On the Riesz Almost Convergent Sequences Space

and Applied Analysis 3 Let us suppose that G is the set of all such matrices obtained by using all possible functions p. Now, right here, let us give a new definition for the set of almost convergent sequences that was introduced by Butković et al. 12 : Lemma 1.1. The set f of all almost convergent sequences is equal to the set ⋂ U∈G cU. Other one of the best known regular matrices is R rnk , the Riesz matrix which is a lower triangular matrix defined by rnk ⎧


Introduction and Preliminaries
Let w be the space of all real or complex valued sequences.Then, each linear subspace of w is called a sequence space.For example, the notations ∞ , c, c 0 , 1 , cs, and bs are used for the sequence spaces of all bounded, convergent, and null sequences, absolutely convergent series, convergent series, and bounded series, respectively.Let λ and μ be two sequence spaces and A a nk an infinite matrix of real or complex numbers a nk , where n, k ∈ N {0, 1, 2, . ..}.Then, A defines a matrix mapping from λ to μ and is denoted by A : λ → μ if for every sequence x x k ∈ λ the sequence Ax { Ax n }, the A-transform of x, is in μ where By λ : μ , we denote the class of matrices A such that A : λ → μ.Thus, A ∈ λ : μ if and only if the series on the right side of 1.1 converges for each n ∈ N and every x ∈ λ, and we have Ax { Ax n } n∈N ∈ μ for all x ∈ λ.The matrix domain λ A of an infinite matrix A in a sequence space λ is defined by x k p n 1 0 uniformly in p .

1.3
If x ∈ f, then x is said to be almost convergent to the generalized limit α.When x ∈ f, we write f − lim x α.Lorentz 11 introduced this concept and obtained the necessary and sufficient conditions for an infinite matrix to contain f in its convergence domain.These conditions on an infinite matrix A a nk consist of the standard Silverman Toeplitz conditions for regularity plus the condition lim n → ∞ k |a nk − a n,k 1 | 0. Such matrices are called strongly regular.One of the best known strongly regular matrices is C, the Cesàro matrix of order one which is a lower triangular matrix defined by for all n, k ∈ N.
A matrix U is called the generalized Cesàro matrix if it is obtained from C by shifting rows.Let p : N → N.Then, U u nk is defined by for all n, k ∈ N.

Abstract and Applied Analysis 3
Let us suppose that G is the set of all such matrices obtained by using all possible functions p. Now, right here, let us give a new definition for the set of almost convergent sequences that was introduced by Butković et al. 12 : Lemma 1.1.The set f of all almost convergent sequences is equal to the set U∈G c U .
Other one of the best known regular matrices is R r nk , the Riesz matrix which is a lower triangular matrix defined by Let K be a subset of N. The natural density δ of K is defined by where the vertical bars indicate the number of elements in the enclosed set.The sequence x x k is said to be statistically convergent to the number l if, for every ε, δ {k : |x k − l| ≥ ε} 0 see 13 .In this case, we write st − lim x l.We will also write S and S 0 to denote the sets of all statistically convergent sequences and statistically null sequences.The statistically convergent sequences were studied by several authors see 13, 14 and others .
Let us consider the following functionals defined on ∞ :

1.8
In 15 , the σ-core of a real bounded sequence x is defined as the closed interval −q σ −x , q σ x and also the inequalities q σ Ax ≤ L x σ-core of Ax ⊆ K-core of x q σ Ax ≤ q σ x σ-core of Ax ⊆ σ-core of x , for all x ∈ ∞ , have been studied.Here the Knopp core, in short K-core, of x is the interval l x , L x .In particular, when σ n n 1, since q σ x L * x , σ-core of x is reduced to the Banach core, in short B-core, of x defined by the interval −L * −x , L * x .
The concepts of B-core and σ-core have been studied by many authors 16, 17 .
Recently, Fridy and Orhan 13 have introduced the notions of statistical boundedness, statistical limit superior or briefly st − lim sup , and statistical limit inferior or briefly st − lim inf , defined the statistical core or briefly st-core of a statistically bounded sequence as the closed interval st − lim inf x, st − lim sup x , and also determined the necessary and sufficient conditions for a matrix A to yield K-core Ax ⊆ st-core x for all x ∈ ∞ .
Let us write Quite recently, B C -core of a sequence has been introduced by the closed intervals −T * −x , T * x and also the inequalities have been studied for all x ∈ ∞ in 18 .

1.11
Therefore, it is easy to see that B R -core of x is if and only if f − lim x .
As known, the method to obtain a new sequence space by using the convergence field of an infinite matrix is an old method in the theory of sequence spaces.However, the study of the convergence field of an infinite matrix in the space of almost convergent sequences is new.

The Sequence Spaces f and f 0
In this section we introduce the new spaces f and f 0 as the sets of all sequences such that their R-transforms are in the spaces f and f 0 , respectively, that is

2.1
With the notation of 1.2 , we can write f f R and f 0 f 0 R .Define the sequence y y k , which will be frequently used, as the R-transform of a sequence x x k , that is, If R C, which is Cesàro matrix, order 1, then the space f and f 0 correspond to the spaces f and f 0 see 18 .Suppose that G {G : G U • R, U ∈ G and R is Riesz matrix}.Then we have the following proposition.
Proof.The proof is similar to the proof of Lemma 1.1 so we omit the details, see 12 .
Consider the function • f : f → R, and define The function • f is a norm and f, • f is BK-space.The proof of this is as follows.
Theorem 2.2.The sets f and f 0 are linear spaces with the coordinate wise addition and scalar multiplication that is the BK-space with the x f Rx f .
Proof.The first part of the theorem can be easily proved.We prove the second part of the theorem.Since 1.2 holds and f and f 0 are the BK-spaces 1 with respect to their natural norm, also the matrix R is normal and gives the fact that the spaces f and f 0 are BK-spaces.
Theorem 2.3.The sequence spaces f and f 0 are linearly isomorphic to the spaces f and f 0 , respectively.
Proof.Since the fact "the spaces f 0 and f 0 are linearly isomorphic" can also be proved in a similar way, we consider only the spaces f and f.In order to prove the fact that f ∼ f, we should show the existence of a linear bijection between the spaces f and f.Consider the transformation T defined, with the notation of 2.2 , from f to f by x → y Tx.The linearity of T is clear.Further, it is trivial that x θ 0, 0, . . .whenever Tx θ and hence T is injective.
Let y y k ∈ f, and define the sequence x x k by Then, we have which shows that x ∈ f.Consequently, we see that T is surjective.Hence, T is a linear bijection that therefore shows that the spaces f and f are linearly isomorphic, as desired.This completes the proof.
Theorem 2.4.The spaces f and f 0 are not solid sequence spaces.

and it is not hard to see by taking into account the definition Riesz matrix that
shows that the multiplication ∞ f of the spaces ∞ and f is not a subset of f and therefore the space f is not solid.The proof for the space f 0 is similar to the proof of the space f, so we omit it.
Theorem 2.5.Let the spaces f and f 0 be given.Then, 1 the inclusion f 0 ⊂ f holds and the space f is not a subset of the space ∞ , Proof. 1 Clearly, the inclusion f 0 ⊂ f holds.Let us consider the sequence given by This shows that to us, the space f is not a subset of the space ∞ .
2 If 1/R n ∈ c and r k ∈ 1 , then for all x ∈ ∞ we have Rx ∈ c.Therefore, since lim Rx f − lim Rx , we see that x ∈ f.
In Theorem 2.6, we will use some similar techniques that are due to M óricz and Rhoades 19 .
Theorem 2.6.Define the sequences α n and β n by for each n ∈ N.
Since the part ii can be proved in a similar way, we prove only part i 2.9 This step completes the proof.Proof.Suppose that lim n → ∞ β 2 n − α 2 n 0. For each n, choose r to satisfy 2 r ≤ n ≤ 2 r 1 .We may write n in a dyadic representation of the form n r i 0 n i 2 i , where each n i is 0 or 1, i 0, 1, 2, . . ., r − 1, and n r 1.Then, since n j ∈ {0, 1}, and hence

2.11
Thus, If T is the lower triangular matrix with nonzero entries t nk n k 2 k 1 / n 1 , then, T is a regular matrix so that lim r → ∞ β 2 r −α 2 r 0. From the equality 2.12 , we see that lim n → ∞ β n − α n 0. Conversely, assume that x ∈ f.Then, since If we take n 2 p , then the proof of sufficiency is obtained.This step completes the proof.

Some Duals of the Spaces f and f 0
In this section, by using techniques in 9 , we have stated and proved the theorems determining the β-and γ-duals of the spaces f 0 and f.For the sequence spaces λ and μ, define the set S λ, μ by With the notation of 3.1 , the α-, β-, and γ-duals of a sequence space λ, which are, respectively, denoted by λ α , λ β , and λ γ , are defined by λ α S λ, 1 , λ β S λ, cs , λ γ S λ, bs .

3.2
The following two lemmas are introduced in 20 which we need in proving Theorems 3.3 and 3.4.

3.4
Theorem 3.3.The γ-duals of the spaces f and f 0 are the set d 1 ∩ d 2 , where

3.5
Proof.Define the matrix T t nk via the sequence a a k ∈ w by for all n, k ∈ N. Here, Δ a k /r k a k /r k − a k 1 /r k 1 .By using 2.2 , we derive that Ty n , n ∈ N .

3.7
From 3.7 , we see that ax a k x k ∈ bs whenever x x k ∈ f if and only if Ty ∈ ∞ whenever y y k ∈ f.Then, we derive by Lemma 3.1 that which yields the desired result f γ f γ 0 Theorem 3.4.Define the set d 3 by Then, f β d 3 ∩ cs.
Proof.Consider equality 3.7 , again.Thus, we deduce that ax a k x k ∈ cs whenever x x k ∈ f if and only if Ty ∈ c whenever y y k ∈ f.It is obvious that the columns of that matrix T , defined by 3.6 , are in the space c.Therefore, we derive the consequence from Lemma 3.2 that f β d 3 ∩ cs.

Some Matrix Mappings Related to the Spaces f and f 0
In this section, we characterize the matrix mappings from f into any given sequence space via the concept of the dual summability methods of the new type introduced by Bas

4.1
It is clear here that the method B is applied to the R-transform of the sequence x x k while the method A is directly applied to the entries of the sequence x x k .So, the methods A and B are essentially different.
Let us assume that the matrix product BR exists, which is a much weaker assumption than the conditions on the matrix B belonging to any matrix class, in general.The methods A and B in 4.1 , 4.2 are called dual summability methods of the new type if z n reduces to t n or t n reduces to z n under the application of formal summation by parts.This leads us to the fact that BR exists and is equal to A and BR x B Rx formally holds if one side exists.This statement is equivalent to the following relation between the entries of the matrices A a nk and B b nk : for all n, k ∈ N. Now, we give the following theorem concerning the dual matrices of the new type.x k ∈ f.Then, Ax exists.Therefore, we obtain from the equality as n → ∞ that Ax By, and this shows that A ∈ f : μ .This completes the proof.
and μ is any given sequence space.Then, D ∈ μ : f if and if only E ∈ μ : f .Proof.Let x x k ∈ μ, and consider the following equality with 4.6 : which yields as m → ∞ that Dx ∈ f whenever x ∈ μ if and if only Ex ∈ f whenever x ∈ μ.This step completes the proof.Now, right here, we give the following propositions that are obtained from Lemmas 3.2 and 3.1 and Theorems 4.1 and 4.2.

Proposition 4.3. Let A
a nk be an infinite matrix of real or complex numbers.Then,

4.8
Proposition 4.4.Let A a nk be an infinite matrix of real or complex numbers.Then, 4.9 Proposition 4.5.Let A a nk be an infinite matrix of real or complex numbers.Then,

4.10
Proposition 4.6.Let A a nk be an infinite matrix of real or complex numbers.Then, 4.11

Core Theorems
In this section, we give some core theorems related to the space f.We need the following lemma due to Das 25 for the proof of next theorem.
Then, it is easy to see that the conditions of Lemma 5.1 are satisfied for the matrix sequence C. Thus, by using the hypothesis, we can write

5.6
Thus, by applying lim sup n → ∞ sup p∈N and using the hypothesis, we have τ * Ax ≤ L x ε.This completes the proof since ε is arbitrary and x ∈ ∞ .
In particular r i 1 for all i since R is reduced to Cesàro matrix, see 18 .

5.11
On the other hand, since Hence, f − lim Ax st − lim x; that is, A ∈ S ∩ m, f reg , which completes the proof.
Similarly, r i 1 for all i since R is reduced to Cesàro matrix, see 18 .Sufficiency: Conversely, assume that A ∈ S ∩ ∞ : f reg and 5.2 hold.If x ∈ ∞ , then β x is finite.Let E be a subset of N defined by E {l : x i > β x ε} for a given ε > 0. Then it is obvious that δ E 0 and x i ≤ β x ε if l / ∈ E.

5.16
By applying the operator lim sup n → ∞ sup p∈N and using the hypothesis, we obtained that τ * Ax ≤ β x ε.Since ε is arbitrary, we conclude that τ * Ax ≤ β x for all x ∈ ∞ , that is, B R − core Ax ⊆ st − core x for all x ∈ ∞ and the proof is complete.Now if r i 1 for all i, then R is reduced to Cesàro matrix and we have B C − core Ax ⊆ st − core x , ∀x ∈ ∞ if and only if A ∈ S ∩ ∞ : f reg 5.17

Theorem 4 . 1 .
Let A a nk and B b nk be the dual matrices of the new type and μ any given sequence space.Then, A ∈ f : μ if and only if B ∈ f : μ and k ∈ N. Proof.Suppose that A a nk and B b nk are dual matrices of the new type, that is to say 4.2 holds and μ is an any given sequence space.Since the spaces f and f are linearly isomorphic, now let A ∈ f : μ and y y k ∈ f.Then, BR exists and a nk k∈N ∈ d 2 ∩ cs, which yields that b nk k∈N ∈ 1 for each n ∈ N. Hence, By exists for each y ∈ f, and thus letting m → ∞ in the equality m, n ∈ N, we have by 4.2 that By Ax, which gives the result B ∈ f : μ .Conversely, let {a nk } k∈N ∈ f β for each n ∈ N and B ∈ f : μ hold, and take any x

Theorem 4 . 2 .
Suppose that the entries of the infinite matrices D d nk and E e nk are connected with the relation

Lemma 5 . 1 . 1 Theorem 5 . 2 .
Let c c nj p < ∞ and lim n → ∞ sup p∈N |c nj p | 0.Then, there is a y y j ∈ ∞ such that y ≤ 1 and lim sup n → ∞ sup p∈N j c nj p y j lim sup n → ∞ sup p∈N j c nj p .5.B R −core Ax ⊆ K−core x for all x ∈ ∞ if and only if A ∈ c : f reg and lim Suppose first that B R − core Ax ⊆ K − core x for all x ∈ ∞ .If x ∈ f, then we have τ * Ax −τ * −Ax .By this hypothesis, we get−L −x ≤ −τ * −Ax ≤ τ * Ax ≤ L x .5.3 If x ∈ c, then L x −L −x lim x.So, we have f − lim Ax τ * Ax −τ * −Ax lim x, which implies that A ∈ c, f reg .Now, let us consider the sequence C c nj p of infinite matrices defined by
p − c nj p .

Theorem 5 . 5 .
A ∈ S ∩ ∞ : f reg if and only if A ∈ c : f reg and lim for every E ⊆ N with natural density zero.Theorem 5.6.B R −core Ax ⊆ st−core x for all x ∈ ∞ if and only if A ∈ S ∩ ∞ : f reg and 5.2 holds.Proof.Necessity: Let B R − core Ax ⊆ st − core x for all x ∈ ∞ .Then, τ * Ax ≤ β x for all x ∈ ∞ , where β x st − lim sup x.Hence, since β x st − lim sup x ≤ L x for all x ∈ ∞ see 13 , we have 5.2 from Theorem 5.2.Furthermore, one can also easily see that−β −x ≤ −τ * −Ax ≤ τ * Ax ≤ β x , that is, st − lim inf x ≤ −τ * −Ax ≤ τ * Ax ≤ st − lim sup x.5.15If x ∈ S ∩ ∞ , then st − lim inf x st − lim sup x st − lim x.Thus, the last inequality implies that st − lim x −τ * −Ax τ * Ax f − lim Ax, that is, A ∈ S ∩ ∞ : f reg .
¸ar 21.Note that some researchers, such as, Bas ¸ar 21 , Bas ¸ar and C ¸olak 22 ,Kuttner 23, and Lorentz and Zeller 24 , worked on the dual summability methods.Now, following Bas ¸ar 21 , we give a short survey about dual summability methods of the new type.
Theorem 5.3.B C −core Ax ⊆ K−core x for all x ∈ ∞ if and only if A ∈ c : f reg and Let A ∈ S ∩ ∞ , f reg .Then, A ∈ c, f reg immediately follows from the fact that c ⊂ S ∩ ∞ .Now, define a sequence t t k for x ∈ ∞ as Sufficiency: Conversely, suppose that A ∈ c, f reg and 5.8 holds.Let x ∈ S ∩ ∞ and st−lim x .Write E {k : |x k − | ≥ ε} for any given ε > 0 so that δ E 0. Since A ∈ c, f reg and f − lim k a nk 1, we have f