On Some New Generalized Difference Sequence Spaces of Non-Absolute Type

In this study, we define a new triangle matrix $\hat{W}=\{w_{nk}^{\lambda}(r,s,t)\}$ which derived by using multiplication of $\lambda=(\lambda_{nk})$ triangle matrix with $B(r,s,t)$ triple band matrix. Also, we introduce the sequence spaces $c_{0}^{\lambda}(\hat{B}),c^{\lambda}(\hat{B}),\ell_{\infty}^{\lambda}(\hat{B})$ and $\ell_{p}^{\lambda}(\hat{B})$ by using matrix domain of this matrix on the sequence spaces $c_{0},c,\ell_{\infty}$ and $\ell_{p}$ of the matrix $\hat{W}$, respectively. Moreover, we show that norm isomorphic to the spaces $c_{0},c,\ell_{\infty}$ and $\ell_{p}$, respectively. Furthermore, we establish some inclusion relations concerning with those spaces and determine $\alpha-,\beta-\gamma-$ duals of those spaces and construct their Schauder basis. Finally, we characterize the classes $(\mu_{1}^{\lambda}(\hat{B}):\mu_{2})$ of infinite matrices, where $\mu_{1}\in\{c,c_{0},\ell_{p}\}$ and $\mu_{2}\in\{\ell_{\infty},c,c_{0},\ell_{p}\}$.

all bounded, convergent, null and absolutely p-summable sequences, respectively, where 1 ≤ p < ∞. Also by bs and cs, we denote the spaces of all bounded and convergent series, respectively. We assume throughout unless stated otherwise that p, q > 1 with p −1 + q −1 = 1 and use the convention that any term with negative subscript is equal to zero. We denote throughout that the collection of all finite subsets of N by F .
Let A = (a nk ) be an infinite matrix of complex numbers a nk , where n, k ∈ N. Then, A defines a matrix mapping from X to Y and is denote by By (X : Y ), denote the class of all matrices A such that A : X → Y . Thus, A ∈ (X : Y ) if and only if the series on the right side (1) converges each n ∈ N and x ∈ X, and we have Ax = {(Ax) n } n∈N ∈ 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. A matrix A = (a nk ) is called a triangle if a nk = 0 for k > n and a nn = 0 for all n, k ∈ N. It is trivial that A(Bx) = (AB)x holds for the triangle matrices A, B and a sequence x. Further, a triangle matrix U uniquely has an inverse U −1 = V which is also triangle matrix. Then, x = U(V x) = V (Ux) holds for all x ∈ ω.
Let us give the definition of some triangle limitation matrices which are needed in the text. Let q = (q k ) be a sequence of positive reals and write Q n = n k=0 q k , (n ∈ N).
Then the Cesàro mean of order one, Riesz mean with respect to the sequence q = (q k ) and Euler mean of order r with 0 < r < 1 are respectively defined by the matrices C = (c nk ), R q = (r q nk ) and E r = (e r nk ); where 0, (k > n), for all k, n ∈ N. We write U for the set of all sequences u = (u k ) such that u k = 0 for all k ∈ N. For u ∈ U, let 1/u = (1/u k ). Let z, u, v ∈ U, and define the summation matrix S = (s nk ), the difference matrix ∆ = (∆ (1) nk ), the generalized weighted mean or factorable matrix G(u, v) = (g nk ), A r u = {a r nk (u)}, ∆ (m) = (∆ (m) nk ) by nk = (−1) n−k , (n − 1 ≤ k ≤ n), 0, (0 ≤ k < n − 1 or k > n), We note that the matrix B(r, s) can be reduced to the difference matrices ∆ in case r = 1 and s = −1.
For a sequence space X, the matrix domain X A of an infinite matrix A is defined by which is a sequence space. If A is triangle, then one can easily observe that the sequence space X A and X are linearly isomorphic, i.e., X A ∼ = X. Although in the most cases the new sequence space X A generated in the limitation matrix A from a sequence space X is the expansion or the contraction of the original space X, it may be observed in some cases that those space overlap. Indeed, one can easily see that the inclusion, X S ⊂ X strictly holds for X ∈ {ℓ ∞ , c, c 0 }. As this, one can deduce that the inclusion X ⊂ X ∆ (1) also strictly holds for X ∈ {ℓ ∞ , c, c 0 , ℓ p }. However, if we define X = c 0 ⊕ span{z} with z = ((−1)) k , i.e., x ∈ X if and only if x := β + αz for some β ∈ c 0 and some α ∈ C, and consider the matrix A with the rows A n defined by A n = (−1) n e (n) for all n ∈ N, we have Ae = z ∈ X but Az = e / ∈ X which lead us to the consequences that z ∈ X\X A and e ∈ X A \X, where e = (1, 1, 1, ...) and e (n) is a sequence whose only non-zero term is a 1 in nth place for each n ∈ N. That is to say that the sequence spaces X A and X overlap but neither contains to other.
Let r, s and t be non-zero real numbers, and define the generalized differ- for all n, k ∈ N. The inverse of B(r, s, t) = {b nk (r, s, t)}, which is denote We should record here that B(r, s, 0) = B(r, s), B(1, −2, 1) = ∆ (2) and B(1, −1, 0) = ∆ (1) . So, the results related to the matrix domain of the triple band matrix B(r, s, t) are more general and more comprehensive than the consequences on the matrix domain of B(r, s), ∆ (2) and ∆ (1) , and include them. We assume throughout that λ = (λ k ) ∞ k=0 is a strictly increasing sequence of positive reals tending to ∞, that is The main purpose of the present paper is to introduce the sequence space µ λ ( B) and to determine the α−, β− and γ− duals of the space, where µ denotes the any of the classical spaces ℓ ∞ , c, c 0 or ℓ p , and B is the triple band matrix B(r, s, t) and the sequence λ = (λ k ) is defined in (5) . Furthermore, the Schauder bases for the spaces c λ 0 ( B), c λ ( B) and ℓ λ p ( B) are given, and some topological properties of the spaces c λ 0 ( B), c λ ( B) and ℓ λ p ( B) are examined. Finally, some classes of matrix mappings on the space µ λ ( B) are characterized.
We say that a sequence x = (x k ) ∈ ω is λ− convergent to the number l ∈ C, In particular, we say that x is a λ− null sequence if Λ n (x) → 0 as n → ∞. Further, we say that x is λ− bounded if sup n∈N |Λ n (x)| < ∞, [27]. Recently, Mursaleen and Noman [27,28] studied the sequence spaces c λ 0 , c λ , ℓ λ ∞ and ℓ λ p of non-absolute type as follows: On the other hand, we define the matrix Λ = ( λ nk ) for all n, k ∈ N by Then, it can be easily seen that the equality holds for all n ∈ N and every x = (x k ) ∈ ω, which leads us together with (2) to the fact that More recently, Sönmez [29] has defined the sequence spaces ℓ ∞ ( B), c( B), c 0 ( B) and ℓ p ( B) as follows: In fact, the sequence spaces ℓ ∞ ( B), c( B), c 0 ( B) and ℓ p ( B) can be consider as the set of all sequences whose B(r, s, t)− transforms are in the spaces ℓ ∞ , c, c 0 and ℓ p , respectively. That is, Now, we introduce the difference sequence spaces ℓ λ ∞ ( B), c λ ( B), c λ 0 ( B) and ℓ λ p ( B) as follows: On the other hand, we define the triangle matrix W = {w λ nk } = Λ B by for all k, n ∈ N. Then , it can be easily seen that the equality holds for all n ∈ N and every x = (x k ) ∈ ω. In fact, the sequence spaces can be consider as the set of all sequences whose W − transforms are in the spaces c 0 , c, ℓ ∞ and ℓ p , respectively. That is, Further, for any sequence x = (x k ) we define the sequence y k (λ) = {y k (λ)} which will be used, as the W -transform of x and so we have Since the proof may also be obtained in the similar way as for the other spaces, to avoid the repetition of the similar statements, we give the proof only for one of those spaces. Now, we may begin with the following theorem which is essential in the study.
Proof. Since (11) holds and c 0 , c and ℓ ∞ are BK− spaces with respect to their natural norms (see [30, pp. 16-17]) and the matrix W is a triangle, Theorem 4.3.12 Wilansky [31, pp. 63] gives the fact that c λ 0 ( B), c λ ( B) and ℓ λ ∞ ( B) are BK− spaces with the given norms. This completes the proof .
. This can be shown that for at least one sequence in those space. Hence µ λ ( B) is the sequence space of non-absolutely type.
Proof. We prove the theorem for the space c λ 0 ( B). To prove our assertion we should show the existence of a linear bijection between the spaces c λ 0 ( B) and c 0 . Let T : c λ 0 ( B) → c 0 be defined by (10) . Then, T (x) = y k (λ) = W (x) ∈ c 0 for every x ∈ c λ 0 ( B) and the linearity of T is clear. Further, it is trivial that x = 0 whenever T x = θ and hence T is injective .
Moreover, let y = (y k ) ∈ c 0 and define the sequence x = x k (λ) by (13) Then we obtain Hence, for every n ∈ N we get by (10) This show that W (x) = y and since y ∈ c 0 , we conclude that W (x) ∈ c 0 . Thus, we deduce that x ∈ c λ 0 ( B) and T x = y. Hence T is surjective. Moreover one can easily see for every 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, as desired.
Let (X, . ) be a normed space. A sequence (b k ) of elements of X is called a Schauder basis for X if and only if, for each x ∈ X there exists a unique sequence (α k ) of scalars such that x = k α k b k , i.e. such that lim n→∞ x − n k=0 α k b k = 0. Because of the isomorphism T , defined in the proof of Theorem 2, is onto the inverse image of the basis of those space c 0 , c and ℓ p are the basis of new spaces c λ 0 ( B), c λ ( B) and ℓ λ p ( B) respectively. Therefore, we have the following: Then, the following statements hold: ..} is a basis for the space c λ ( B) and any x ∈ c λ ( B) has a unique representation of the norm

The Inclusion Relations
In the present section, we prove some inclusion relations concerning with the spaces c λ .
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 Then we obtain by (10)  Proof. Suppose that r + s + t = 0 and x ∈ c. Then Bx = B(r, s, t)(x) = (rx k + sx k−1 + tx k−2 ) ∈ c 0 and hence B(x) ∈ c λ 0 since the inclusion c 0 ⊂ c λ 0 [27]. This show that x ∈ c λ 0 ( B) ,i.e, c ⊂ c λ 0 ( B) holds. Further consider the sequence y = (y k ) defined by y k = ln(k + 3) for all k ∈ N. Then it is trivial that y / ∈ c . On the other hand, it can easily seen that By ∈ c 0 . Hence By ∈ c λ 0 which means that y ∈ c λ 0 ( B). Thus the sequence y ∈ c λ 0 ( B)\c. Hence, the inclusion c ⊂ c λ 0 ( B) is strict. This completes the proof.
Proof. Let ℓ ∞ be a subset of c λ 0 ( B) . Then we obtain that W (x) ∈ c 0 for every x ∈ ℓ ∞ and the matrix W = {w λ nk } is in the class (ℓ ∞ : c 0 ). By using Lemma 1 it follows that lim n→∞ k |w λ nk | = 0.
Proof. To prove the validity of the inclusion ℓ ∞ ⊂ ℓ λ ∞ ( B), it suffices to show that, for every x ∈ ℓ λ ∞ ( B), there exists a positive real number K such that so that x ∈ ℓ λ ∞ ( B) and hence, ℓ ∞ ⊂ ℓ λ ∞ ( B). Furthermore, we consider x = (x k ) defined by x k = k j=0 d kj for all k ∈ N. Then we have W (x) = 1. Thus, we deduce that . On the other hand, we know that the inclusion ℓ ∞ ⊂ c λ 0 ( B) is strict from Theorem 6. Since . Further, to show that this inclusion is strict, we consider the sequence x = (x k ) defined by x = (1, 1, 1, ...) and assume that r + s + t = 1. Then, we have Hence, we have x / ∈ ℓ λ p ( B), which we want to show. We shall firstly give the definition of α-, β-and γ-duals of a sequences space and after quote the lemmas due to Stieglitz and Tietz [32] which are needed in proving the theorems given in Section 4 and 5.
For the sequence spaces λ and µ, define the set S(λ, µ) by With the notation of (22), the α-, β-and γ-duals of a sequences space λ, which are respectively denoted by λ α = S(λ, ℓ 1 ), λ β = S(λ, cs) and λ γ = S(λ, bs).  Now we consider the following sets: where the matrices F λ = (f λ nk ) and V λ = (v λ nk ) are defined as follows, for all k, n ∈ N and the g k (n) is defined as follows Proof. We prove the theorem for the space c λ 0 ( B). Let a = (a n ) ∈ w. Then, we obtain the equality a n x n = n k=0 d nk k j=k−1 (−1) k−j λ j λ k − λ k−1 a n y i = F λ n (y) for all n ∈ N, (27) by relation (13). Thus we observe by (27) that ax = (a n x n ) ∈ ℓ 1 whenever x = (x k ) ∈ c λ 0 ( B) if and only if F λ y ∈ ℓ 1 whenever y = (y k ) ∈ c 0 . This means that the sequence a = (a n ) ∈ {c λ 0 ( B)} α if and only if F λ ∈ (c 0 : ℓ 1 ). Therefore we obtain by Lemma 2 with F λ instead of A that a = (a n ) ∈ {c λ 0 ( B)} α if and only if sup K∈F n k∈K f λ nk < ∞ which leads us to the consequence that {c λ 0 ( B)} α = f λ 1 . This completes the proof.
Proof. Consider the equality n k=0 Then we deduce by (28) that ax = (a k x k ) ∈ cs whenever x = (x k ) ∈ c λ 0 ( B) if and only if V λ y ∈ c whenever y = (y k ) ∈ c 0 . This means that a = (a k ) ∈ {c λ 0 ( B)} β if and only if V λ ∈ (c 0 : c). Therefore, by using Lemma 3, we obtain : Hence, we conclude that Finally, we ended up this section with the following theorem which determines the γ-duals of sequence spaces c λ 0 ( B), c λ ( B), ℓ λ ∞ ( B) and ℓ λ p ( B) .

Some Matrix Transformations Related To
The Spaces c λ 0 ( B), c λ ( B), ℓ λ ∞ ( B) and ℓ λ p ( B) In this final section, we state some results which characterize various matrix mappings on the spaces c λ 0 ( B), c λ ( B), ℓ λ ∞ ( B) and ℓ λ p ( B). We shall write throughout for brevity that for k < m. and for all k, m, n ∈ N provided the convergence of the series.
We shall begin with lemmas which are needed in the proof of our theorems.
a nk = a n exists for each fixed n ∈ N (37) ∞ j=k d jk a nj ∈ cs; (n ∈ N) (38) (a n ) ∈ ℓ p (39) Proof. If the conditions (34)-(39) hold and x = (x k ) be any sequence in the space c λ ( B) then by using Theorem 9 we have that {a nk } k∈N ∈ {c λ ( B)} β for all n ∈ N. Hence, A−transform of x exists,i.e., Ax exists. Furthermore, since the associated sequence y = (y k ) is in the space c, we may write lim k y k = l for some suitable l. Also, the matrix A = ( g nk ) is in the class (c : ℓ p ) by Lemma 10 and condition (34) where 1 < p < ∞. Now, we may consider the m th partial sum of the series k a nk x k which is derived by using the relation (13): a nm y m ; for all n, m ∈ N (40) Then, since y ∈ c and A ∈ (c : ℓ p ); Ay exists and so the series k g nk y k converges for every n ∈ N. Also, it follows by (35) we have that lim m→∞ g nk (m) = g nk . Therefore, if we pass to limit in (40) as m → ∞, then we obtain by (37) that k a nk x k = k g nk y k + la n for all n ∈ N (41) which can be written as follows : A n (x) = A n (y) + la n for all n ∈ N.
On the other hand, since c λ ( B) and ℓ p are BK−spaces, we have by Lemma 9 that there is a constant M > 0 such that holds for all x ∈ c λ ( B). Now F ∈ F . Then, the sequence (14) for every fixed k ∈ N.
Since W (b (k) (λ)) = e (k) for each fixed k ∈ N, we have Furthermore, for every n ∈ N, we obtain by (14) that Hence, since the inequality (43) is satisfied for the sequence z ∈ c λ ( B) ; we have for any F ∈ F that n k∈F g nk which shows the necessity of (34). Thus, it follows by Lemma10 that A = ( g nk ) ∈ (c : ℓ p ). Moreover, we consider the sequence x = x k defined by (13) for every k ∈ N and suppose that y = (y k ) ∈ c\c 0 . Then, since x ∈ c λ ( B) such that y = W (x) by (12), the transforms Ax and Ay exists. Hence, the series k a nk x k and g nk y k converges for every n ∈ N. So we infer that lim m→∞ m−1 k=0 g nk (m)y k = g nk y k ; (n ∈ N).
Consequently, we obtain from (40) that a nm y m exists for each fixed n ∈ N.
Hence, we deduce that a nm exists for each fixed n ∈ N.
which leads us to the necessity of (37) and so the relation (40) holds, where l = lim k y k . Finally, since Ax ∈ ℓ p and Ay ∈ ℓ p ; the necessity of (39) is immediate by (40). This completes the proof.
Theorem 12. In order that A = (a nk ) ∈ (c λ ( B) : ℓ ∞ ) where A = (a nk ) be an infinite matrix , it is necessary and sufficient that (37) and (38) hold, and Proof. This is an immediate consequence of Lemma 7 and Theorem 11.
Proof. This may be obtained by proceedings as in Theorem 11 , above. So, we omit the detail. Hence, Ax exists . We also observe from (44) and (48) holds for every k ∈ N. So (α k ) ∈ ℓ 1 and hence the series k α k (y k − l) converges, where y = (y k ) ∈ c is the sequence connected with x = (x k ) by the relation (12) such that lim k y k = l. Also, it is obvious by combining Lemma 4 with the conditions (44), (48) and (49) that the matrix A = ( g nk ) is in the class (c : c). Now, by following the similar way used in the proof of Theorem11, we obtain that the relation (41) holds, which can be written as follows: k a nk x k = k g nk (y k − l) + l k g nk + la n for all n ∈ N (50) If we pass the limit in (50) as n → ∞ we have that which shows that Ax ∈ c, i.e., A ∈ (c λ ( B) : c). Conversely, suppose that A ∈ (c λ ( B) : c). Since the inclusion c ⊂ ℓ ∞ holds; A ∈ (c λ ( B) : ℓ ∞ ). Therefore, the necessity of conditions (37), (38) and (44) are obvious from Theorem12. Furthermore, consider the sequence b (k) (λ) = {b (k) n (λ)} n∈N ∈ c λ ( B) defined by (14) for every fixed k ∈ N. Then, one can see that Ab (k) (λ) = { g nk } n∈N and hence { g nk } n∈N ∈ c for every k ∈ N which shows that the necessity of (48). Let z = k b (k) (λ). Then , since the linear transformation T : c λ ( B) → c, defined as in the proof of Theorem2 by analogy, is continuous and W (b (k) (λ)) = e (k) for each fixed k ∈ N, we obtain that W n (z) = k W n (b (k) (λ)) = k δ nk = 1 for each n ∈ N which shows that W n (z) = e ∈ c and hence z ∈ c λ ( B). On the other hand, since c λ ( B) and c are the BK−spaces , Lemma 9 implies the continuity of the matrix mapping A : c λ ( B) → c. Thus, we have for every n ∈ N that This show the necessity of (49). Now, it follows by (44), (48) and (49) with Lemma4 that A = ( g n k ) ∈ (c : c). So by (37), (38) and relation (42) holds for all x ∈ c λ ( B) and y ∈ c.
Finally, the necessity of (47) is immediately by (42) since Ax ∈ c and Ax ∈ c. This completes the proof. Proof. This is obtained in the similar way used in the proof of Theorem14 with Lemma 11 instead of Lemma 4 and so we omit the detail. Proof. This is an immediate consequence of Lemma 3, Theorem 9 and Theorem 13(ii).
Proof. This is an immediate consequence of Lemma 12, Theorem 9 and Theorem 16.
Conversely, suppose that A = (a nk ) ∈ (ℓ λ p ( B) : ℓ ∞ ). Thus, Ax exists and bounded for all x ∈ ℓ λ p ( B). Also, {a nk } k∈N ∈ {ℓ λ p ( B)} β for all n ∈ N which is implies the necessity of (36) and (46). If we define the sequences g n such that g n = { g nk } k∈N then we have that g n ℓq < ∞ So, bearing in mind (46) one can easily obtain the relation (52) by using the relation (40). On the other hand, the sequences a n = {a nk } k∈N define the continuous linear functionals on the space ℓ λ p ( B) as follows a nk x k ; (n ∈ N).
Since the spaces ℓ λ p ( B) and ℓ λ p are linear isomorphic; we have that f n = g n ℓq .
Since m k=0 | g nk | q for all m > 0. In this situation we see by passing to the limit in last inequality as n → ∞ that m k=0 Because of (53) holds for all integer m > 0; we have that ∞ k=0 |α k | q 1/q < ∞.
We remember that x = (x k ) and y = (y k ) are associated sequences by the relation (13) where y = (y k ) ∈ ℓ p for x = (x k ) ∈ ℓ λ p ( B) . Let ε be any positive number. Then, there exists a number N such that ∞ k=N |y k | q On the other hand, there is an integer N 1 such that N k=0 { g nk − α k }y k ≤ ε 2 whenever n ≥ N 1 . Therefore, we obtain