RECENT RESULTS ON THE DOMAIN OF THE SOME LIMITATION METHODS IN THE SEQUENCE SPACES f 0 AND f

Transcription

1 ISSN: Received: Year: 205, Number: 2, Pages: Accepted: Original Article ** RECENT RESULTS ON THE DOMAIN OF THE SOME LIMITATION METHODS IN THE SEQUENCE SPACES f 0 AND f Seran Demiriz,* <serandemiriz@gmailcom> Department of Mathematics, University of Gaziosmanpaşa, Toat, Turey Abstract In the present paper, we summarize the literature on the sequence spaces almost A null and almost A convergent derived by using the domain of the certain A limitation matrix Moreover, we introduce the spaces f(r, s, t) and f 0 (r, s, t) and examine some properties of this spaces Keywords Almost convergence, Matrix domain, Generalized means, Matrix transformations Introduction By a sequence space, we mean any vector subspace of ω, the space of all real or complex valued sequences x = (x ) The well-nown sequence spaces that we shall use throughout this paper are as following: l : the space of all bounded sequences, c: the space of all convergent sequences, c 0 : the space of all null sequences, bs: the space of all sequences which forms bounded series, cs: the space of all sequences which forms convergent series, l : the space of all sequences which forms absolutely convergent series, l p : the space of all sequences which forms p-absolutely convergent series, where < p < Let λ, µ be two sequence spaces and A = (a n ) be an infinite matrix of real or complex numbers a n, where n, N Then, we say that A defines a matrix mapping from λ into µ, and we denote it by writing A : λ µ, if for every sequence x = (x ) λ the sequence Ax = (Ax) n, the A-transform of x, is in µ; where (Ax) n = a n x, (n N) () For simplicity in notation, here and in what follows, the summation without limits runs from 0 to By (λ : µ), we denote the class of all matrices A such that A : λ µ Thus, A (λ : µ) if and only if the series on the right side of () converges for each n N and every x λ, and we have ** Edited by Oatay Muhtaroğlu (Area Editor) and Naim Çağman (Editor-in-Chief) * Corresponding Author

2 Journal of New Theory 2 (205) Ax = (Ax) n n N µ for all x λ A sequence x is said to be A-summable to α if Ax converges to α which is called as the A-limit of x If a normed sequence space λ contains a sequence (b n ) with the property that for every x λ there is a unique sequence of scalars (α n ) such that lim x (α 0b 0 + α b + + α n b n ) = 0, then (b n ) is called a Schauder basis (or briefly basis) for λ The series α b which has the sum x is then called the expansion of x with respect to (b n ), and written as x = α b The β dual of a subset X of ω is defined by X β = a = (a ) ω : ax = (a x ) cs for all x = (x ) X The shift operator P is defined on ω by (P x) n = x n+ for all n N A Banach limit L is defined on l, as a non-negative linear functional, such that L(P x) = L(x) and L(e) = A sequence x = (x ) l is said to be almost convergent to the generalized limit α if all Banach limits of x are α [], and denoted by f lim x = α Let P j be the composition of P with itself j times and define t mn (x) for a sequence x = (x ) by t mn (x) = m + m (P j x) n for all m, n N Lorentz [] proved that f lim x = α if and only if lim t mn (x) = α, uniformly in n It is wellnown that a convergent sequence is almost convergent such that its ordinary and generalized limits are equal By f and f 0, we denote the space of all almost convergent sequences and almost convergent to zero sequences, respectively, ie, and f = x = (x ) ω : α C lim f 0 = x = (x ) ω : lim m m x n+ = α uniformly in n m + x n+ = 0 uniformly in n m + A matrix A = (a n ) is called a triangle if a n = 0 for > n and a nn 0 for all n N It is trivial that A(Bx) = (AB)x holds for triangle matrices A, B and a sequence x Further, a triangle matrix U uniquely has an inverse U = V that is also a triangle matrix Then, x = U(V x) = V (Ux) holds for all x ω We write by U and U 0 for the sets of all sequences with non-zero terms and non-zero first terms, respectively For u U, let /u = (/u n ) Let us give the definition of some triangle limitation matrices which are needed in the text Let q = (q ) be a sequence of positive reals and write Q n = q, (n N) Then the Cesàro mean of order one, Riesz mean with respect to the sequence q = (q ) and A r mean with 0 < r < are respectively defined by the matrices C = (c n ), R q = (r q n ) and Ar = (a ); where c n = n +, (0 n), 0, ( > n), r q n = q, Q n (0 n), 0, ( > n), and + r a =, (0 n), + n 0, ( > n),

3 Journal of New Theory 2 (205) for all, n N Additionally, the Euler mean of order r and the weighted mean matrix and the double band matrix are respectively defined by the matrices E r = (e ), G(u, v) = (g n) and B(r, s) = b n (r, s); where ( n ) e r ( r) n r, (0 n) un v n = and g n =, (0 n), 0, ( > n) 0, ( > n), and r, ( = n), b n (r, s) = s, ( = n ), 0, otherwise for all, n N and u, v U and r, s R\0 For a sequence space λ, the matrix domain λ A of an infinite matrix A is defined by λ A = x = (x ) ω : Ax λ, (2) which is a sequence space Although in the most cases the new sequence space λ A generated in the limitation matrix A from a sequence space λ is the expansion or the contraction of the original space λ, it may be observed in some cases that those spaces overlap Indeed, one can deduce that the inclusions λ S λ strictly holds for λ l, c, c 0 As this, one can deduce that the inclusions l p bv p and λ λ also strictly hold for λ c, c 0, where p and the space (l p ) () = bv p has been studied by Başar and Altay [2], (see also Çola and Et and Malowsy [3]) However, if we define λ = c 0 z with z = (( ) ), that is, x λ if and only if x = s+αz for some s c 0 and some α C, and consider the matrix A with the rows A n defined by A n = ( ) n e (n) for all n N, we have Ae = z λ but Az = e / λ which lead us to the consequences that z λ \ λ A and e λ A \ λ, where e (n) denotes the sequence whose only non-zero term is a in n th place for each n N and e = (,,, ) That is to say that the sequence spaces λ A and λ overlap but neither contains the other The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has been recently employed by Wang [4], Ng and Lee [5], Aydın and Başar [6], Altay and Başar [7], and Altay et all [8] They introduced the sequence spaces (l ) Nq and c Nq in [4], (l p ) C = X p in [5], (c 0 ) A r = a r 0 and c A r = a r c in [6], (c 0 ) E r = e r 0 and c E r = e r c in [7], (l p ) E r = e r p and (l ) E r = e r in [8]; where p < In this study, we summarize some nowledge in the existing literature on the almost A null and almost A convergent sequence spaces derived by using the domain A limitation matrix Additionally, we introduce the new sequence spaces f 0 (r, s, t) and f(r, s, t) and examine some properties of these sequence spaces 2 Domain of the A limitation matrix in the sequence spaces f 0 and f In this section, we shortly give the nowledge on the sequence spaces derived by the A limitation matrix from well-nown almost convergent and almost null sequence spaces For the concerning literature about the domain µ A of an infinite limitation matrix A in a sequence space µ, Table may be useful µ A µ A refer to f 0, f B(r, s) f, f0 [9] f 0, f C f, f0 [4] f 0, f R q f R q, f 0 R q [5] f 0, f A r a r f, ar f 0 [6] f 0, f G(u, v) f 0 (G), f(g) [7] f 0, f E r f(e), f 0 (E) [8] f 0, f B(r, s, t) f(b), f 0 (B) [9] f 0, f A λ A λ (f 0 ), A λ (f) [20] Table : The domains of the certain A limitation matrix in the sequence spaces f 0 and f

4 Journal of New Theory 2 (205) The matrix domain of a certain limitation method on the sequence spaces f 0 and f firstly were studied by Başar and Kirişçi [9] Başar and Kirişçi introduced the sequence spaces f 0 and f in [9] as follows: f 0 : = f : = x = (x ) ω : lim m x = (x ) ω : α C lim sx +j + rx +j m + m = 0 uniformly in, sx +j + rx +j m + = α uniformly in It is trivial that the sequence spaces f 0 and f are the domain of the matrix B(r, s) in the spaces f 0 and f, respectively Thus, with the notation of (2) we can redefine the spaces f 0 and f by f 0 := f 0 B(r,s) and f := fb(r,s) Define the sequence y = (y ) by the B(r, s) transform of a sequence x = (x ), ie, y := sx + rx for all N (3) Since the matrix B(r, s) is triangle, one can easily observe that x = (x ) X if and only if y = (y ) X, where the sequences x = (x ) and y = (y ) are connected with the relation (3), and X denotes any of the sequence spaces f 0 and f Therefore, one can easily see that the linear operator T : X X, T x = y = B(r, s)x which maps every sequence x in X to the associated sequence y in X, is bijective and norm preserving, where x X = B(r, s)x X This gives the fact that X and X are norm isomorphic Başar and Kirişçi [9] proved that the sequence space f is a BK space with the norm and non-separable closed subspace of l So, the sequence space f has no Schauder basis Jarrah and Malowsy [2] showed that the matrix domain λ A of a normed sequence space λ has a basis whenever A = (a n ) is triangle Then; our corollary concerning the space f 0 and f is about their Schauder basis: Corollary 2 [9, Corollary 42] The space f has no Schauder basis The gamma- and beta-duals of the spaces f 0 and f are determined Also, some matrix transformations on these sequence spaces are characterized Quite recently, E E Kara and K Elmaa gaç [2] introduced the sequence space ĉ u as follows: ĉ u = x = (x ) ω : α C lim m u +j x +j + u +j x +j m + = α uniformly in It is trivial that the sequence space ĉ u is the domain of the matrix A u = (a u n ) in the space f, where the matrix A u = (a u n ) is defined by a u ( ) n = n u, n n, 0, 0 < n or > n, for all, n N Also, they show that ĉ u is linearly isomorphic to the space ĉ Further, they compute the β dual of the space ĉ u and characterize the classes of infinite matrices related to sequence space ĉ u 3 Spaces of Ā(r, s, t) almost null and Ā(r, s, t) almost convergent sequences In this section, we study some properties of the spaces of the Ā(r, s, t) almost null and Ā(r, s, t) almost convergent sequences

5 Journal of New Theory 2 (205) For any sequences s, t ω, the convolution s t is a sequence defined by (s t) n = s n t ; Throughout this section, let r, t U and s U 0 sequence x = ( x n ) of generalized means of x by x n = (n N) For any sequence x = (x n ) ω, we define the s n t x ; (n N), (4) that is x n = (s tx) n / for all n N Further, we define the infinite matrix Ā(r, s, t) of generalized means by sn t Ā(r, s, t), 0 n, n = (5) 0, > n for all n, N Then, it follows by (4) that x is the Ā(r, s, t) transform of x, that is x = (Ā(r, s, t)x) for all x ω It is obvious by (5) that Ā(r, s, t) is a triangle Moreover, it can easily be seen that Ā(r, s, t) is regular if and only if s n i = o( ) for each i N, n s n t = O( ) and (s t) n / (n ) The above definition of the matrix Ā(r, s, t) of generalized means given by (5) includes the following special cases: () If = (s t) n 0 for all n, then Ā(r, s, t) reduces to the matrix (N, s, t) of generalized Nörlund means [22, 23] In particular, if t = e then Ā(r, s, t) reduces to the familiar matrix of Nörlund means [30, 4] (2) If α > 0, r = Γ(α++)!Γ(α+), s = Γ(α+)!Γ(α) and t = for all, then Ā(r, s, t) reduces to the matrix (C, α) of Cesàro means of order α [24, 25] In particular, if α = then Ā(r, s, t) reduces to the famous matrix (C, ) of arithmetic means [5, 26] (3) If 0 < α <, r =!, s = ( α)! and t = α! for all, then Ā(r, s, t) reduces to the matrix (E, α) of Euler means of order α [7, 0, 8] (4) If t n > 0 and = n t for all n, then Ā(r, s, t) reduces to the matrix (N, t) of weighted means [2, 27] (5) If 0 < α <, r = +, s = and t = + α for all, then Ā(r, s, t) reduces to the matrix A α studied by Aydın and Başar [6, 28] (6) If s = e (0) and t = e, then Ā(r, s, t) reduces to the diagonal matrix D /r studied by de Malafosse [29] Now, since Ā(r, s, t) is a triangle, it has a unique inverse which is also a triangle More precisely, by maing a slight generalization of a wor done in [30], we put D (s) 0 = /s 0 and D (s) n = s n+ 0 s s s 2 s s s 3 s 2 s s 0 0 s n s n 2 s n 3 s n 4 s 0 s n s n s n 2 s n 3 s Then the inverse of Ā(r, s, t) is the triangle B = ( b n ) n, defined by ( ) n D (s) bn = n r, (0 n), t n 0, ( > n), ; (n =, 2, ) for all n, N For an arbitrary subset X of ω, the set X(r, s, t) has recently been introduced in [3] as the matrix domain of the triangle Ā(r, s, t) in X We introduce the sequence spaces f(r, s, t) and f 0 (r, s, t) as the sets of all sequences whose Ā(r, s, t) transforms are in the spaces f 0 and f, that is

6 Journal of New Theory 2 (205) f 0 (r, s, t) = f(r, s, t) = x = (x ) ω : lim m + x = (x ) ω : l C lim m n+j m + s n+j t x m n+j = 0 uniformly in n, s n+j t x = l uniformly in n With the notation of (2), we can redefine the spaces f(r, s, t) and f 0 (r, s, t) as follows: f(r, s, t) = f Ā(r,s,t) and f0 (r, s, t) = f 0 Ā(r,s,t) It is worth mentioning that the general forms of the well-nown matrices of Nörlund, Cesàro, Euler and weighted means can be obtained as special cases of the matrix Ā(r, s, t) of generalized means Therefore, all of the sequence spaces in Tablo can be obtained by special choice from the sequence spaces f(r, s, t) and f 0 (r, s, t) which are defined by using matrix domain of the matrix Ā(r, s, t) Theorem 3 The sequence spaces f(r, s, t) and f 0 (r, s, t) are BK spaces with the same norm given by x f(r,s,t) = Ā(r, s, t)x f = sup m,n N t mn (Ā(r, s, t)x), (6) where for all m, n N t mn (Ā(r, s, t)x) = m + = m + (Ā(r, s, t)x) n+j m n+j s n+j t x Proof f 0 and f endowed with the norm are BK spaces [24, Example 732 (b)] and Ā(r, s, t) is a triangle matrix, Theorem 432 of Wilansy [32, p6] gives the fact that f(r, s, t) and f 0 (r, s, t) are BK spaces with the norm f(r,s,t) Remar 32 It can easily be seen that the absolute property does not hold on the spaces f(r, s, t) and f 0 (r, s, t), that is x f(r,s,t) x f(r,s,t) for at least one sequence x in each of these spaces, where x = ( x ) Thus, the spaces f(r, s, t) and f 0 (r, s, t) are BK spaces of non-absolute type Theorem 33 The sequence spaces f(r, s, t) and f 0 (r, s, t) are norm isomorphic to the spaces f and f 0, respectively Proof Since the fact f 0 (r, s, t) = f 0 can be similarly proved, we consider only the case f(r, s, t) = f To prove this, we should show the existence of a linear bijection between the spaces f(r, s, t) and f which preserves the norm Consider the transformation T defined, with the notation of (4), from f(r, s, t) to f by x x = T x = Ā(r, s, t)x The linearity of T is clear Further, it is trivial that x = θ whenever T x = θ and hence T is injective Let us tae any x = ( x ) f and define the sequence x = (x n ) by Then, it is immediate that n+j x n = t n ( ) n D (s) n r x ; for all n N (7) s n+j t x = n+j = x n+j s n+j t t ( ) i D (s) i r i x i i=0

7 Journal of New Theory 2 (205) which gives by a short calculation that Therefore, we have lim m + m + m n+j m Ā(r, s, t)x n+j = lim s n+j t x = m + m + m x n+j m x n+j = l uniformly in n This means that x f(r, s, t) and hence T is surjective Thus, one can easily see from (6) that T is a norm preserving transformation This completes the proof Remar 34 It is nown from Corollary of Başar and Kirişçi [9] that the Banach space f has no Schauder basis It is also nown from Theorem 23 of Jarrah and Malowsy [2] that the domain λ A of a matrix A in a normed sequence space λ has a basis if and only if λ has a basis whenever A = (a n ) is a triangle Combining these two facts one can immediately conclude that both the space f(r, s, t) and the space f 0 (r, s, t) have no Schauder basis Now, we give the beta- and gamma-duals of the sequence spaces f(r, s, t) and f 0 (r, s, t) For this, we need the following lemmas: Lemma 35 [] A = (a n ) (f : l ) if and only if a n < (8) sup n N Lemma 36 [] A = (a n ) (f : c) if and only if (8) holds, and there are α, α C such that lim a n = α for each N, (9) a n = α, (0) lim lim (a n α ) = 0 () Theorem 37 Define the sets F (r, s, t), F 2 (r, s, t), F 3 (r, s, t), F 4 (r, s, t), F 5 (r, s, t) as follows: F (r, s, t) = F 2 (r, s, t) = F 3 (r, s, t) = F 4 (r, s, t) = F 5 (r, s, t) = a = (a ) ω : sup n N a = (a ) ω : lim j= a = (a ) ω : lim a = (a ) ω : lim a = (a ) ω : lim j= ( ) j D (s) j t r a j <, j ( ) j D (s) j t r a j j [ n j= =n+ Then, the β dual of the sequence space f(r, s, t) is j= 5 F i (r, s, t) i= exists, ] ( ) j D (s) j t r a j exists, j t j ( ) j D (s) j r a j = 0 j=n+ ( ā j α ) = 0,

8 Journal of New Theory 2 (205) Proof Let a = (a ) ω and consider the equality a x = = [ t [ n j= ( ) j D (s) j r j x j ]a t j ( ) j D (s) j r a j ] x = F (r, s, t) x n, (2) where F (r, s, t) = f n (r, s, t) is defined by ( ) f j D (s) j n (r, s, t) = t r a j (0 n), j j= 0 ( > n) (3) for all n, N Thus, we deduce from Lemma 36 with (2) that ax = (a x ) cs whenever x = (x ) f(r, s, t) if and only if F (r, s, t) x c whenever x = ( x ) f, where F (r, s, t) = f n (r, s, t) is defined by (3) Therefore, we derive from (8), (9), (0) and () that sup n N j= t j ( ) j D (s) j r a j <, which shows that lim j= t j ( ) j D (s) j r a j = α for each fixed N, lim lim j= [ n j= f(r, s, t) β = t j ( ) j D (s) j r a j = α, t j ( ) j D (s) j r a j ] = 0 5 F i (r, s, t) i= Theorem 38 The γ dual of the sequence spaces f(r, s, t) and f 0 (r, s, t) is the set F (r, s, t) Proof This is similar to the proof of Theorem 37 with Lemma 35 instead of Lemma 36 So, we omit the detail 4 Matrix Transformations Related to The Sequence Space f(r, s, t) In the present section, we characterize the matrix transformations from f(r, s, t) into any given sequence space µ Since f(r, s, t) = f, it is trivial that the equivalence x f(r, s, t) if and only if x f holds Theorem 4 Suppose that the entries of the infinite matrices A = (a n ) and E = (e n ) are connected with the relation e n = ( ) j D (s) j t r a nj (4) j j= for all n, N and µ is any given sequence space Then A ( f(r, s, t) : µ) if and only if a n N f(r, s, t) β for all n N and E (f : µ)

9 Journal of New Theory 2 (205) Proof Let µ be any given sequence space Suppose that (4) holds between A = (a n ) and E = (e n ), and tae into account that the spaces f(r, s, t) and f are linearly isomorphic Let A ( f(r, s, t) : µ) and tae any x = ( x ) f Then EĀ(r, s, t) exists and a n N 5 i= F i(r, s, t) which yields that e n N l for each n N Hence, E x exists and thus e n x = a n x for all n N We have that E x = Ax which leads us to the consequence E (f : µ) Conversely, let a n N f(r, s, t) β for each n N and E (f : µ) hold, and tae any x = (x ) f(r, s, t) Then, Ax exists Therefore, we obtain from the equality m m [ m a n x = ( ) j D (s) j t r a nj ] x j j= for all n N, as m that E x = Ax and this shows that A ( f(r, s, t) : µ) This step completes the proof By changing the roles of the spaces f(r, s, t) and µ in Theorem (4), we have: Theorem 42 Suppose that the elements of the infinite matrices A = (a n ) and C = (c n ) are connected with the relation c n = s n j t j a j for all n, N Let µ be any given sequence space Then, A = (a n ) (µ : f(r, s, t)) if and only if C (µ : f) Proof Let z = (z ) µ and consider the following equality m c n z = ( m ) s n j t j a j z for all m, n N, which yields as m that (Cz) n = Ā(r, s, t)(az) n for all n N Therefore, one can observe from here that Az f(r, s, t) whenever z µ if and only if Cz f whenever z µ This completes the proof Of course, Theorems 4 and 42 have several consequences depending on the choice of the sequence space µ Define a(n, ), a(n,, m) and a n for all, m, n N as follows; a(n, ) = a j, a(n,, m) = m + m a n+j, and a n = a n a n,+ Prior to giving some results as an application of this idea, we give the following basic lemma, which is the collection of the characterizations of matrix transformations related to almost convergence: Lemma 43 Let A = (a n ) be an infinite matrix Then, the following statements hold: (i) [33, J P Duran]A = (a n ) (l : f) if and only if (8) holds and α C f lim a n = α for all N, (5) α C lim a(n,, m) α = 0 uniformly in n (6) also hold (ii) [33, J P Duran]A = (a n ) (f : f) if and only if (8) and (5) hold, and α C f lim a n = α, (7)

10 Journal of New Theory 2 (205) α C lim [ ] a(n,, m) α = 0 uniformly in n (8) also hold (iii) [34, J P King]A = (a n ) (f : f) if and only if (8), (5) and (7) hold (iv) [35, Başar and Çola]A = (a n ) (cs : f) if and only if (5) holds, and sup n N a n < (9) also holds (v) [36, Başar and Sola]A = (a n ) (bs : f) if and only if (5) and (9) hold, and lim a n = 0 for all n N, (20) q α C lim q q + [ ] a(n + i, ) α = 0 uniformly in n (2) i=0 also hold (vi) [37, Başar]A = (a n ) (f : cs) if and only if the following conditions hold: a(n, ) <, (22) sup n N α C n a n = α for all N, (23) α C a n = α, (24) n α C lim [ ] a(n, ) α = 0 (25) Now, we can give the following two corollaries as a direct consequence of Theorems 4 and 42 and Lemma 43: Corollary 44 The following statements hold: (i) A = (a n ) ( f(r, s, t) : l ) if and only if a n N f(r, s, t) β and (8) holds with e n instead of a n (ii) A = (a n ) ( f(r, s, t) : c) if and only if a n N f(r, s, t) β and (8), (9), (0) and () hold with e n instead of a n (iii) A = (a n ) ( f(r, s, t) : bs) if and only if a n N f(r, s, t) β and (22) holds with e n instead of a n (iv) A = (a n ) ( f(r, s, t) : cs) if and only if a n N f(r, s, t) β and (22), (23), (24) and (25) hold with e n instead of a n (v) A = (a n ) ( f(r, s, t) : f) if and only if a n N f(r, s, t) β and (8), (5), (7) and (8) hold with e n instead of a n Corollary 45 The following statements hold: (i) A = (a n ) (l : f(r, s, t)) if and only if (8), (5) and (7) hold with e n instead of a n (ii) A = (a n ) (f : f(r, s, t)) if and only if (8), (5), (7) and (8) hold with e n instead of a n (iii) A = (a n ) (c : f(r, s, t)) if and only if (8), (5) and (7) hold with e n instead of a n (iv) A = (a n ) (bs : f(r, s, t)) if and only if (5), (9), (20) and (2) hold with en instead of a n (v) A = (a n ) (cs : f(r, s, t)) if and only if (5) and (9) hold with e n instead of a n Acnowledgement We wish to than the referees for their valuable suggestions and comments which improved the paper considerably

11 Journal of New Theory 2 (205) References [] GG Lorentz, A contribution to the theory of divergent sequences, Acta Math 80(948), [2] F Başar, B Altay, On the space of sequences of p bounded variation and related matrix mappings, Urainian Math J 55()(2003), [3] R Çola, M Et, E Malowsy, Some topics of sequence spaces, in: Lecture Notes in Mathematics, Fırat Univ Press, Turey, 2004, pp-63 Fırat Univ Elazı g, 2004, ISBN: [4] C S Wang, On Nörlund sequence spaces, Tamang J Math 9(978), [5] P N Ng, PY Lee, Cesàro sequence spaces of non-absolute type, Comment Math Prace Mat 20(2)(978), [6] C Aydın, F Başar, On the new sequence spaces which include the spaces c 0 and c, Hoaido Math J 33(2) (2004), [7] B Altay, F Başar, Some Euler sequence spaces of non-absolute type, Urainian Math J 57()(2005), -7 [8] B Altay, F Başar, M Mursaleen, On the Euler sequence spaces which include the spaces l p and l I, Inform Sci 76(0)(2006), [9] F Başar, M Kirişci, Almost convergence and generalized difference matrix, Comput Math Appl 6(20), [0] M Mursaleen, F Başar, B Altay, On the Euler sequence spaces which include the spaces l p and l II, Nonlinear Anal TMA, 65(3)(2006), [] J A Sıddıqi, Infinite matrices summing every almost periodic sequences, Pasific J Math 33()(97), [2] A M Jarrah, E Malowsy, BK-spaces, bases and linear operators, Ren Circ Mat Palermo II 52(990), 77-9 [3] F Başar, Summability theory and its applications, Bentham Science Publishers, Istanbul, 202, e-boos, Monographs [4] K Kayaduman and M Şengönül, The spaces of Cesàro almost convergent sequences and core theorems, Acta Math Sci 32B(6) (202), [5] M Şengönül and K Kayaduman, On the Riesz almost convergent sequences space, Abstr Appl Anal (202), Article ID 69694, 8pp [6] A Karaisa and Ümit Karabıyı, Almost sequence spaces by the domain of the matrix Ar, Abstr Appl Anal, (203), Article ID 78373, 8pp [7] M Kirişçi, Almost convergence and generalized weighted mean II, Journal of Inequalities and Applications (204), 204:93 [8] M Kirişçi, The spaces of Euler almost null and Euler almost convergent sequences, Commun Fac Sci Univ An Series A, 62(2) (203), [9] A Sönmez, Almost convergence and triple band matrix, Mathematical and Computer Modelling, 57(203), [20] M Yeşilayagil, F Başar, Spaces of A λ almost null and A λ almost convergence sequences, Journal of Egyptian Mathematical Society (204), [2] E E Kara, K Elmaa gaç, On the u difference almost sequence space and matrix transformations, Int J Mod Math Sci, 8() (203), [22] Y Ouyama, On absolute generalized Nörlund summability of orthogonal series, Tamang J Math, 33(2) (2002), 6-65 [23] U Stadtmüller, A Tali, A note on families of generalized Nörlund matrices as bounded operators on l p, Proc Estonian Acad Sci 58(3) (2009), 37-45

12 Journal of New Theory 2 (205) [24] J Boos, Classical and Modern Methods in Summability, Oxford University Press, New Yor, 2000 [25] H Bor, On a summability factor theorem, Commun Math Anal, () (2006), 46-5 [26] M Şengönül, F Başar, Some new Cesàro sequence spaces of non-absolute type which include the spaces c 0 and c, Soochow J Math, 3() (2005), 07-9 [27] E Malowsy, V Raočević, An introduction into the theory of sequence spaces and measures of noncompactness, Zb Rad (Beogr), 9(7) (2000), [28] C Aydın, F Başar, Some new sequence spaces which iclude the spaces l p and l, Demonstratio Math, 38(3) (2005), [29] B de Malafosse, The Banach algebra B(X), where X is a BK space and applications, Mat Vesni, 57 (2005), 4-60 [30] F M Mears, The inverse Nörlund mean, Ann Math, 44(3) (943), [3] M Mursaleen, A K Noman, On generalized means and some related sequence spaces, Comput Math Appl 6(4) (20), [32] A Wilansy, Summability Through Functional Analysis, Mathematical Studies, North Holland, Amsterdam, 984 [33] J P Duran, Infinite matrices and almost convergence, Math Z 28 (972), [34] J P King, Almost summable sequences, Proc Am Math Soc, 7 (966), [35] F Başar, R Çola, Almost conservative matrix transformations, Tur J Math 3(3) (989), 9-00 [36] F Başar, I Sola, Almost-coercive matrix transformations, Rend Math Appl (2) (99), [37] F Başar, Strongly-conservative sequence-to-series matrix transformations, Erc Uni Fen Bil Derg 5(2) (989),