Abstract

We have discussed some important problems about the spaces and of Cesàro sigma convergent and Cesàro null sequence.

1. Introduction and Preliminaries

In the theory of the sequence spaces, by using the matrix domain of a particular limitation method, so many sequence spaces have been built and published in famous maths journals. By reviewing the literature, one can reach them easily (for instance, see Başar et al. [1], Kirişçi and Başar [2], Şengönül and Başar [3], Altay [4], Mohiuddine and Alotaibi [5], and numerous). As known, the method to obtain a new sequence space by using convergence field of an infinite matrix is an old method in the theory of sequence spaces. But, the study of convergence field of an infinite matrix in the space of -convergent sequences is quite new. For example, quite recently, Kayaduman and Şengönül introduced the spaces and consisting of the sequences such that , and gave some important results on those spaces in [6]. Furthermore, in [7], Şengönül and Kayaduman have introduced the spaces and consisting of the sequences such that , .

After here, we will pass to the preliminaries for our study. We will denote the space of all real or complex valued sequences by and we will write , , , , , and for the spaces of all bounded, convergent, null sequences, absolutely convergent series, convergent series, and bounded series, respectively. Each linear subspace of is called a sequence space. Let and be two sequence spaces and be an infinite matrix of real or complex numbers , where . Then, we can say that defines a matrix mapping from to , and we denote it by writing , if for every sequence , the sequence , that is -transform of , in where For simplicity in notation, here and in what follows, the summation without limits runs from to . By , we denote the class of matrices such that . Thus, if and only if the series on the right side of (1) converges for each and every , we have for all . The matrix domain   of an infinite matrix in a sequence space is defined by If we take , then is called convergence domain of . We write the limit of as , and is called regular if for each convergent sequence .

The sets and are Banach spaces with the norm . Let be a one-to-one mapping from into itself. A continuous linear functional on is said to be an invariant mean or a -mean if and only if(i) for all sequence with nonnegative terms,(ii), where ,(iii) for all . Throughout this paper, we consider the mapping such that for all positive integers and , where is the th iterate of at . Thus, a -mean extends the limit functional on in the sense that for all (see [8]). Consequently, , where is the set of bounded sequences all of whose -means are equal; that is,  where  [9]. We say that a bounded sequence is -convergent if . In case , a -mean is often called a Banach limit and is reduced to the set of almost sequences, introduced by Lorentz (see [10]). If , write . By , we denote the set of -convergent sequences with -limit zero. It is well known [11] that if and only if .

Let be a subset of . The natural density of is defined by , where the vertical bars indicate the number of elements in the enclosed set. The sequence is said to be statistically convergent to the number if for every ,   (see [12]). In this case, we write . We will also write and to denote the sets of all statistically convergent sequences and statistically null sequences. Let us consider the following functionals defined on : In [13], the -core of a real bounded sequence is defined as the closed interval , and also the inequalities (-core of -core of ), (-core of -core of ), for all , have been studied. Here, the Knopp core, in short -core of , is the interval (see [14]). When , since , -core of is reduced to the Banach core, in short -core of defined by the interval , (see [15]). The concepts of -core and -core have been studied by many authors [8, 15–19] and Fridy and Orhan [12] have introduced the notions of statistical boundedness, statistical limit superior (or briefly ), and statistical limit inferior (or briefly ), defined that the statistical core (or briefly -core) of a statistically bounded sequence is the closed interval , and also determined necessary and sufficient conditions for a matrix to yield -core -core for all .

In this paper, we define the spaces of Cesàro sigma convergent and Cesàro null sequences and give some interesting theorems.

2. Some New Type Sigma Convergent Sequence Spaces

Definition 1. A bounded sequence is said to be Cesàro sigma convergent to the number if and only if uniformly in , and the set of all such sequences is denoted with . If , then we write instead of ; that is, where , respectively.

Definition 2. A bounded sequence is said to be Cesàro sigma bounded if and only if , and the set of all such sequences is denoted with ; that is,

With the notation of (2), we can write and , where denotes the Cesàro matrix of one order. Define the sequence , which will be frequently used as the -transform of a sequence ; that is, Clearly, if we take , then the spaces and are reduced to the spaces and , respectively. Also, we note that the inclusions hold.

Now, we begin with the following theorem.

Theorem 3. The sequence spaces and are linearly isomorphic to the spaces and , respectively; that is, and .

Proof. We consider only the spaces and . In order to prove the fact , we should show the existence of a linear bijection between the spaces and . Consider the transformation of defined with the notation of (8) from to by . The linearity of is clear. Further, it is trivial that whenever and hence is injective. Let and define the sequence by Then, we have which shows that . Consequently, we see that is surjective. Hence, is linear bijection which therefore shows that the spaces and are linearly isomorphic, as desired. This completes the proof. The fact that the spaces and are linearly isomorphic can also be proved by the similar way, so we omit it.

Remark 4 (see [20]). The spaces and are BK-spaces with the norm .

Theorem 5. The sets , , and are linear spaces with the coordinatewise addition and scalar multiplication which are BK-spaces with the norm

Proof. The proof of first part of the theorem is easy. We will prove second part of the theorem. Since (8) holds, the spaces and are BK-spaces with the norm , (see Remark 4), the matrix is normal and Theorem 4.3.2 of Wilansky [21] gives the fact that the spaces and are BK-spaces.

It is known that, if and are normed spaces, then the set is subspace of and the set is normed space with the norm where is a linear transformation from to .

Theorem 6. The graph of the transformation is closed subspace in .

Proof. We know that the spaces and are Banach spaces (see, Theorem 5 and Remark 4) and the transformation is linear and continuous from to . Let us suppose that the sequence is convergent to in for and . With this supposition, and from the equality we see that and as . Also, since is continuous and from the definition of the sequential continuous, we obtain that and this completes the proof.

Following Başar [22], we start with giving short knowledge on the dual summability methods of the new type.

Let us suppose that the infinite matrices and map the sequences and which are connected by the relation (8) to the sequences and , respectively; that is, It is clear here that the method is applied to the -transform of the sequence while the method is directly applied to the entries of the sequence . So, the methods and are essentially different. Let us assume that the matrix product exists which is a much weaker assumption than the conditions on the matrix belonging to any matrix class, in general. The methods and in (14) are called dual summability methods of the new type if reduces to (or reduces to ) under the application of formal summation by parts. This leads us to the fact that exists and is equal to and formally holds, if one side exists. This statement is equivalent to the following relation between the entries of the matrices and : for all .

Lemma 7 (see [9]). , if and only if

Lemma 8 (see [9]). , if and only if (16) and (17) hold, and

Lemma 9. , if and only if (16) and (17) hold, and where is identity matrix.

Now, we give the following theorem concerning to the dual matrices of the new type.

Theorem 10. Suppose that the entries of the infinite matrices and are connected with the relation and let be any given sequence space. Then, , if and only if .

Proof. Let and consider the following equality with (21): which yields as that , whenever , if and if only , whenever . This step completes the proof.

If we take , then Theorem 10 is reduced to Theorem 5.2 of Kayaduman and Şengönül, [6].

Now, right here, we have stated two theorem which are natural consequences of the Lemma 7, Lemma 8, and Theorem 10.

Theorem 11. Let be an infinite matrix real or complex numbers. Then, if and only if(1),(2),(3),
where is the identity matrix.

Theorem 12. Let be an infinite matrix real or complex numbers. Then, if and only if(4),(5), exists for each fixed ,(6), uniformly in .

Proof. The proof is clear from Theorem 10 and Lemma 7.

Theorem 13. Let be an infinite matrix real or complex numbers. Then, if and only if(7),(8) exists for each fixed ,(9).

Proof. The proof is clear from Theorem 10 and Lemma 8.

Furthermore, from the Lemma 3.2 of [23], we have the following proposition.

Proposition 14. Suppose that the entries of the infinite matrices and are connected with the relation and let be any given sequence space. Then, if , then(10) exists for every , (11), (12).

4. Some Inequalities

In this section, we use matrices classes , , and to show the following inequalities: , , and , which are analogues of Knopp's core theorem.

Definition 15. Let . Then, -core of is defined by the closed interval , where Therefore, it is easy to see that -core of is if and only if .

We need the following lemma due to Das [17] for the proof of next theorem.

Lemma 16. Let and let . Then, there is a such that and

Lemma 17. Let and be sublinear functionals on a linear space . Then, if and only if for all .

Theorem 18. for all if and only if and where for all .

Proof. Consider the following.
Sufficiency. Since , it is known that for any given , there exists a positive integer such that whenever . Now, let us write Then, by hypothesis, the first and the last sums on the right-hand side of (27) tend to zero, as , uniformly in . Therefore, we obtain by applying the operator to the equality (27) that Since (26) holds, we have .
Necessity. We observe by inserting in place of in the inequality that If , then and so which means that . Hence, . It is clear that uniformly in . On the other hand, if we choose the sequence of matrices defined by , for all , then clearly satisfies the conditions of Lemma 16. So, there exists a with and Thus, we derive by inserting this in the hypothesis that Combining this result by the fact in (30), we obtain the required condition.

In the special case , we also have the following.

Theorem 19 (see [6]). for all if and only if and

Corollary 20. for all if and only if and (26) holds.

Proof. Consider the following.
Necessity. By the similar way used in the proof of necessity of Theorem 18, one can see that . On the other hand, since for any sequence , condition (26) follows from Theorem 18.
Sufficiency. Suppose that and (26) hold. Since implies that , from Theorem 18, On the other hand, since for , we have Now, combining (34) and (35), we obtain for all . Since [13], the proof is completed.

In the special case , we also have the following theorem.

Theorem 21 (see [6]). One can see that for all if and only if and (33) holds.

Corollary 22. if and only if and for every with natural density zero.

Proof. Let . Then, immediately follows from the fact that . Now, define a sequence via as where is any subset of with . Then, and , so we have . On the other hand, since , the matrix , defined by for all , must belong to the class . Hence, the necessity of (36) follows from (4) part of Theorem 12. Conversely, suppose that and (36) holds. Let and let . Write for any given , so that . Since and , we have On the other hand, since the condition (36) implies that Hence, ; that is, , which completes the proof.

In the special case , we also have the following theorem.

Theorem 23 (see [6]). Consider that if and only if and for every with natural density zero.

Corollary 24. Consider that for all if and only if and (26) holds.

Proof. Consider the following.
Necessity. Firstly, assume that for all where . Hence, since for all (see [12]), we have (26) from Theorem 18. Furthermore, one can also easily see that ; that is, . If , then . Thus, the last inequality implies that ; that is, .
Sufficiency. Let and let (26) holds. If , then is finite. Let be a subset of defined by for a given . Then, it is obvious that and if . For any real number , we write and whence , and . Now, we can write Applying the operator , we obtained from hypothesis that . This completes the proof since is arbitrary.

In the special case , we also have the following theorem.

Theorem 25 (see [6]). Consider that for all if and only if and (33) holds.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.