#### Abstract

We define a new triangle matrix by the composition of the matrices and . Also, we introduce the sequence spaces , and by using matrix domain of the matrix on the classical sequence spaces , and , respectively, where . Moreover, we show that the space is norm isomorphic to for . Furthermore, we establish some inclusion relations concerning those spaces and determine -, -, and -duals of those spaces and construct the Schauder bases , and . Finally, we characterize the classes of infinite matrices where and .

#### 1. Preliminaries, Background, and Notation

By a sequence space, we understand a linear subspace of the space of all complex sequences which contains , the set of all finitely nonzero sequences. We write , and for the classical sequence spaces of all bounded, convergent, null, and absolutely -summable sequences, respectively, where . Also by and , we denote the spaces of all bounded and convergent series, respectively. We assume throughout unless stated otherwise that with 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 by .

Let be an infinite matrix and two sequence spaces. Then, defines a matrix mapping from to and is denoted by if for every sequence the sequence , the -transform of , is in , where By , denote the class of all matrices such that . Thus, if and only if the series on the right hand side of (1) converges for each and , and we have for all . A sequence is said to be -summable to if converges to , which is called the -limit of .

A matrix is called a triangle if for and for all . It is trivial that holds for the triangle matrices and a sequence . Further, a triangle matrix has a unique inverse which is also a triangle matrix. Then, holds for all .

Let us give the definition of some triangle limitation matrices which are needed in the text. Let be a sequence of positive reals and write Then the CesÃ ro mean of order one, Riesz mean with respect to the sequence , and Euler mean of order with are, respectively, defined by the matrices ,â€‰â€‰, and , where for all . We write for the set of all sequences such that for all . For , let . Let , and define the summation matrix , the difference matrix , and the generalized weighted mean or factorable matrix ,â€‰â€‰,â€‰â€‰ by for all , where and depend only on and , respectively. Let and be nonzero real numbers, and define the generalized difference matrix by for all . We note that if we choose and then the matrix is reduced to the backward difference.

For a sequence space , the matrix domain of an infinite matrix in the space is defined by which is a sequence space. If is triangle, then one can easily observe that the sequence spaces and are linearly isomorphic; that is, .

Following BaÅŸar [1, page 51], we note that although in the most cases the new sequence space generated by the limitation matrix from a sequence space is the expansion or the contraction of the original space , it may be observed in some cases that those spaces overlap. Indeed, one can easily see that the inclusion, , strictly holds for . As this, one can deduce that the inclusion also strictly holds for . However, if we define with , that is, if and only if for some and some , and consider the matrix with the rows defined by for all , we have but which lead us to the consequences that and , where and is a sequence whose only nonzero term is a in th place for each . That is to say, the sequence spaces and 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 recently been employed by Wang [2], Ng and Lee [3], Malkowsky [4], Altay and BaÅŸar [5], Malkowsky and SavaÅŸ [6], BaÅŸarÄ±r [7], AydÄ±n and BaÅŸar [8], BaÅŸar et al. [9], ÅžengÃ¶nÃ¼l and BaÅŸar [10], Altay [11], Polat and BaÅŸar [12], and Malkowsky et al. [13]. and are the transposes of the matrices , and , respectively, and are the spaces consisting of the sequences such that in the spaces and for , respectively, studied by BaÅŸarÄ±r [7]. More recently, the generalized difference matrix has been used by KiriÅŸÃ§i and BaÅŸar [14] for generalizing the difference spaces , and . Finally, the new technique for deducing certain topological properties, for example, -,â€‰â€‰-, andâ€‰â€‰-properties, and determining the - and -duals of the domain of a triangle matrix in a sequence space has been given by Altay and BaÅŸar [15].

Let , and be nonzero real numbers, and define the generalized difference matrix by for all . The inverse of is given by We should record here that , , and . So, the results related to the matrix domain of the triple band matrix are more general and more comprehensive than the consequences on the matrix domain of ,â€‰â€‰ and and include them. We assume throughout that is a strictly increasing sequence of positive reals tending to ; that is

The main purpose of the present paper is to introduce the sequence spaces and to determine the -,â€‰â€‰-, and -duals of this space, where denotes throughout any of the classical spaces ,â€‰â€‰, or , and is the triple band matrix and the sequence is defined in (9). Furthermore, the Schauder bases for the spaces ,â€‰â€‰ and are given, and some topological properties of the spaces ,â€‰â€‰, and are examined. Finally, some classes of matrix mappings on the spaces are characterized.

The paper is organized as follows. In Section 2, the -spaces ,â€‰â€‰,â€‰â€‰, and of generalized difference sequence spaces are introduced and the Schauder bases of the spaces ,â€‰â€‰, and are given. In Section 3, some inclusion relations concerning these spaces are examined. In Section 4, the -,â€‰â€‰-, and -duals of the generalized difference sequence spaces of nonabsolute type are determined. In Section 5, the classes of infinite matrices are characterized, where and .

#### 2. The Difference Sequence Spaces of Nonabsolute Type

The difference sequence spaces have been studied by several authors in different ways [16â€“34]. In the present section, we introduce the spaces ,â€‰â€‰,â€‰â€‰, and and show that these spaces are -spaces of nonabsolute type which are norm isomorphic to the spaces ,â€‰â€‰,â€‰â€‰, and , respectively. Furthermore, we give the bases of the spaces ,â€‰â€‰, and .

We say that a sequence is -convergent to the number , called the -limit of , if as where In particular, we say that is a -null sequence if as . Further, we say that is -bounded if , [35]. Recently, Mursaleen and Noman [35, 36] studied the sequence spaces ,â€‰â€‰,â€‰â€‰, and of nonabsolute type as follows: On the other hand, we define the matrix for all by Then, the space can be restated with the notation of (6) that

More recently, SÃ¶nmez [37] has defined the sequence spaces , and as follows: In fact, the sequence spaces , and can be considered as the set of all sequences whose -transforms are in the spaces , and , respectively. That is,

Now, we introduce the difference sequence spaces , and as follows: Now, we define the triangle matrix ; that is, for all . Further, for any sequence we define the sequence which will be used, as the -transform of ; that is, In fact, the sequence spaces , and can be considered as the set of all sequences whose -transforms are in the spaces , and , respectively. That is,

Since the proof may also be obtained in a similar way as 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.

Theorem 1. *(i) The difference sequence spaces , , and are -spaces with the norm ; that is,
**
(ii) Let . Then is a -space with the norm ; that is,
*

*Proof. *Since (19) holds and , and are -spaces with respect to their natural norms (see [1, pages 16-17]) and the matrix is a triangle, Theorem 4.3.12 of Wilansky [38, page 63] gives the fact that , ,â€‰â€‰, and are -spaces with the given norms. This completes the proof.

*Remark 2. *Let . Then the absolute property does not hold on the space ; that is, . This can be shown for at least one sequence in those spaces. Hence is the sequence space of nonabsolute type.

Theorem 3. *The sequence spaces , , , and of nonabsolute type are norm isomorphic to the spaces , , , and , respectively; that is, , , , and .*

*Proof. *We prove the theorem for the space . To prove our assertion we should show the existence of a linear bijection between the spaces and . Let be defined by (18). Then, for every and the linearity of is clear. Further, it is trivial that whenever and hence is injective.

Moreover, let and we define the sequence by
Then we obtain
Hence, we get for every that
This shows that and since , we conclude that . Thus, we deduce that and . Hence is surjective.

Moreover one can easily see for every that
which means that is norm preserving. Consequently is a linear bijection which shows that the spaces and are linearly isomorphic, as desired.

Let be a normed space. A sequence of points of is called a Schauder basis for if and only if, for each , there exists a unique sequence of scalars such that ; that is,

Because of the isomorphism , which is defined in the proof of Theorem 3, the inverse image of the basis of the spaces , and are the basis of new spaces , and , respectively. Therefore, we have the following.

Theorem 4. *Let for all and . Define the sequence for every fixed by
**Then, the following statements hold.*(i)*The sequence is a basis for the spaces and ; any or has a unique representation of the form .*(ii)*The sequence is a basis for the space and any has a unique representation of the form , where .*

#### 3. The Inclusion Relations

In the present section, we prove some inclusion relations concerning the spaces , and .

Theorem 5. *The inclusion strictly holds.*

*Proof. *It is obvious that the inclusion holds. Further to show that this inclusion is strict, consider the sequence defined by
for all . Then
Then one can easily see that , where . Thus, this sequence is in but not in . Hence, the inclusion is strict and this completes the proof of the theorem.

Theorem 6. *If then the inclusion strictly holds.*

*Proof. *Suppose that and . Then and hence since the inclusion holds [35]. This shows that ; that is, . Further consider the sequence defined by for all . Then it is trivial that . On the other hand, it can easily be seen that . Hence, . Thus the sequence . Hence, the inclusion is strict. This completes the proof.

Lemma 7. * if and only ifâ€‰â€‰.*

Theorem 8. *Let be defined by
**
for all . Then the inclusion strictly holds if and only if .*

*Proof. *Let be a subset of . Then we obtain that for every and the matrix is in the class . By using Lemma 7 it follows that
Now, by taking into account the definition of matrix given by (17), we have for every that
By Lemma 7, we have that
Now, we have for every that
and since by (33) and (34), we obtain that by (35)
which shows that where the sequence is defined by
for all .

Conversely, we suppose that . Then we have (37). Further, for every , we derive that
Then, (37) and (39) together imply that (35) holds. On the other hand, we have for every that
Therefore, it follows by (35)
Particularly if (i) , , , then we obtain , which shows that (33) holds. (ii) , , , and then we obtain , which shows that (34) holds. Thus, we deduce by relation (32) that (31) holds. This leads us with Lemma 7 to the consequence that . Hence, the inclusion holds. Finally, it is obvious that the sequence , defined in the proof of Theorem 6, is in but not in , so the inclusion is strict.

Theorem 9. *The inclusion strictly holds.*

*Proof. *To prove the validity of the inclusion , it suffices to show that, for every , there exists a positive real number such that . Let . Then, we have
so that and hence . Furthermore, we consider defined by for all . Then we have . Thus, we deduce that . Hence .

On the other hand, we know from Theorem 8 that the inclusion is strict. Since , the inclusion strictly holds.

#### 4. The -, -, and -Duals of the Spaces of Nonabsolute Type

In this section, we determine the -, -, and -duals of the generalized difference sequence spaces , and of nonabsolute type.

We will firstly give the definition of -, -, and -duals of sequence spaces and secondly we give the lemmas due to [39] which are needed in proving the theorems given in Sections 4 and 5.

For the sequence spaces and , define the set by

With the notation of (43), the -, -, and -duals , and of a sequence space are defined by

Lemma 10. *(i) if and only if
**
(ii) if and only if
*

Lemma 11. * if and only if
*

Lemma 12. * if and only if (47) and (48) hold, and
*

Lemma 13. * if an only if (47) holds and
*

Lemma 14. * if and only if (47) holds and
*

Lemma 15. * if and only if (48) holds.*

Lemma 16. * if and only if (51) holds.*

Now we consider the following sets: