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

Abstract

The aim of the present paper is to introduce the spaces [superscript] c 0 λ[/superscript] (B ) and [superscript] c λ[/superscript] (B ) of generalized difference sequences which generalize the paper due to Mursaleen and Noman (2010). These spaces are the BK-spaces of non-absolute type and norm isomorphic to the spaces [subscript] c 0[/subscript] and c , respectively. Furthermore, we derive some inclusion relations determine the α - , β - , and γ - duals of those spaces, and construct their Schauder bases. Finally, we characterize some matrix classes from the spaces [superscript] c 0 λ[/superscript] (B ) , and [superscript] c λ[/superscript] (B ) to the spaces [subscript] [cursive l] p[/subscript] , [subscript] c 0[/subscript] , and c.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Abdulcabbar Sönmez 1 and Feyzi Basar 2

Recommended by Stevo Stevic

1, Department of Mathematics, Faculty of Sciences, Erciyes University, Melikgazi, 38039 Kayseri, Turkey
2, Department of Mathematics, Faculty of Art and Sciences, Fatih University, The Hadimköy Campus, Büyükçekmece, 34500 Istanbul, Turkey

Received 17 February 2012; Accepted 10 October 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. 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 [arrow right] ... defined by _{p i} (x ) = _{x i} is continuous for all i ∈ ... , where ... denotes the complex field and ... = {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 [varphi] , the set of all finitely nonzero sequences. We write _{[cursive l] ∞} , 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 ∈ ... . Also by _{[cursive l] 1} and _{[cursive l] p} , we denote the spaces of all absolutely and p -absolutely convergent series, respectively, which are BK -spaces with the usual norm defined by ||x _{|| p} = ^{( _{∑ k} ... | _{x k} | p ) 1 /p} , where 1 ...4;p < ∞ . For simplicity in notation, here and in what follows, the summation without limits runs 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 ∈ ... . Then, we say that A defines a matrix mapping from X into Y and we denote it by writing A :X [arrow right]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 [figure omitted; refer to PDF] By (X :Y ) , we denote the class of all infinite matrices A = ( _{a nk} ) such that A :X [arrow right]Y . Thus A ∈ (X :Y ) if and only if the series on the right side of ( 1.1) converges for each 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 [figure omitted; refer to PDF] We denote the collection of all finite subsets of ... by ... . Also, we write ^{e (k )} for the sequence whose only nonzero term is a 1 in the k th place for each k ∈ ... .

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 ( _{[cursive l] ∞ ) N q} and _{c N q} in [ 14], ( _{[cursive l] p ) C 1} = _{X p} and ( _{[cursive l] ∞ ) C 1} = _{X ∞} in [ 10], _{μ G} =Z (u ,v ; μ ) in [ 9], ( _{[cursive l] ∞ ) ^{R t}} = ^{r ∞ t} , _{c ^{R t}} = ^{r c t} and _{( c 0 ) ^{R t}} = ^{r 0 t} in [ 8], ( _{[cursive l] p ) ^{R t}} = ^{r p t} in [ 2], ( _{c 0 ) ^{E t}} = ^{e 0 r} and _{c ^{E r}} = ^{e c r} in [ 3], ( _{[cursive l] p ) ^{E r}} = ^{e p r} and ( _{[cursive l] ∞ ) ^{E r}} = ^{e ∞ r} in [ 4], ( _{c 0 ) ^{A r}} = ^{a 0 r} and _{c ^{A r}} = ^{a c r} in [ 5], [ _{c 0} (u ,p ) _{] ^{A r}} = ^{a 0 r} (u ,p ) and [c (u ,p ) _{] ^{A r}} = ^{a c r} (u ,p ) in [ 6], ( _{[cursive l] p ) ^{A r}} = ^{a p r} and ( _{[cursive l] ∞ ) ^{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 , _{[cursive l] p} } , ν ∈ { _{[cursive l] ∞} ,c , _{c 0} , _{[cursive l] p} } and 1 ...4;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 Basarir [ 15]. In the present paper, following [ 2- 14], we introduce the difference sequence spaces ^{c 0 λ} (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 0 λ} (B ) and ^{c λ} (B ) .

The rest of this paper is organized, as follows.

In Section 2, the BK -spaces ^{c 0 λ} (B ) and ^{c λ} (B ) of generalized difference sequences are introduced. Section 3is devoted to inclusion relations concerning with the spaces ^{c 0 λ} (B ) and ^{c λ} (B ) . In Sections 4and 5, the Schauder bases of the spaces ^{c 0 λ} (B ) and ^{c λ} (B ) are given and the α -, β -, and γ -duals of the generalized difference sequence spaces ^{c 0 λ} (B ) and ^{c λ} (B ) of non-absolute type are determined, respectively. In Section 6, the classes ( ^{c λ} (B ) : _{[cursive l] p} ) , ( ^{c 0 λ} (B ) : _{[cursive l] p} ) , ( ^{c λ} (B ) :c ) , ( ^{c λ} (B ) : _{c 0} ) , ( ^{c 0 λ} (B ) :c ) , and ( ^{c 0 λ} (B ) : _{c 0} ) of matrix transformations are characterized, where 1 ...4;p ...4; ∞ . 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.

2. The Difference Sequence Spaces ^{c 0 λ} (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 λ} ( 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, [figure omitted; refer to PDF]

Recently, Mursaleen and Noman [ 22] studied the sequence spaces ^{c 0 λ} and ^{c λ} of non-absolute type, and later they introduced the difference sequence spaces ^{c 0 λ} ( Δ ) and ^{c λ} ( Δ ) in [ 21] of non-absolute type as follows: [figure omitted; refer to PDF] 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 0 λ} ( Δ ) and ^{c λ} ( Δ ) by [figure omitted; refer to PDF] where Δ denotes the band matrix representing the difference operator, that is, Δx = ( _{x k} - _{x k -1} ) ∈ ω for x = ( _{x k} ) ∈ ω .

Let r and s be nonzero real numbers and define the generalized difference matrix B (r ,s ) = { _{b nk} (r ,s ) } by [figure omitted; refer to PDF] for all k ,n ∈ ... . The B (r ,s ) -transform of a sequence x = ( _{x k} ) is [figure omitted; refer to PDF] 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 Basar and Altay [ 18] and Aydin and Basar [ 17], we proceed slightly differently to Kizmaz [ 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 λ} (B ) and ^{c λ} (B ) , which are the generalization of the spaces ^{c 0 λ} ( Δ ) and ^{c λ} ( Δ ) introduced by Mursaleen and Noman [ 21], as follows: [figure omitted; refer to PDF] With the notation of ( 1.2), we can redefine the spaces ^{c 0 λ} (B ) and ^{c λ} (B ) as [figure omitted; refer to PDF] 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 0 λ} (B ) and ^{c λ} (B ) are linear spaces with coordinatewise addition and scalar multiplication, that is, ^{c 0 λ} (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 [figure omitted; refer to PDF] for all n ,k ∈ ... . With a direct calculation we derive the equality [figure omitted; refer to PDF] and every x = ( _{x k} ) ∈ ω which leads us together with ( 1.2) to the fact that [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] 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: [figure omitted; refer to PDF] 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.

Theorem 2.1.

The difference sequence spaces ^{c 0 λ} (B ) and ^{c λ} (B ) are BK-spaces with the norm ||x _{|| ^{c 0 λ} (B )} = ||x _{|| ^{c λ} (B )} = || Λ ^ (x ) _{|| ∞} , that is, [figure omitted; refer to PDF]

Proof.

Since ( 2.10) holds and _{c 0} and c are BK -spaces with respect to their natural norms (see [ 23, pages 217-218]) and the matrix Λ ^ is a triangle, Theorem 4.3 .12 of Wilansky [ 24, page 63] gives the fact that ^{c 0 λ} (B ) and ^{c λ} (B ) are BK -spaces with the given norms. This completes the proof.

Remark 2.2.

One can easily check that the absolute property does not hold on the spaces ^{c 0 λ} (B ) and ^{c λ} (B ) , that is, ||x _{|| ^{c 0 λ} (B )} ...0; || | x | _{|| ^{c 0 λ} (B )} and ||x _{|| ^{c λ} (B )} ...0; || | x | _{|| ^{c λ} (B )} for at least one sequence in the spaces ^{c 0 λ} (B ) and ^{c λ} (B ) , and this shows that ^{c 0 λ} (B ) and ^{c λ} (B ) are the sequence spaces of non-absolute type, where | x | = ( | _{x k} | ) .

Now, we give the final theorem of this section.

Theorem 2.3.

The sequence spaces ^{c 0 λ} (B ) and ^{c λ} (B ) of non-absolute type are norm isomorphic to the spaces _{c 0} and c , respectively, that is, ^{c 0 λ} (B ) [congruent with] _{c 0} and ^{c λ} (B ) [congruent with]c .

Proof.

To prove this, we should show the existence of a linear bijection between the spaces ^{c 0 λ} (B ) and _{c 0} . Consider the transformation T defined, with the notation of ( 2.11), from ^{c 0 λ} (B ) to _{c 0} by x ...y ( λ ) . Then, Tx =y ( λ ) = Λ ^ (x ) ∈ _{c 0} for every x ∈ ^{c 0 λ} (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 [figure omitted; refer to PDF] Then, we obtain [figure omitted; refer to PDF] Hence, for every n ∈ ... , we get by ( 2.9) [figure omitted; refer to PDF] This shows that Λ ^ (x ) =y and since y ∈ _{c 0} , we conclude that Λ ^ (x ) ∈ _{c 0} . Thus, we deduce that x ∈ ^{c 0 λ} (B ) and Tx =y . Hence T is surjective.

Moreover, one can easily see for every x ∈ ^{c 0 λ} (B ) that [figure omitted; refer to PDF] which means that T is norm preserving. Consequently T is a linear bijection which show that the spaces ^{c 0 λ} (B ) and _{c 0} are linearly isomorphic.

It is clear that if the spaces ^{c 0 λ} (B ) and _{c 0} are replaced by the spaces ^{c λ} (B ) and c , respectively, then we obtain the fact that ^{c λ} (B ) [congruent with]c . This completes the proof.

3. The Inclusion Relations

In the present section, we establish some inclusion relations concerning with the spaces ^{c 0 λ} (B ) and ^{c λ} (B ) . We may begin with the following theorem.

Theorem 3.1.

The inclusion ^{c 0 λ} (B ) ⊂ ^{c λ} (B ) strictly holds.

Proof.

It is obvious that the inclusion ^{c 0 λ} (B ) ⊂ ^{c λ} (B ) holds. Further to show that this inclusion is strict, consider the sequence x = ( _{x k} ) defined by _{x k} = ^{∑ j =0 k} ... ^{( -s / r ) j} /r for all k ∈ ... . Then, we obtain by ( 2.9) that [figure omitted; refer to PDF] which shows that Λ ^ (x ) =e and hence Λ ^ (x ) ∈c \ _{c 0} , where e = (1,1 ,1 , ... ) . Thus, the sequence x is in ^{c λ} (B ) but not in ^{c 0 λ} (B ) . Hence, the inclusion ^{c 0 λ} (B ) ⊂ ^{c λ} (B ) is strict and this completes the proof.

Theorem 3.2.

If s +r =0 , then the inclusion c ⊂ ^{c 0 λ} (B ) strictly holds.

Proof.

Suppose that s +r =0 and x ∈c . Then B (r ,s )x = (r _{x k} +s _{x k -1} ) ∈ _{c 0} and hence B (r ,s )x ∈ ^{c 0 λ} , since the inclusion _{c 0} ⊂ ^{c 0 λ} . This shows that x ∈ ^{c 0 λ} (B ) . Consequently, the inclusion c ⊂ ^{c 0 λ} (B ) holds. Further consider the sequence y = ( _{y k} ) defined by _{y k} = k +1 for all k ∈ ... . 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 0 λ} which means that y ∈ ^{c 0 λ} (B ) . Thus, the sequence y is in ^{c 0 λ} (B ) but not in c . We therefore deduce that the inclusion c ⊂ ^{c 0 λ} (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

Corollary 3.3.

The inclusions _{c 0} ⊂ ^{c 0 λ} (B ) and c ⊂ ^{c λ} (B ) strictly hold.

Further, it is obvious that the sequence y , defined in the proof of Theorem 3.2, is in ^{c 0 λ} (B ) but not in _{[cursive l] ∞} . This leads us to the following result.

Corollary 3.4.

Although the spaces _{[cursive l] ∞} and ^{c 0 λ} (B ) overlap, the space _{[cursive l] ∞} does not include the space ^{c 0 λ} (B ) .

Now, to prove the next theorem, we need the following lemma [ 24, page 4].

Lemma 3.5.

A ∈ ( _{[cursive l] ∞} : _{c 0} ) if and only if _{lim n [arrow right] ∞ ∑ k} ... | _{a nk} | =0 .

Theorem 3.6.

The inclusion _{[cursive l] ∞} ⊂ ^{c 0 λ} (B ) strictly holds if and only if z ∈ ^{c 0 λ} , where the sequence z = ( _{z k} ) is defined by [figure omitted; refer to PDF]

Proof.

Suppose that the inclusion _{[cursive l] ∞} ⊂ ^{c 0 λ} (B ) holds. Then we obtain that Λ ^ (x ) ∈ _{c 0} for every x ∈ _{[cursive l] ∞} and hence the matrix Λ ^ = ( _{λ ^ nk} ) is in the class ( _{[cursive l] ∞} : _{c 0} ) . Thus it follows by Lemma 3.5that [figure omitted; refer to PDF] Now, by taking into account the definition of the matrix Λ ^ = ( _{λ ^ nk} ) given by ( 2.8), we have for every n ∈ ... that [figure omitted; refer to PDF] Thus, the condition ( 3.3) implies both [figure omitted; refer to PDF] Now we have for every n ...5;1 that [figure omitted; refer to PDF] and since _{lim n [arrow right] ∞} ( _{λ n -1} / _{λ n} ) =1 by ( 3.5), we obtain by ( 3.6) that [figure omitted; refer to PDF] which shows that z = ( _{z k} ) ∈ ^{c 0 λ} .

Conversely, suppose that z = ( _{z k} ) ∈ ^{c 0 λ} . Then we have ( 3.8). Further, for every n ...5;1 , we derive that [figure omitted; refer to PDF] Then, ( 3.9) and ( 3.8) together imply that ( 3.6) holds. On the other hand, we have for every n ...5;1 that [figure omitted; refer to PDF] Therefore, it follows by ( 3.6) that _{lim n [arrow right] ∞} [r _{λ n -1} +s ( _{λ n} - _{λ 0} ) ] / _{λ n} =0 . Particularly, if we take r =1 and s = -1 , then we have _{lim n [arrow right] ∞} [ _{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 3.5to the consequence that Λ ^ ∈ ( _{[cursive l] ∞} : _{c 0} ) . Hence, the inclusion _{[cursive l] ∞} ⊂ ^{c 0 λ} (B ) holds and is strict by Corollary 3.4. This completes the proof.

4. The Bases for the Spaces ^{c 0 λ} (B ) and ^{c λ} (B )

In the present section, we give two sequences of the points of the spaces ^{c 0 λ} (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 [figure omitted; refer to PDF] 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 0 λ} (B ) to _{c 0} in the proof of Theorem 2.3is 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 0 λ} (B ) . Therefore, we have the following.

Theorem 4.1.

Let _{α k} ( λ ) = _{Λ ^ k} (x ) for all k ∈ ... and l = _{lim k [arrow right] ∞ Λ ^ k} (x ) . Define the sequence ^{b ( k )} ( λ ) = ^{{ b n ( k ) ( λ ) } k =0 ∞} for every fixed k ∈ ... by [figure omitted; refer to PDF] Then, the following statements hold.

(a) The sequence ^{{ b ( k ) ( λ ) } k =0 ∞} is a basis for the space ^{c 0 λ} (B ) and any x ∈ ^{c 0 λ} (B ) has a unique representation of the form x = _{∑ ... k α k} ( λ ) ^{b ( k )} ( λ ) .

(b) The sequence { b , ^{b ( 0 )} ( λ ) , ^{b ( 1 )} ( λ ) , ... } is a basis for the space ^{c λ} (B ) and any x ∈ ^{c λ} (B ) has a unique representation of the form x =lb + _{∑ ... k} [ _{α k} ( λ ) -l ] ^{b (k )} ( λ ) , where b = ( _{b k} ) = ^{{ ∑ j =0 k ... ( -s /r ) j /r } k =0 ∞} .

Finally, it easily follows from Theorem 2.1that ^{c 0 λ} (B ) and ^{c λ} (B ) are the Banach spaces with their natural norms. Then by Theorem 4.1we obtain the following.

Corollary 4.2.

The difference sequence spaces ^{c 0 λ} (B ) and ^{c λ} (B ) are seprable.

5. The α -, β -, and γ -Duals of the Spaces ^{c 0 λ} (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 0 λ} (B ) and ^{c λ} (B ) of non-absolute type.

For arbitrary sequence spaces X and Y , the set M (X ,Y ) defined by [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 5.5to 5.8.

Lemma 5.1.

A = ( _{a nk} ) ∈ ( _{c 0} : _{[cursive l] 1} ) = (c : _{[cursive l] 1} ) if and only if [figure omitted; refer to PDF]

Lemma 5.2.

A = ( _{a nk} ) ∈ ( _{c 0} :c ) if and only if [figure omitted; refer to PDF]

Lemma 5.3.

A = ( _{a nk} ) ∈ (c :c ) if and only if ( 5.4) and ( 5.5) hold, and [figure omitted; refer to PDF]

Lemma 5.4.

A = ( _{a nk} ) ∈ (c : _{[cursive l] ∞} ) = ( _{c 0} : _{[cursive l] ∞} ) if and only if ( 5.5) holds.

Now, we prove the following result.

Theorem 5.5.

The α -dual of the spaces ^{c 0 λ} (B ) and ^{c λ} (B ) is the set [figure omitted; refer to PDF] where the matrix ^{H λ} = ( ^{h nk λ} ) is defined via the sequence a = ( _{a n} ) ∈ ω by [figure omitted; refer to PDF] for all n ,k ∈ ... .

Proof.

Let a = ( _{a n} ) ∈ ω . Then, by bearing in mind the relations ( 2.11) and ( 2.14), it is immediate that the equality [figure omitted; refer to PDF] holds for all n ∈ ... . Thus, we observe by ( 5.9) that ax = ( _{a n x n} ) ∈ _{[cursive l] 1} whenever x = ( _{x k} ) ∈ ^{c 0 λ} (B ) or ^{c λ} (B ) if and only if ^{H λ} y ∈ _{[cursive l] 1} whenever y = ( _{y k} ) ∈ _{c 0} or c . This means that the sequence a = ( _{a n} ) ∈ ^{{ c 0 λ (B ) } α} or a = ( _{a n} ) ∈ ^{{ c λ (B ) } α} if and only if ^{H λ} ∈ ( _{c 0} : _{[cursive l] 1} ) = (c : _{[cursive l] 1} ) . We therefore obtain by Lemma 5.4with ^{H λ} instead of A that a = ( _{a n} ) ∈ ^{{ c 0 λ (B ) } α} = ^{{ c λ (B ) } α} if and only if [figure omitted; refer to PDF] which leads us to the consequence that ^{{ c 0 λ (B ) } α} = ^{{ c λ (B ) } α} = ^{h 1 λ} . This completes the proof.

Theorem 5.6.

Define the sets ^{h 2 λ} , ^{h 3 λ} , ^{h 4 λ} , and ^{h 5 λ} , as follows: [figure omitted; refer to PDF] Then ^{{ c 0 λ (B ) } β} = ^{h 2 λ} ∩ ^{h 3 λ} ∩ ^{h 4 λ} and ^{{ c λ (B ) } β} = ^{h 2 λ} ∩ ^{h 3 λ} ∩ ^{h 4 λ} ∩ ^{h 5 λ} .

Proof.

Consider the equality [figure omitted; refer to PDF] where the matrix ^{T λ} = ( ^{t nk λ} ) is defined by [figure omitted; refer to PDF] for all k ,n ∈ ... . Then, we deduce by ( 5.12) that ax = ( _{a k x k} ) ∈cs whenever x = ( _{x k} ) ∈ ^{c 0 λ} (B ) if and only if ^{T λ} y ∈c whenever y = ( _{y k} ) ∈ _{c 0} . This means that a = ( _{a k} ) ∈ ^{{ c 0 λ (B ) } β} if and only if ^{T λ} ∈ ( _{c 0} :c ) . Therefore, by using Lemma 5.2, we derive from ( 5.4) and ( 5.5) that [figure omitted; refer to PDF] Therefore, we conclude that ^{{ c 0 λ (B ) } β} = ^{h 2 λ} ∩ ^{h 3 λ} ∩ ^{h 4 λ} .

Similarly, we deduce from Lemma 5.3with ( 5.12) that a = ( _{a k} ) ∈ ^{{ c λ (B ) } β} if and only if ^{T λ} ∈ (c :c ) . Therefore, we derive from ( 5.4) and ( 5.5) that ( 5.14), ( 5.15) hold.

Further, with a simple calculation one can easily see that the equality [figure omitted; refer to PDF] holds for all n ∈ ... . Consequently, from ( 5.6) we obtain that [figure omitted; refer to PDF] Hence, we deduce that ^{{ c λ (B ) } β} = ^{h 2 λ} ∩ ^{h 3 λ} ∩ ^{h 4 λ} ∩ ^{h 5 λ} . This completes the proof.

Remark 5.7.

We may note by combining ( 5.17) with the conditions ( 5.15) that { ^{∑ j =0 k} ... ^{( -s / r ) j} _{a k} /r } ∈bs for every sequence a = ( _{a k} ) ∈ ^{{ c 0 λ (B ) } β} .

Finally, we close this section with the following theorem which determines the γ -dual of the spaces ^{c 0 λ} (B ) and ^{c λ} (B ) :

Theorem 5.8.

The γ -duals of the spaces ^{c 0 λ} (B ) and ^{c λ} (B ) are the set ^{h 3 λ} ∩ ^{h 4 λ} .

Proof.

The proof of this result follows the same lines that in the proof of Theorem 5.6using Lemma 5.4instead of Lemma 5.2.

6. Certain Matrix Mappings Related to the Spaces ^{c 0 λ} (B ) and ^{c λ} (B )

In this final section, we characterize the matrix classes ( ^{c λ} (B ) : _{[cursive l] p} ) , ( ^{c 0 λ} (B ) : _{[cursive l] p} ) , ( ^{c λ} (B ) :c ) , ( ^{c λ} (B ) : _{c 0} ) , ( ^{c 0 λ} (B ) :c ) , and ( ^{c 0 λ} (B ) : _{c 0} ) , where 1 ...4;p ...4; ∞ . Also, by means of a given basic lemma, we derive the characterizations of some other classes involving the Euler, difference, Riesz, and Cesàro sequence spaces.

For an infinite matrix A = ( _{a nk} ) , we write for brevity that [figure omitted; refer to PDF] for all k ,m ,n ∈ ... provided the convergence of the series.

The following lemmas will be needed in proving our main results.

Lemma 6.1 (see [ 24, page 57]).

The matrix mappings between the BK-spaces are continuous.

Lemma 6.2 (see [ 25, pages 7-8]).

A = ( _{a nk} ) ∈ (c : _{[cursive l] p} ) if and only if [figure omitted; refer to PDF]

Lemma 6.3 (see [ 25, page 5]).

A = ( _{a nk} ) ∈ (c : _{c 0} ) if and only if ( 5.5) holds and [figure omitted; refer to PDF]

Lemma 6.4 (see [ 25, page 5]).

A = ( _{a nk} ) ∈ ( _{c 0} : _{c 0} ) if and only if ( 5.5) and ( 6.3) hold.

Now, we give the following results on the matrix transformations.

Theorem 6.5.

Let A = ( _{a nk} ) be an infinite matrix over the complex field. Then, the following statements hold.

: (i) Let (1 ...4;p < ∞ ) . Then, A ∈ ( ^{c λ} (B ) : _{[cursive l] p} ) if and only if [figure omitted; refer to PDF]

: (ii) A ∈ ( ^{c λ} (B ) : _{[cursive l] ∞} ) if and only if ( 6.7) and ( 6.8) hold, and [figure omitted; refer to PDF]

Proof.

Suppose that the conditions ( 6.4)-( 6.9) hold and take any x = ( _{x k} ) ∈ ^{c λ} (B ) . Then, we have by Theorem 5.6that _{{ a nk } k ∈ ...} ∈ ^{{ c λ (B ) } β} for all n ∈ ... and this implies that the A -transform of x exists. Also, it is clear that the associated sequence y = ( _{y k} ) is in the space c and hence _{y k} [arrow right]l as k [arrow right] ∞ for some suitable l . Further, it follows by combining Lemma 6.2with ( 6.5) that the matrix A ^ = ( _{a ^ nk} ) is in the class (c : _{[cursive l] p} ) , where 1 ...4;p < ∞ .

Let us now consider the following equality derived by using the relation ( 2.11) from the m th partial sum of the series _{∑ k} ... _{a nk x k} : [figure omitted; refer to PDF] Then, since y ∈c and A ^ ∈ (c : _{[cursive l] p} ) , the series _{∑ k} ... _{a ^ nk y k} converges for every n ∈ ... . Furthermore, it follows by ( 6.4) that the series ^{∑ j =k ∞} ... ^{( -s / r ) n -j} _{a nj} converges for all n ,k ∈ ... and hence _{a ^ nk} (m ) [arrow right] _{a ^ nk} as m [arrow right] ∞ . Therefore, if we pass to limit in ( 6.12) as m [arrow right] ∞ then we obtain by ( 6.8) that [figure omitted; refer to PDF] which can be written as follows: [figure omitted; refer to PDF] This yields by taking p -norm that [figure omitted; refer to PDF] which leads us to the consequence that Ax ∈ _{[cursive l] p} . Hence, A ∈ ( ^{c λ} (B ) : _{[cursive l] p} ) .

Conversely, suppose that A ∈ ( ^{c λ} (B ) : _{[cursive l] p} ) , where 1 ...4;p < ∞ . Then _{{ a nk } k ∈ ...} ∈ ^{{ c λ (B ) } β} for all n ∈ ... which implies with Theorem 5.6that the conditions ( 6.6) and ( 6.7) are necessary.

On the other hand, since ^{c λ} (B ) and _{[cursive l] p} are BK -spaces, we have by Lemma 6.1that there is a constant M >0 such that [figure omitted; refer to PDF] holds for all x ∈ ^{c λ} (B ) . Now, K ∈ ... . Then, the sequence z = _{∑ k ∈K} ... ^{b (k )} ( λ ) is in ^{c λ} (B ) , where the sequence ^{b (k )} ( λ ) = _{{ ^{b n ( k )} ( λ ) } n ∈ ...} is defined by ( 4.2) for every fixed k ∈ ... .

Since Λ ^ ( ^{b (k )} ( λ ) ) = ^{e (k )} for each fixed k ∈ ... , we have [figure omitted; refer to PDF]

Furthermore, for every n ∈ ... , we obtain by ( 4.2) that [figure omitted; refer to PDF] Hence, since the inequality ( 6.16) is satisfied for the sequence z ∈ ^{c λ} (B ) , we have for any K ∈ ... that [figure omitted; refer to PDF] which shows the necessity of ( 6.5). Thus, it follows by Lemma 6.2that A ^ = ( _{a ^ nk} ) ∈ (c : _{[cursive l] p} ) .

Now, let y = ( _{y k} ) ∈c \ _{c 0} and consider the sequence x = ( _{x k} ) defined by ( 2.14) for every k ∈ ... . Then, x ∈ ^{c λ} (B ) such that y = Λ ^ (x ) , that is, the sequences x and y are connected by the relation ( 2.11). Therefore, Ax and A ^ y exist. This leads us to the convergence of the series _{∑ k} ... _{a nk x k} and _{∑ k} ... _{a ^ nk y k} for every n ∈ ... . We thus deduce that [figure omitted; refer to PDF] Consequently, we obtain from ( 6.12) as m [arrow right] ∞ that [figure omitted; refer to PDF] and since y = ( _{y k} ) ∈c \ _{c 0} , we conclude that [figure omitted; refer to PDF] which shows the necessity of ( 6.8). Then relation ( 6.14) holds.

Finally, since Ax ∈ _{[cursive l] p} and A ^ y ∈ _{[cursive l] p} , the necessity of ( 6.9) is immediate by ( 6.14). This completes the proof of Part (i) of the theorem.

Since Part (ii) can be proved by using the similar way that used in the proof of Part (i) with Lemma 5.4instead of Lemma 6.2, we leave the details to the reader.

Remark 6.6.

It is clear by ( 6.10) that the limit [figure omitted; refer to PDF] exists for each n ∈ ... . This just tells us that condition ( 6.10) implies condition ( 6.6).

Now, we may note that ( _{c 0} : _{[cursive l] p} ) = (c : _{[cursive l] p} ) for 1 ...4;p ...4; ∞ , (see [ 25, pages 7-8]). Thus, by means of Theorem 5.6and Lemmas 6.2and 5.4, we immediately conclude the following theorem.

Theorem 6.7.

Let A = ( _{a nk} ) be an infinite matrix over the complex field. Then, the following statements hold.

: (i) Let 1 ...4;p < ∞ . Then, A ∈ ( ^{c 0 λ} (B ) : _{[cursive l] p} ) if and only if ( 6.5) and ( 6.6) hold, and [figure omitted; refer to PDF]

: (ii) A ∈ ( ^{c 0 λ} (B ) : _{[cursive l] ∞} ) if and only if ( 6.10) and ( 6.24) hold.

Proof.

It is natural that the present theorem can be proved by the same technique used in the proof of Theorem 6.5, above, and so we omit the proof.

Theorem 6.8.

A = ( _{a nk} ) ∈ ( ^{c λ} (B ) :c ) if and only if ( 6.7), ( 6.8), and ( 6.10) hold, and [figure omitted; refer to PDF]

Proof.

Suppose that A satisfies the conditions ( 6.7), ( 6.8), ( 6.10), ( 6.25), ( 6.26), and ( 6.27) and take any x ∈ ^{c λ} (B ) . Then, since ( 6.10) implies ( 6.6), we have by Theorem 5.6that _{{ a nk } k ∈ ...} ∈ ^{{ c λ (B ) } β} for all n ∈ ... and hence Ax exists. We also observe from ( 6.10) and ( 6.26) that [figure omitted; refer to PDF] holds for every k ∈ ... . This implies that ( _{α k} ) ∈ _{[cursive 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 ( 2.11) such that _{y k} [arrow right]l as k [arrow right] ∞ . Further it is obvious by combining Lemma 5.3with the condition ( 6.10), ( 6.26), and ( 6.27) that the matrix A ^ = ( _{a ^ nk} ) is in the class (c :c ) .

Now reasoning as in the proof of Theorem 6.5, we obtain that the relation ( 6.13) holds which can be written as follows: [figure omitted; refer to PDF] In this situation, we see by passing to the limit in ( 6.29) as n [arrow right] ∞ that the first term on the right tends to _{∑ k} ... _{α k} ( _{y k} -l ) by ( 6.10) and ( 6.26). The second term tends to l α by ( 6.27) and the last term to la by ( 6.25). Consequently, we have [figure omitted; refer to PDF] which shows that Ax ∈c , that is to say that A ∈ ( ^{c λ} (B ) :c ) .

Conversely, Suppose that A ∈ ( ^{c λ} (B ) :c ) . Then, since the inclusion c ⊂ _{[cursive l] ∞} holds, it is trivial that A ∈ ( ^{c λ} (B ) : _{[cursive l] ∞} ) . Therefore, the necessity of the conditions ( 6.7), ( 6.8), and ( 6.10) are obvious from Theorem 6.5. Further, consider the sequence ^{b ( k )} ( λ ) = _{{ ^{b n (k )} ( λ ) } n ∈ ...} ∈ ^{c λ} (B ) defined by ( 4.2) for every fixed k ∈ ... . Then, it is easily seen that A ^{b (k )} ( λ ) = _{{ a ^ nk } n ∈ ...} and hence _{{ a ^ nk } n ∈ ...} ∈c for every k ∈ ... which shows the necessity of ( 6.26). Let z = _{∑ k} ... ^{b ( k )} ( λ ) . Then, since the linear transformation T : ^{c λ} (B ) [arrow right]c , defined as in the proof of Theorem 2.3by analogy, is continuous and Λ ^ ( ^{b (k )} ( λ ) ) = ^{e (k )} for each fixed k ∈ ... , we obtain that [figure omitted; refer to PDF] which shows that Λ ^ (z ) =e ∈c and hence z ∈ ^{c λ} (B ) . On the other hand, since ^{c λ} (B ) and c are the BK -spaces, Lemma 6.1implies the continuity of the matrix mapping A : ^{c λ} (B ) [arrow right]c . Thus, we have for every n ∈ ... that [figure omitted; refer to PDF] This shows the necessity of ( 6.27).

Now, it follows by ( 6.10), ( 6.26), and ( 6.27) with Lemma 5.3that A ^ = ( _{a ^ nk} ) ∈ (c :c ) . So by ( 6.7), and ( 6.8), relation ( 6.14) holds for all x ∈ ^{c λ} (B ) and y ∈c , and x and y are connected by relation ( 2.11), where _{y k} [arrow right]l ( k [arrow right] ∞ ).

Lastly, since Ax ∈c and A ^ x ∈c ; the necessity of ( 6.25) is immediate by ( 6.14). This step concludes the proof.

Theorem 6.9.

A = ( _{a nk} ) ∈ ( ^{c λ} (B ) : _{c 0} ) if and only if ( 6.7), ( 6.8), and ( 6.10) hold, and [figure omitted; refer to PDF]

Proof.

Since the present theorem can be proved by the similar way used in the proof of Theorem 6.8with Lemma 6.3instead of Lemma 5.3, we omit the detailed proof.

Theorem 6.10.

A = ( _{a nk} ) ∈ ( ^{c 0 λ} (B ) :c ) if and only if ( 6.10), ( 6.24), and ( 6.26) hold.

Proof.

This is similarly obtained by using Lemma 5.2, Theorem 5.6, and Part (ii) of Theorem 6.7.

Theorem 6.11.

A = ( _{a nk} ) ∈ ( ^{c 0 λ} (B ) : _{c 0} ) if and only if ( 6.10) and ( 6.24) hold, and (6.26) also holds with _{α k} =0 for all k ∈ ... .

Proof.

This is immediate by Lemma 6.4, Theorems 5.6and 6.10.

7. Conclusion

Let ν denotes any of the classical sequence spaces _{[cursive l] ∞} , c , or _{c 0} . Then, ν ( Δ ) consisting of the sequences x = ( _{x k} ) such that Δx = ( _{x k} - _{x k +1} ) ∈ ν is called as the difference sequence space which was introduced by Kizmaz [ 19]. Kizmaz [ 19] proved that ν ( Δ ) is a Banach space with the norm ||x _{|| Δ} = | _{x 1} | + || Δx _{|| ∞} , where x = ( _{x k} ) ∈ ν ( Δ ) and the inclusion relation ν ⊂ ν ( Δ ) strictly holds. He also determined the α -, β -, and γ -duals of the difference spaces and characterized the classes ( ν ( Δ ) : μ ) and ( μ : ν ( Δ ) ) of infinite matrices, where ν , μ ∈ { _{[cursive l] ∞} ,c } . Following Kizmaz [ 19], Sarigöl [ 26] extended the difference space ν ( Δ ) to the space ν ( _{Δ r} ) defined by [figure omitted; refer to PDF] and computed the α -, β -, and γ -duals of the space ν ( _{Δ r} ) , where ν ∈ { _{[cursive l] ∞} ,c , _{c 0} } . It is easily seen that ν ( _{Δ r} ) ⊂ ν ( Δ ) , if 0 <r <1 and ν ( Δ ) ⊂ ν ( _{Δ r} ) , if r <0 . Recently, the difference spaces b _{v p} consisting of the sequences x = ( _{x k} ) such that ( _{x k} - _{x k -1} ) ∈ _{[cursive l] p} have been studied in the case 0 <p <1 by Altay and Basar [ 27] and in the case 1 ...4;p ...4; ∞ by Basar and Altay [ 18], Çolak et al. [ 28], and Malkowsky et al. [ 29]. Quite recently, Mursaleen and Noman have introduced the spaces ^{c λ} and ^{c 0 λ} of λ -convergent and λ -null sequences and nextly studied the difference spaces ^{c λ} ( Δ ) and ^{c 0 λ} ( Δ ) in [ 21, 22], respectively. Of course, there is a wide literature concerning the difference sequence spaces. By the domain of the generalized difference matrix B (r ,s ) in the spaces of λ -convergent and λ -null sequences we have generalized the difference spaces ^{c λ} ( Δ ) and ^{c 0 λ} ( Δ ) defined by Mursaleen and Noman [ 21]. Since the generalized difference matrix B (r ,s ) reduces, in the special case r =1 , s = -1 , to the usual difference matrix Δ ; our results are more general and more comprehensive than the corresponding results of Mursaleen and Noman [ 21].

Since the difference spaces of λ -bounded and absolutely λp -summable sequences are not studied, the domain of both the difference matrix Δ , and the generalized difference matrix B (r ,s ) in those spaces are still open. So, it is meaningful to fill this gap.

Acknowledgments

The authors have benefited a lot from discussion on the main results of the earlier version of the paper with Professor Bilâl Altay, Department of Mathematical Education, Faculty of Education, I nönü University, Turkey. So, they wish to express their gratitude for his many helpful suggestions and interesting comments.

References

[1] B. Choudhary, S. Nanda Functional Analysis with Applications , pp. xii+344, John Wiley & Sons, New Delhi, India, 1989.

[2] B. Altay, F. Basar, "On the paranormed Riesz sequence spaces of non-absolute type," Southeast Asian Bulletin of Mathematics , vol. 26, no. 5, pp. 701-715, 2003.

[3] B. Altay, F. Basar, "Some Euler sequence spaces of nonabsolute type," Ukrainian Mathematical Journal , vol. 57, no. 1, pp. 3-17, 2005.

[4] B. Altay, F. Basar, M. Mursaleen, "On the Euler sequence spaces which include the spaces _lp and _{l ∞} . I," Information Sciences , vol. 176, no. 10, pp. 1450-1462, 2006.

[5] C. Aydin, F. Basar, "On the new sequence spaces which include the spaces _c0 and c ," Hokkaido Mathematical Journal , vol. 33, no. 2, pp. 383-398, 2004.

[6] C. Aydin, F. Basar, "Some new paranormed sequence spaces," Information Sciences , vol. 160, no. 1-4, pp. 27-40, 2004.

[7] C. Aydin, F. Basar, "Some new sequence spaces which include the spaces _lP and _{l ∞} ,", vol. 38, no. 3, pp. 641-656, 2005.

[8] E. Malkowsky, "Recent results in the theory of matrix transformations in sequence spaces," Matematichki Vesnik , vol. 49, no. 3-4, pp. 187-196, 1997.

[9] E. Malkowsky, E. Savas, "Matrix transformations between sequence spaces of generalized weighted means," Applied Mathematics and Computation , vol. 147, no. 2, pp. 333-345, 2004.

[10] P. N. Ng, P. Y. Lee, "Cesàro sequence spaces of non-absolute type," Commentationes Mathematicae. Prace Matematyczne , vol. 20, no. 2, pp. 429-433, 1978.

[11] M. Sengönül, F. Basar, "Some new Cesàro sequence spaces of non-absolute type which include the spaces _c0 and c ," Soochow Journal of Mathematics , vol. 31, no. 1, pp. 107-119, 2005.

[12] A. Sönmez, "Some new sequence spaces derived by the domain of the triple band matrix," Computers & Mathematics with Applications , vol. 62, no. 2, pp. 641-650, 2011.

[13] A. Sönmez, "Almost convergence and triple band matrix," Mathematical and Computer Modelling . In press

[14] C. S. Wang, "On Nörlund sequence spaces," Tamkang Journal of Mathematics , vol. 9, no. 2, pp. 269-274, 1978.

[15] M. Basarir, "On some new sequence spaces and related matrix transformations," Indian Journal of Pure and Applied Mathematics , vol. 26, no. 10, pp. 1003-1010, 1995.

[16] B. Altay, F. Basar, E. Malkowsky, "Matrix transformations on some sequence spaces related to strong Cesàro summability and boundedness," Applied Mathematics and Computation , vol. 211, no. 2, pp. 255-264, 2009.

[17] C. Aydin, F. Basar, "Some new difference sequence spaces," Applied Mathematics and Computation , vol. 157, no. 3, pp. 677-693, 2004.

[18] F. Basar, B. Altay, "On the space of sequences of p -bounded variation and related matrix mappings," Ukrainian Mathematical Journal , vol. 55, no. 1, pp. 136-147, 2003.

[19] H. Kizmaz, "On certain sequence spaces," Canadian Mathematical Bulletin , vol. 24, no. 2, pp. 169-176, 1981.

[20] E. Malkowsky, M. Mursaleen, S. Suantai, "The dual spaces of sets of difference sequences of order m and matrix transformations," Acta Mathematica Sinica , vol. 23, no. 3, pp. 521-532, 2007.

[21] M. Mursaleen, A. K. Noman, "On some new difference sequence spaces of non-absolute type," Mathematical and Computer Modelling , vol. 52, no. 3-4, pp. 603-617, 2010.

[22] M. Mursaleen, A. K. Noman, "On the spaces of λ -convergent and bounded sequences," Thai Journal of Mathematics , vol. 8, no. 2, pp. 311-329, 2010.

[23] I. J. Maddox Elements of Functional Analysis , pp. xii+242, Cambridge University Press, Cambridge, UK, 1988., 2nd.

[24] A. Wilansky Summability Throught Functional Analysis , vol. 85, of North-Holland Mathematics Studies, pp. xii+318, North-Holland, Oxford, UK, 1984.

[25] M. Stieglitz, H. Tietz, "Matrixtransformationen von Folgenräumen. Eine Ergebnisübersicht," Mathematische Zeitschrift , vol. 154, no. 1, pp. 1-16, 1977.

[26] M. A. Sarigöl, "On difference sequence spaces," Journal of Karadeniz Technical University , vol. 10, pp. 63-71, 1987.

[27] B. Altay, F. Basar, "The fine spectrum and the matrix domain of the difference operator Δ on the sequence space _lp , (0 <P <1 ) ," Communications in Mathematical Analysis , vol. 2, no. 2, pp. 1-11, 2007.

[28] R. Çolak, M. Et, E. Malkowsky Some Topics of Sequence Spaces , of Lecture Notes in Mathematics, Firat University Press, 2004.

[29] E. Malkowsky, V. Rakocevic, S. Zivkovic, "Matrix transformations between the sequence space b ^vp and certain BK spaces," Bulletin. Classe des Sciences Mathématiques et Naturelles. Sciences Mathématiques , no. 27, pp. 33-46, 2002.

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