Article
Kyungpook Mathematical Journal 2018; 58(2): 271-289
Published online June 23, 2018
Copyright © Kyungpook Mathematical Journal.
On Some Spaces Isomorphic to the Space of Absolutely q -summable Double Sequences
Hüsamettin Çapan and Feyzi Başar*
Graduate School of Natural and Applied Sciences, İstanbul University, Beyazıt Campus, 34134 - Vezneciler/İstanbul, Turkey, e-mail : husamettincapan@gmail.com, İnönü University, 44280 - Malatya, Turkey, e-mail : feyzibasar@gmail.com
Received: July 15, 2017; Accepted: June 6, 2018
Abstract
Let 0 <
Keywords: summability theory, double sequence, difference sequence space, double series, alpha-dual, beta-dual, matrix domain of 4-dimensional matrices, matrix transformations
1. Introduction
We denote the set of all complex valued double sequences by Ω which forms a vector space with coordinatewise addition and scalar multiplication. Any vector subspace of Ω is called as
By ℳ
which is a Banach space with the norm ‖ · ‖∞; where ℕ = {0, 1, 2,…}.
If for every
It is well-known that in single sequence spaces a convergent single sequence is bounded. But, in double sequence spaces a
Throughout the text the summation without limits runs from 0 to ∞, for example ∑
We denote the space of all absolutely
If we take
Let
for all
Let
It is easy to see for any two spaces
Let
We define the
We say with the notation (
For all
We shall write throughout for simplicity in notation for all
The four dimensional backward difference matrix Δ = (
for all
for all
for all
for all
It is worth mentioning here that Altay and Başar [2] have defined the spaces and by using summation matrix
In this study, we introduce the spaces and of all double sequences whose Δ-transforms and
One can easily observe that the sets and are the domain of the backward difference matrix Δ and summation matrix
for 0
for 1
2. New Sequence Spaces
In the present section, we examine some topological properties of the spaces and and also give important inclusion theorems related to them.
Theorem 2.1
We will only show with 0
Let 0
This completes the proof.
Since and , we can give following theorem without proof.
Theorem 2.2
(i)
Let 0< q < 1. Then, and are complete q-normed spaces with and , respectively. (ii)
Let 1 ≤q < ∞. Then, and are Banach spaces with and , respectively.
Now, we define the double sequences
for all
Definition 2.3
([18, p. 225]) A double sequence space
If
Theorem 2.4
Let
Let us define the double sequence
for all
To show is not monotone take
Theorem 2.5
Let
for 0
for 1
Since backward difference matrix Δ and summation matrix
Theorem 2.6
Theorem 2.7
It is immediate that and . Consider the sequence
for all
is convergent, that is,
that the series
diverges which gives the fact that . Therefore, .
Theorem 2.8
One can easily see that and . Consider the sequence Δ
i.e., , but Δ
Theorem 2.9
It is clear that and . Define
for all
is convergent. On the other hand, we see from
that the series
is divergent. Hence, .
Theorem 2.10
It is easy to see that and . If we consider the sequence
Let 0
Theorem 2.11
Theorem 2.12
It is clear that
for all
one can conclude that and . Hence, the spaces and
3. Dual Spaces
In this section, we give the
Theorem 3.1
Let 0
(i)
that is,
(ii)
that
for all
then we obtain that
which leads us to the fact that
i.e., . Nevertheless, by choosing
i.e.,
Now, by using the facts
We give the following lemma which is needed in proving the
Lemma 3.2
([17])
Now, we may give the beta-duals of the new spaces with respect to the
Let us define the sets and via the double sequence
Theorem 3.3
We will determine the necessary and sufficient conditions in order to the sequence
for all
Let us define the sequence by the relation (
where the four-dimensional matrix
for all
for all
This completes the proof.
Theorem 3.4
We prove the theorem it by the similar way used in the proof of Theorem 3.3.
Consider by (
where the four-dimensional matrix
for all
i.e.,
we have (
This completes the proof.
4. Matrix Transformations
In the present section, we characterize the classes (ℒ
Theorem 4.1
Let us consider
for all
Conversely, suppose that the condition (
for any fixed
This completes the proof.
Theorem 4.2
We obtain the necessity of the condition (
Let us define by (
for all
for all
Theorem 4.3
Take by the relation
for each
which gives that if and only if
for each
By using Theorem 4.2 and Theorem 4.3, we can give the following two theorems without proof.
Theorem 4.4
Theorem 4.5
Theorem 4.6
Let
for all
which lead us to the fact
for all
This completes the proof.
As a consequence of Theorem 4.6, we can give the following corollary.
Corollary 4.7
(i)
E = (e mnkl ) ∈ (ℒq : (Δ))if and only if the conditions ( 3.2 ) and (3.3 ) hold with fmnkl instead of a mnkl . (ii)
F = (e mnkl ) ∈ (ℒq : )if and only if the conditions ( 3.2 ) and (3.3 ) hold with emnkl instead of a mnkl . (iii)
E = (e mnkl ) ∈ (ℒq : )if and only if the condition ( 4.1 ) holds with fmnkl instead of a mnkl . (iv)
F = (e mnkl ) ∈ (ℒq : )if and only if the condition ( 4.1 ) holds with emnkl instead of a mnkl . (v)
E = (e mnkl ) ∈ ( )if and only if the condition ( 4.2 ) holds with fmnkl instead of a mnkl . (vi)
F = (e mnkl ) ∈ ( )if and only if the condition ( 4.2 ) holds with emnkl instead of a mnkl . (vii)
E = (e mnkl ) ∈ ( )if and only if the conditions ( 4.3 )–(4.6 ) hold with fmnkl instead of a mnkl . (viii)
F = (e mnkl ) ∈ ( )if and only if the conditions ( 4.3 )–(4.6 ) hold with emnkl instead of a mnkl . (ix)
E = (e mnkl ) ∈ ( )if and only if the conditions ( 4.7 ) and (4.8 ) hold with fmnkl instead of a mnkl . (x)
F = (e mnkl ) ∈ ( )if and only if the conditions ( 4.7 ) and (4.8 ) hold with emnkl instead of a mnkl . (xi)
E = (e mnkl ) ∈ ( : (Δ))if and only if the conditions ( 3.2 ) and (3.3 ) hold with fmnkl instead of a mnkl . (xii)
F = (e mnkl ) ∈ ( )if and only if the conditions ( 3.2 ) and (3.3 ) hold with emnkl instead of a mnkl .
5. Conclusion
As the domain of backward difference matrix in the space
Recently, the space ℒ
Additionally, one can generalize the main results of the present paper related to the space by using the four dimensional triangle matrix
for all
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