Journal of Inequalities in Pure and
Applied Mathematics
THE DUAL SPACES OF THE SETS OF DIFFERENCE
SEQUENCES OF ORDER m
volume 7, issue 3, article 101,
2006.
Ç.A. BEKTAŞ AND M. ET
Department of Mathematics
Firat University
Elazig, 23119, TURKEY.
Received 16 December, 2005;
accepted 07 January, 2006.
Communicated by: A.G. Babenko
EMail: cbektas@firat.edu.tr
EMail: mikailet@yahoo.com
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2000
Victoria University
ISSN (electronic): 1443-5756
247-04
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Abstract
The idea of difference sequence spaces was introduced by Kızmaz [5] and
the concept was generalized by Et and Çolak [3]. Let p = (pk ) be a bounded
sequence of positive real numbers and v = (vk ) be any fixed sequence of nonzero complex numbers. If x = (xk ) is any sequence of complex numbers we
m
write ∆m
v x for the sequence of the m -th order differences of x and ∆v (X) =
m
{x = (xk ) : ∆v x ∈ X} for any set X of sequences. In this paper we determine
the α -, β - and γ - duals of the sets ∆m
v (X) which are defined by Et et al. [2]
for X = `∞ (p), c(p) and c0 (p). This study generalizes results of Malkowsky [9]
in special cases.
The Dual Spaces of the Sets of
Difference Sequences of Order
m
Ç.A. Bektaş and M. Et
2000 Mathematics Subject Classification: 40C05, 46A45.
Key words: Difference sequences, α−, β− and γ−duals.
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Contents
1
Introduction, Notations and Known Results . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
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J. Ineq. Pure and Appl. Math. 7(3) Art. 101, 2006
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1.
Introduction, Notations and Known Results
Throughout this paper ω denotes the space of all scalar sequences and any subspace of ω is called a sequence space. Let `∞ , c and c0 be the linear space
of bounded, convergent and null sequences with complex terms, respectively,
normed by
kxk∞ = sup |xk | ,
k
where k ∈ N = {1, 2, 3, . . . }, the set of positive integers. Furthermore, let
p = (pk ) be bounded sequences of positive real numbers and
pk
`∞ (p) = x ∈ ω : sup |xk | < ∞ ,
k
n
o
c(p) = x ∈ ω : lim |xk − l|pk = 0 , for some l ∈ C ,
k→∞
n
o
pk
c0 (p) = x ∈ ω : lim |xk | = 0
k→∞
(for details see [6], [7], [11]).
Let x and y be complex sequences, and E and F be subsets of ω. We write
\
M (E, F ) =
x−1 ∗ F = {a ∈ ω : ax ∈ F for all x ∈ E }
[12].
x∈E
Ç.A. Bektaş and M. Et
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In particular, the sets
E α = M (E, l1 ),
The Dual Spaces of the Sets of
Difference Sequences of Order
m
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E β = M (E, cs) and E γ = M (E, bs)
are called the α−, β− and γ− duals of E, where l1 , cs and bs are the sets of all
convergent, absolutely convergent and bounded series, respectively. If E ⊂ F ,
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then F η ⊂ E η for η = α, β, γ. It is clear that E α ⊂ (E α )α = E αα . If E = E αα ,
then E is an α -space. In particular, an α -space is called a Köthe space or a
perfect sequence space.
Throughout this paper X will be used to denote any one of the sequence
spaces `∞ , c and c0 .
Kızmaz [5] introduced the notion of difference sequence spaces as follows:
X(∆) = {x = (xk ) : (∆xk ) ∈ X}.
Later on the notion was generalized by Et and Çolak in [3], namely,
The Dual Spaces of the Sets of
Difference Sequences of Order
m
X(∆m ) = {x = (xk ) : (∆m xk ) ∈ X}.
Subsequently difference sequence spaces have been studied by Malkowsky and
Parashar [8], Mursaleen [10], Çolak [1] and many others.
Let v = (vk ) be any fixed sequence of non-zero complex numbers. Et and
Esi [4] generalized the above sequence spaces to the following ones
∆m
v (X)
= {x = (xk ) :
(∆m
v xk )
m−1
where ∆0v x = (vk xk ), ∆m
xk −
v x = (∆v
P
m
i m
(−1)
v
x
.
Recently
Et
et
al.
[2]
k+i k+i
i=0
i
m
∆v (X) to the sequence spaces
∈ X},
∆m−1
xk+1 )
v
∆m
v xk
such that
=
generalized the sequence spaces
m
∆m
v (X(p)) = {x = (xk ) :(∆v xk )∈ X(p)}
and showed that these spaces are complete paranormed spaces paranormed by
g(x) =
m
X
i=1
Ç.A. Bektaş and M. Et
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|xi vi | + sup
k
pk /M
|∆m
v xk |
,
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where H = supk pk and M = max (1, H).
m
Let us define the operator D : ∆m
v X(p) → ∆v X(p) by
Dx = (0, 0, . . . , xm+1 , xm+2 , . . . ),
where x = (x1 , x2 , x3 , . . . ). It is trivial that D is a bounded linear operator on
∆m
v X(p). Furthermore the set
m
D[∆m
v X(p)] = D∆v X(p)
= {x = (xk ) : x ∈ ∆m
v X(p), x1 = x2 = · · · = xm = 0}
m
is a subspace of ∆m
v X(p). D∆v X(p) and X(p) are equivalent as topological
spaces, since
(1.1)
The Dual Spaces of the Sets of
Difference Sequences of Order
m
Ç.A. Bektaş and M. Et
m
∆m
v : D∆v X(p) → X(p)
m
0
defined by ∆m
v x = y = (∆v xk ) is a linear homeomorphism. Let [X(p)] and
m
0
m
[D∆v X(p)] denote the continuous duals of X(p) and D∆v X(p), respectively.
It can be shown that
0
0
T : [D∆m
v X(p)] → [X(p)] ,
−1
f∆ → f∆ ◦ (∆m
= f,
v )
0
0
is a linear isometry. So [D∆m
v X(p)] is equivalent to [X(p)] .
Lemma 1.1 ([5]). Let (tn ) be a sequence of positive numbers increasing monotonically to infinity, then
P
P
< ∞,
i) If supn | ni=1 ti ai | < ∞, then supn tn ∞
a
k
k=n+1
P
P
ii) If k tk ak is convergent, then limn→∞ tn ∞
k=n+1 ak = 0.
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2.
Main Results
In this section we determine the α−, β− and γ− duals of ∆m
v X(p).
Theorem 2.1. For every strictly positive sequence p = (pk ), we have
α
α
(i) [∆m
v l∞ (p)] = D1 (p),
αα
(ii) [∆m
= D1αα (p)
v l∞ (p)]
where
D1α (p)
=
∞
\
(
a = (ak ) :
N =2
∞
X
k=1
−1
|ak | |vk |
k−m
X
j=1
)
k−j−1
1/pj
N
<∞
m−1
The Dual Spaces of the Sets of
Difference Sequences of Order
m
Ç.A. Bektaş and M. Et
and
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

"k−m #−1
∞


[
X k−j−1
D1αα (p) =
a = (ak ) : sup |ak | |vk |
N 1/pj
<∞ .


m−1
k≥m+1
Contents
j=1
N =2
Proof.
(i) Let a ∈
Since
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D1α (p)
and x ∈
∆m
v l∞ (p). We choose N
>
pn
max(1, supn |∆m
v an | ).
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k−m
X
j=1
m X
k−j−1
k−j−1
1/pj
N
>
N 1/pj ≥ 1
m−1
m−j
j=1
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for arbitrary N > 1 (k = 2m, 2m + 1, . . . ) and |∆m−j
xj | ≤ M (1 ≤ j ≤
v
α
m) for some constant M , a ∈ D1 (p) implies
∞
X
−1
|ak | |vk |
k=1
m X
k − j − 1 m−j ∆v xj < ∞.
m−j
j=1
Then
k−m
X
−1
m k−j −1
|ak xk | =
|ak | |vk |
(−1)
∆m
v xj
m−1
j=1
k=1
k=1
!
m
X
k
−
j
−
1
+
(−1)m−j
∆m−j
x
j
v
m−j
j=1
∞
k−m
X
X k − j − 1
−1
N 1/pj
≤
|ak | |vk |
m−1
j=1
k=1
∞
m X
X
k − j − 1 m−j −1
∆v xj
+
|ak | |vk |
m−j
j=1
k=1
∞
X
∞
X
Conversely let a ∈
/ D1α (p). Then we have
k=1
|ak ||vk |−1
Ç.A. Bektaş and M. Et
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< ∞.
∞
X
The Dual Spaces of the Sets of
Difference Sequences of Order
m
k−m
X
j=1
k−j−1
N 1/pj = ∞
m−1
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for some integer N > 1. We define the sequence x by
xk = vk
−1
k−m
X
j=1
k−j−1
N 1/pj
m−1
(k = m + 1, m + 2, . . . ).
Then it is easy to see that x ∈ ∆m
v l∞ (p) and
P
α
a∈
/ [∆m
v l∞ (p)] . This completes the proof of (i).
|ak xk | = ∞. Hence
k
α
α
(ii) Let a ∈ D1αα (p) and x ∈ [∆m
v l∞ (p)] = D1 (p), by part (i). Then for
some N > 1, we have
"k−m #−1
∞
∞
X
X
X k − j − 1
|ak xk | =
|ak | |vk |
N 1/pj
m
−
1
j=1
k=m+1
k=m+1
"k−m #
X k − j − 1
× |xk | |vk |−1
N 1/pj
m
−
1
j=1

"k−m #−1 
X k − j − 1

≤ sup |ak | |vk |
N 1/pj
m
−
1
k≥m+1
j=1
×
∞
X
k=m+1
< ∞.
|xk | |vk |−1
k−m
X
j=1
k−j−1
N 1/pj
m−1
The Dual Spaces of the Sets of
Difference Sequences of Order
m
Ç.A. Bektaş and M. Et
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Conversely let a ∈
/ D1αα (p). Then for all integers N > 1, we have
"k−m #−1
X k − j − 1
1/pj
N
= ∞.
sup |ak | |vk |
m−1
k≥m+1
j=1
We recall that
k−m
X
j=1
k−j−1
yj = 0
m−1
(k < m + 1)
for arbitrary yj .
Hence there is a strictly increasing sequence (k(s)) of integers k(s) ≥
m + 1 such that
−1

k(s)−m X k(s) − j − 1
s1/pj  > sm+1
|ak(s) | |v k(s) | 
m
−
1
j=1
(s = m + 1, m + 2, . . . ).
We define the sequence x by

 |ak(s) |−1 , (k = k(s))
xk =

0,
(k 6= k(s)) (k = m + 1, m + 2, . . . )
Then for all integers N > m + 1, we have
∞
k−m
∞
X
X k − j − 1
X
−1
|xk | |vk |
s−(m+1) < ∞.
N 1/pj <
m
−
1
s=m+1
j=1
k=1
The Dual Spaces of the Sets of
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m
Ç.A. Bektaş and M. Et
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P∞
P
α
/
Hence x ∈ [∆m
and ∞
v l∞ (p)]
N =1 1 = ∞. Hence a ∈
k=1 |ak xk | =
αα
m
[∆v l∞ (p)] . The proof is completed.
Theorem 2.2. For every strictly positive sequence p = (pk ), we have
α
α
(i) [∆m
v c0 (p)] = M0 (p),
αα
(ii) [∆m
= M0αα (p)
v c0 (p)]
The Dual Spaces of the Sets of
Difference Sequences of Order
m
where
M0α (p) =
∞
[
(
a∈ω:
N =2
∞
X
|ak | |vk |−1
k=1
k−m
X
j=1
)
k−j−1
N −1/pj < ∞
m−1
Ç.A. Bektaş and M. Et
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M0αα (p) =
∞
\
N =2


"k−m #−1


X k − j − 1
a ∈ ω : sup |ak | |vk |
N −1/pj
<∞ .


m−1
k≥m+1
j=1
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Proof.
(i) Let a ∈ M0α (p) and x ∈ ∆m
v c0 (p). Then there is an integer k0 such that
pk
−1
sup |∆m
x
|
≤
N
,
where
N is the number in M0α (p). We put
v k
k>k0
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M = max
1≤k≤k0
pk
|∆m
v xk | ,
n = min pk ,
1≤k≤k0
L = (M + 1)N
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and define the sequence y by yk = xk · L−1/n (k = 1, 2, . . . ). Then it is
pk
easy to see that supk |∆m
≤ N −1 .
v yk |
Since
k−m
X
j=1
m X
k−j−1
k−j−1
−1/pj
N
>
N −1/pj
m−1
m
−
j
j=1
for arbitrary N > 1 (k = 2m, 2m + 1, . . . ), a ∈ M0α (p) implies
∞
X
|ak | |vk |
m X
j=1
k=1
k−j−1
|∆m−j
yj | < ∞.
v
m−j
The Dual Spaces of the Sets of
Difference Sequences of Order
m
Ç.A. Bektaş and M. Et
Then
∞
X
|ak xk | = L1/n
k=1
∞
X
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|ak yk |
k=1
∞
X
k−m
X
k−j−1
−1
1/n
N −1/pj
≤L
|ak | |vk |
m
−
1
j=1
k=1
∞
m X
X
k−j−1
1/n
−1
+L
|ak | |vk |
|∆m−j
yj |
v
m
−
j
j=1
k=1
< ∞.
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So we have a ∈
α
[∆m
v c0 (p)] .
Therefore
M0α (p)
⊂
α
[∆m
v c0 (p)] .
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Conversely, let a ∈
/ M0α (p). Then we can determine a strictly increasing
sequence (k(s)) of integers such that k(1) = 1 and
k(s+1)−1
M (s) =
X
−1
|ak | |vk |
k−m
X
j=1
k=k(s)
k−j−1
(s+1)−1/pj > 1
m−1
(s = 1, 2, . . . ).
We define the sequence x by

xk = vk −1 
s−1 k(l+1)−1
X
X k − j − 1
l=1
m−1
j=k(l)
+
k−m
X
j=k(s)
(l + 1)−1/pj
Ç.A. Bektaş and M. Et

k−j−1
(s + 1)−1/pj 
m−1
(k(s) ≤ k ≤ k(s + 1) − 1; s = 1, 2, . . . ).
1
s+1
hence x ∈ ∆m
v c0 (p), and
(ii) Omitted.
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Then it is easy to see that
pk
|∆m
=
v xk |
The Dual Spaces of the Sets of
Difference Sequences of Order
m
(k(s) ≤ k ≤ k(s + 1) − 1; s = 1, 2, . . . )
P∞
k=1
|ak xk | ≥
P∞
s=1
α
= ∞, i.e. a ∈
/ [∆m
v c0 (p)] .
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Theorem 2.3. For every strictly positive sequence p = (pk ), we have
α
α
[∆m
v c(p)] = M (p)
(
= M0α (p) ∩
a∈ω:
∞
X
|ak | |vk |−1
k−m
X
k=1
j=1
)
k−j−1
<∞ .
m−1
Proof. Let a ∈ M α (p) and x ∈ ∆m
v c(p). Then there is a complex number l
pk
such that |∆m
x
−
l|
→
0
(k
→
∞).
We define y = (yk ) by
v k
yk = xk +
vk−1 l(−1)m+1
k−m
X
j=1
k−j−1
m−1
(k = 1, 2, . . . ).
Then y ∈ ∆m
v c0 (p) and
∞
X
∞
X
The Dual Spaces of the Sets of
Difference Sequences of Order
m
Ç.A. Bektaş and M. Et
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k−m
X
k−j−1
|∆m
|ak xk | ≤
|ak | |vk |
v yj |
m−1
j=1
k=1
k=1
∞
m X
X
k−j−1
−1
+
|∆m−j
yj |
|ak | |vk |
v
m−j
j=1
k=1
k−m
∞
X k − j − 1
X
−1
+ |l|
|ak | |vk |
<∞
m
−
1
j=1
k=1
−1
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by Theorem 2.2(i) and since a ∈ M α (p).
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α
m
α
α
Now let a ∈ [∆m
v c(p)] ⊂ [∆v c0 (p)] = M0 (p) by Theorem 2.2(i). Since
the sequence x defined by
xk =
(−1)m vk−1
k−m
X
j=1
k−j−1
m−1
(k = 1, 2, . . . )
is in ∆m
v c(p), we have
∞
X
|ak |
k−m
X
j=1
k=1
k−j−1
< ∞.
m−1
The Dual Spaces of the Sets of
Difference Sequences of Order
m
Ç.A. Bektaş and M. Et
Theorem 2.4. For every strictly positive sequence p = (pk ), we have
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β
β
(i) [D∆m
v `∞ (p)] = M∞ (p),
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γ
γ
(ii) [D∆m
v `∞ (p)] = M∞ (p)
where
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(
β
M∞
(p) =
\
N >1
∞
X
k−m
X
k−j−1
−1
ak v k
a∈ω:
N 1/pj converges and
m
−
1
j=1
k=1
)
∞
k−m+1
X
X k − j − 1
|bk |
N 1/pj < ∞ ,
m
−
2
j=1
k=1
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(
γ
M∞
(p)
=
\
N >1
and bk =
P∞
j=k+1
n
X
k−m
X
k−j−1
a ∈ ω : sup |
N 1/pj | < ∞,
m
−
1
n
j=1
k=1
)
k−m+1
∞
X k − j − 1
X
|bk |
N 1/pj < ∞
m−2
j=1
k=1
ak vk−1
vj−1 aj (k = 1, 2, . . . ).
Proof.
(i) If x ∈
D∆m
v `∞ (p)
then there exists a unique y = (yk ) ∈ `∞ (p) such that
k−m
X
−1
m k−j −1
xk = vk
(−1)
yj
m−1
j=1
for sufficiently large k, for instance k > m by (1.1). Then there is an
pk
β
integer
N > max{1, supk |∆m
v xk | }. Let a ∈ M∞ (p), and suppose that
−1
= 1. Then we may write
−1
!
n
n
k−m
X
X
X
k
−
j
−
1
ak x k =
ak vk−1
(−1)m
yj
m
−
1
j=1
k=1
k=1
n−m
k X
X
k+m−j−2
m
= (−1)
bk+m−1
yj
m−2
j=1
k=1
n−m
X
m n−j −1
− bn
(−1)
yj .
m−1
j=1
The Dual Spaces of the Sets of
Difference Sequences of Order
m
Ç.A. Bektaş and M. Et
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Since
∞
X
|bk+m−1 |
m−2
j=1
k=1
the series
k X
k+m−j−2
∞
X
bk+m−1
k X
k+m−j−2
j=1
k=1
N 1/pj < ∞,
m−2
yj
is absolutely convergent. Moreover by Lemma 1.1(ii), the convergence of
∞
X
ak vk−1
k−m
X
j=1
k=1
k−j−1
N 1/pj
m−1
implies
lim bn
n→∞
Hence
n−m
X
j=1
n−j−1
N 1/pj = 0.
m−1
P∞
m
m
β
k=1 ak xk is convergent for all x ∈ D∆v `∞ (p), so a ∈ [D∆v `∞ (p)] .
P∞
β
Conversely let a ∈ [D∆m
v `∞ (p)] . Then
k=1 ak xk is convergent for
m
each x ∈ D∆v `∞ (p). If we take the sequence x = (xk ) defined by

k≤m
 0,
xk =
 −1 Pk−m k−j−1 1/pj
N
, k>m
vk
j=1
m−1
The Dual Spaces of the Sets of
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m
Ç.A. Bektaş and M. Et
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then we have
∞
X
ak vk−1
k−m
X
j=1
k=1
Thus the series
plies that
P∞
k=1
∞
X
k−j−1
1/pj
N
=
ak xk < ∞.
m−1
k=1
ak vk−1
lim bn
n→∞
Pk−m
j=1
n−m
X
j=1
k−j−1
m−1
N 1/pj is convergent. This im-
n−j−1
N 1/pj = 0
m−1
by Lemma 1.1(ii).
P∞
Pk−m+1
β
β
Now let a ∈ [D∆m
v `∞ (p)] −M∞ (p). Then
k=1 |bk |
j=1
is divergent, that is,
∞
k−m+1
X
X k − j − 1
|bk |
N 1/pj = ∞.
m
−
2
j=1
k=1
We define the sequence x = (xk ) by

 0,
xk =
Pi−m+1
 −1 Pk−1
vk
i=1 sgn bi
j=1
The Dual Spaces of the Sets of
Difference Sequences of Order
m
k−j−1
m−2
N 1/pj
k≤m
i−j−1
m−2
N 1/pj , k > m
where ak > 0 for all k or ak < 0 for all k. It is trivial that x = (xk ) ∈
D∆m
v `∞ (p). Then we may write for n > m
n
X
k=1
ak x k = −
m
X
k=1
bk−1 ∆v xk−1 −
n−m
X
k=1
Ç.A. Bektaş and M. Et
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bk+m−1 ∆v xk+m−1 − bn xn vn .
J. Ineq. Pure and Appl. Math. 7(3) Art. 101, 2006
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Since (bn xn vn ) ∈ c0 , now letting n → ∞we get
∞
X
ak x k = −
k=1
=
∞
X
bk+m−1 ∆v xk+m−1
k=1
∞
X
k X
k+m−j−2
k=1
j=1
|bk+m−1 |
m−2
N 1/pj = ∞.
β
β
This is a contradiction to a ∈ [D∆m
v `∞ (p)] . Hence a ∈ M∞ (p).
(ii) Can be proved by the same way as above, using Lemma 1.1(i).
The Dual Spaces of the Sets of
Difference Sequences of Order
m
Ç.A. Bektaş and M. Et
η
m
η
Lemma 2.5. [D∆m
v `∞ (p)] = [D∆v c(p)] for η = β or γ.
The proof is obvious and is thus omitted.
Theorem 2.6. Let c+
0 denote the set of all positive null sequences.
(a) We put
(
M3β (p)
=
∞
X
k−m
X
k−j−1
a∈ω:
N 1/pj uj converges and
m
−
1
j=1
k=1
)
∞
k−m+1
X
X k − j − 1
|bk |
N 1/pj uj < ∞, ∀u ∈ c+
.
0
m
−
2
j=1
k=1
ak vk−1
β
β
Then [D∆m
v c0 (p)] =M3 (p).
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(b) We put
(
M4γ (p)
=
n
X
k−m
X
k−j−1
N 1/pj uj | < ∞,
a ∈ ω : sup |
m
−
1
n
j=1
k=1
)
∞
k−m+1
X k − j − 1
X
1/pj
+
|bk |
N
uj < ∞, ∀u ∈ c0 .
m−2
j=1
k=1
ak vk−1
γ
γ
Then [D∆m
v c0 (p)] =M4 (p).
Proof. (a) and (b) can be proved in the same manner as Theorem 2.4, using
Lemma 1.1(i) and (ii).
Lemma 2.7.
ii)
Ç.A. Bektaş and M. Et
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η
m
η
i) [∆m
v `∞ (p)] = [D∆v `∞ (p)] ,
η
[∆m
v c(p)]
The Dual Spaces of the Sets of
Difference Sequences of Order
m
=
η
[D∆m
v c(p)] ,
η
m
η
iii) [∆m
v c0 (p)] = [D∆v c0 (p)]
for η = β or γ.
The proof is omitted.
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References
[1] R. ÇOLAK, Lacunary strong convergence of difference sequences with
respect to a modulus function, Filomat, 17 (2003), 9–14.
[2] M. ET, H. ALTINOK AND Y. ALTIN, On some generalized sequence
spaces, Appl. Math. Comp., 154 (2004), 167–173.
[3] M. ET AND R. ÇOLAK, On some generalized difference sequence spaces,
Soochow J. Math., 21(4) (1995), 377-386.
[4] M. ET AND A. ESI, On Köthe-Toeplitz duals of generalized difference
sequence spaces, Bull. Malays. Math. Sci. Soc., (2) 23 (2000), 25–32.
The Dual Spaces of the Sets of
Difference Sequences of Order
m
Ç.A. Bektaş and M. Et
[5] H. KIZMAZ, On certain sequence spaces, Canad. Math. Bull., 24 (1981),
169–176.
[6] C.G. LASCARIDES AND I.J. MADDOX, Matrix transformations between
some classes of sequences, Proc. Cambridge Philos. Soc., 68 (1970), 99–
104.
[7] I.J. MADDOX, Continuous and Köthe-Toeplitz duals of certain sequence
spaces, Proc. Cambridge Philos. Soc., 65 (1967), 471–475.
[8] E. MALKOWSKY AND S.D. PARASHAR, Matrix transformations in
spaces of bounded and convergent difference sequences of order m, Analysis, 17 (1997), 87–97.
[9] E. MALKOWSKY, Absolute and ordinary Köthe-Toeplitz duals of some
sets of sequences and matrix transformations, Publ. Inst. Math., (Beograd)
(N.S.) 46(60) (1989), 97–103.
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[10] MURSALEEN, Generalized spaces of difference sequences, J. Math.
Anal. Appl., 203 (1996), 738–745.
[11] S. SIMONS, The sequence spaces `(pv ) and m(pv ), Proc. London Math.
Soc., (3) 15 (1965), 422–436.
[12] A. WILANSKY, Summability through Functional Analysis, NorthHolland Mathematics Studies, 85 (1984).
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m
Ç.A. Bektaş and M. Et
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