# On the Riesz Almost Convergent Sequences Space

Authors: Mehmet Şengönül; Kuddusi Kayaduman
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The purpose of this paper is to introduce new spacesf̂ and f̂0 that consist of all sequences whose Riesz transforms of order one are in the spaces f and f0, respectively. We also show that f̂ and f̂0 are linearly isomorphic to the spaces f and f0, respectively. The β- and γ-duals of the spaces f̂ and f̂0 are computed. Furthermore, the classes (f̂:μ) and (μ:f̂) of infinite matrices are characterized for any given sequence space μ and determine the necessary and sufficient conditions on a matrix A to satisfy BR-core(Ax)⊆K-core(x), BR-core(Ax)⊆st-core(x) for all x∈ℓ∞.


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## Body

## 1. Introduction and Preliminaries

Letw 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 ℓ∞, c, c0, ℓ1, 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=(ank) an infinite matrix of real or complex numbers ank, 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=(xk)∈λ the sequence Ax={(Ax)n}, the A-transform of x, is in μ where(1.1)(Ax)n=∑kankxk,(n∈N).
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 domainλA of an infinite matrix A in a sequence space λ is defined by(1.2)λA={x=(xk)∈w:Ax∈λ}.
If we take λ=c, then cA is called, convergence domain of A, and we write the limit of Ax as limAx=limn→∞∑kankxk. Further A is called regular if limAx=limx for each convergent sequence x.Letλ be a sequence space. Then λ is called solid if and only if ℓ∞λ⊂λ, [1]. The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by Altay and Başar [2], Başarir [3], Aydın and Başar [4], Kirişçi and Başar [5], Şengönül and Başar [6], Polat and Başar [7], and Malkowsky et al. [8]. Finally, the new technique for deducing certain topological properties, such as AB-, KB-, and AD-properties, and solidity and monotonicity, and determining the α-, β- and γ-duals of the domain of a triangle matrix in a sequence space is given by Altay and Başar [9].Furthermore, quite recently, Kirişçi and Başar [10] introduced the new sequence space f̂ derived from the space f of almost convergent sequences by means of the domain of the generalized difference matrix B(r,s).Define the setsf and f0 by(1.3)f={x=(xk)∈w:∃α∈C∋lim⁡n→∞⁡∑k=0nxk+pn+1=αuniformlyinp},f0={x=(xk)∈w:lim⁡n→∞∑k=0nxk+pn+1=0uniformlyinp}.Ifx∈f, then x is said to be almost convergent to the generalized limit α. When x∈f, we write f-limx=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=(ank) consist of the standard Silverman Toeplitz conditions for regularity plus the condition limn→∞∑k|ank-an,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(1.4)cnk={1n+1,0≤k≤n,0,k>n,for alln,k∈ℕ.A matrixU is called the generalized Cesàro matrix if it is obtained from C by shifting rows. Let p:ℕ→ℕ. Then, U=(unk) is defined by(1.5)unk={1n+1,p(n)≤k≤p(n)+n,0,otherwise,for alln,k∈ℕ.Let us suppose thatG 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 setf of all almost convergent sequences is equal to the set ⋂U∈GcU.Other one of the best known regular matrices isR=(rnk), the Riesz matrix which is a lower triangular matrix defined by(1.6)rnk={rkRn,0≤k≤n,0,k>n,for alln,k∈ℕ, where (rk) is real sequence with rk≥0(k∈ℕ) and Rn=r0+r1+⋯+rn.LetK be a subset of ℕ. The natural density δ of K is defined by(1.7)δ(K)=lim⁡n→∞⁡1n|{k≤n:k∈K}|,where the vertical bars indicate the number of elements in the enclosed set. The sequencex=(xk) is said to be statistically convergent to the number l if, for every ɛ, δ({k:|xk-l|≥ɛ})=0 (see [13]). In this case, we write st-limx=l. We will also write 𝒮 and 𝒮0 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 onℓ∞:(1.8)l(x)=liminf⁡k→∞⁡xk,L(x)=limsup⁡k→∞⁡xk,qσ(x)=limsup⁡p→∞⁡sup⁡n∈N⁡1p+1∑i=0pxσi(n),L*(x)=limsup⁡p→∞⁡sup⁡n∈N⁡1p+1∑i=0pxn+i.In [15], the σ-core of a real bounded sequence x is defined as the closed interval [-qσ(-x),qσ(x)] and also the inequalities qσ(Ax)≤L(x) (σ-core of Ax⊆K-core of x) qσ(Ax)≤qσ(x) (σ-core of Ax⊆σ-core of x), for all x∈ℓ∞, 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 qσ(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 ofB-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-limsup), and statistical limit inferior (or briefly st-liminf), defined the statistical core (or briefly st-core) of a statistically bounded sequence as the closed interval [st-liminfx,st-limsupx], and also determined the necessary and sufficient conditions for a matrix A to yield K-core(Ax)⊆st-core(x) for all x∈ℓ∞.Let us write(1.9)T*(x)=limsup⁡n→∞sup⁡p∈N∑k=0n1n+1∑j=0kxj+pk+1.Quite recently,BC-core of a sequence has been introduced by the closed intervals [-T*(-x),T*(x)] and also the inequalities(1.10)T*(Ax)≤L(x),L(Ax)≤T*(x),T*(Ax)≤T*(x),T*(Ax)≤st-limsup⁡xhave been studied for allx∈ℓ∞ in [18].Definition 1.2.
Letx∈ℓ∞. Then, BR-core of x is defined by the closed interval [-τ*(-x),τ*(x)], where
(1.11)τ*(x)=limsup⁡n→∞sup⁡p∈N∑j=0n1n+1∑i=0jrixi+pRj,-τ*(-x)=liminf⁡n→∞sup⁡p∈N∑j=0n1n+1∑i=0jrixi+pRj.
Therefore, it is easy to see that BR-core of x is ℓ if and only if f̂-limx=ℓ.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.


## 2. The Sequence Spacesf̂ and f̂0

In this section we introduce the new spacesf̂ and f̂0 as the sets of all sequences such that their R-transforms are in the spaces f and f0, respectively, that is(2.1)f̂={x=(xk)∈w:∃α∈C∋lim⁡n→∞∑j=0n1n+1∑i=0jrixi+pRj=αuniformlyinp},f̂0={x=(xk)∈w:lim⁡n→∞∑j=0n1n+1∑i=0jrixi+pRj=αuniformlyinp}.With the notation of (1.2), we can write f̂=fR and f̂0=(f0)R. Define the sequence y=(yk), which will be frequently used, as the R-transform of a sequence x=(xk), that is,(2.2)yn=∑k=0nrkxkRn∀n∈N.
If R=C, which is Cesàro matrix, order 1, then the space f̂ and f̂0 correspond to the spaces f̃ and f̃0 (see [18]).Suppose thatG′={𝒢:𝒢=U∘R,U∈GandRisRieszmatrix}. Then we have the following proposition.Proposition 2.1.
f̂=⋂T∈G′cT dir.Proof.
The proof is similar to the proof of Lemma1.1 so we omit the details, (see [12]).Consider the function∥·∥f̂:f̂→ℝ, and define(2.3)‖x‖f̂=sup⁡n|∑j=0n1n+1∑i=0jrixi+pRj|.
The function ∥·∥f̂ is a norm and (f̂,∥·∥f̂) is BK-space. The proof of this is as follows.Theorem 2.2.
The setsf̂ and f̂0 are linear spaces with the coordinate wise addition and scalar multiplication that is the BK- space with the ∥x∥f̂=∥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 f̂0 are the BK-spaces [1] with respect to their natural norm, also the matrix R is normal and gives the fact that the spaces f̂ and f̂0 are BK-spaces.Theorem 2.3.
The sequence spacesf̂ and f̂0 are linearly isomorphic to the spaces f and f0, respectively.Proof.
Since the fact “the spacesf̂0 and f0 are linearly isomorphic” can also be proved in a similar way, we consider only the spaces f̂ and f. In order to prove the fact that f̂≅f, we should show the existence of a linear bijection between the spaces f̂ and f. Consider the transformation T defined, with the notation of (2.2), from f̂ 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.
Lety=(yk)∈f̂, and define the sequence x=(xk) by
(2.4)xk=1rk(Rkyk-Rk-1yk-1),(k∈N).
Then, we have
(2.5)fR-lim⁡x=lim⁡n→∞∑j=0n1n+1∑i=0jrixi+pRjuniformlyinp=lim⁡n→∞∑j=0n1n+1∑i=0jri((1/ri)(Riyi+p-Ri-1yi+p-1))Rjuniformlyinp=lim⁡n→∞⁡1n+1∑j=0nyj+puniformlyinp=f-lim⁡yuniformlyinp,
which shows that x∈f̂. Consequently, we see that T is surjective. Hence, T is a linear bijection that therefore shows that the spaces f̂ and f are linearly isomorphic, as desired. This completes the proof.Theorem 2.4.
The spacesf̂ and f̂0 are not solid sequence spaces.Proof.
If we takeu=(uk)=(R0/r0,-(R0/r0+R1/r1),(R1/r1+R2/r2),…,(-1)k(Rk/rk+Rk+1/rk+1),…) and v=(vk)=(1,-1,1,…,(-1)k,…), then we see that u∈f̂ and v∈ℓ∞. Let uv be t, that is, (uv)k=tk. Then, since (tk)=(R0/r0,(R0/r0+R1/r1),(R1/r1+R2/r2),…,(Rk/rk+Rk+1/rk+1),…) and it is not hard to see by taking into account the definition Riesz matrix that fR-limt=limn→∞∑j=0n1/(n+1)∑i=0j(riti+p)/Rj=∞. This shows that the multiplication ℓ∞f̂ of the spaces ℓ∞ and f̂ is not a subset of f̂ and therefore the space f̂ is not solid. The proof for the space f̂0 is similar to the proof of the space f̂, so we omit it.Theorem 2.5.
Let the spacesf̂ and f̂0 be given. Then,(1)
the inclusionf̂0⊂f̂  holds and the space f̂ is not a subset of the space ℓ∞,(2)
if(1/Rn)∈c and (rk)∈ℓ1, then  ℓ∞⊂f̂  strictly holds.Proof.
(1) Clearly, the inclusionf̂0⊂f̂ holds. Let us consider the sequence given by
(2.6)uk={-(Rkrk+Rk+1rk+1),kisodd,Rkrk+Rk+1rk+1,kiseven,
Since (Rk)→∞(k→∞), the sequence (uk) is not a bounded sequence. But clearly u∈f̂. This shows that to us, the space f̂ is not a subset of the space ℓ∞.
(2) If(1/Rn)∈c and (rk)∈ℓ1, then for all x∈ℓ∞ we have Rx∈c. Therefore, since lim(Rx)=f-lim(Rx), we see that x∈f̂.In Theorem2.6, we will use some similar techniques that are due to Móricz and Rhoades [19].Theorem 2.6.
Define the sequences(αn) and (βn) by
(2.7)αn=inf⁡p>0∑j=0n1n+1∑i=0jrixi+pRj,βn=sup⁡p>0∑j=0n1n+1∑i=0jrixi+pRj
for all n∈ℕ. Then, αn≤βn for each n∈ℕ and(i)
the sequence(α2n) is nondecreasing,(ii)
the sequence(β2n) is nonincreasing.Proof.
It is trivial that(2.8)αn=inf⁡p>0∑j=0n1n+1∑i=0jrixi+pRj≤sup⁡p>0∑j=0n1n+1∑i=0jrixi+pRj=βn
for each n∈ℕ.
Since the part (ii) can be proved in a similar way, we prove only part (i)(2.9)α2n+1=inf⁡p>0∑j=pp+2n+112n+1+1∑i=0jrixiRj=inf⁡p>012n+1+1[2n+12n+1(∑j=pp+2n+1∑i=0jrixiRj)]=inf⁡p>012n+1+1[12n+1(∑j=pp+2n+1∑i=0jrixiRj)(2n+1)]=inf⁡p>012n+1+1[12n+1(∑j=pp+2n∑i=0jrixiRj)(2n+1)+12n+1(∑j=p+2n+1p+2n+1∑i=0jrixiRj)(2n+1)]≥12n+1+1[(2n+1)α2n+(2n+1)α2n]=α2n.
This step completes the proof.Theorem 2.7.
limp→∞(β2p-α2p)=0 if and only if x∈f̂.Proof.
Suppose thatlimn→∞(β2n-α2n)=0. For each n, choose r to satisfy 2r≤n≤2r+1. We may write n in a dyadic representation of the form n=∑i=0rni2i, where each ni is 0 or 1, i=0,1,2,…,r-1, and nr=1. Then,
(2.10)1n+1∑j=pp+n∑i=0jrixiRj=1n+1∑j=pp+∑i=0rni2i∑i=0jrixiRj=1n+1[120+1(∑j=pp+n020∑i=0jrixiRj)(20+1)+121+1(∑j=n121n222∑i=0jrixiRj)(21+1)+⋯+12r+1(∑j=nr-12r-1nr2r∑i=0jrixiRj)(2r+1)]=1n+1[(2r+1)α2r+(2r-1+1)α2r-1+⋯+20α20]≥1n+1∑j=0rnj(2j+1)αj
since nj∈{0,1}, and hence
(2.11)inf⁡p>0∑j=pp+n1n+1∑i=0jrixiRj=αn≥1n+1∑j=0rnj(2j+1)α2j,sup⁡p>0∑j=pp+n1n+1∑i=0jrixiRj=βn≤1n+1∑j=0rnj(2j+1)α2j.
Thus,
(2.12)0≤βn-αn≤1n+1∑j=0rnj(2j+1)(β2j-α2j).
If T is the lower triangular matrix with nonzero entries tnk=nk(2k+1)/(n+1), then, T is a regular matrix so that limr→∞(β2r-α2r)=0. From the equality (2.12), we see that limn→∞(βn-αn)=0.
Conversely, assume thatx∈f̂. Then, since
(2.13)lim⁡n→∞∑j=pp+n1n+1∑i=0jrixiRj=α
implies
(2.14)lim⁡n→∞αn=inf⁡p>0lim⁡n→∞∑j=pp+n1n+1∑i=0jrixiRj=α,lim⁡n→∞βn=sup⁡p>0lim⁡n→∞∑j=pp+n1n+1∑i=0jrixiRj=α,
we have
(2.15)lim⁡n→∞αn-lim⁡n→∞βn=lim⁡n→∞(αn-βn)=0.
If we take n=2p, then the proof of sufficiency is obtained. This step completes the proof.


## 3. Some Duals of the Spacesf̂ and f̂0

In this section, by using techniques in [9], we have stated and proved the theorems determining the β- and γ-duals of the spaces f̂0 and f̂. For the sequence spaces λ and μ, define the set S(λ,μ) by(3.1)S(λ,μ)={z=(zk)∈w:xz=(xkzk)∈μ∀x=(xk)∈λ}.
With the notation of (3.1), the α-, β-, and γ-duals of a sequence space λ, which are, respectively, denoted by λα, λβ, and λγ, are defined by(3.2)λα=S(λ,l1),λβ=S(λ,cs),λγ=S(λ,bs).The following two lemmas are introduced in [20] which we need in proving Theorems 3.3 and 3.4.Lemma 3.1.
A∈(f:ℓ∞) if and only if
(3.3)sup⁡n∈N∑k|ank|<∞.Lemma 3.2.
A∈(f:c) if and only if
(3.4)lim⁡n→∞∑kank=a,lim⁡n→∞ank=ak;k∈N,lim⁡n→∞∑k|Δ(ank-ak)|=0.Theorem 3.3.
Theγ-duals of the spaces f̂ and f0̂ are the set d1∩d2, where
(3.5)d1={a=(ak)∈w:∑k|RkΔ(akrk)|<∞},d2={a=(ak)∈w:{anrnRn}∈l∞}.Proof.
Define the matrixT=(tnk) via the sequence a=(ak)∈w by
(3.6)tnk={RkΔ(akrk),0≤k≤n-1,anrnRn,n=k,0,otherwise,
for all n,k∈ℕ. Here, Δ(ak/rk)=(ak/rk-ak+1/rk+1). By using (2.2), we derive that
(3.7)∑k=0nakxk=∑k=0n-1RkΔ(akrk)yk+anrnRnyn=(Ty)n,(n∈N).
From (3.7), we see that ax=(akxk)∈bs whenever x=(xk)∈f̂ if and only if Ty∈ℓ∞ whenever y=(yk)∈f. Then, we derive by Lemma 3.1 that
(3.8)∑k=0n|RkΔ(akrk)|<∞,{anrnRn}∈l∞,
which yields the desired result f̂γ=f̂0γ=d1∩d2.Theorem 3.4.
Define the setd3 by
(3.9)d3={a=(ak)∈w:∑k|Δ[RkΔ(akrk)-ak]|<∞}.
Then, f̂β=d3∩cs.Proof.
Consider equality (3.7), again. Thus, we deduce that ax=(akxk)∈cs whenever x=(xk)∈f̂ if and only if Ty∈c whenever y=(yk)∈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 f̂β=d3∩cs.


## 4. Some Matrix Mappings Related to the Spacesf̂ and f̂0

In this section, we characterize the matrix mappings fromf̂ into any given sequence space via the concept of the dual summability methods of the new type introduced by Başar [21].Note that some researchers, such as, Başar [21], Başar and Çolak [22], Kuttner [23], and Lorentz and Zeller [24], worked on the dual summability methods. Now, following Başar [21], we give a short survey about dual summability methods of the new type.Let us suppose that the infinite matricesA=(ank) and B=(bnk) map the sequences x=(xk) and y=(yk), which are connected by the relation (2.2) to the sequences z=(zn) and t=(tn), respectively, that is,(4.1)zn=(Ax)n=∑kankxk,(n∈N),tn=(By)n=∑kbnkyk,(n∈N).
It is clear here that the method B is applied to the R-transform of the sequence x=(xk) while the method A is directly applied to the entries of the sequence x=(xk). So, the methods A and B are essentially different.Let us assume that the matrix productBR 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 zn reduces to tn (or tn reduces to zn) 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=(ank) and B=(bnk):(4.2)ank:=∑j=k∞rjRjbnjorbnk:=Rk(ankrk-an,k+1rk+1)=RkΔ(ankrk)
for all n,k∈ℕ.Now, we give the following theorem concerning the dual matrices of the new type.Theorem 4.1.
LetA=(ank) and B=(bnk) be the dual matrices of the new type and μ any given sequence space. Then, A∈(f̂:μ) if and only if B∈(f:μ) and
(4.3){(Rnrn)ank}n∈N∈c0
for every fixed k∈ℕ.Proof.
Suppose thatA=(ank) and B=(bnk) 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 f̂ and f are linearly isomorphic, now let A∈(f̂:μ) and y=(yk)∈f. Then, BR exists and (ank)k∈ℕ∈d2∩cs, which yields that (bnk)k∈ℕ∈ℓ1 for each n∈ℕ. Hence, By exists for each y∈f, and thus letting m→∞ in the equality
(4.4)∑k=0mbnkyk=∑k=0m∑j=kmrjRjbnjxk
for all m,n∈ℕ, we have by (4.2) that By=Ax, which gives the result B∈(f:μ).
Conversely, let{ank}k∈ℕ∈f̂β for each n∈ℕ and B∈(f:μ) hold, and take any x=(xk)∈f̂. Then, Ax exists. Therefore, we obtain from the equality
(4.5)∑k=0mankxk=∑k=0m-1RkΔankrkyk+anmrnym=∑k=0mbnkyk;(n∈N),
as n→∞ that Ax=By, and this shows that A∈(f̂:μ). This completes the proof.Theorem 4.2.
Suppose that the entries of the infinite matricesD=(dnk) and E=(enk) are connected with the relation
(4.6)enk=∑j=0nrjdjkRj,(n,k∈N)
and μ is any given sequence space. Then, D∈(μ:f̂) if and if only E∈(μ:f).Proof.
Letx=(xk)∈μ, and consider the following equality with (4.6):
(4.7)∑j=0nrjRj∑k=0mdjkxk=∑k=0menkxk;(m,n,k∈N),
which yields as m→∞ that Dx∈f̂ 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 Lemmas3.2 and 3.1 and Theorems 4.1 and 4.2.Proposition 4.3.
LetA=(ank) be an infinite matrix of real or complex numbers. Then,
(4.8)A=(ank)∈(f̂:l∞)⟺{(1)lim⁡n→∞∑k|Δ(∑j=k∞rjanjRj-ak)|=0,(2){ank}∈f̂β∀n∈N.Proposition 4.4.
LetA=(ank) be an infinite matrix of real or complex numbers. Then,
(4.9)A=(ank)∈(f̂:c)⟺{(3)lim⁡n→∞∑k∑j=k∞rjanjRj=a,(4)lim⁡n→∞∑j=k∞rjanjRj=akfor eachk∈N,(5)lim⁡n→∞∑k|Δ(∑j=k∞rjanjRj-ak)|=0,(6){ank}k∈N∈f̂β∀n∈N.Proposition 4.5.
LetA=(ank) be an infinite matrix of real or complex numbers. Then,
(4.10)A=(ank)∈(l∞:f̂)⟺{(7)sup⁡n∈N∑k|∑j=k∞rjanjRj|<∞,(8)f-lim⁡n→∞∑j=k∞rjanjRj=αkexists for each fixedk∈N,(9)lim⁡m→∞∑k|∑i=0m1m+1∑j=k∞rjan+i,jRj-αk|=0uniformly inn.Proposition 4.6.
LetA=(ank) be an infinite matrix of real or complex numbers. Then,
(4.11)A=(ank)∈(c:f̂)⟺{(10)sup⁡n∈N∑k|∑j=k∞rjanjRj|<∞,(11)f-lim⁡n→∞∑j=k∞rjanjRj=αkexists for each fixedk∈N,(12)f-lim⁡n→∞∑k∑j=k∞rjanjRj=α.


## 5. Core Theorems

In this section, we give some core theorems related to the spacef̂. We need the following lemma due to Das [25] for the proof of next theorem.Lemma 5.1.
Let∥c∥=∥cnj(p)∥<∞ and limn→∞supp∈ℕ|cnj(p)|=0. Then, there is a y=(yj)∈ℓ∞ such that ∥y∥≤1 and
(5.1)limsup⁡n→∞sup⁡p∈N∑jcnj(p)yj=limsup⁡n→∞sup⁡p∈N∑j|cnj(p)|.Theorem 5.2.
BR-core(Ax)⊆K-core(x) for all x∈ℓ∞ if and only if A∈(c:f̂)reg and
(5.2)lim⁡n→∞sup⁡p∈N∑k1n+1|∑j=0n1Rj∑i=0jriai+p,k|=1.Proof.
Necessity: Suppose first that BR-core(Ax)⊆K-core(x) for all x∈ℓ∞. If x∈f̂, then we have τ*(Ax)=-τ*(-Ax). By this hypothesis, we get
(5.3)-L(-x)≤-τ*(-Ax)≤τ*(Ax)≤L(x).
If x∈c, then L(x)=-L(-x)=limx. So, we have f̂-limAx=τ*(Ax)=-τ*(-Ax)=limx, which implies that A∈(c,f̂)reg.
Now, let us consider the sequenceC=(cnj(p)) of infinite matrices defined by
(5.4)cnj(p)=1n+1∑j=0n1Rj∑i=0jriai+p,k∀n,i,p∈N.
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
(5.5)1≤liminf⁡n→∞sup⁡p∈N∑j|cnj(p)|≤limsup⁡n→∞sup⁡p∈N∑j|cnj(p)|=limsup⁡n→∞sup⁡p∈N∑jcnj(p)yj=τ*(Ay)≤L(y)≤‖y‖≤1.
This gives the necessity of (5.2).
Sufficiency: Conversely, let A∈(c:f̂)reg and (5.2) hold for all x∈ℓ∞. For any real number λ̃ we write λ̃+=max{λ̃,0} and λ̃-=max{-λ̃,0}; then |λ̃|=λ̃++λ̃- and λ̃=λ̃+-λ̃-. Therefore, for any given ɛ>0, there is a j0∈ℕ such that xj<L(x)+ɛ for all j>j0. Now, we can write
(5.6)∑jcnj(p)xj=∑j<j0cnj(p)xj+∑j≥j0(cnj(p))+xj-∑j≥j0(cnj(p))-xj≤‖x‖∑j<j0|cnj(p)|+[L(x)+ɛ]∑j|cnj(p)|+‖x‖∑j[|cnj(p)|-cnj(p)].
Thus, by applying limsupn→∞supp∈ℕ and using the hypothesis, we have τ*(Ax)≤L(x)+ɛ. This completes the proof since ɛ is arbitrary and x∈ℓ∞.In particularri=1 for all i since R is reduced to Cesàro matrix, see [18].Theorem 5.3.
BC-core(Ax)⊆K-core(x) for all x∈ℓ∞ if and only if A∈(c:f̃)reg and
(5.7)lim⁡n→∞sup⁡p∈N∑i1n+1|∑k=0n1k+1∑j=0kaj+p,i|=1.Theorem 5.4.
A∈(𝒮∩ℓ∞:f̂)reg if and only if A∈(c:f̂)reg and
(5.8)lim⁡n→∞∑k∈E1n+1|∑j=0n1Rj∑i=0jriai+p,k|=0uniformly inp
for every E⊆ℕ with natural density zero.Proof.
Necessity: Let A∈(S∩ℓ∞,f̂)reg. Then, A∈(c,f̂)reg immediately follows from the fact that c⊂S∩ℓ∞. Now, define a sequence t=(tk) for x∈ℓ∞ as
(5.9)tk={xk,k∈E,0,k∉E,
where E is any subset of ℕ with δ(E)=0. Then, st-limtk=0 and t∈S0, and so we have At∈f̂0. On the other hand, since At=∑k∈Eanktk, the matrix B=(bnk) defined by
(5.10)bnk={ank,k∈E,0,k∉E,
for all n, must belong to the class (ℓ∞,f̂0). Hence, the necessity of (5.8) follows from Proposition 4.5.
Sufficiency: Conversely, suppose that A∈(c,f̂)reg and (5.8) holds. Let x∈S∩ℓ∞ and st-limx=ℓ. Write E={k:|xk-ℓ|≥ɛ} for any given ɛ>0 so that δ(E)=0. Since A∈(c,f̂)reg and f̂-lim∑kank=1, we have
(5.11)f̂-lim⁡(Ax)=f̂-lim⁡(∑kank(xk-l)+l∑kank)=f̂-lim⁡(∑kank(xk-l)+l)=lim⁡n→∞sup⁡p∈N∑k1n+1∑j=0n1Rj∑i=0jriai+p,k(xk-l)+l.
On the other hand, since
(5.12)|∑k1n+1∑j=0n1Rj∑i=0jriai+p,k(xk-l)|≤‖x‖∑k∈E1n+1|∑j=0n1Rj∑i=0jriai+p,k|+ɛ‖A‖,
condition (5.8) implies that
(5.13)lim⁡n→∞∑k1n+1∑j=0n1Rj∑i=0jriai+p,k(xk-l)=0uniformlyinp.
Hence, f̂-lim(Ax)=st-limx; that is, A∈(S∩m,f̂)reg, which completes the proof.Similarly,ri=1 for all i since R is reduced to Cesàro matrix, see [18].Theorem 5.5.
A∈(S∩ℓ∞:f̃)reg if and only if A∈(c:f̃)reg and
(5.14)lim⁡n→∞∑i∈E1n+1|∑k=0n1k+1∑j=0kaj+p,i|=0uniformly inp
for every E⊆ℕ with natural density zero.Theorem 5.6.
BR-core(Ax)⊆st-core(x) for all x∈ℓ∞ if and only if A∈(𝒮∩ℓ∞:f̂)reg and (5.2) holds.Proof.
Necessity: Let BR-core(Ax)⊆st-core(x) for all x∈ℓ∞. Then, τ*(Ax)≤β(x) for all x∈ℓ∞, where β(x)=st-limsupx. Hence, since β(x)=st-limsupx≤L(x) for all x∈ℓ∞ (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,
(5.15)st-liminf⁡x≤-τ*(-Ax)≤τ*(Ax)≤st-limsup⁡x.
If x∈𝒮∩ℓ∞, then st-liminfx=st-limsupx=st-limx. Thus, the last inequality implies that st-limx=-τ*(-Ax)=τ*(Ax)=f̂-limAx, that is, A∈(𝒮∩ℓ∞:f̂)reg.
Sufficiency: Conversely, assume that A∈(𝒮∩ℓ∞:f̂)reg and (5.2) hold. If x∈ℓ∞, then β(x) is finite. Let E be a subset of ℕ defined by E={l:xi>β(x)+ɛ} for a given ɛ>0. Then it is obvious that δ(E)=0 and xi≤β(x)+ɛ if l∉E.
For any real numberλ̃ we write λ̃+=max{λ̃,0} and λ̃-=max{-λ̃,0} whence |λ̃|=λ̃++λ̃-, λ̃=λ̃+-λ̃- and |λ̃|-λ̃=2λ̃-. Now, we can write
(5.16)∑jcnj(p)xj=∑j<j0cnj(p)xj+∑j≥j0cnj(p)xj=∑j<j0cnj(p)xj+∑j≥j0cnj+(p)xj-∑j≥j0cnj-(p)xj≤‖x‖∑j<j0|cnj(p)|+∑j≥j0j∉Ecnj+(p)xj+∑j≥j0j∈Ecnj+(p)xj+‖x‖∑j≥j0[|cnj(p)|-cnj(p)]≤‖x‖∑j<j0|cnj(p)|+[β(x)+ɛ]∑j≥j0j∉E|cnj(p)|+‖x‖∑j≥j0j∈E|cnj(p)|+‖x‖∑j≥j0[|cnj(p)|-cnj(p)].
By applying the operator limsupn→∞supp∈ℕ and using the hypothesis, we obtained that τ*(Ax)≤β(x)+ɛ. Since ɛ is arbitrary, we conclude that τ*(Ax)≤β(x) for all x∈ℓ∞, that is, BR-core(Ax)⊆st-core(x) for all x∈ℓ∞ and the proof is complete. Now if ri=1 for all i, then R is reduced to Cesàro matrix and we have
(5.17)BC-core(Ax)⊆st-core(x),∀x∈l∞ifandonlyifA∈(S∩l∞:f̃)reg
and (5.7) holds, see [18].



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*Source: 691694-2012-05-09.xml*