Generalized Difference Spaces of Non-Absolute Type of Convergent and Null Sequences

and Applied Analysis 3 2. The Difference Sequence Spaces c 0 B and c λ B of Non-Absolute Type The difference sequence spaces have been studied by several authors in different ways see e.g. 12, 16–21 . In the present section, we introduce the spaces c 0 Δ , and c λ Δ , and show that these spaces are BK-spaces of non-absolute type which are norm isomorphic to the spaces c0 and c, respectively. We assume throughout that λ λk ∞ k 0 is a strictly increasing sequence of positive reals tending to∞, that is, 0 < λ0 < λ1 < · · · , lim k→∞ λk ∞. 2.1 Recently, Mursaleen and Noman 22 studied the sequence spaces c 0 and c λ of nonabsolute type, and later they introduced the difference sequence spaces c 0 Δ and c λ Δ in 21 of non-absolute type as follows: c 0 Δ { x xk ∈ ω : lim n→∞ 1 λn n ∑ k 0 λk − λk−1 xk − xk−1 0 } , c Δ { x xk ∈ ω : lim n→∞ 1 λn n ∑ k 0 λk − λk−1 xk − xk−1 exists } . 2.2 Here and after, we use the convention that any termwith a negative subscript is equal to zero, for example, λ−1 0 and x−1 0. With the notation of 1.2 we can redefine the spaces c 0 Δ and c Δ by c 0 Δ ( c 0 ) Δ , c Δ ( c ) Δ , 2.3 where Δ denotes the band matrix representing the difference operator, that is, Δx xk − xk−1 ∈ ω for x xk ∈ ω. Let r and s be nonzero real numbers and define the generalized difference matrix B r, s {bnk r, s } by bnk r, s : ⎧ ⎪ ⎨ ⎪ ⎩ r, k n, s, k n − 1, 0, otherwise, 2.4 for all k, n ∈ N. The B r, s -transform of a sequence x xk is B r, s k x rxk sxk−1, ∀k ∈ N. 2.5 We note that the matrix B r, s can be reduced to the difference matrices Δ in case r 1 and s −1. So, the results related to the matrix domain of the matrix B r, s are more general and more comprehensive than the consequences of the matrices domain of Δ and include them. 4 Abstract and Applied Analysis Now, following Başar and Altay 18 and Aydın and Başar 17 , we proceed slightly differently to Kızmaz 19 and the other authors following him and employ a technique of obtaining a new sequence space by means of the matrix domain of a triangle limitation method. We thus introduce the difference sequence spaces c 0 B and c λ B , which are the generalization of the spaces c 0 Δ and c λ Δ introduced by Mursaleen and Noman 21 , as follows: c 0 B { x xk ∈ ω : lim n→∞ 1 λn n ∑ k 0 λk − λk−1 rxk sxk−1 0 } , c B { x xk ∈ ω : lim n→∞ 1 λn n ∑ k 0 λk − λk−1 rxk sxk−1 exists } . 2.6 With the notation of 1.2 , we can redefine the spaces c 0 B and c λ B as c 0 B ( c 0 )


Introduction
By ω, we denote the space of all complex valued sequences. Any vector subspace of ω is called a sequence space. A sequence space E with a linear topology is called a K-space provided each of the maps p i : E → C defined by p i x x i is continuous for all i ∈ N, where C denotes the complex field and N {0, 1, 2, . . .}. A K-space is called an FK-space provided E is a complete linear metric space. An FK-space whose topology is normable is called a BK-space see 1, pages 272-273 which contains φ, the set of all finitely nonzero sequences. We write ∞ , c and c 0 for the classical sequence spaces of all bounded, convergent, and null sequences, respectively, which are BK-spaces with the usual sup-norm defined by x ∞ sup |x k |, where, here and in the sequel, the supremum is taken over all k ∈ N. Also by 1 and 2 Abstract and Applied Analysis from 0 to ∞. Also by bs and cs, we denote the spaces of all bounded and convergent series, respectively.
Let X and Y be two sequence spaces, and let A a nk be an infinite matrix of complex numbers a nk , where n, k ∈ N. Then, we say that A defines a matrix mapping from X into Y and we denote it by writing A : X → Y , if for every sequence x x k ∈ X the sequence Ax {A n x }, A-transform of x, exists and is in Y , where A n x : k a nk x k , ∀n ∈ N. 1.1 By X : Y , we denote the class of all infinite matrices A a nk such that A : X → Y . Thus A ∈ X : Y if and only if the series on the right side of 1.1 converges for each n ∈ N and every x ∈ X, and Ax ∈ Y for all x ∈ X. A sequence x ∈ ω is said to be A-summable to l if Ax converges to l, which is called the A-limit of x.
The domain X A of an infinite matrix A in a sequence space X is defined by We denote the collection of all finite subsets of N by F. Also, we write e k for the sequence whose only nonzero term is a 1 in the kth place for each k ∈ N.
The approach of constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by several authors, for example, 2-14 . They introduced the sequence spaces ∞ N q and c N q in 14 , p C 1 X p and ∞ C 1 X ∞ in 10 , μ G Z u, v; μ in 9 , ∞ R t r t ∞ , c R t r t c and c 0 R t r t 0 in 8 , p R t r t p in 2 , c 0 E t e r 0 and c E r e r c in 3 , p E r e r p and ∞ E r e r ∞ in 4 , c 0 A r a r 0 and c A r a r c in 5 , c 0 u, p A r a r 0 u, p and c u, p A r a r c u, p in 6 , p A r a r p and ∞ A r a r ∞ in 7 and c 0 C 1 c 0 and c C 1 c in 11 , ν B r,s,t ν B in 12 , and f B r,s,t f B in 13 ; where, N q , C 1 , R t , and E r denote the Nörlund, Cesáro, Riesz, and Euler means, respectively, A r , G, and B r, s, t are, respectively, defined in 5, 9, 12 , μ ∈ {c 0 , c, p }, ν ∈ { ∞ , c, c 0 , p } and 1 ≤ p < ∞. Also c 0 u, p and c u, p denote the sequence spaces generated from the Maddox's spaces c 0 p and c p by Başarir 15 . In the present paper, following 2-14 , we introduce the difference sequence spaces c λ 0 B and c λ B of non-absolute type and derive some related results. We also establish some inclusion relations. Furthermore, we determine the α-, β-, and γ-duals of those spaces and construct their bases. Finally, we characterize some classes of infinite matrices concerning the spaces c λ 0 B and c λ B . The rest of this paper is organized, as follows. In Section 2, the BK-spaces c λ 0 B and c λ B of generalized difference sequences are introduced. Section 3 is devoted to inclusion relations concerning with the spaces c λ 0 B and c λ B . In Sections 4 and 5, the Schauder bases of the spaces c λ 0 B and c λ B are given and the α-, β-, and γ-duals of the generalized difference sequence spaces c λ 0 B and c λ B of non-absolute type are determined, respectively. In Section 6, the classes c λ B : p , c λ 0 B : p , c λ B : c , c λ B : c 0 , c λ 0 B : c , and c λ 0 B : c 0 of matrix transformations are characterized, where 1 ≤ p ≤ ∞. Also, by means of a given basic lemma, the characterizations of some other classes involving the Euler, difference, Riesz, and Cesàro sequence spaces are derived. In the final section of the paper, we note the significance of the present results in the literature related with difference sequence spaces and record some further suggestions.

The Difference Sequence Spaces c λ 0 B and c λ B of Non-Absolute Type
The difference sequence spaces have been studied by several authors in different ways see e.g. 12, 16-21 . In the present section, we introduce the spaces c λ 0 Δ , and c λ Δ , and show that these spaces are BK-spaces of non-absolute type which are norm isomorphic to the spaces c 0 and c, respectively.
We assume throughout that λ λ k ∞ k 0 is a strictly increasing sequence of positive reals tending to ∞, that is, Recently, Mursaleen and Noman 22 studied the sequence spaces c λ 0 and c λ of nonabsolute type, and later they introduced the difference sequence spaces c λ 0 Δ and c λ Δ in 21 of non-absolute type as follows:

2.2
Here and after, we use the convention that any term with a negative subscript is equal to zero, for example, λ −1 0 and x −1 0. With the notation of 1.2 we can redefine the spaces c λ 0 Δ and c λ Δ by where Δ denotes the band matrix representing the difference operator, that is, Δx We note that the matrix B r, s can be reduced to the difference matrices Δ in case r 1 and s −1. So, the results related to the matrix domain of the matrix B r, s are more general and more comprehensive than the consequences of the matrices domain of Δ and include them. Now, following Başar and Altay 18 and Aydın and Başar 17 , we proceed slightly differently to Kızmaz 19 and the other authors following him and employ a technique of obtaining a new sequence space by means of the matrix domain of a triangle limitation method.
We thus introduce the difference sequence spaces c λ 0 B and c λ B , which are the generalization of the spaces c λ 0 Δ and c λ Δ introduced by Mursaleen and Noman 21 , as follows:

2.6
With the notation of 1.2 , we can redefine the spaces c λ 0 B and c λ B as where B denotes the generalized difference matrix B r, s {b nk r, s } defined by 2.4 . It is immediate by 2.7 that the sets c λ 0 B and c λ B are linear spaces with coordinatewise addition and scalar multiplication, that is, c λ 0 B and c λ B are the sequence spaces of generalized differences.
On the other hand, we define the triangle matrix Λ λ nk by λ nk : for all n, k ∈ N. With a direct calculation we derive the equality and every x x k ∈ ω which leads us together with 1.2 to the fact that Further, for any sequence x x k we define the sequence y λ {y k λ } which will be frequently used as the Λ-transform of x, that is, y λ Λ x and so we have Abstract and Applied Analysis 5 Where, here and in what follows, the summation running from 0 to k −1 is equal to zero when k 0. Moreover, it is clear by 2.9 that the relation 2.11 can be written as follows: We assume throughout that the sequences x x k and y y k are connected by the relation 2.11 . Now, we may begin with the following theorem which is essential in the text.
Proof. Proof. To prove this, we should show the existence of a linear bijection between the spaces c λ 0 B and c 0 . Consider the transformation T defined, with the notation of 2.11 , from c λ 0 B to c 0 by x → y λ . Then, Tx y λ Λ x ∈ c 0 for every x ∈ c λ 0 B and the linearity of T is clear. Further, it is trivial that x θ whenever Tx θ and hence T is injective.
Furthermore, let y y k ∈ c 0 and define the sequence x {x k λ } by 2.14 Then, we obtain Abstract and Applied Analysis Hence, for every n ∈ N, we get by 2.9

2.16
This shows that Λ x y and since y ∈ c 0 , we conclude that Λ x ∈ c 0 . Thus, we deduce that x ∈ c λ 0 B and Tx y. Hence T is surjective. Moreover, one can easily see for every x ∈ c λ 0 B that which means that T is norm preserving. Consequently T is a linear bijection which show that the spaces c λ 0 B and c 0 are linearly isomorphic. It is clear that if the spaces c λ 0 B and c 0 are replaced by the spaces c λ B and c, respectively, then we obtain the fact that c λ B ∼ c. This completes the proof.

The Inclusion Relations
In the present section, we establish some inclusion relations concerning with the spaces c λ 0 B and c λ B . We may begin with the following theorem. Proof. It is obvious that the inclusion c λ 0 B ⊂ c λ B holds. Further to show that this inclusion is strict, consider the sequence x x k defined by x k k j 0 −s/r j /r for all k ∈ N. Then, we obtain by 2.9 that  Proof. Suppose that s r 0 and x ∈ c. Then B r, s x rx k sx k−1 ∈ c 0 and hence B r, s x ∈ c λ 0 , since the inclusion c 0 ⊂ c λ 0 . This shows that x ∈ c λ 0 B . Consequently, the inclusion c ⊂ c λ 0 B holds. Further consider the sequence y y k defined by y k √ k 1 for all k ∈ N. Then, it is trivial that y / ∈ c. On the other hand, it can easily seen that B r, s y ∈ c 0 . Hence, B r, s y ∈ c λ 0 which means that y ∈ c λ 0 B . Thus, the sequence y is in c λ 0 B but not in c. We therefore deduce that the inclusion c ⊂ c λ 0 B is strict. This completes the proof.
On the other hand, we recall that if A ∈ c : c and B ∈ c : c , then AB ∈ c : c , namely, Λ λ nk is stronger than the ordinary convergence, hence we have the following Further, it is obvious that the sequence y, defined in the proof of Theorem 3.2, is in c λ 0 B but not in ∞ . This leads us to the following result.
Proof. Suppose that the inclusion ∞ ⊂ c λ 0 B holds. Then we obtain that Λ x ∈ c 0 for every x ∈ ∞ and hence the matrix Λ λ nk is in the class ∞ : c 0 . Thus it follows by Lemma 3.5 that lim n → ∞ k λ nk 0.

3.3
Now, by taking into account the definition of the matrix Λ λ nk given by 2.8 , we have for every n ∈ N that and since lim n → ∞ λ n−1 /λ n 1 by 3.5 , we obtain by 3.6 that which shows that z z k ∈ c λ 0 . Conversely, suppose that z z k ∈ c λ 0 . Then we have 3.8 . Further, for every n ≥ 1, we derive that

3.9
Then, 3.9 and 3.8 together imply that 3.6 holds. On the other hand, we have for every n ≥ 1 that

3.10
Therefore, it follows by 3.6 that lim n → ∞ rλ n−1 s λ n − λ 0 /λ n 0. Particularly, if we take r 1 and s −1, then we have lim n → ∞ λ n − λ n−1 − λ 0 /λ n 0 which shows that 3.5 holds. Thus, we deduce by the relation 3.4 that 3.3 holds. This leads us with Lemma 3.5 to the consequence that Λ ∈ ∞ : c 0 . Hence, the inclusion ∞ ⊂ c λ 0 B holds and is strict by Corollary 3.4. This completes the proof.

The Bases for the Spaces c λ 0 B and c λ B
In the present section, we give two sequences of the points of the spaces c λ 0 B and c λ B which form the bases for those spaces.
If a normed sequence space X contains a sequence b n with the property that for every x ∈ X there is a unique sequence of scalars α n such that Abstract and Applied Analysis 9 then b n is called a Schauder basis or briefly basis for X. The series k α k b k which has the sum x is then called the expansion of x with respect to b n and is written as x k α k b k . Now, since the transformation T defined from c λ 0 B to c 0 in the proof of Theorem 2.3 is an isomorphism, the inverse image of the basis {e k } ∞ k 0 of the space c 0 is the basis for the new space c λ 0 B . Therefore, we have the following.

4.2
Then, the following statements hold.
a The sequence {b k λ } ∞ k 0 is a basis for the space c λ 0 B and any x ∈ c λ 0 B has a unique representation of the form

} is a basis for the space c λ B and any x ∈ c λ B has a unique representation of the form
Finally, it easily follows from Theorem 2.1 that c λ 0 B and c λ B are the Banach spaces with their natural norms. Then by Theorem 4.1 we obtain the following.

The α-, β-, and γ-Duals of the Spaces c λ 0 B and c λ B
In this section, we state and prove the theorems determining the α-, β-, and γ-duals of the generalized difference sequence spaces c λ 0 B and c λ B of non-absolute type. For arbitrary sequence spaces X and Y , the set M X, Y defined by is called the multiplier space of X and Y . One can easily observe for a sequence space Z with Y ⊂ Z and Z ⊂ X that the inclusions M X, Y ⊂ M X, Z and M X, Y ⊂ M Z, Y hold, respectively. With the notation of 5.1 , 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 , X γ M X, bs .

5.2
It is clear that X α ⊂ X β ⊂ X γ . Also it can be obviously seen that the inclusions X α ⊂ Y α , X β ⊂ Y β , and X γ ⊂ Y γ hold whenever Y ⊂ X. Now, we may begin with quoting the following lemmas see 25 which are needed to prove Theorems 5.5 to 5.8. where the matrix H λ h λ nk is defined via the sequence a a n ∈ ω by λ n r λ n − λ n−1 a n , k n, 0, k > n

5.8
for all n, k ∈ N.
Proof. Let a a n ∈ ω. Then, by bearing in mind the relations 2.11 and 2.14 , it is immediate that the equality a n x n holds for all n ∈ N. Thus, we observe by 5.9 that ax a n x n ∈ 1 whenever x x k ∈ c λ 0 B or c λ B if and only if H λ y ∈ 1 whenever y y k ∈ c 0 or c. This means that the sequence a a n ∈ {c λ 0 B } α or a a n ∈ {c λ B } α if and only if H λ ∈ c 0 : 1 c : 1 . We therefore obtain by Lemma

Abstract and Applied Analysis
Proof. Consider the equality n k 0 1 r λ n λ n − λ n−1 a n y n n−1 k 0 a k n y k 1 r λ n λ n − λ n−1 a n y n T λ n y ∀n ∈ N,

5.12
where the matrix T λ t λ nk is defined by t λ nk : Finally, we close this section with the following theorem which determines the γ-dual of the spaces c λ 0 B and c λ B : for all k, m, n ∈ N provided the convergence of the series.
The following lemmas will be needed in proving our main results. ∈ cs for each fixed n ∈ N, 6.7 lim k → ∞ λ k r λ k − λ k−1 a nk a n for each fixed n ∈ N, 6.8 a n ∈ p . 6.9 (ii) A ∈ c λ B : ∞ if and only if 6.7 and 6.8 hold, and sup n∈N k | a nk | < ∞, 6.10 a n ∈ ∞ . 6.11 Proof. Suppose that the conditions 6.4 -6.9 hold and take any x x k ∈ c λ B . Then, we have by Theorem 5.6 that {a nk } k∈N ∈ {c λ B } β for all n ∈ N and this implies that the Atransform of x exists. Also, it is clear that the associated sequence y y k is in the space c and hence y k → l as k → ∞ for some suitable l. Further, it follows by combining Lemma 6.2 with 6.5 that the matrix A a nk is in the class c : p , where 1 ≤ p < ∞.