Some New Type Sigma Convergent Sequence Spaces and Some New Inequalities

We have discussed some important problems about the spaces V~σ and V~0σ of Cesàro sigma convergent and Cesàro null sequence.


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 spaces̃and̃0 consisting of the sequences = ( ) such that (∑ =0 ( /( + 1))) ∈ , 0 and gave some important results on those spaces in [6]. Furthermore, in [7], Şengönül and Kayaduman have introduced the spaceŝand̂0 consisting of the sequences = ( ) such that (∑ =0 ( / )) ∈ , 0 .
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 ℓ ∞ , , 0 , ℓ 1 , , 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 , ∈ N = {0, 1, 2, . . .}. 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 0 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 ∈ N and every ∈ , we have = {( ) } ∈N ∈ 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 lim = lim → ∞ ∑ , and is called regular if lim = lim for each convergent sequence . The sets ℓ ∞ and are Banach spaces with the norm ‖ ‖ = sup | |. Let be a one-to-one mapping from N into itself. (iii) ( ( ) ) = ( ) for all ∈ ℓ ∞ . Throughout this paper, we consider the mapping such that ( ) ̸ = for all positive integers ≥ 0 and ≥ 1, where ( ) is the th iterate of at . Thus, a -mean extends the limit functional on in the sense that ( ) = lim for all ∈ (see [8]). Consequently, ⊂ , where is the set of bounded sequences all of whose -means are equal; that is, [9]. We say that a bounded sequence = ( ) is -convergent if ∈ . In case ( ) = + 1, 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 withlimit zero. It is well known [11] that ∈ ℓ ∞ if and only if − ∈ .
Let be a subset of N. The natural density of is defined by ( ) = lim → ∞ (1/ )|{ ≤ : ∈ }|, 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 , ({ : | − | ≥ }) = 0 (see [12]). In this case, we write -lim = . We will also write and 0 to denote the sets of all statistically convergent sequences and statistically null sequences. Let us consider the following functionals defined on ℓ ∞ : 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 ∈ ℓ ∞ , have been studied. Here, the Knopp core, in short -core of , is the interval [ ( ), ( )] (see [14]). When ( ) = + 1, 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][16][17][18][19] and Fridy and Orhan [12] have introduced the notions of statistical boundedness, statistical limit superior (or briefly -lim sup), and statistical limit inferior (or briefly -lim inf), defined that the statistical core (or briefly -core) of a statistically bounded sequence is the closed interval [ -lim inf , -lim sup ], and also determined necessary and sufficient conditions for a matrix to yield -core( ) ⊆ -core( ) for all ∈ ℓ ∞ . In this paper, we define the spaces of Cesàro sigma convergent and Cesàro null sequences and give some interesting theorems.
The Scientific World Journal 3 Now, we begin with the following theorem.
Proof. We consider only the spaces̃and . In order to prove the fact̃≅ , we should show the existence of a linear bijection between the spaces̃and . Consider the transformation of defined with the notation of (8) fromt o by → = . The linearity of is clear. Further, it is trivial that = = (0, 0, ...) 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 spaces̃and are linearly isomorphic, as desired. This completes the proof. The fact that the spaces̃0 and 0 are linearly isomorphic can also be proved by the similar way, so we omit it.
Remark 4 (see [20]). The spaces and 0 are BK-spaces with the norm ‖ ⋅ ‖ ℓ ∞ . Theorem 5. The sets̃∞,̃, and̃0 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 0 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 spaces̃and̃0 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 iñ× .
Proof. We know that the spaces̃and are Banach spaces (see, Theorem 5 and Remark 4) and the transformation is linear and continuous from̃to . 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.

Some Matrix Mappings Related to
the SpaceF ollowing 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 , ∈ N.
Lemma 7 (see [9]). ∈ (ℓ ∞ , ), if and only if The Scientific World Journal Lemma 8 (see [9]). ∈ ( , ), if and only if (16) and (17) where is identity matrix. Now, we give the following theorem concerning to the dual matrices of the new type.
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 ( ) = + 1, 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.
where is the identity matrix.
Proof. The proof is clear from Theorem 10 and Lemma 8.
and let be any given sequence space. Then, if ∈ (̃: ), then

Some Inequalities
In this section, we use matrices classes Therefore, it is easy to see that -core of is if and only if -lim = .
We need the following lemma due to Das [17] for the proof of next theorem.
Proof. Consider the following.
Sufficiency. Since ∈ ℓ ∞ , it is known that for any given > 0, there exists a positive integer 0 such that ≤ ( ) + whenever ≥ 0 . 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 lim sup → ∞ sup ∈N to the equality (27) that Since (26) holds, we havẽ( ) ≤ ( ).
Combining this result by the fact in (30), we obtain the required condition.
In the special case ( ) = + 1, we also have the following.
In the special case ( ) = +1, we also have the following theorem.
In the special case ( ) = +1, we also have the following theorem.
Proof. Consider the following.
In the special case ( ) = +1, we also have the following theorem.