On the β-Dual of Banach-Space-Valued Difference Sequence Spaces
The main object of the paper is to introduce Banach-space-valued difference sequence spaces ℓ ∞(X, Δ), c(X, Δ); and c0(X, Δ) as a generalization of the well-known difference sequence spaces of Kizmaz. We obtain a set of sufficient conditions for (A k ) ∈ E β (X, Δ); where E ∈ {ℓ ∞, c, c 0} and (A...
Gespeichert in:
| Datum: | 2013 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2013
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2497 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508396722061312 |
|---|---|
| author | Bhardwaj, V. K. Gupta, S. Бхардвай, В. К. Гупта, С. |
| author_facet | Bhardwaj, V. K. Gupta, S. Бхардвай, В. К. Гупта, С. |
| author_sort | Bhardwaj, V. K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:16:44Z |
| description | The main object of the paper is to introduce Banach-space-valued difference sequence spaces ℓ ∞(X, Δ), c(X, Δ); and c0(X, Δ) as a generalization of the well-known difference sequence spaces of Kizmaz. We obtain a set of sufficient conditions for (A k ) ∈ E β (X, Δ); where E ∈ {ℓ ∞, c, c 0} and (A k ) is a sequence of linear operators from a Banach space X into another Banach space Y: Necessary conditions for (A k ) ∈ E β (X, Δ) are also investigated. |
| first_indexed | 2026-03-24T02:24:33Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.9
V. K. Bhardwaj (Kurukshetra Univ., India),
S. Gupta (Arya P. G. College, India)
ON β-DUAL OF BANACH-SPACE-VALUED DIFFERENCE SEQUENCE SPACES
ПРО β-ДУАЛЬНI ПРОСТОРИ ДИФЕРЕНЦIАЛЬНИХ ПОСЛIДОВНОСТЕЙ
IЗ ЗНАЧЕННЯМИ У БАНАХОВИХ ПРОСТОРАХ
The main object of the paper is to introduce Banach-space-valued difference sequence spaces `∞(X,∆), c(X,∆), and
c0(X,∆) as a generalization of the well-known difference sequence spaces of Kizmaz. We obtain a set of sufficient
conditions for (Ak) ∈ Eβ(X,∆), where E ∈ {`∞, c, c0} and (Ak) is a sequence of linear operators from a Banach space
X into another Banach space Y. Necessary conditions for (Ak) ∈ Eβ(X,∆) are also investigated.
Основна мета статтi — ввести простори диференцiальних послiдовностей `∞(X,∆), c(X,∆), значення яких лежать
у банаховому просторi, i c0(X,∆), як узагальнення добре вiдомих просторiв диференцiальних послiдовностей
Кiзмаза. Встановлено низку достатнiх умов для (Ak) ∈ Eβ(X,∆), де E ∈ {`∞, c, c0}, а (Ak) — послiдовнiсть
лiнiйних операторiв iз банахового простору X в iнший банахiв простiр Y. Дослiджено також i необхiднi умови для
(Ak) ∈ Eβ(X,∆).
1. Introduction and background. Let X, Y be Banach spaces with zero element θ and ‖ · ‖ denote
the norm in either X or Y. Let B(X,Y ) be the Banach space of bounded linear operators on X into
Y with the usual operator norm. S =
{
x ∈ X : ‖x‖ ≤ 1
}
is the closed unit sphere in X. By s(X)
we mean the space of all X-valued sequences x = (xk), where xk ∈ X, for each k ∈ N, the set
of positive integers. In case X = C, the space of complex numbers, s(X) reduces to s, the space
of all scalar sequences. `∞(X), c(X) and c0(X) denote the Banach spaces of bounded, convergent
and null X-valued sequences respectively, normed by ‖x‖∞ = supk ‖xk‖. Let A = (Ak) denote a
sequence of linear but not necessarily bounded operators on X into Y. If E is any nonempty subset
of s(X), then the α- and β-duals of E were defined by Maddox [11] as follows:
Eα =
{
(Ak) :
∑
k
‖Akxk‖ <∞, for all x = (xk) ∈ E
}
,
Eβ =
{
(Ak) :
∑
k
Akxk converges in Y for all x = (xk) ∈ E
}
.
All sums without limits will be taken from k = 1 to k = ∞. Since Y is complete, we have
Eα ⊂ Eβ. The α- and β- duals of E may be regarded as generalized Köthe – Toeplitz duals, since in
case X = Y = C, when the (Ak) may be identified with complex numbers ak, the duals reduce to
the classical spaces first considered by Köthe and Toeplitz (see, for instance, [8]).
Maddox [11] determined Köthe – Toeplitz duals, in the operator case, for the sequence spaces
`∞(X), c(X) and c0(X). The results indicate the gap between the operator and the ordinary scalar
c© V. K. BHARDWAJ, S. GUPTA, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8 1145
1146 V. K. BHARDWAJ, S. GUPTA
case. For example, in the scalar case, it is well known that `β∞ = cβ = cβ0 = `1 (the space of
absolutely summable sequences of scalars). However, for the operator case it is possible only to assert
that `β∞(X) ⊂ cβ(X) ⊂ cβ0 (X). But, as far as, α-duals are concerned, Maddox [11] showed that
`α∞(X) = cα(X) = cα0 (X) which is natural extension of the scalar case where `α∞ = cα = cα0 = `1.
Inspired from the work of Maddox, many mathematicians have contributed in the determination
of generalized Köthe – Toeplitz duals of various vector valued sequence spaces (see, for instance, [14,
16, 17] where many more references can be found).
The concept of difference sequence spaces was introduced by Kizmaz [9] as follows:
E(∆) =
{
x = (xk) ∈ s : (∆xk) ∈ E
}
,
where E ∈ {`∞, c, c0} and ∆xk = xk − xk+1, for all k ∈ N. For a detailed account of differ-
ence sequence spaces one may refer to [1 – 7, 9, 12, 13]. It is well known [3, 4, 6, 9, 13] that
`α∞(∆) = cα(∆) = cα0 (∆) = D1, where D1 =
{
a = (ak) :
∑
k|ak| < ∞
}
, and `β∞(∆) =
= cβ(∆) = D2, where D2 =
{
a = (ak) :
∑
kak is convergent,
∑∣∣∣∑∞
v=k+1
av
∣∣∣ <∞} whereas
cβ0 (∆) =
{
a = (ak) :
∑
ak
(∑k
j=0
vj
)
converges for all v ∈ c+
0 }
⋂
{a = (ak) :
∑∣∣∣∑∞
j=k
aj
∣∣∣ <
< ∞
}
, where c+
0 denotes the set of all positive sequences in c0. Thus [13] (Theorem 3) `β∞(∆) =
= cβ(∆) 6= cβ0 (∆).
The main object of this paper is to introduce the Banach-space-valued difference sequence spaces
`∞(X,∆), c(X,∆), and c0(X,∆) as a generalization of the classical difference sequence spaces
of Kizmaz. We obtain a set of sufficient conditions for (Ak) ∈ Eβ(X,∆), where E ∈ {`∞, c, c0}.
Necessary conditions for (Ak) ∈ Eβ(X,∆) have also been investigated.
The following definition and well-known lemmas are required for establishing the results of this
paper.
Let (Tk) = (T1, T2, T3, . . .) be a sequence in B(X,Y ). Then the group norm of (Tk) is defined
by
∥∥(Tk)
∥∥ = sup
∥∥∥∥∑n
k=1
Tkxk
∥∥∥∥ where the supremum is taken over all n ∈ N and all xk in
S. This concept was introduced by Robinson [15] and was termed as group norm by Lorentz and
Macphail [10].
We write Rn for the n th tail of the sequence (Tk), i. e., Rn = (Tn, Tn+1, Tn+2, . . .).
Lemma 1.1 [11]. If (Tk) be a sequence in B(X,Y ), then∥∥∥∥∥
n+p∑
k=n
Tkxk
∥∥∥∥∥ ≤ ‖Rn‖max
{
‖xk‖ : n ≤ k ≤ n+ p
}
for any xk and all n ∈ N and all nonnegative integers p.
Lemma 1.2 [16]. If (Tk) is a sequence in B(X,Y ), then exactly one of the following is true:
(i) ‖Rn‖ =∞ for all n ≥ 1,
(ii) ‖Rn‖ <∞ for all n ≥ 1.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
ON β-DUAL OF BANACH-SPACE-VALUED DIFFERENCE SEQUENCE SPACES 1147
We now introduce the following sequence spaces:
c0(X,∆) =
{
x = (xk) ∈ s(X) : (∆xk) ∈ c0(X)
}
,
c(X,∆) =
{
x = (xk) ∈ s(X) : (∆xk) ∈ c(X)
}
,
`∞(X,∆) =
{
x = (xk) ∈ s(X) : (∆xk) ∈ `∞(X)
}
.
If we take X = C, then we obtain the familiar difference sequence spaces c0(∆), c(∆) and
`∞(∆) of Kizmaz [9], respectively.
It is easy to see that these sequence spaces are BK spaces with the norm ‖x‖∆ = ‖x1‖+‖∆x‖∞,
x = (xk) ∈ E(X,∆), ∆x = (∆xk) = (xk − xk+1) for E ∈ {`∞, c, c0}.
2. Main results. We start this section by investigating necessary conditions for (Ak) ∈ Eβ(X,∆)
where E ∈ {`∞, c, c0.}. It is also shown that these conditions do not turn out to be sufficient.
Theorem 2.1 (Necessity). If (Ak) ∈ cβ0 (X,∆), then there exists m ∈ N such that:
(i) Ak ∈ B(X,Y ) for all k ≥ m,
(ii)
∥∥Rm(λ)
∥∥ <∞ for some λ > 1, where Rm(λ) =
(
mλ−mAm, (m+1)λ−(m+1)A(m+1), . . .
)
,
(iii)
∥∥Rn(λ)
∥∥→ 0 as n→∞.
Proof. Suppose that (Ak) ∈ cβ0 (X,∆) but no m ∈ N exists for which Ak ∈ B(X,Y ) for all
k ≥ m. Proceeding as in [11] (Proposition 3.1), we get a strictly increasing sequence (ki) of natural
numbers and a sequence (zi) in S such that ‖Akizi‖ > i for each i ≥ 1.
Define
xk =
zi
i
, for k = ki, i ≥ 1,
θ, otherwise.
Then (xk) ∈ c0(X) ⊂ c0(X,∆) but ‖Akxk‖ > 1 for infinitely many k, which is a contradiction to
the fact that
∑
Akxk converges. Hence the Ak’s are ultimately bounded.
Next suppose that (ii) fails, i.e., ‖Rm(λ)‖ =∞ for all λ > 1. By Lemma 1.2, we have
∥∥Rn(λ)
∥∥ = sup
p∈N,zk∈S
∥∥∥∥∥
n+p∑
k=n
kλ−kAkzk
∥∥∥∥∥ =∞
for all n ≥ m and for all λ > 1. Then there exists a subsequence m = n(1) < n(2) < . . . of natural
numbers and a sequence (zk) in S such that
∥∥∥∑n(i+1)
k=1+n(i)
kλ−kAkzk
∥∥∥ > 1 for each i ≥ 1 and for
all λ > 1.
Define
xk =
kλ
−kzk, for n(i) < k ≤ n(i+ 1), i ≥ 1,
θ, otherwise.
Then we have (xk) ∈ c0(X) ⊂ c0(X,∆) but
∥∥∥∑n(i+1)
k=1+n(i)
Akxk
∥∥∥ > 1 for each i ≥ 1 showing that∑
Akxk does not converge in Y which is again a contradiction.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1148 V. K. BHARDWAJ, S. GUPTA
Let, if possible, (iii) fails, say lim supn
∥∥Rn(λ)
∥∥ = 3p > 0. Following Maddox [11] (Propo-
sition 3.3), there exist natural numbers n(1) ≥ m(1) > m and zm(1), . . . , zn(1) in S such that∥∥∥∑n(1)
k=m(1)
kλ−kAkzk
∥∥∥ > p. Choose m(2) > n(1) such that
∥∥Rm(2)(λ)
∥∥ > 2p. Then there
exist n(2) ≥ m(2) and zm(2), . . . , zn(2) in S such that
∥∥∥∑n(2)
k=m(2)
kλ−kAkzk
∥∥∥ > p. Proceeding
in this way, we define xk = θ
(
1 ≤ k < m(1)
)
, xk = kλ−kzk
(
m(1) ≤ k ≤ n(1)
)
, xk = θ(
n(1) < k < m(2)
)
, xk = kλ−kzk
(
m(2) ≤ k ≤ n(2)
)
, etc. Then (xk) ∈ c0(X) ⊂ c0(X,∆) but∑
Akxk diverges, which gives a contradiction.
Remark 2.1. It is clear that the conditions of Theorem 2.1 are also necessary for (Ak) ∈
∈ Eβ(X,∆), where E = `∞ or c.
Remark 2.2. The conditions of Theorem 2.1 are not sufficient for (Ak) ∈ cβ0 (X,∆) and hence
for (Ak) ∈ Eβ(X,∆), where E = `∞ or c, as is clear from the following example.
Example 2.1. Let X = Y = c0. Define Ak : X → Y as Ak(x) = (0, 0, . . . , k−1xk, 0, 0, . . .)
with k−1xk in the k-position, where x = (xk) ∈ c0. Then Ak ∈ B(X,Y ) for all k ∈ N, and for any
n ∈ N and λ > 1,
‖Rn(λ)‖ =
∥∥∥(nλ−nAn, (n+ 1)λ−(n+1)A(n+1), . . .)
∥∥∥ =
= supxk∈S,p∈N
∥∥∥∥∥
n+p∑
k=n
(kλ−kAk)xk
∥∥∥∥∥ ≤ 1
λn
so that
∥∥Rn(λ)
∥∥→ 0 as n→∞.
Let 0 6= x ∈ C and define a sequence (xn) whose each term xn is itself the sequence
(x, 2x, 3x, . . .). Then (∆xn) = (xn − xn+1)n∈N =
(
(0, 0, . . .), (0, 0, . . .), (0, 0, . . .) . . .
)
which con-
verges to (0, 0, . . .) as n → ∞ so that (∆xn) ∈ c0(c0) and hence (xn) ∈ c0(c0,∆). However∑n
k=1
Akxk = (x, x, x, . . . x, 0, . . .) with entry x in the first n positions and 0 elsewhere and so∑
k
Akxk is not convergent.
Although the conditions of Theorem 2.1 are not sufficient for (Ak) ∈ Eβ(X,∆) where E ∈
∈ {`∞, c, c0}, it is quite interesting to note that if we take λ = 1 in condition (ii) and conditions (i)
and (iii) remaining the same, we get a set of sufficient conditions as proved below.
Theorem 2.2 (Sufficiency). (Ak) ∈ cβ0 (X,∆) if there exists m ∈ N such that:
(i) Ak ∈ B(X,Y ) for all k ≥ m,
(ii) ‖Rm‖ <∞, where Rm =
(
mAm, (m+ 1)A(m+1), . . .
)
,
(iii) ‖Rn‖ → 0 as n→∞.
Proof. Let x = (xk) ∈ c0(X,∆). Then (xk − xk+1) ∈ c0(X) and so supk ‖xk − xk+1‖ < ∞.
Now
∥∥x1 − xk+1
∥∥ =
∥∥∥∥∥
k∑
v=1
(xv − xv+1)
∥∥∥∥∥ ≤
k∑
v=1
‖xv − xv+1‖ = O(k)
and so ‖xk‖ ≤ ‖∆xk‖+ ‖xk+1 − x1‖+ ‖x1‖, for every k, which implies that supk k
−1‖xk‖ <∞.
Let ε > 0 be given. For n ≥ m and a nonnegative integer p, by Lemma 1.1 we have
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
ON β-DUAL OF BANACH-SPACE-VALUED DIFFERENCE SEQUENCE SPACES 1149∥∥∥∥∥
n+p∑
k=n
Akxk
∥∥∥∥∥ =
∥∥∥∥∥
n+p∑
k=n
kAk(k
−1xk)
∥∥∥∥∥ ≤ ‖Rn‖maxn≤k≤n+p k
−1‖xk‖ ≤ ‖Rn‖M,
where M = supk k
−1‖xk‖.
We can choose k1 ≥ m such that ‖Rk‖ <
ε
M
for all k ≥ k1. Consequently we have∥∥∥∑n+p
k=n
Akxk
∥∥∥ < ε for all n ≥ k1 and a nonnegative integer p, whence the completeness of Y
implies that
∑
Akxk converges.
Remark 2.3. It is clear that the conditions of Theorem 2.2 are also sufficient for (Ak) ∈
∈ Eβ(X,∆), where E = `∞ or c.
Remark 2.4. The conditions of Theorem 2.2 are not necessary for (Ak) ∈ cβ0 (X,∆) otherwise
`β∞(X,∆) = cβ(X,∆) = cβ0 (X,∆), contrary to the case when X = C since `β∞(∆) = cβ(∆) 6=
6= cβ0 (∆), as mentioned before.
Remark 2.5. Although it seems that the conditions of the Theorem 2.2 are also not necessary
for (Ak) ∈ Eβ(X,∆) where E = `∞ or c, but we have not been able to prove it and hence is an
open problem.
3. Some further generalizations. The difference sequence spaces of Kizmaz were generalized
by Et and Çolak [5] as follows:
Let r be a nonnegative integer. Then E(∆r) =
{
x = (xk) : (∆rxk) ∈ E
}
for E ∈ {`∞, c, c0},
where ∆0x = (xk) and ∆rxk = ∆r−1xk −∆r−1xk+1, for all k ∈ N. The sequence spaces E(∆r)
are BK spaces normed by
‖x‖∆ =
r∑
i=1
|xi|+ ‖∆rx‖∞, E ∈ {`∞, c, c0}.
Analogously, we define the following X-valued generalized difference sequence spaces
E(X,∆r) = {x = (xk) ∈ s(X) : (∆rxk) ∈ E(X)} for E ∈ {`∞, c, c0}. Obviously, taking X = C
we have E(X,∆r) = E(∆r). Proceeding on the lines similar to the scalar case it is not a big issue
to see that E(X,∆r) are BK spaces with norm ‖x‖∆ =
∑r
i=1
‖xi‖ + ‖∆rx‖∞, E ∈ {`∞, c, c0}
and to have the following simple but useful lemma.
Lemma 3.1. supk ‖∆rxk‖ <∞ implies supk k
−r‖xk‖ <∞.
Theorem 3.1 (Necessity). If (Ak) ∈ cβ0 (X,∆r), then there exists m ∈ N such that:
(i) Ak ∈ B(X,Y ) for all k ≥ m,
(ii) ‖Rm(λ)‖ <∞ for some λ > 1, where Rm(λ) =
(
mrλ−mAm, (m+1)rλ−(m+1)A(m+1), . . .
)
,
(iii) ‖Rn(λ)‖ → 0 as n→∞.
The proof is similar to that of Theorem 2.1 and hence is omitted.
Remark 3.1. The conditions of Theorem 3.1 are also necessary for (Ak) ∈ Eβ(X,∆r), where
E = `∞ or c.
Remark 3.2. From Example 2.1, it is clear that the conditions of Theorem 3.1 are not sufficient
for (Ak) ∈ Eβ(X,∆r), where E ∈ {`∞, c, c0}.
Using Lemma 3.1 and applying the same technique as in Theorem 2.2, we have the following
theorem.
Theorem 3.2 (Sufficiency). (Ak) ∈ cβ0 (X,∆r) if there exists m ∈ N such that:
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
1150 V. K. BHARDWAJ, S. GUPTA
(i) Ak ∈ B(X,Y ) for all k ≥ m,
(ii) ‖Rm‖ <∞, where Rm =
(
mrAm, (m+ 1)rA(m+1), . . .
)
,
(iii) ‖Rn‖ → 0 as n→∞.
Remark 3.3. The conditions of Theorem 3.2 are not necessary for (Ak) ∈ cβ0 (X,∆r) otherwise
cβ0 (X,∆r) = cβ(X,∆r) = `β∞(X,∆r), contrary to the case when X = C and r = 1.
Remark 3.4. To see that the conditions of Theorem 3.2 are not necessary for (Ak) ∈ Eβ(X,∆r),
where E = `∞ or c, is an open problem.
The difference sequence spaces of Kizmaz were also generalized by Gnanaseelan and Srivas-
tava [7] as follows:
Let v = (vk) be a sequence of non-zero complex numbers such that
|vk|
|vk+1|
= 1 +O
(1
k
)
for each k,
k−1|vk|
∑k
i=1
|v−1
i | = O(1),(
k|v−1
k |
)
is a sequence of positive numbers increasing monotonically to infinity.
Then E(∆v) = {x = (xk) : (∆vxk) ∈ E} for E ∈ {`∞, c, c0}, where ∆vxk = vk(xk − xk+1),
for all k ∈ N.
We define E(X,∆v) = {x = (xk) ∈ s(X) : (∆vxk) ∈ E(X)} for E ∈ {`∞, c, c0}. Obviously,
taking X = C and v = (vk) = (1, 1, 1, . . .), we get back the classical spaces of Kizmaz.
The following extension of Lemma 1 of [7] is a useful tool for obtaining the sufficient conditions
for (Ak) ∈ Eβ(X,∆v), where E ∈ {`∞, c, c0}.
Lemma 3.2. supk ‖vk(xk − xk+1)‖ <∞ implies supk k
−1‖vkxk‖ <∞.
Arguing in the same way as in Theorem 2.1, we have the following theorem.
Theorem 3.3 (Necessity). If (Ak) ∈ cβ0 (X,∆v), then there exists m ∈ N such that
(i) Ak ∈ B(X,Y ) for all k ≥ m,
(ii) ‖Rm(λ, v)‖ < ∞ for some λ > 1, where Rm(λ, v) =
(
mλ−mv−1
m Am, (m + 1)λ−(m+1) ×
× v−1
m+1A(m+1), . . .
)
,
(iii) ‖Rn(λ, v)‖ → 0 as n→∞.
Remark 3.5. The conditions of Theorem 3.3 are also necessary for (Ak) ∈ Eβ(X,∆v), where
E = `∞ or c.
Remark 3.6. In view of Example 2.1, we see that the conditions of Theorem 3.3 are not suffi-
cient for (Ak) ∈ Eβ(X,∆v) where E ∈ {`∞, c, c0}.
Arguing in the same way as in Theorem 2.2 and using Lemma 3.2, we have the following
theorem.
Theorem 3.4 (Sufficiency). (Ak) ∈ cβ0 (X,∆v) if there exists m ∈ N such that:
(i) Ak ∈ B(X,Y ) for all k ≥ m,
(ii)
∥∥Rm(v)
∥∥ <∞, where Rm(v) =
(
mv−1
m Am, (m+ 1)v−1
m+1A(m+1), . . .
)
,
(iii)
∥∥Rn(v)
∥∥→ 0 as n→∞.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
ON β-DUAL OF BANACH-SPACE-VALUED DIFFERENCE SEQUENCE SPACES 1151
Remark 3.7. The conditions of Theorem 3.4 are not necessary for (Ak) ∈ cβ0 (X,∆v) otherwise
cβ0 (X,∆v) = cβ(X,∆v) = `β∞(X,∆v), contrary to the case where X = C and v = (vk) =
= (1, 1, 1, . . .).
Remark 3.8. It is an open problem to see the necessity of conditions of Theorem 3.4 for
(Ak) ∈ Eβ(X,∆v), where E = `∞ or c.
1. Altay B., Başar F. The fine spectrum and the matrix domain of the difference operator ∆ on the sequence space
`p // Commun. Math. Anal. – 2007. – 2, № 2. – P. 1 – 11.
2. Başar F., Altay B. On the space of sequences of p-bounded variation and related matrix mappings // Ukr. Math. J. –
2003. – 55, № 1. – P. 136 – 147.
3. Bektaş Ç. A., Et M., Çolak R. Generalized difference sequence spaces and their dual spaces // J. Math. Anal. and
Appl. – 2004. – 292. – P. 423 – 432.
4. Bektaş Ç. A., Çolak R. On some generalized difference sequence spaces // Thai J. Math. – 2005. – 3, № 1. – P. 83 – 98.
5. Et M., Çolak R. On some generalized difference sequence spaces // Soochow J. Math. – 1995. – 21, № 4. – P. 377 – 386.
6. Et M., Esi A. On Köthe – Toeplitz duals of generalized difference sequence spaces // Bull. Malaysian Math. Sci. Soc.
(Second Ser.). – 2000. – 23. – P. 1 – 8.
7. Gnanaseelan C., Srivastva P. D. The α-, β-, γ-duals of some generalized difference sequence spaces // Indian J.
Math. – 1996. – 38, № 2. – P. 111 – 120.
8. Kamthan P. K., Gupta M. Sequence spaces and series. – New York: Marcel Dekker, 1981.
9. Kizmaz H. On certain sequence spaces // Can. Math. Bull. – 1981. – 24. – P. 169 – 176.
10. Lorentz G. G., Macphail M. S. Unbounded operators and a theorem of A. Robinson // Trans. Roy. Soc. Can. –
1952. – 46. – P. 33 – 37.
11. Maddox I. J. Infinite matrices of operators // Lecture Notes Math. – Berlin etc.: Springer, 1980.
12. Malkowsky E., Mursaleen M., Suantai S. The dual spaces of sets of difference sequences of order m and matrix
transformations // Acta math. sinica. – 2007. – 23, № 3. – P. 521 – 532.
13. Malkowsky E., Parashar S. D. Matrix transformations in spaces of bounded and convergent difference sequences of
order m // Analysis. – 1997. – 17, № 1. – P. 87 – 97.
14. Rath N. Operator duals of some sequence spaces // Indian J. Pure and Appl. Math. – 1989. – 20, № 9. – P. 953 – 963.
15. Robinson A. On functional transformations and summability // Proc. London Math. Soc. – 1950. – 52. – P. 132 – 160.
16. Srivastava J. K., Srivastava B. K. Generalized sequence space c0(X,λ, p) // Indian J. Pure and Appl. Math. –
1996. – 27, № 1. – P. 73 – 84.
17. Suantai S., Sanhan W. On β-dual of vector valued sequence spaces of Maddox // Int. J. Math. and Math. Sci. –
2002. – 30, № 7. – P. 383 – 392.
Received 29.03.12,
after revision — 06.04.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 8
|
| id | umjimathkievua-article-2497 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:24:33Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f4/973d5eea13423317b5737e757e1cbcf4.pdf |
| spelling | umjimathkievua-article-24972020-03-18T19:16:44Z On the β-Dual of Banach-Space-Valued Difference Sequence Spaces Про β-дуальні простори диференціальних послiдовностей із значеннями у банахових просторах Bhardwaj, V. K. Gupta, S. Бхардвай, В. К. Гупта, С. The main object of the paper is to introduce Banach-space-valued difference sequence spaces ℓ ∞(X, Δ), c(X, Δ); and c0(X, Δ) as a generalization of the well-known difference sequence spaces of Kizmaz. We obtain a set of sufficient conditions for (A k ) ∈ E β (X, Δ); where E ∈ {ℓ ∞, c, c 0} and (A k ) is a sequence of linear operators from a Banach space X into another Banach space Y: Necessary conditions for (A k ) ∈ E β (X, Δ) are also investigated. Основна мета статті — ввести простори диференціальних послідовностей ℓ ∞(X, Δ), c(X, Δ); значення яких лежать у банаховому просторі, i c0(X, Δ), як узагальнення добре відомих просторів диференщальних послідовностей Кізмаза. Встановлено низку достатніх умов для (A k ) ∈ E β (X, Δ); де E ∈ {ℓ ∞, c, c 0} а (A k ) — послідовність лінійних операторів із банахового простору X в інший банахів простір Y. Досліджено також і необхідні умови для (A k ) ∈ E β (X, Δ) . Institute of Mathematics, NAS of Ukraine 2013-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2497 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 8 (2013); 1145–1151 Український математичний журнал; Том 65 № 8 (2013); 1145–1151 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2497/1759 https://umj.imath.kiev.ua/index.php/umj/article/view/2497/1760 Copyright (c) 2013 Bhardwaj V. K.; Gupta S. |
| spellingShingle | Bhardwaj, V. K. Gupta, S. Бхардвай, В. К. Гупта, С. On the β-Dual of Banach-Space-Valued Difference Sequence Spaces |
| title | On the β-Dual of Banach-Space-Valued Difference Sequence Spaces |
| title_alt | Про β-дуальні простори диференціальних послiдовностей із значеннями у банахових просторах |
| title_full | On the β-Dual of Banach-Space-Valued Difference Sequence Spaces |
| title_fullStr | On the β-Dual of Banach-Space-Valued Difference Sequence Spaces |
| title_full_unstemmed | On the β-Dual of Banach-Space-Valued Difference Sequence Spaces |
| title_short | On the β-Dual of Banach-Space-Valued Difference Sequence Spaces |
| title_sort | on the β-dual of banach-space-valued difference sequence spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2497 |
| work_keys_str_mv | AT bhardwajvk onthebdualofbanachspacevalueddifferencesequencespaces AT guptas onthebdualofbanachspacevalueddifferencesequencespaces AT bhardvajvk onthebdualofbanachspacevalueddifferencesequencespaces AT guptas onthebdualofbanachspacevalueddifferencesequencespaces AT bhardwajvk probdualʹníprostoridiferencíalʹnihposlidovnostejízznačennâmiubanahovihprostorah AT guptas probdualʹníprostoridiferencíalʹnihposlidovnostejízznačennâmiubanahovihprostorah AT bhardvajvk probdualʹníprostoridiferencíalʹnihposlidovnostejízznačennâmiubanahovihprostorah AT guptas probdualʹníprostoridiferencíalʹnihposlidovnostejízznačennâmiubanahovihprostorah |