The Sequence Space bv and Some Applications

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1 Mathematica Aeterna, Vol. 4, 2014, no. 3, The Sequence Space bv and Some Applications Murat Kirişci Department of Mathematical Education, Hasan Ali Yücel Education Faculty, Istanbul University, Vefa, 34470, Fatih, Istanbul, Turkey Abstract In this work, we give well-known results related to some properties, dual spaces and matrix transformations of the sequence space bv and introduce the matrix domain of space bv with arbitrary triangle matrix A. Afterward, wechoosethematrix Aas Cesàromeanof orderone, generalized weighted mean and Riesz mean and compute α, β, γ duals of these spaces. And also, we characterize the matrix classes of the new spaces. Mathematics Subject Classification: Primary 46A45; Secondary 46B45, 46A35. Keywords: Matrix transformations, sequence space, BK-space, α, β, γ duals, Schauder basis. 1 Introduction The set of all sequences denotes with ω := C N := x = (x k ) : x : N C,k x k := x(k), where C denotes the complex field and N = 0,1,2,... Each linear subspace of ω (with the induced addition and scalar multiplication) is called a sequence space. We will write φ,l,c and c 0 for the sets of all finite, bounded, convergent and null sequences, respectively. It obviously that these sets are subsets of ω. A sequence, whose k th term is x k, is denoted by x or (x k ). By e and e (n),(n = 0,1,2,...),wedenotethesequencessuchthate k = 1fork = 0,1,2,..., and e n (n) = 1 and e (n) k = 0 for k n. A coordinate space (or K space) is a vector space of numerical sequences, where addition and scalar multiplication are defined pointwise. That is, a sequence space X with a linear topology is called a K-space provided each of the maps p i : X C defined by p i (x) = x i is continuous for all i N.

2 208 Murat Kirişci A BK space is a K space, which is also a Banach space with continuous coordinate functionals f k (x) = x k, (k = 1,2,...). A K space K is called an FK space providedx isacompletelinearmetricspace. AnFK space whose topology is normable is called a BK space. A sequence (b n ),(n = 0,1,2,...) in a linear metric X is called a Schauder basis if for each x X there exists a unique sequence (α n ),(n = 0,1,2,...) of scalars such that x = n=0 α nb n. An FK space X is said to have AK property, if φ X and e (n) is a basis for X and φ = spane (n), the set of all finitely non-zero sequences. The series α k b k which has the sum x is then called the expansion of x with respect to (b n ), and written as x = α k b k. An FK space X is said to have AK property, if φ X and e k is a basis for X, where e k is a sequence whose only non-zero term is a 1 in k th place for each k N and φ = spane k, the set of all finitely non-zero sequences. Let X is a sequence space and A is an infinite matrix. The sequence space X A = x = (x k ) ω : Ax X (1) is called the matrix domain of X which is a sequence space(for several examples of matrix domains, see [6] p ). 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 u,v U, (t k ) be a sequence of positive and write T n = n k=0 t k. Now, we define the difference matrix = (δ nk ), the matrix C = (c nk ) of the Cesaro mean of order one, the generalized weighted mean or factorable matrix G(u,v) = (g nk ) and the matrix R t = (rnk t ) of the Riesz mean by δ nk = ( 1) n k, (n 1 k n) 0, (0 k < n 1 or k > n) (2) c nk = 1 n+1, (0 k n) 0, (k > n) g nk = un v k, (0 k n) 0, (k > n) r t nk = tk /T n, (0 k n) 0, (k > n) for all k,n N; where u n depends only on n and v k only on k.

3 The Sequence Space bv and Some Applications 209 In this work, we give well-known results related to some properties, dual spaces and matrix transformations of the sequence space bv and introduce the matrix domain of space bv with arbitrary triangle matrix A. Afterward, we choose the matrix A as Cesàro mean of order one, generalized weighted mean and Riesz mean and compute α,β γ duals of these spaces. And also, we characterize the matrix classes of the spaces bv(c), bv(g), bv(r). 2 Well-Known Results In this section, we will give some well-known results and will define a new form of the sequence space bv with arbitrary triangle A. The space of all sequences of bounded variation defined by bv = x = (x k ) ω : x k x k 1 <, k=1 which is a BK space under the norm x bv = x 0 + k=1 x k x k 1 for x bv. The space bv 0 denotes bv 0 = bv c 0. It is clear that bv = bv 0 +e. Also the inclusions l 1 bv 0 bv = bv 0 +e c are strict. Let X beasequence space and denotes thematrix asdefined by (2). The matrix domain X for X = l,c,c 0 is called the difference sequence spaces, which was firstly defined and studied by Kizmaz [22]. If we choose X = l 1, the space l 1 ( ) is called the space of all sequences of bounded variation and denote by bv. In [8], Başar and Altay have defined the sequence space bv p for 1 p which consists of all sequences such that -transforms of them are in l p and have studied several properties of these spaces. Başar and Altay[8] proved that the space bv p is a BK space with the norm x bvp = x lp and linearly isomorphic to l p for 1 p. The inclusion relations for the space bv p are given in [8] as below: (i) The inclusion l p bv p strictly holds for 1 p. (ii) Neither of the spaces bv p and l includes the other one, where 1 < p <. (iii) If 1 p < s, then bv p bv s. Define a sequence b (k) = b (k) n n N of elements of the space bv p for every fixed k N by 0 for n < k and 1 for n k. Then the sequence b (k) n n N is a Schauder basis for the space bv p and every sequence x bv p has a unique representation x = k ( x) kb (k) for all k N. The space bv has no Schauder basis[8].

4 210 Murat Kirişci Let X and Y be two sequence spaces, and A = (a nk ) be an infinite matrix of complex numbers a nk, where k,n 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 = (Ax) n, the A-transform of x, is in Y; where (Ax) n = k a nk x k for each n N. (3) For simplicity in notation, here and in what follows, the summation without limits runs from 0 to. By (X : Y), we 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 of (3) converges for each n N and each x X and we have Ax = (Ax) n n N Y for all x X. A sequence x is said to be A-summable to l if Ax converges to l which is called the A-limit of x. If X,Y ω and z any sequence, we can write z 1 X = x = (x k ) ω : xz X and M(X,Y) = x X x 1 Y. If we choose Y = l 1,cs,bs, then we obtain the α,β,γ duals of X, respectively as X α = M(X,l 1 ) = a = (a k ) ω : ax = (a k x k ) l 1 for all x X X β = M(X,cs) = a = (a k ) ω : ax = (a k x k ) cs for all x X X γ = M(X,bs) = a = (a k ) ω : ax = (a k x k ) bs for all x X. The α, β, γ duals of classical sequence spaces are defined by c α 0 = cα = l α = l 1, l α 1 = l, cs α = bs α = bv α = bv α 0 = l 1 c β 0 = c β = l β = l 1, l β 1 = l, cs β = bv, bs β = bv 0, bv β = cs, bv β 0 = bs c γ 0 = c γ = l γ = l 1, l γ 1 = l, cs γ = bs γ = bv, bv γ = bv γ 0 = bs. The continuous dual of X is the space of all continuous linear functionals on X and is denote by X. The continuous dual X with the norm. defined by f = sup f(x) : x = 1 for f X. The spaces c and c 0 are norm isomorphic with l 1. Also, a β = a l1 for all a l β. The following theorem is the most important result in the theory of matrix transformations: Theorem 2.1. [38, Theorem 4.2.8] Matrix transformations between BK spaces are continuous.

5 The Sequence Space bv and Some Applications 211 In [5], Başar and Altay developed very useful tools for duals and matrix transformations of sequence spaces as below: Theorem 2.2. [8, Lemma 5.3] Let X,Y be any two sequence spaces, A be an infinite matrix and U a triangle matrix matrix.then, A (X : Y U ) if and only if UA (X : Y). Theorem 2.3. [5, Theorem 3.1] B U µ = (b nk ) be defined via a sequence a = (a k ) ω and inverse of the triangle matrix U = (u nk ) by b nk = n a j v jk j=k for all k,n N. Then, and λ β U = a = (a k) ω : B U (λ : c) λ γ U = a = (a k) ω : B U (λ : l ). The following theorem proved by Zeller [39]: Theorem 2.4. Let X be an FK space whose topology is given by means of the seminorms q n n=1 and let A be an infinite matrix. Then X A is an F K space when topologized by x x j (j = 1,2,...) n x sup n a ij x j (i = 1,2,...) j=1 x q n (Ax) (n = 1,2,...). Following theorems are given by Bennet [19]. Theorem 2.5. An FK space X contains l 1 if and only if e (j) : j = 1,2,... is a bounded subsets of X. Theorem 2.6. Let A be a matrix and X an FK sace. Then A maps l 1 into X if and only if the columns of A belong to X and form a bounded subset there. Theorem 2.7. An FK space contains bv 0 (bv) if and only if (e E) and n j=1 e(j) : j = 1,2,... is a bounded subsets of X.

6 212 Murat Kirişci Theorem 2.8. Let A be a matrix and X an FK sace. Then A maps bv 0 into X if and only if the columns of A belong to X and their partial sums form a bounded subset there. The important class of co-null F K space was introduced by Snyder [36]: An FK space X containing ϕ e is said to be co-null if j=1 ej = e weakly in X. Therefore, using the definition of co-null and Theorem 2.7, we have: Corollary 2.9. [19] Any co-null F K space must contain bv. Theorem [38, Theorem 4.3.2] Let (X,. ) be a BK space. Then X β is a BK space with a β = supsup n n k=0 a kx k : x = 1. Theorem [38, Theorem 7.2.9] The inclusion X β X holds in the following sense: Let the : X β X be defined by (a) = â : X C,(a X β ) where â(x) = k=0 a kx k for all x X. Then is an isomorphism into X. If X has AK, then the map is onto X. Now, we define the space bv with a lower triangle matrix A = (a nk ) for all k,n N as below: bv(a) = x = (x k ) ω : Ax bv. (4) The space bv is BK space with the norm x bv = x l1 and A is a triangle matrix. Then, from Theorem in [38], we say that the space bv(a) is a BK space with the norm x bv(a) = Ax bv. If A is a triangle, then one can easily observe that the sequence X A and X are linearly isomorphic,i.e., X A = X. Therefore, the spaces bv and bv(a)are norm isomorphic to the spaced l 1 and bv, respectively. Now we give some results: Lemma Let A = (a nk ) be an infinite matrix. Then, the following statements hold: (i) A (l 1 : l ) if and only if sup a nk <. (5) k,n N (ii) A (l 1 : c) if and only if (5) holds, and there are α k, C such that (iii) A (l 1 : l 1 ) if and only if lim a nk = α k for each k N. (6) n sup k N a nk <. (7) n

7 The Sequence Space bv and Some Applications Applications In this section, we give some results related to the space bv(a) as choose wellknown matrices instead of arbitrary matrix A. 3.1 The space bv(c) We give the matrix C 1 = (c nk ) instead of the matrix A = (a nk ), in (4). Then, we obtain the space bv(c) as below: bv(c) = x = (x k ) ω : 1 k x j 1 k 1 x j <. (8) k +1 k k j=0 j=0 Using the notation (1), we can denotes the space bv(c) as bv(c) = (bv) C1 = (l 1 ).C1, where Φ = φ nk =.C 1 defined by φ nk = 1 (0 k < n); (n+1).n φ nk = 1 (k = n) and φ n nk = 0 (k > n) for all n. The Φ transform of the sequence x = (x k ) defined by y k = x k k +1 1 k(k +1) k x j. (9) Theorem 3.1. Define a sequence t (k) = t (k) n n N of elements of the space bv(c) for every fixed k N by t (k) n = j=0 (k +1), (n = k) 1, (n < k) 0, (n > k) Therefore, the sequence t (k) k N is a basis for the space bv(c) and any x bv(c) has a unique representation of the form x = k (Φx) k t (k). This theorem can be proved as Theorem 3.1 in [8], we omit details. Theorem 3.2. The α dual of the space bv(c) is the set d 1 = a = (a k ) ω : sup b nk < k N where the matrix B = (b nk ) is defined via the sequence a = (a n ) ω by (n+1)a n (n = k), a k (k < n) and 0 (k > n). n

8 214 Murat Kirişci Proof. Let a = (a n ) ω. Using the relation (9), we obtain a n x n = n n 1 y k +(n+1)y n a n = (By) n (10) k=0 It follows from (10) that ax = (a n x n ) l 1 whenever x bv(c) if and only if By l 1 whenever y l 1. We obtain that a [bv(c)] α whenever x bv(c) if and only if B (l 1 : l 1 ). Therefore, we get by Lemma 2.12 (iii) with B instead of A that a [bv(c)] α if and only if sup k N n b nk <. This gives us the result that [bv(c)] α = d 1. Theorem 3.3. Define the sets by d 2 = d 3 = a = (a k ) ω : limd nk exists for each k N n a = (a k ) ω : sup d nk < k where the matrix D = (d nk ) is defined via the sequence a = (a n ) ω by (n + 1)a n (n = k), n j=k a j (k > n) and 0 (k < n). Then, [bv(c)] β = d 2 d 3. Proof. Consider the equation n n a k x k = k=0 [ n k 1 ] a k y j +(k +1)y k = (Dy) n (n N). (11) k=0 j=0 Therefore, we deduce from Lemma 2.12 (ii) with (11) that ax = (a n x n ) cs whenever x bv(c) if and only if Dy c whenever y l 1. From (5) and (6), we have lim n d nk = α k and sup k which shows that [bv(c)] β = d 2 d 3. Theorem 3.4. [bv(c)] γ = d 3. d nk < Proof. We obtain from Lemma 2.12 (i) with (11) that ax = (a n x n ) bs whenever x bv(c) if and only if Dy l whenever y l 1. Then, we see from (5) that [bv(c)] γ = d 3. n

9 The Sequence Space bv and Some Applications 215 In this subsection to use, we define the matrices for brevity that k 1 ã nk = j=0 for all k,n N. a nj +(k +1)a nk and bnk = a nk n+1 1 n(n+1) Theorem 3.5. Suppose that the entries of the infinite matrices A = (a nk ) and E = (e nk ) are connected with the relation n j=0 a jk e nk = ã nk (12) for all k,n N and Y be any given sequence space. Then, A (bv(c) : Y) if and only if a nk k N [bv(c)] β for all n N and E (l 1 : Y). Proof. Let Y be any given sequence space. Suppose that (12) holds between A = (a nk ) and E = (e nk ), and take into account that the spaces bv(c) and l 1 are norm isomorphic. Let A (bv(c) : Y) and take any y = (y k ) l 1. Then ΦE exists and a nk k N bv(c) β which yields that (12) is necessary and e nk k N (l 1 ) β for each n N. Hence, Ey exists for each y l 1 and thus e nk y k = a nk x k for all n N, k k we obtain that Ey = Ax which leads us to the consequence E (l 1 : Y). Conversely, let a nk k N bv(c) β for each n N and E (l 1 : Y) hold, and take any x = (x k ) bv(c). Then, Ax exists. Therefore, we obtain from the equality m a nk x k = k=0 [ m k 1 ] a nk y j +(k +1)y k = k=0 j=0 m k=0 ( m 1 a nj y k )+(m+1)a nm y m for all m,n N as m that Ax = Ey and this shows that E (l 1 : Y). This completes the proof. j=k Theorem 3.6. Suppose that the entries of the infinite matrices B = ( b nk ) and F = (f nk ) are connected with the relation f nk = b nk (13) for all k,n N and Y be any given sequence space. Then, B (Y : bv(c)) if and only if F (Y : l 1 ).

10 216 Murat Kirişci Proof. Let z = (z k ) Y and consider the following equality [ m m a nk bnk z k = n+1 1 n 1 a jk ]z k for all, m,n N n(n+1) k=0 k=0 j=0 which yields that as m that (Fz) n = Φ(Bz) n for all n N. Therefore, one can observe from here that Bz bv(c) whenever z Y if and only if Fz l 1 whenever z Y. 3.2 The space bv(g) In (4), if we choose the matrix G = G(u,v) = (g nk ) instead of the matrix A = (a nk ), then, we obtain the space bv(g) as below: bv(g) = x = (x k ) ω : k k 1 u k v j x j u k 1 v j x j <. (14) k j=0 j=0 Using the notation (1), we can denotes the space bv(g) as bv(g) = (bv) G(u,v) = (l 1 ).G(u,v), where Γ = γ nk =.G(u,v) defined by γ nk = (u n u n 1 )v k (1 k < n); γ nk = u n v n (k = n) and γ nk = 0 (k > n) for all n. The Γ transform of the sequence x = (x k ) defined by k 1 y k = (u k u k 1 )v j x j +u k v k x k. (15) j=0 (16) The Theorem 3.7 and Theorem 3.8 can be proved as Theorem 2.8 in [3],[4] and Theorem 3.6 in [25], respectively. Theorem 3.7. Define a sequence s (k) = s (k) n n N of elements of the space bv(g) for every fixed k N by 1 1, (1 < k < n) s (k) n = v n ( ) u k 1 u k 1 1 u nv n, (n = k) 0, (k > n) Therefore, the sequence s (k) k N is a basis for the space bv(g) and any x bv(g) has a unique representation of the form x = k (Γx) k s (k).

11 The Sequence Space bv and Some Applications 217 Theorem 3.8. We define the matrix H = (h nk ) as ( 1 1 v k u k 1 u k 1 )a n, (1 k < n) h nk = a n u nv n, (k = n) 0, (k > n) (17) for all k,n N, where u,v U,a = (a k ) ω. The α dual of the space bv(g) is the set d 5 = a = (a k ) ω : sup h nk <. N F Using the Theorem 2.3 and the matrix (17), we can give β and γ duals of the space bv(g) as below: k Corollary 3.9. Let u,v U for all k N. Then, ( 1 1 bv(g) β = a = (a k ) ω : 1 )a k l 1 v k u k u k 1 and bv(g) γ = ( 1 1 z = (z k ) ω : 1 )a k l 1 v k u k u k 1 and and ( an u n v n ( an u n v n ) c ) l. Following theorems can be proved as Theorem 4.2 and 4.3 in [25]. Theorem Suppose that the entries of the infinite matrices A = (a nk ) and B = (b nk ) are connected with the relation a nk = (u j u j 1 )v k b nj or b nk = 1 ( 1 1 ) a nk v k u k u k 1 j=k for all k,n N and Y be any given sequence space. Then A (bv(g) : Y) if and only if a nk k N bv(g) β for all n N and B (l 1 : Y). Theorem Suppose that the entries of the infinite matrices A = (a nk ) and T = (t nk ) are connected with the relation n t nk = (u n u n 1 )v j a jk j=0 for all k,n N and Y be any given sequence space. Then, A (Y : bv(g)) if and only if T (Y : l 1 ).

12 218 Murat Kirişci 3.3 The space bv(r) If we choose u n = 1/Q n,v k = q k in the space bv(g) defined by (14), we obtain the space bv(r): bv(r) = x = (x k ) ω : k k 1 (q j /Q k )x j (q j /Q k 1 )x j <. (18) k j=0 j=0 we can denotes the space bv(r) as bv(r) = (bv) R t = (l 1 ).R t as the spaces bv(g), where Σ = σ nk =.R t defined by σ nk = (1/Q n 1/Q n 1 )q k (1 k < n); σ nk = q n /Q n (k = n) and σ nk = 0 (k > n) for all n. The Σ transform of the sequence x = (x k ) defined by k 1 y k = j=0 ( 1 1 ) q j x j + q k x k. (19) Q k Q k 1 Q k Following theorems can be proved by Theorem 2.9, Theorem 2.7 in [1], respectively. Theorem Define a sequence p (k) = p (k) n n N of elements of the space bv(r) for every fixed k N by p (k) n = (Q k Q k 1 ) q n, (1 k < n) Q n q n, (n = k) 0, (k > n) Therefore, the sequence p (k) k N is a basis for the space bv(r) and any x bv(r) has a unique representation of the form x = k (Σx) k p (k). Theorem We define the matrix M = (m nk ) as (Q n Q n 1 ) q k a n, (1 k < n) Q m nk = n q n a n, (k = n) 0, (k > n) (20) for all k,n N. The α dual of the space bv(r) is the set d 6 = a = (a k ) ω : sup m nk <. N F k

13 The Sequence Space bv and Some Applications 219 Using the Theorem 2.3 and the matrix (20), we can give β and γ duals of the space bv(r) as below: Corollary bv(r) β (Qn Q n 1 ) = a = (a k ) ω : and bv(r) γ = q k a n l 1 (Qn Q n 1 ) z = (z k ) ω : a n l 1 q k and and ( ) Qn a n c q n ( ) Qn a n l. q n Theorem Suppose that the entries of the infinite matrices A = (a nk ) and B = (b nk ) are connected with the relation a nk = j=k ( 1 1 ) Q n Q n 1 q j b nj or b nk = (Q n Q n 1 ) a nk q k for all k,n N and Y be any given sequence space. Then A (bv(r) : Y) if and only if a nk k N bv(r) β for all n N and B (l 1 : Y). Theorem Suppose that the entries of the infinite matrices A = (a nk ) and W = (w nk ) are connected with the relation n ( 1 w nk = 1 ) q j a jk Q n Q n 1 k=0 for all k,n N and Y be any given sequence space. Then, A (Y : bv(r)) if and only if W (Y : l 1 ). 4 Conclusion The matrix domain X for X = l,c,c 0 is called the difference sequence spaces, which was firstly defined and studied by Kızmaz [22]. If we choose X = l 1, thespacel 1 ( ) iscalledthe space of all sequences of bounded variation and denote by bv. The space bv p consisting of all sequences whose differences are in the space l p. The space bv p was introduced by Baar and Altay [8]. More recently, the sequence spaces bv are studied in [8], [9], [19], [20], [21], [26], [31]. Several authors studied deal with the sequence spaces which obtained with the domain of the triangle matrices. It can be seen that the matrix domains

14 220 Murat Kirişci for the matrix C 1 which known as Cesàro mean of order one in [7], [24], [33], [35]; for the generalized weighted mean in [3], [4], [14], [15], [23], [25], [29], [32], [34]; for the Riesz mean in [1], [2], [10], [11], [13], [16], [17], [18], [27], [28]. Further, different works related to the matrix domain of the sequence spaces can be seen in [6]. In this work, we give well-known results related to some properties, dual spaces and matrix transformations of the sequence space bv and introduce the matrix domain of space bv with arbitrary triangle matrix A. Afterward, we choose the matrix A as Cesàro mean of order one, generalized weighted mean and Riesz mean and compute α,β,γ duals of those spaces. And also, we characterize the matrix classes of the spaces bv(c), bv(g), bv(r). As a natural continuation of this paper, one can study the domain of different matrices instead of A. Additionally, sequence spaces in this paper can be defined by a index p for 1 p < and a bounded sequence of strictly positive real numbers (p k ) for 0 < p k 1 and 1 < p k < and the concept almost convergence. And also it may be characterized several classes of matrix transformations between new sequence spaces in this work and sequence spaces which obtained with the domain of different matrices. ACKNOWLEDGEMENTS. This work was supported by Scientific Research Projects Coordination Unit of Istanbul University. Project number References [1] B. Altay and F. Başar, On the paranormed Riesz sequence spaces of nonabsolute type, Southeast Asian Bull. Math., 26 (2002), [2] B. Altay and F. Başar, Some paranormed Riesz sequence spaces of nonabsolute type, Southeast Asian Bull. Math., 30 (2006), [3] B. Altay and F. Başar, Some paranormed sequence spaces of non-absolute type derived by weighted mean, J. Math. Analysis and Appl., 319 (2006), [4] B. Altay and F. Başar, Generalization of the sequence space l(p) derived by weighted mean, J. Math. Analysis and Appl., 330 (2007), [5] B. Altay and F. Başar, Certain topological properties and duals of the domain of a triangle matrix in a sequence spaces, J. Math. Analysis and Appl., 336 (2007), [6] F. Başar, Summability Theory and its Applications, Bentham Science Publishers, e-books, Monographs, (2011).

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