Gen. Math. Notes, Vol. 27, No. 2, April 2015, pp.37-46
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
Some Generalized Difference
Sequence Spaces of Non-Absolute Type
Sinan Ercan1 and Çiğdem A. Bektaş2
1
Department of Mathematics, Firat University, Elaziğ, Turkey
E-mail: sinanercan45@gmail.com
2
Department of Mathematics, Firat University, Elaziğ, Turkey
E-mail: cigdemas78@hotmail.com
(Received: 4-3-15 / Accepted: 12-4-15)
Abstract
m
m
In this paper, we introduce the spaces `∞ (∆m
λ ), c(∆λ ) and c0 (∆λ ), which
are BK-spaces of non-absolute type and we prove that these spaces are linearly
isomorphic to the spaces `∞ , c and c0 , respectively. Moreover, we give some
inclusion relations and compute the α−, β− and γ−duals of these spaces. We
m
also determine the Schauder basis of the c(∆m
λ ) and c0 (∆λ ).
Keywords: Sequence spaces of non-absolute type, BK-spaces, Difference
Sequence Spaces.
1
Introduction
A sequence space is defined to be a linear space of real or complex sequences.
Let w denote the spaces of all complex sequences. If x ∈ w, then we simply
write x = (xk ) instead of x = (xk )∞
k=0 .
Let X be a sequence space. If X is a Banach space and
τk : X → C, τk (x) = xk (k = 1, 2, ...)
is a continuous for all k, X is called a BK−space.
We shall write `∞ , c and c0 for the sequence spaces of all bounded, convergent and null sequences, respectively, which are BK−spaces with the norm
given by kxk∞ = supk |xk | for all k ∈ N.
38
Sinan Ercan et al.
For a sequence space X, the matrix domain XA of an infinite matrix A
defined by
XA = {x = (xk ) ∈ w : Ax ∈ X}
(1)
which is a sequence space.
We shall denote the collection of all finite subsets of N by F.
M. Mursaleen and A. K. Noman [9] introduced the sequence spaces `λ∞ , cλ
and cλ0 as the sets of all λ − bounded, λ − convergent and λ − null sequences,
respectively, that is
`λ∞ = {x ∈ w : sup |Λn (x)| < ∞}
n
c
λ
n→∞
cλ0
where Λn (x) =
1
λn
n
P
= {x ∈ w : lim Λn (x) exists}
= {x ∈ w : lim Λn (x) = 0}
n→∞
(λk − λk−1 ) xk , k ∈ N.
k=0
M. Mursaleen and A. K. Noman [10] also introduced the sequence spaces
cλ (∆) and cλ0 (∆), respectively, that is
cλ (∆) = {x ∈ w : lim Λ̄n (x) exists}
n→∞
cλ0
where Λ̄n (x) =
1
λn
n
P
(∆) = {x ∈ w : n→∞
lim Λ̄n (x) = 0}.
(λk − λk−1 ) (xk − xk−1 ), k ∈ N.
k=0
H. Ganie and N. A. Sheikh [2] introduced the spaces c0 (∆λu ) and c(∆λu ) as
follows:
b (x) = 0}
c0 (∆λu ) = {x ∈ w : n→∞
lim Λ
n
b (x) exists}
c(∆λu ) = {x ∈ w : n→∞
lim Λ
n
b (x) =
where Λ
n
2
1
λn
Pn
k=0 (λk
− λk−1 )uk (xk − xk−1 ), k ∈ N.
m
m
The Sequence Spaces `∞(∆m
λ ) , c(∆λ ) and c0 (∆λ )
of Non-Absolute Type
m
m
We define the sequence spaces `∞ (∆m
λ ) , c(∆λ ) and c0 (∆λ ) as follows;
`∞ (∆m
λ ) = x ∈ w : sup Λ̃n (x) < ∞
n
c(∆m
lim Λ̃n (x) exists
λ ) = x ∈ w : n→∞
c0 (∆m
λ ) = x ∈ w : lim Λ̃n (x) = 0
n→∞
39
Some Generalized Difference...
1
λn
Pn
− λk−1 )∆m xk , k, m ∈ N. ∆ denotes the difference
!
Pm
m
v
0
m
xk−v .
operator. i.e., ∆ xk = xk , ∆xk = xk −xk−1 and ∆ xk = v=0 (−1)
v
λ = (λk )∞
k=0 is a strictly increasing sequence of positive reals tending to infinity,
that is 0 < λ0 < λ1 < ... and λk → ∞ as k → ∞.
Here and in sequel, we use the convention that any term with a negative
subscript is equal to naught. e.g. λ−1 = 0 and x−1 = 0.
If we take m = 1 sequence spaces which we defined reduces to `λ∞ (∆), cλ (∆)
and cλ0 (∆).
We define the matrix Λ̃ = λ̃nk for all n, k ∈ N by
where Λ̃n (x) =
k=0 (λk
λ̃nk =

n

 P
i=k


m
i−k
!
(−1)i−k
λi −λi−1
,
λn
0,
k≤n
.
n<k
Λ̃ = λ̃nk equality can be eaisly seen from
n
1 X
Λ̃n (x) =
(λk − λk−1 ) ∆m xk
λn k=0
(2)
for all m, n ∈ N and every x = (xk ) ∈ w. Then it leads us together with (1)
to the fact that
m
m
`∞ (∆m
λ ) = (`∞ )Λ̃ , c0 (∆λ ) = (c0 )Λ̃ , c (∆λ ) = (c)Λ̃ .
The matrix Λ̃ = λ̃nk is a triangle, i.e., λ̃nn 6= 0 and λ̃nk = 0 (k > n) for
all n, k ∈ N. Further, for any sequence x = (xk ) we define the sequence
y (λ) = {yk (λ)} as the Λ̃-transform of x, i.e., y (λ) = Λ̃(x) and so we have that
y (λ) =
k X
k
X
i−j
(−1)
j=0 i=j
m
i−j
!
!
λi − λi−1
xj
λk
(3)
for k ∈ N. Here and in what follows, the summation running from 0 to k − 1
is equal to zero when k = 0.
m
m
Theorem 2.1 `∞ (∆m
λ ), c0 (∆λ ) and c(∆λ ) are BK-spaces with the norm
kxk(`∞ ) = Λ̃n (x)
Λ̃
∞
= sup Λ̃n (x) .
(4)
n
Proof: We know that candc0 are BK−spaces with their natural norms
from [5]. (3) holds and Λ̃ = λ̃nk is a triangle matrix and from Theorem 4.3.12
m
m
of Wilansky [1], we derive that `∞ (∆m
λ ), c0 (∆λ ) and c (∆λ ) are BK−spaces.
This completes the proof.
40
Sinan Ercan et al.
m
Remark 2.2 The absolute property does not hold on the `∞ (∆m
λ ), c0 (∆λ ) and
m
c(∆λ ) spaces. For instance, if we take |x| = (|xk |) we hold kxk(`∞ ) 6=
Λ̃
k|x|k(`∞ ) .Thus, the space `∞ (∆m
), c0 (∆m
) and c(∆m
) are BK-space of nonλ
λ
λ
Λ̃
absolute type.
m
m
Theorem 2.3 The sequence spaces `∞ (∆m
λ ), c0 (∆λ ) and c(∆λ ) of nonabsolute type are linearly isomorphic to the spaces `∞ , c0 and c, respectively,
m ∼
m ∼
∼
that is `∞ (∆m
λ ) = `∞ , c0 (∆λ ) = c0 and c(∆λ ) = c.
∼
Proof: We only consider c0 (∆m
λ ) = c0 and others will prove similarly.
To prove the theorem we must show the existence of linear bijection operator
between c0 (∆m
λ ) and c0 . Hence, let define the linear operator with the notation
(3), from c0 (∆m
λ ) and c0 by x → y (λ) = T x.
Then T x = y (λ) = Λ̃ (x) ∈ c0 for every x ∈ c0 (∆m
λ ). Also, the linearity
of T is clear. Further, it is trivial that x = 0 whenever T x = 0. Hence T is
injective.
Let y = (yk ) ∈ c0 and define the sequence x = {x (λ)} by
xk (λ) =
k
X
j=0
m+k−j−1
k−j
!
j
X
(−1)j−i
i=j−1
λi
yi .
λj − λj−1
(5)
and we have
∆m xk =
k
X
(−1)k−i
i=k−1
λi
yi .
λk − λk−1
(6)
Thus, for every k ∈ N, we have by (2) that
Λ̃n (x) =
n
k
n
X
1 X
1 X
(−1)k−i λi yi =
(λk yk − λk−1 yk−1 ) = yn
λn k=0 i=k−1
λn k=0
(7)
This shows that Λ̃(x) = y and since y ∈ c0 , we obtain that Λ̃(x) ∈ c0 . Thus we
deduce that x ∈ c0 (∆m
λ ) and T x = y. Hence T is surjective.
Further, we have for every x ∈ c0 (∆m
λ ) that
kT xkc0 = kT xk`∞ = ky (λ)k`∞ = Λ̃(x)
`∞
= kxk(c
0 )Λ̃
which means that c0 (∆m
λ ) and c0 is linearly isomorphic.
3
The Inclusion Relations
m
Theorem 3.1 The inclusion c0 (∆m
λ ) ⊂ c (∆λ ) strictly holds.
(8)
41
Some Generalized Difference...
m
Proof: It is clear that c0 (∆m
λ ) ⊂ c (∆λ ) . To show strict, consider the
m
sequence x = (xk ) defined by xk = k for all k ∈ N. Then we obtain that
Λ̃n (x) =
n
1 X
(λk − λk−1 ) ∆m xk = m!
λn k=0
(9)
for n ∈ N which shows that Λ̃(x) ∈ c − c0 . Thus, the sequence x is in c (∆m
λ)
m
m
m
but not in c0 (∆λ ) . Hence the inclusion c0 (∆λ ) ⊂ c (∆λ ) is strict and this
completes the proof.
Theorem 3.2 The inclusion c ⊂ c0 (∆m
λ ) strictly holds.
Proof: Let x ∈ c. Then Λ̃(x) ∈ c0 . This shows that x ∈ c0 (∆m
λ ) . Hence,
m
the inclusion
√ c ⊂ c0 (∆λ ) holds. Then, consider the sequence y = (yk ) defined
by yk = k + 1 for k ∈ N. It is trivial that y ∈
/ c. On the other hand, it can
)
.Consequently,
the sequence y is
easily be seen that Λ̃(y) ∈ c0 and y ∈ c0 (∆m
λ
m
in c0 (∆λ ) but not in c. We therefore deduce that the inclusion c ⊂ c0 (∆m
λ ) is
strict. This concludes proof.
Theorem 3.3 The inclusion c ∆m−1
⊂ c (∆m
λ ) holds.
λ
Proof: Let x ∈ c ∆m−1
. Then we have
λ
Λ̃n (x) =
n
1 X
(λk − λk−1 ) ∆m−1 xk → l
λn k=0
(k → ∞) .
(10)
Furthermore, we obtain that x ∈ c (∆m
λ ) from the following inequality,
m−1
m
hence the inclusion c ∆λ
⊂ c (∆λ ) holds.
P
1
n
λ
k=0 (λk
n
− λk−1 ) ∆m xk ≤ λ1n
+ λ1n
Pn
k=0
Pn
) ∆m−1 xk − l
k=0 (λk − λk−1
(λk − λk−1 ) ∆m−1 xk−1 − l → 0.
(11)
Theorem 3.4 The inclusion `∞ (∆m−1
) ⊂ `∞ (∆m
λ ) strictly holds.
λ
Proof: Let x ∈ `∞ (∆m−1
). Then we have
λ
Λ̃n (x)
=
n
1 X
(λk
λn
k=0
− λk−1 ) ∆m−1 xk ≤K
(12)
for K > 0. We obtain the following equality that x ∈ `∞ (∆m
λ ), hence the
m
inclusion `∞ (∆m−1
)
⊂
`
(∆
)
holds.
∞
λ
λ
n
1 X
(λk
λn
k=0
− λk−1 )∆
m
xk ≤
n
1 X
(λk
λn
k=0
m−1
− λk−1 )∆
n
1 X
xk +
(λk
λn
m−1
− λk−1 )∆
k=0
(13)
To show strict, we consider x = (xk ) defined by x = (k m ) , then we obtain
m−1
x ∈ `∞ (∆m
).
λ ) − `∞ (∆λ
xk−1 .
42
4
Sinan Ercan et al.
m
The Bases for the Spaces c (∆m
λ ) and c0 (∆λ )
If a normed sequence space X contains a sequence (bn ) with the property that
for every x ∈ X there is a unique sequence (αn ) of scalars such that
lim kx − (α0 b0 + α1 b1 + ... + αn bn )k = 0.
(14)
n
P
Then (bn ) is called a Schauder basis (or briefly basis) for X. The series αk bk
which has the sum x is then called the expansion of x with respect to (bn ) ,
P
and written as x = αk bk .
k
o∞
n
Theorem 4.1 Define the sequence b(k) (λ, m) = b(k)
n (λ, m)
fixed k, m ∈ N and by
b(k)
n
(λ, m) =






m+n−k−1
!
λk
λk −λk−1
n−k
−
m+n−k−2
for every
!
λk
,
λk+1 −λk
n−k−1
λk
,
λk −λk−1





k=0
n>k
n=k
n<k
0,
. (15)
o∞
n
m
Then, the sequence b(k)
n (λ, m) k=0 is a basis for the space c0 (∆λ ) and every
x ∈ c0 (∆m
λ ) has a unique representation of the form
x=
X
αk (λ) b(k) (λ, m)
(16)
k
where αk (λ) = Λ̃k (x) for all k ∈ N.
n
o
Theorem 4.2 The sequence b, b(0) (λ, m) , b(1) (λ, m) , ... is a basis for the
m
space c (∆m
λ ) and every x ∈ c (∆λ ) has a unique representation of the form
x = lb +
X
[αk (λ) − l] b(k) (λ, m) ;
(17)
k
where αk (λ) = Λ̃k (x) for all k ∈ N, the sequence b = (bk ) is defined by
bk =
k
X
j=0
m+k−j−1
k−j
!
.
(18)
m
Corollary 4.3 The difference sequence spaces c (∆m
λ ) and c0 (∆λ ) are seperable.
43
Some Generalized Difference...
5
The α−, β− and γ−Duals of the Spaces c (∆m
λ)
and c0 (∆m
λ)
In this section, we introduce and prove the theorems determining the α−, β−
m
and γ− duals of the difference sequence spaces c (∆m
λ ) and c0 (∆λ ) of nonabsolute type.
For arbitrary sequence spaces X and Y ,the set M (X, Y ) defined by
M (X, Y ) = {a = (ak ) ∈ w : ax = (ak xk ) ∈ Y f or all x = (xk ) ∈ X}
(19)
is called the multipier space of X and Y.
With the notation of (19); the α−, β− and γ−duals of a sequence space
X, which are respectively denoted by X α , X β and X γ are defined by
X α = M (X, `1 ) , X β = M (X, cs) and X γ = M (X, bs) .
(20)
Now, we may begin with lemmas which are needed in proving theorems.
Lemma 5.1 A ∈ (c0 : `1 ) = (c : `1 ) if and only if
XX
sup
ank K∈F n k∈K
< ∞.
(21)
Lemma 5.2 A ∈ (c0 : c) if and only if
lim ank exists f or each k ∈ N,
(22)
n
sup
X
n
k
|ank | < ∞.
(23)
Lemma 5.3 A ∈ (c : c) if and only if (22) and (23) hold, and
lim
X
n
ank exists.
(24)
k
Lemma 5.4 A ∈ (c0 : `∞ ) = (c : `∞ ) if and only if (23) holds.
Lemma 5.5 A ∈ (`∞ : c) if and only if (22) holds and
lim
n→∞
X
k
|ank | =
X
|αk | .
(25)
k
m
Theorem 5.6 The α−dual of the space c0 (∆m
λ ) and c (∆λ ) is the set
bλ1 =


a = (ak ) ∈ w :

X X
bnk (λ, m)
sup
K∈F n k∈K


<∞ ;

where the matrix B λ = bλm
is defined via the sequence a = (ak ) by
nk
(26)
44
Sinan Ercan et al.
bn(k)
(λ, m) =
 "





m+n−k−1
!
λk
λk −λk−1
n−k
m+n−k−2
−
!
#
λk
λk+1 −λk
n−k−1
an , n > k
.
n=k
n<k
(27)
λn
a ,
λn −λn−1 n





0,
Proof: Let a = (ak ) ∈ w. Then, we obtain the equality
n
X
!
k
X
λj
yj = Bnλ (y) , (n ∈ N) .
n−k
λ
−
λ
k
k−1
k=0
j=k−1
(28)
Thus, we observe by (28) that ax = (ak xk ) ∈ `1 whenever x = (xk ) ∈ c0 (∆m
λ)
λ
or c (∆m
)
if
and
only
if
B
y
∈
`
whenever
y
=
(y
)
∈
c
or
c.
This
means
that
1
k
0
λ
m
the the sequence a = (ak ) is in the α−dual of the spaces c0 (∆m
λ ) or c (∆λ ) if
λ
and only if B ∈ (c0 : `1 ) = (c : `1 ) . We therefore obtain by Lemma 5.1 with
α
m α
B λ instead of A that a ∈ {c0 (∆m
λ )} = {c (∆λ )} if and only if
ak x k =
m+n−k−1
(−1)k−j
X X
sup
b
(λ,
m)
nk
K∈F n k∈K
< ∞.
(29)
α
λ
m α
Which leads us to the consequence that {c0 (∆m
λ )} = {c (∆λ )} = b1 . This
concludes proof.
Theorem 5.7 Define the sets


bλ2 = a = (ak ) ∈ w :
∞
X
m+n−j−1
aj exists f or each k ∈ N.
n−j
j=k
(
bλ3
= a = (ak ) ∈ w : sup
bλ5 =


= a = (ak ) ∈ w :
a = (ak ) ∈ w : lim
n→∞

sup n∈N λn
n X
k
X
k=0 j=0
(
bλ6
n−1
X
(30)
)
|gk (n)| < ∞.
(31)
n∈N k=0
(
bλ4


!
= a = (ak ) ∈ w : lim
n→∞
)
λn
an < ∞.
− λn−1 m+k−j−1
k−j
X tλnk =
k
!
ak exists.
)
X
λ
lim t n→∞ nk (32)


(33)

(34)
k
where the matrice T λ = tλnk is defined as follow:
tλnk



ak (n) ,
k<n
λn
a
,
k
=n
=
n
λ −λ

 n n−1
0,
k>n
(35)
45
Some Generalized Difference...
for all k, n ∈ N and the ak (n) is defined by

n
X
1
ak (n) = λk 
λk − λk−1 j=k
m+j−k−1
!
j−k
n
X
1
aj −
λk+1 − λk j=k
m+j−k−2
!
j−k−1

aj  y k
(36)
β
λ
λ
λ
m β
λ
λ
λ
λ
for k < n. Then {c0 (∆m
)}
=
b
∩
b
∩
b
,
{c
(∆
)}
=
b
∩
b
∩
b
∩
b
λ
2
3
4
λ
2
3
4
5 and
m β
λ
λ
λ
{`∞ (∆λ )} = b2 ∩ b4 ∩ b6 .
Proof: We have from (5)
n
X
ak x k =

k
n
X
X

k=0
k=0
=
=
k−j
j=0

n−1
X
m+k−j−1
n
P

 j=k
λk 


k=0
n−1
X
k=0
j
X
!
(−1)j−i
i=j−1
m+j−k−1
!
j−k
λk − λk−1
ak (n) yk +

aj
−
λi
y i  ak
λj − λj−1
n
P
m+j−k−2
j=k+1
j−k−1
!

aj 
λk+1 − λk

 yk


+
an λn
yn
λn − λn−1
an λ n
yn = T λ y ; (n ∈ N) .
n
λn − λn−1
Then we derive that ax = (ak xk ) ∈ cs whenever x = (xk ) ∈ c0 (∆m
λ ) if and only
β
λ
if T y ∈ c whenever y = (yk ) ∈ c0 . This means that a = (ak ) ∈ {c0 (∆m
λ )} if
and only if T λ ∈ (c0 : c) . Therefore, by using Lemma 5.2, we obtain
∞
X
j=k
m+k−j−1
k−j
sup
!
n−1
X
aj exists f or each k ∈ N,
(37)
|ak (n)| < ∞
(38)
n∈N k=0
and
sup
n−1
X
k∈N k=0
λk
λk
ak < ∞.
− λk−1
(39)
β
λ
λ
λ
Hence we conclude that {c0 (∆m
λ )} = b2 ∩ b3 ∩ b4 .
γ
m γ
m γ
λ
λ
Theorem 5.8 {c0 (∆m
λ )} = {c (∆λ )} = {`∞ (∆λ )} = b3 ∩ b4 .
Proof: It can be proved similalry as the proof of the Theorem 5.7 with
Lemma 5.4 instead of Lemma 5.2.
Acknowledgements: We thank the anonymous referees for their comments and suggestions that improved the presentation of this paper.
46
Sinan Ercan et al.
References
[1] A. Wilansky, Summability Through Functional Analysis, North-Holland
Mathematics Studies, Elsevier Science Publishers, Amsterdam, New York,
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