato in [7].A weight sequencev
=
{
v
(
i
)
} is a positive decreasing sequence such that v
(
1
)
=
1, lim
⁡
i
→
∞
v
(
i
)
=
0 and lim
⁡
i
→
∞
V
(
i
)
=
∞, where V
(
i
)
=
∑
n
=
1
i
v
(
n
) for every i
∈
N. Popa [8] defined the generalized Lorentz sequence space d
(
v
,
p
) for 0
<
p
<
∞ as follows:
(5)
d
(
v
,
p
)
=
{
(
∑
i
=
1
∞
‍
|
x
π
(
i
)
|
p
v
(
i
)
)
1
/
p
x
=
{
x
i
}
∈
w
:
∥
x
∥
v
,
p
00000
=
sup
⁡
π
(
∑
i
=
1
∞
‍
|
x
π
(
i
)
|
p
v
(
i
)
)
1
/
p
<
∞
}
,
where π ranges over all permutations of the positive integers and v
=
{
v
(
i
)
} is a weight sequence. It is known that d
(
v
,
p
)
⊂
c
0 and hence for each x
∈
d
(
v
,
p
) there exists a nonincreasing rearrangement {
x
*
}
=
{
x
i
*
} of x and
(6)
∥
x
∥
v
,
p
=
(
∑
n
=
1
∞
‍
|
x
i
*
|
p
v
(
i
)
)
1
/
p
(see [8, 9]).Let(
X
,
∥
·
∥
) be a Banach space and let v
=
{
v
(
n
)
} be a weight sequence. We introduce the vector-valued generalized Lorentz difference sequence space d
(
v
,
Δ
,
p
) for 0
<
p
<
∞. The space d
(
v
,
Δ
,
p
) is the collection of all X-valued 0-sequences {
x
n
}  (
{
x
n
}
∈
c
0
{
X
}
) such that
(7)
(
∑
n
=
1
∞
‍
[
∥
Δ
x
ϕ
(
n
)
∥
]
p
v
(
n
)
)
1
/
p
is finite, where {
∥
Δ
x
ϕ
(
n
)
∥
} is nonincreasing rearrangement of {
∥
Δ
x
n
∥
} and Δ
x
ϕ
(
n
)
=
x
ϕ
(
k
)
-
x
ϕ
(
k
+
1
) for all k
∈
N.We will need the following lemmas.Lemma 1 (see [10]).
Let{
c
i
*
} and {
c
*
i
} be the nonincreasing and nondecreasing rearrangements of a finite sequence {
c
i
}
1
≤
i
≤
n of positive numbers, respectively. Then for two sequences {
a
i
}
1
≤
i
≤
n and {
b
i
}
1
≤
i
≤
n of positive numbers we have
(8)
∑
i
‍
a
i
*
·
*
b
i
≤
∑
i
‍
a
i
·
b
i
≤
∑
i
‍
a
i
*
·
b
i
*
.Lemma 2 (see [7]).
Let{
x
i
(
μ
)
} be an X-valued double sequence such that lim
⁡
i
→
∞
x
i
(
μ
)
=
0 for each μ
∈
N and let {
x
i
} be an X-valued sequence such that lim
⁡
μ
→
∞
x
i
(
μ
)
=
x
i (uniformly in i). Then lim
⁡
i
→
∞
x
i
=
0 and for each i
∈
N
(9)
∥
x
ϕ
(
i
)
∥
≤
lim
⁡
μ
→
∞
¯
∥
x
ϕ
μ
(
i
)
(
μ
)
∥
,
where {
∥
x
ϕ
(
i
)
∥
} and {
∥
x
ϕ
μ
(
i
)
(
μ
)
∥
}
i are the nonincreasing rearrangements of {
∥
x
i
∥
} and {
∥
x
i
(
μ
)
∥
}
i, respectively.


## 2. Main Results

Theorem 3.
The spaced
(
v
,
Δ
,
p
) for 0
<
p
<
∞ is a linear space over the field K
=
R or C.Proof.
Letx
,
y
∈
d
(
v
,
Δ
,
p
) and let {
∥
Δ
x
ϕ
(
n
)
∥
}
,
{
∥
Δ
y
η
(
n
)
∥
} and {
∥
Δ
x
ψ
(
n
)
+
Δ
y
ψ
(
n
)
∥
} be the nonincreasing rearrangements of the sequences {
∥
Δ
x
n
∥
}
,
{
∥
Δ
y
n
∥
} and {
∥
Δ
x
n
+
Δ
y
n
∥
}, respectively. Since v is nonincreasing, by Lemma 1 we have
(10)
∑
n
=
1
∞
‍
∥
Δ
x
ψ
(
n
)
+
Δ
y
ψ
(
n
)
∥
p
v
(
n
)
≤
D
∑
n
=
1
∞
‍
(
∥
Δ
x
ψ
(
n
)
∥
p
v
(
n
)
+
∥
Δ
y
ψ
(
n
)
∥
p
v
(
n
)
)
≤
D
{
∑
n
=
1
∞
‍
∥
Δ
x
ϕ
(
n
)
∥
p
v
(
n
)
+
∑
n
=
1
∞
‍
∥
Δ
y
η
(
n
)
∥
p
v
(
n
)
}
<
∞
,
where D
=
max
⁡
{
1
,
2
p
-
1
}. Let α
∈
K. Hence we get
(11)
∑
n
=
1
∞
‍
∥
Δ
(
α
x
)
ϕ
(
n
)
∥
p
v
(
n
)
=
∑
n
=
1
∞
‍
∥
α
Δ
x
ϕ
(
n
)
∥
p
v
(
n
)
=
|
α
|
p
∑
n
=
1
∞
‍
∥
Δ
x
ϕ
(
n
)
∥
p
v
(
n
)
<
∞
.
This shows thatx
+
y
∈
d
(
v
,
Δ
,
p
)
,
α
x
∈
d
(
v
,
Δ
,
p
) and so d
(
v
,
Δ
,
p
) is a linear space.Theorem 4.
The spaced
(
v
,
Δ
,
p
) for 1
≤
p
<
∞ is normed space with the norm
(12)
∥
x
∥
v
,
Δ
,
p
=
∥
x
ϕ
(
1
)
∥
+
(
∑
n
=
1
∞
‍
∥
Δ
x
ϕ
(
n
)
∥
p
v
(
n
)
)
1
/
p
,
where {
∥
Δ
x
ϕ
(
n
)
∥
} denotes the nonincreasing rearrangements of {
∥
Δ
x
n
∥
}.Proof.
It is clear that∥
0
∥
v
,
Δ
,
p
=
0. Let ∥
x
∥
v
,
Δ
,
p
=
0. Then we have x
ϕ
(
1
)
=
0 and Δ
x
ϕ
(
k
)
=
x
ϕ
(
k
)
-
x
ϕ
(
k
+
1
)
=
0 for all k
∈
N. Hence we get x
=
0.
Letx
,
y
∈
d
(
v
,
Δ
,
p
). Since weight sequence v is decreasing, by Lemma 1 we have
(13)
∥
x
+
y
∥
v
,
Δ
,
p
=
∥
x
ψ
(
1
)
+
y
ψ
(
1
)
∥
+
(
∑
n
=
1
∞
‍
∥
Δ
x
ψ
(
n
)
+
Δ
y
ψ
(
n
)
∥
p
v
(
n
)
)
1
/
p
≤
∥
x
ψ
(
1
)
∥
+
∥
y
ψ
(
1
)
∥
+
(
∑
n
=
1
∞
‍
∥
Δ
x
ψ
(
n
)
∥
p
v
(
n
)
)
1
/
p
+
(
∑
n
=
1
∞
‍
∥
Δ
y
ψ
(
n
)
∥
p
v
(
n
)
)
1
/
p
≤
∥
x
ϕ
(
1
)
∥
+
∥
y
η
(
1
)
∥
+
(
∑
n
=
1
∞
‍
∥
Δ
x
ϕ
(
n
)
∥
p
v
(
n
)
)
1
/
p
+
(
∑
n
=
1
∞
‍
∥
Δ
y
η
(
n
)
∥
p
v
(
n
)
)
1
/
p
=
∥
x
∥
v
,
Δ
,
p
+
∥
y
∥
v
,
Δ
,
p
,
where {
∥
Δ
x
ϕ
(
n
)
∥
}
,
{
∥
Δ
y
η
(
n
)
∥
} and {
∥
Δ
x
ψ
(
n
)
+
Δ
y
ψ
(
n
)
∥
} denote the nonincreasing rearrangements of {
∥
Δ
x
n
∥
}
,
{
∥
Δ
y
n
∥
} and {
∥
Δ
x
n
+
Δ
y
n
∥
}, respectively.
Letλ be an element in K and let x be a vector in d
(
v
,
Δ
,
p
). Hence we have
(14)
∥
λ
x
∥
v
,
Δ
,
p
=
∥
(
λ
x
)
ϕ
(
1
)
∥
+
(
∑
n
=
1
∞
‍
∥
Δ
(
λ
x
)
ϕ
(
n
)
∥
p
v
(
n
)
)
1
/
p
=
|
λ
|
∥
x
ϕ
(
1
)
∥
+
|
λ
|
(
∑
n
=
1
∞
‍
∥
Δ
x
ϕ
(
n
)
∥
p
v
(
n
)
)
1
/
p
=
|
λ
|
∥
x
∥
v
,
Δ
,
p
.Theorem 5.
The spaced
(
v
,
Δ
,
p
) for 1
≤
p
<
∞ is complete with respect to its norm.Proof.
Let{
x
(
s
)
} be an arbitrary Cauchy sequence in d
(
v
,
Δ
,
p
) with x
(
s
)
=
{
x
n
(
s
)
}
n
=
1
∞ for all s
∈
N. Then we have
(15)
lim
⁡
s
,
t
→
∞
∥
x
(
s
)
-
x
(
t
)
∥
v
,
Δ
,
p
=
0
.
Hence we obtain lim
⁡
s
,
t
→
∞
∥
Δ
x
π
s
,
t
(
n
)
(
s
)
-
Δ
x
π
s
,
t
(
n
)
(
t
)
∥
=
0 for each n
∈
N and so {
x
n
(
s
)
}, for a fixed n
∈
N, is a Cauchy sequence in X.
Then, there existsx
n
∈
X such that x
n
(
s
)
→
x
n as s
→
∞. Let x
=
{
x
n
}. Since lim
⁡
n
→
∞
x
n
(
s
)
=
0 for each   s
∈
N, by Lemma 2 we have lim
⁡
n
→
∞
x
n
=
0. Therefore we can choose the nonincreasing rearrangement {
∥
Δ
x
π
t
(
n
)
-
Δ
x
π
t
(
n
)
(
t
)
∥
}
n of {
∥
Δ
x
n
-
Δ
x
n
(
t
)
∥
}
n. Also, for an arbitrary ε
>
0 there exists N
∈
N such that
(16)
∥
x
π
s
,
t
(
1
)
(
s
)
-
x
π
s
,
t
(
1
)
(
t
)
∥
+
(
∑
n
=
1
∞
‍
∥
Δ
x
π
s
,
t
(
n
)
(
s
)
-
Δ
x
π
s
,
t
(
n
)
(
t
)
∥
p
v
(
n
)
)
1
/
p
<
ε
for s
,
t
>
N. Hence we get
(17)
∥
x
π
s
,
t
(
1
)
(
s
)
-
x
π
s
,
t
(
1
)
(
t
)
∥
<
ε
,
∑
n
=
1
∞
‍
∥
Δ
x
π
s
,
t
(
n
)
(
s
)
-
Δ
x
π
s
,
t
(
n
)
(
t
)
∥
p
v
(
n
)
<
ε
p
for s
,
t
>
N. Let t be an arbitrary positive integer with t
>
N and fixed. If we put
(18)
Δ
y
n
(
s
)
=
Δ
x
n
(
s
)
-
Δ
x
n
(
t
)
,
Δ
y
n
=
Δ
x
n
-
Δ
x
n
(
t
)
,
then we have
(19)
lim
⁡
n
→
∞
Δ
y
n
(
s
)
=
0
for
each
s
∈
N
,
lim
⁡
s
→
∞
Δ
y
n
(
s
)
=
Δ
y
n
(
uniformly
in
n
)
.
Thus by Lemma 2 we get
(20)
∥
Δ
y
ϕ
(
n
)
∥
≤
lim
⁡
s
→
∞
¯
∥
Δ
y
ϕ
s
(
n
)
(
s
)
∥
for each n
∈
N; that is,
(21)
∥
Δ
x
π
t
(
n
)
-
Δ
x
π
t
(
n
)
(
t
)
∥
≤
lim
⁡
s
→
∞
¯
∥
Δ
x
π
s
,
t
(
n
)
(
s
)
-
Δ
x
π
s
,
t
(
n
)
(
t
)
∥
for each n
∈
N. Hence, by (17), (21) we get
(22)
∥
x
-
x
(
t
)
∥
v
,
Δ
,
p
=
∥
x
π
t
(
1
)
-
x
π
t
(
1
)
(
t
)
∥
+
(
∑
n
=
1
∞
‍
[
∥
Δ
x
π
t
(
n
)
-
Δ
x
π
t
(
n
)
(
t
)
∥
]
p
v
(
n
)
)
1
/
p
≤
∥
x
π
t
(
1
)
-
x
π
t
(
1
)
(
t
)
∥
+
(
∑
n
=
1
∞
‍
(
lim
⁡
s
→
∞
¯
∥
Δ
x
π
s
,
t
(
n
)
(
s
)
-
Δ
x
π
s
,
t
(
n
)
(
t
)
∥
)
p
v
(
n
)
)
1
/
p
=
∥
x
π
t
(
1
)
-
x
π
t
(
1
)
(
t
)
∥
+
lim
⁡
s
→
∞
¯
(
∑
n
=
1
∞
‍
(
∥
Δ
x
π
s
,
t
(
n
)
(
s
)
-
Δ
x
π
s
,
t
(
n
)
(
t
)
∥
)
p
v
(
n
)
)
1
/
p
<
2
ε
.
Also, since d
(
v
,
Δ
,
p
) is a linear space we have {
x
n
}
=
{
x
n
-
x
n
(
N
)
}
+
{
x
n
(
N
)
}
∈
d
(
v
,
Δ
,
p
). Hence the space d
(
v
,
Δ
,
p
) is complete with respect to its norm.Theorem 6.
Let0
<
p
<
∞. Then, the inclusion d
(
v
,
p
)
⊂
d
(
v
,
Δ
,
p
) holds.Proof.
Letx
∈
d
(
v
,
p
). Then we have
(23)
(
∑
n
=
1
∞
‍
∥
x
ϕ
(
n
)
∥
p
v
(
n
)
)
1
/
p
<
∞
,
where {
∥
x
ϕ
(
n
)
∥
} denotes the nonincreasing rearrangements of {
∥
x
n
∥
}. Since v
(
n
) is decreasing, by Lemma 1 we get
(24)
(
∑
n
=
1
∞
‍
∥
Δ
x
ψ
(
n
)
∥
p
v
(
n
)
)
=
∑
n
=
1
∞
‍
∥
x
ψ
(
n
)
-
x
ψ
(
n
+
1
)
∥
p
v
(
n
)
≤
K
∑
n
=
1
∞
‍
(
∥
x
ψ
(
n
)
∥
p
+
∥
x
ψ
(
n
+
1
)
∥
p
)
v
(
n
)
≤
K
(
∑
n
=
1
∞
‍
∥
x
ϕ
(
n
)
∥
p
v
(
n
)
+
∑
n
=
1
∞
‍
∥
x
ϕ
(
n
+
1
)
∥
p
v
(
n
)
)
<
∞
,
where {
∥
Δ
x
ψ
(
n
)
∥
} denotes the nonincreasing rearrangements of {
∥
Δ
x
n
∥
} and K
=
max
⁡
{
1
,
2
p
-
1
}. This completes the proof.Theorem 7.
If1
≤
p
<
q
<
∞, then d
(
v
,
Δ
,
p
)
⊂
d
(
v
,
Δ
,
q
).Proof.
Letx
∈
d
(
v
,
Δ
,
p
). Since v
(
n
) is decreasing we have
(25)
(
∑
n
=
1
∞
‍
∥
Δ
x
ϕ
(
n
)
∥
p
v
(
n
)
)
1
/
p
≥
(
∑
n
=
1
k
‍
∥
Δ
x
ϕ
(
n
)
∥
p
v
(
n
)
)
1
/
p
≥
∥
Δ
x
ϕ
(
k
)
∥
(
∑
n
=
1
k
‍
v
(
n
)
)
1
/
p
≥
∥
Δ
x
ϕ
(
k
)
∥
(
v
(
k
)
)
1
/
p
k
1
/
p
for every k
∈
N. Hence we get
(26)
∥
Δ
x
ϕ
(
k
)
∥
≤
(
v
(
k
)
)
-
1
/
p
k
-
1
/
p
(
∥
Δ
x
ϕ
(
1
)
∥
+
∑
n
=
1
∞
‍
∥
Δ
x
ϕ
(
n
)
∥
p
v
(
n
)
)
1
/
p
≤
(
v
(
k
)
)
-
1
/
p
∥
x
∥
v
,
Δ
,
p
for every k
∈
N. Thus
(27)
∑
n
=
1
∞
‍
∥
Δ
x
ϕ
(
n
)
∥
q
v
(
n
)
=
∑
n
=
1
∞
‍
∥
Δ
x
ϕ
(
n
)
∥
q
-
p
∥
Δ
x
ϕ
(
n
)
∥
p
v
(
n
)
≤
∑
n
=
1
∞
‍
(
(
v
(
n
)
)
-
1
/
p
∥
x
∥
v
,
Δ
,
p
)
q
-
p
∥
Δ
x
ϕ
(
n
)
∥
p
v
(
n
)
≤
(
∥
x
∥
v
,
Δ
,
p
)
q
-
p
∑
n
=
1
∞
‍
∥
Δ
x
ϕ
(
n
)
∥
p
v
(
n
)
<
∞
.
This implies that x
∈
d
(
v
,
Δ
,
q
).


## 3. Conclusion

If we putΔ
m
x instead of Δ
x, where m
∈
N and Δ
0
x
k
=
{
x
k
}, Δ
x
k
=
x
k
-
x
k
+
1, Δ
m
x
k
=
Δ
m
-
1
x
k
-
Δ
m
-
1
x
k
+
1
=
∑
v
=
1
m
(
-
1
)
v
(
m
v
)
x
k
+
v for all k
∈
N in the definition of d
(
v
,
Δ
,
p
), we obtain generalized Lorentz difference sequence space d
(
v
,
Δ
m
,
p
) of order m. It can be shown that the sequence space d
(
v
,
Δ
m
,
p
) is a Banach space with norm
(28)
∥
x
∥
v
,
Δ
m
,
p
=
∑
n
=
1
m
‍
∥
x
ϕ
(
n
)
∥
+
(
∑
n
=
1
∞
‍
∥
Δ
m
x
ϕ
(
n
)
∥
p
v
(
n
)
)
1
/
p
,
where {
∥
Δ
m
x
ϕ
(
n
)
∥
} denotes the nonincreasing rearrangements of {
∥
Δ
m
x
n
∥
} and properties in this work.



---

*Source: 545346-2014-09-21.xml*