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On Köthe-Toeplitz Duals of Generalized Difference Sequence Spaces

Authors:
Mikail Et at Fırat University
Mikail Et
Ayhan Esi at Turgut Özal University
Ayhan Esi
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Abstract

In this paper, we define the sequence spaces ) , c f " and 0 c be the linear spaces of bounded, convergent, and null sequences ) ( k x x with complex terms, respectively, normed by k k x x sup f
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BULLETIN of the Bull. Malaysian Math. Sc. Soc. (Second Series) 23 (2000) 25-32
MALAYSIAN
MATHEMATICAL
SCIENCES
SOCIETY
On Köthe-Toeplitz Duals of Generalized
Difference Sequence Spaces
MIKAIL ET AND AYHAN ESI
Department of Mathematics, Fırat University, 23169-Elazig Turkiye (TR)
e-mail: mikailet@hotmail.com
e-mail: aesi23@hotmail.com
Abstract. In this paper, we define the sequence spaces
)(,)( c
m
v
m
v
''
f
" and ,)(,)( N' mc
o
m
v
and give some topological properties, inclusion relations of these sequence spaces, compute their
continuous and Köthe-Toeplitz duals. The results of this paper, in a particular case, include the
corresponding results of Kızmaz [5] , Çolak [1], [2], Et-Çolak [4], and Çolak
et al. [3].
1. Introduction
Let
,, c
f
" and
0
c
be the linear spaces of bounded, convergent, and null sequences
)(
k
xx with complex terms, respectively, normed by
kk
xx sup
f
where
^`
,2,1 Nk , the set of positive integers.
Kızmaz [5] defined the sequence spaces
^`
ff
' ' "" xx :)()(
k
x ,
^`
cxc
k
' ' xx :)()( ,
^`
00
:)()( cxc
k
' ' xx
where
,)()(
1
' '
kkk
xxxx and showed that these are Banach spaces with norm
f
' x
1
1
xx .
Then Çolak [1] defined the sequence space
^`
XxxX
kvkv
'
' :)()( x , where
)()(
11
'
kkkkkv
xvxvx and X is any sequence space, and investigated some
topological properties of this space.
M. Et and A. Esi
26
Recently Et and Çolak [4] generalized the above sequence spaces to the following
sequence spaces.
,}:)({)(
ff
' ' "" xx
m
k
m
x
}:)({)( cxc
m
k
m
' ' xx ,
}:)({)(
00
cxc
m
k
m
' ' xx
where
,Nm
,)()(),(),(
1
11
1
0
'' ' ' ' '
k
m
k
m
k
mm
kkk
xxxxxx xxx
and
vk
m
v
v
k
m
x
v
m
x
¦
¸
¸
¹
·
¨
¨
©
§
'
0
)1( 
These are Banach spaces with norm
f
'
'
¦
xx
m
m
i
i
x
1
.
It is trivial that
,)()(,)()(,)()(
111
00
ff
''''''
mmmmmm
cccc "" and
)()()(
0
mmm
cc '''
f
" are satisfied and strict [4]. For convenience we denote
these spaces
 ,)()(,)()(
mmmm
cc ' '' '
ff
"" and ).()(
00
mm
cc ' '
Throughout the paper we write
¦
k
for
¦
f
1k
and
n
lim for
fon
lim .
Let
)(
k
vv be any fixed sequence of nonzero complex numbers. Now we define
^`
ff
' ' "" xx
m
vk
m
v
x :)()(
^`
cxc
m
vk
m
v
' ' xx :)()( (1.1)
^
`
00
:)()( cxc
m
vk
m
v
' ' xx
where
,Nm
kvkkv
xxv ' ' ,)(
0
x
k
m
vkkkk
xxvxv '
,)(
11
)(
1
11
''
k
m
vk
m
v
xx
,
and so that
ikik
m
i
i
k
m
v
xv
i
m
x
¦
¸
¸
¹
·
¨
¨
©
§
'
0
)1(
On Köthe-Toeplitz Duals of Generalized Difference Sequence Spaces
27
It is trivial that ,)(
f
' "
m
v
)(c
m
v
' and )(
0
c
m
v
' are linear spaces. If we take
),1,1()(
k
v and 1 m in (1.1), then we obtain )(,)( c
'
'
f
" and )(
0
c' . Also if we
take
1 m and ),1,1()(
k
v in (1.1), then we obtain ,)(
f
' "
v
)(c
v
' and ),(
0
c
v
'
and ,)(
f
' "
m
)(c
m
' and ,)(
0
c
m
' respectively.
2. Main results
Theorem 2.1. The sequence spaces )(,)( c
m
v
m
v
''
f
" and )(
0
c
m
v
' are Banach spaces
normed by
f
¦
'
m
i
m
vii
v
vx
1
xx .
(2.1)
Proof. Omitted.
Let
X stand for ,
f
" c and
0
c and let us define the operator
)()(: XXD
m
v
m
v
'o'
by ,),,,,0,0(
21
mm
xxDx where ),,,(
321
xxx x . It is trivial that D is a
bounded linear operator on
.)(X
m
v
' Furthermore the set
>@
^`
0),(:)()()(
21
' ' '
m
m
vk
m
v
m
v
xxxXxXDXD xx
is a subspace of
)(X
m
v
' and
f
' xx
m
v
v
in )(XD
m
v
' . )(XD
m
v
' and X are
equivalent as topological space since
XXD
m
v
m
v
o'' )(: , defined by )(
k
m
v
m
v
xy ' ' x (2.2)
is a linear homeomorphism [7].
Let
X
c
and ])([
c
' XD
m
v
denote the continuous duals of X and )(XD
m
v
' ,
respectively. It can be shown that
fffXXDT
m
v
m
v
'o
c
o
c
'
''
10
)(,])([:
is a linear isometry. So ])([
c
' XD
m
v
is equivalent to Xc [7].
M. Et and A. Esi
28
Corollary 2.2.
(i) )(c
m
v
' and )(
0
c
m
v
' are closed subspaces of ,)(
f
' "
m
v
(ii) )(c
m
v
' and )(
0
c
m
v
' are separable spaces,
(iii)
,)(
f
' "
m
v
)(c
m
v
' and )(
0
c
m
v
' are BK-spaces with the same norm as in (2.1),
(iv)
,)(
f
' "
m
v
)(c
m
v
' and )(
0
c
m
v
' are not sequence algebras.
3. Dual spaces
In this section we give Köthe-Toeplitz duals of
,)(
f
' "
m
v
)(c
m
v
' and ).(
0
c
m
v
' Now we
give the following lemmas.
Lemma 3.1. )(
f
' "
m
v
x if and only if
(i)
f'
k
m
vk
xk
11
sup .
(ii) f''
1
111
)1(sup
k
m
vk
m
vk
xkkx .
Proof. Omitted.
Lemma 3.2. f'
kv
i
k
xksup implies f
kk
i
k
xvk
)1(
sup for all .Ni
Proof.
Omitted.
Lemma 3.3. f'
k
im
v
i
k
xksup implies f'
k
im
v
i
k
xk
)1()1(
sup for all
Nmi , and .1 mi d
Proof. If
kv
x' is replaced with
k
im
v
x
' in Lemma.3.2, the result is immediate.
Lemma 3.4. f'
k
m
vk
xk
11
sup implies f
kk
m
k
xvksup .
Proof. For 1 i in Lemma.3.3, we obtain f'
k
m
vk
xk
11
sup implies
f'
k
m
vk
xk
22
sup . Again, for 2 i in Lemma 3.3, we obtain
f'
k
m
vk
xk
22
sup implies f'
k
m
vk
xk
33
sup . Continuing this procedure,
for
,1 mi we arrive f'
kv
m
k
xk
)1(
sup implies .sup f
kv
m
k
xvk
On Köthe-Toeplitz Duals of Generalized Difference Sequence Spaces
29
Lemma 3.5. )(
f
' "
m
v
x implies .sup f
kk
m
k
xvk
Proof. Proof follows from Lemma.3.1 and Lemma.3.4.
Definition 3.6. [6] Let X be a sequence space and define
^`
XxaaaX
kkkk
f¦ xallfor,:)(
D
,
then
D
X
is called Köthe-Toeplitz dual of X. If ,YX then .
DD
XY It is clear that
.)(
DDDD
XXX If
DD
X
X
then X is called an
D
-space. In particular, an
D
-space is a Köthe space or a perfect sequence space.
Theorem 3.7. Let }||:)({
1
1
f¦
kk
m
kk
vakaaU and
^`
,sup:)(
2
f
kk
m
kk
vakaaU then
i)
10
))(())(())(( Ucc
m
v
m
v
m
v
' ' '
f
DDD
"
ii)
20
))(())(())(( Ucc
m
v
m
v
m
v
' ' '
f
DDDDDD
"
Proof. Omitted.
Corollary 3.8. )(,)( c
m
v
m
v
''
f
" and )(
o
m
v
c' are not perfect.
Corollary 3.9. If we take ),1,1()(
k
v and ,1 m in Theorem 3.7, then we obtain for
f
"X or c.
(i
^`
,:)())(( f¦ '
k
m
kk
m
akaaX
D
(ii)
^`
,sup:)())(( f '
k
m
kk
m
akaaX
DD
(iii)
^
`
.:)())((
1
f¦ '
kkkkv
vakaaX
D
Corollary 3.10. If we take )(
m
kv in Theorem 3.7, then we obtain
(i)
1
))(())(())(( "" ' ' '
f
DDD
o
m
v
m
v
m
v
cc ,
(ii)
ff
' ' ' ""
DDDDDD
))(())(())((
o
m
v
m
v
m
v
cc
.
M. Et and A. Esi
30
4. Inclusions theorems
In this section we give inclusion relation of these spaces. Firstly, we note that )(X
m
v
'
and )(X
m
' overlap but neither one contains the other, for cX ,
f
" and .
0
c For
example, we choose,
)(
m
k x
and ,)(kv then
,)(
f
' "
m
x
but
,)(
f
' "
m
v
x
conversely if we choose )(
1
m
kx and )(
1
kv then ,)(
f
' "
m
x but
.)(
f
' "
m
v
x
Theorem 4.1.
(i)
)()(
1
XX
m
v
m
v
''
and the inclusion is strict, for cX ,
f
" and
0
c ,
(ii)
)()()(
0 f
''' "
m
v
m
v
m
v
cc and the inclusion is strict.
Proof .
(i) We give the proof for
f
"X only. Let ).(
f
' "
m
v
x Since
1111
1
''d''d'
kk
m
kk
m
kk
m
kk
m
kk
m
vxvxvxvxvx
we obtain ).(
1
f
' "
m
v
x This inclusion is strict since the sequence )(
m
k x
belongs to ,)(
1
f
' "
m
v
but does not belong to ,)(
f
' "
m
v
where .)(kv
(ii) Proof is trivial.
Theorem 4.2. Let )(
k
uu and )(
k
vv be any fixed sequences of nonzero complex
numbers, then
(i) If
,sup
1
f
kk
m
k
uvk then
,)()(
ff
'' ""
m
u
m
v
(ii) If
,)(
1
foo
kuvk
kk
m
" for some " , then ,)()( cc
m
u
m
v
''
(iii) If ,)(0
1
foo
kuvk
kk
m
then
.)()(
00
cc
m
u
m
v
''
Proof.
(i)
f
kk
m
k
uvk
1
sup and assume that .)(
f
' "
m
v
x Since
On Köthe-Toeplitz Duals of Generalized Difference Sequence Spaces
31
¦
'
¸
¸
¹
·
¨
¨
©
§
'' '
1
0
1
)(
1
)1())(()(
m
i
ikik
i
u
mm
u
ux
i
m
xx
«
¬
ª
¸
¸
¹
·
¨
¨
©
§
d
¦
ikik
m
ikik
m
m
i
xvikuvik
i
m
)()(
1
1
1
0
@
111
1
1
)1()1(
ikik
m
ikik
m
xvikuvik
we obtain
.)(
f
' "
m
u
x If we take ),1,1( v and ),1,1( u in Theorem
4.2, then we have the corollaries, respectively.
Corollary 4 3.
(i) If
,sup f
k
m
k
vk then ,)()(
ff
'' ""
m
v
m
(ii) If ,)( foo kvk
k
m
" for some ," then ,)()( cc
m
v
m
''
(iii) If ,)(0 foo kvk
k
m
then )()(
00
cc
m
v
m
'' .
Corollary 4 4.
(i) If ,sup
1
f
k
m
k
vk then ,)()(
ff
'' ""
mm
v
(ii) If
,)(
1
foo
kvk
k
m
" for some ," then ,)()( cc
mm
v
''
(iii) If
,)(0
1
foo
kvk
k
m
then .)()(
00
cc
mm
v
''
If we take )(
m
kx in [3], then we obtain the following sequence spaces.
i)
,}||sup:)({ f
f k
m
kk
vkvvv
ii) ,}(||:)({ foo kvkvvv
k
m
kc
" for some }" ,
iii)
,})(0||:)({
0
foo kvkvvv
k
m
k
ic) ,}||sup:)({
11
f
f
k
m
kk
vkvvv
iic) ,)(||:)({
11
foo
kvkvvv
k
m
kc
" for some ,}"
iiic) })(0||:)({
11
foo
kvkvvv
k
m
ko
.
It is trivial that the sequence spaces
0
and, vvv
cf
are BK-spaces with the norm
.||sup
k
m
k
vkv The
K
-duals of these sequence spaces are also readily obtained by
[3] , where
E
D
K
, and .
J
M. Et and A. Esi
32
Theorem 4.5. Let X stand for
0
and, vvv
cf
, then OXX
1
.
Proof. We give the proof for
f
vX only. Let
1
ff
vvv and 0z
k
v for all k,
then there are constants
0,
21
!MM such that
1
|| Mvk
k
m
d and
2
1
|| Mvk
k
m
d
for
all
.Nk This implies
21
2
MMk
m
d for all k, a contradiction, since .1tm

Theorem 4.6.
.)()(
0
cc
m
v
m
v
' '
ff
""
Proof. Let
.)(c
m
v
'
f
"x
Then
f
"x and
11
11
''
kk
m
kk
m
vxvx
,)( foo k" .),0(,
11
11
foo ''
kvxvx
kkkk
m
kk
m
HH
" This implies that
.
1
1
11
11
11
11
¦
''
n
k
knn
mm
nvxnvxn
H
"
This yields
0 " and .)(
0
cx
m
v
'
f
"
References
1. R. Çolak, On Some Generalized Sequence Spaces, Commun. Fac. Sci. Univ. Ank. Series A
1
,
38 (1989), 35-46.
2. R. Çolak, On Invariant Sequence Spaces, Erciyes Univ. Journal of Sci. 5 (1989), 881-887.
3. R. Çolak, P.D. Srivastava and S. Nanda, On Certain Sequence Spaces and Their Köthe
Toeplitz Duals, Rendiconti di Mat. 13 (1993), 27-39.
4. M. Et and R. Çolak, On Some Generalized Difference Sequence Spaces, Soochow Journal of
Mathematics 21 (1995), 377-386.
5. H. Kızmaz, On Certain Sequence Spaces, Canad. Math. Bull. 24 (1981), 169-176.
6. P.K. Kampthan and M. Gupta, Sequence Spaces and Series, Marcel Dekker Inc. New York,
1981.
7. I.J. Maddox, Elements of Functional Analysis, Cambridge Univ. Press, 1970.
Keywords and phrases: difference sequence spaces, Köthe-Toeplitz dual.
1991 Mathematics Subject Classification: 40A05, 40C05, 46A45.
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In this paper, we introduce bounded, convergent, null and eventually alternating triple sequence spaces, i.e.3_(l_∞ ),3_c, 〖 3〗_(c_0 ) and respectively. Then we find -dual of order r (0 of these spaces. Further, we check whether they are perfect or not.
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... Recently, the literature focused on the creation of new sequence spaces through the matrix domain and the investigation of their algebraic and topological properties, and the study of matrix transformations has expanded. To enhance comprehension of the theory concerning sequence spaces, you can refer to these valuable articles (see Altay et al. [7], Gürdal [8], Şahiner and Gürdal [9], Gürdal and Şahiner [10], Et and Esi [11], Aiyub et al. [12], and many others). ...
On Some New Sequence Spaces and Their Duals
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Full-text available
  • Apr 2024
In this study, we defined some new sequence spaces using regular Tribonacci matrix. We examined some properties of these spaces such as completeness, Schauder basis. We have identified α−,β−, and γ−duals of the newly created spaces.
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... Kızmaz [2] was firstly introduced the concept of the difference operator in the sequence spaces. Further Et and Ç olak [3] generalized the idea of difference sequence spaces of Kızmaz.Besides this topic was studied by many authors ( [4], [5]) . Now, the difference matrix ∆ = (δ nk ) defined by ...
Duals of generalized Orlicz Hilbert sequence spaces and matrix transformations
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Full-text available
  • Aug 2023
In this paper, wedefine the sequence spaces hc (?(m)v,M,u,p), h0 (?(m) v,M,u,p) and h?(?(m)v,M,u,p) resulting from the infinite Hilbert matrix and the Musielak-Orlicz function. We give some topological properties and inclusion relations of these newly created spaces. We also identified ??, ?? and ??duals of the spaces. Finally, we tried to characterize some matrix transformations between these spaces.
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Further Results on (Δ_s^j,f)-Lacunary Statistical Convergence of Double Sequences of order α
Article
  • Jun 2024
This paper defines the space S_(θ_uv)^α (Δ_s^j,f), encompassing all sequences that are (Δ_s^j,f)-lacunary statistically convergent of order α, utilizing an unbounded modulus function f, a double lacunary sequence θ_uv={(k_u,l_v )}, a generalized difference operator Δ_s^j, and a real number α ∈ (0,1]. Additionally, the space ω_(θ_uv)^α (Δ_s^j,f) is introduced to include all sequences that are strongly (Δ_s^j,f)-lacunary summable of order α. The paper investigates properties associated with these spaces, and under specific conditions, inclusion relations between the spaces S_(θ_uv)^α (Δ_s^j,f) and ω_(θ_uv)^α (Δ_s^j,f) are established.
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On ρ-Statistically Convergence Defined by a Modulus Function in Fuzzy Difference Sequences
Article
  • Apr 2024
In this study, we first introduced the definition ∆_ρ^m-statistical convergence for sequences of fuzzy numbers using the generalized difference operator ∆^m. Furthermore, we defined the strong N_F^ρ (∆^m,q)-summable sequence set and the strong N_F^ρ (∆^m,f,q)-summable sequence set for fuzzy difference sequences aided by a modulus function f. Subsequently, we provided certain inclusion theorems between these sets and the S_F^ρ (∆^m ) set.
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On Some Generalized Sequence Spaces
Article
Full-text available
  • Jan 1989
We introduce a general sequence space Δ V (X)={x=(x k ): (v k x k -v k+1 x k+1 )∈X}, where X is any sequence space. We establish some inclusion relations, topological results, in general case and we characterize the continuous, α-, β- and γ-duals of Δ V (X) for various sequence spaces X. The results of this paper, in a particular case, include the corresponding results of H. Kizmaz [Can. Math. Bull. 24, No. 2, 169-176 (1981; Zbl 0454.46010)].
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On Certain Sequence Spaces
Article
  • Jun 1981
  • CAN MATH BULL
In this paper define the spaces l ∞ (Δ), c(Δ), and c 0 (Δ), where for instance l ∞ (Δ) = {x=(x k ):sup k |x k -x k + l |< ∞} , and compute their duals (continuous dual, α-dual, β-dual and γ-dual). We also determine necessary and sufficient conditions for a matrix A to map l ∞ (Δ) or c(Δ) into l ∞ or c , and investigate related questions.
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On certain sequence spaces and their Köthe-Toeplitz duals
Article
  • Jan 1993
We introduce a general sequence space X v , where X is any sequence space and establish some inclusion relations, topological results. Furthermore we give α- and β-duals of sequence spaces [ℓ(p)] v , [c 0 (p)] v , [ℓ ∞ (p)] v and [c(p)] v together with α-duals of sequence spaces ℓ(p), c 0 (p), ℓ ∞ (p) and c(p). The perfectness of sequence spaces [ℓ(p)] v , [ℓ ∞ (p)] v and [c 0 (p)] v is also examined in this paper. Our results include the corresponding results of Dutta, Srivastava, Gurusingh and others.
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On Invariant Sequence Spaces
  • Jan 1989
  • 881-887
  • R Çolak
R. Çolak, On Invariant Sequence Spaces, Erciyes Univ. Journal of Sci. 5 (1989), 881-887.
Keywords and phrases: difference sequence spaces
  • Jan 1970
  • I J Maddox
I.J. Maddox, Elements of Functional Analysis, Cambridge Univ. Press, 1970. Keywords and phrases: difference sequence spaces, Köthe-Toeplitz dual.