# Almost Sequence Spaces Derived by the Domain of the MatrixAr

Authors: Ali Karaisa; Ümıt Karabıyık
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

By usingAr, we introduce the sequence spaces afr, af0r, and afsr of normed space and BK-space and prove that afr,af0r, and afsr are linearly isomorphic to the sequence spaces f, f0, and fs
, respectively. Further, we give some inclusion relations concerning the spaces afr, af0r, and the nonexistence of Schauder basis of the spaces fs
and afsr is shown. Finally, we determine the β- and γ-duals of the spaces afr and afsr. Furthermore, the characterization of certain matrix classes on new almost convergent sequence and series spaces has exhaustively been examined.


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## Body

## 1. Preliminaries, Background and Notation

Byw, we will denote the space of all real or complex valued sequences. Any vector subspace of w is called sequence space. We will write ℓ∞, c0, c, and ℓp for the spaces of all bounded, null, convergent, and absolutely p-summable sequences, respectively, which are BK-space with the usual sup-norm defined by ∥x∥∞=supk|xk| and ∥x∥ℓp=(∑k‍|xk|p)1/p, for 1<p<∞, where, here and in what follows, the summation without limits runs from 0 to ∞. Further, we will write bs, cs for the spaces of all sequences associated with bounded and convergent series, respectively, which are BK-spaces with their natural norm [1].Letμ and γ be two sequence spaces and A=(ank) an infinite matrix of real or complex numbers ank, where n,k∈ℕ. Then, we say that A defines a matrix mapping from μ into γ and we denote it by writing that A:μ→γ and if for every sequence x=(xk)∈μ the sequence Ax=(Ax)n, the A-transform of x is in γ, where(1)(Ax)n=∑k‍ankxk,(n∈ℕ).The notation(μ:γ) denotes the class of all matrices A such that A:μ→γ. Thus, A∈(μ:γ) if and only if the series on the right hand side of (1) converges for each n∈ℕ and every x∈μ and we have Ax={(Ax)n}n∈ℕ∈γ for all x∈μ. The matrix domain μA of an infinite matrix A in a sequence space μ is defined by
(2)μA={x=(xk)∈ω:Ax∈μ}.The approach constructing a new sequence space by means of the matrix domain of a particular triangle has recently been employed by several authors in many research papers. For example, they introduced the sequence spaces(c)C1=c~ in [2], (ℓp)Ar=apr and (ℓ∞)Ar=a∞r in [3], μG=Z(u,v;μ) in [4], (c0)Λ=c0λ and cΛ=cλ in [5], and (ℓp)Er=epr and (ℓ∞)Er=e∞r in [6]. Recently, matrix domains of the generalized difference matrix B(r,s) and triple band matrix B(r,s,t) in the sets of almost null and almost convergent sequences have been investigated by Başar and Kirişçi [7] and Sönmez [8], respectively. Later, Kayaduman and Şengönül introduced some almost convergent spaces which are the matrix domains of the Riesz matrix and Cesàro matrix of order 1 in the sets of almost null and almost convergent sequences (see [9, 10]).We now focus on the sets of almost convergent sequences. A continuous linear functionalϕ on ℓ∞ is called a Banach limit if (i) ϕ(x)⩾0 for x=(xk) and xk⩾0 for every k, (ii) ϕ(xσ(k))=ϕ(xk), where σ is shift operator which is defined on ω by σ(k)=k+1, and (iii) ϕ(e)=1, where e=(1,1,1,…). A sequence x=(xk)∈ℓ∞ is said to be almost convergent to the generalized limit α if all Banach limits of x are α [11] and denoted by f-limx=α. In other words, f-limxk=α uniformly in n if and only if
(3)limm→∞1m+1∑k=0m‍xk+nuniformlyinn.
The characterization given above was proved by Lorentz in [11]. We denote the sets of all almost convergent sequences f and series fs by
(4)f={x=(xk)∈ω:limm→∞tmn(x)=αuniformlyinn},
where
(5)tmn(x)=∑k=0m‍1m+1xk+n,t-1,n=0,fs={∑j=0n+k‍x=(xk)∈ω:0000∃l∈ℂ∋limm→∞∑k=0m‍∑j=0n+k‍xjm+1=l000000000000000000uniformlyinn∑k=0m‍∑j=0n+k‍}.
We know that the inclusions c⊂f⊂ℓ∞ strictly hold. Because of these inclusions, norms ∥·∥f and ∥·∥∞ of the spaces f and ℓ∞ are equivalent. So the sets f and f0 are BK-spaces with the norm ∥x∥f=supm,n|tmn(x)|.The rest of this paper is organized, as follows. We give foreknowledge on the main argument of this study and notations in this section. In Section2, we introduce the almost convergent sequence and series spaces afsr and afr which are the matrix domains of the Ar matrix in the almost convergent sequence and series spaces fs and f, respectively. In addition, we give some inclusion relations concerning the spaces afr, af0r, and the non-existence of Schauder basis of the spaces fs and afsr is shown to give certain theorems related to behavior of some sequences. In Section 3, we determine the beta- and gamma-duals of the spaces afr and afsr and characterize the classes (γ:afr), (afr:μ), (δ:afsr) and (afr:θ), where γ∈{c(p),c0(p),ℓ∞(p),cs,bs,fs,f,c,ℓ∞}, μ∈{cs,c,ℓ∞}, δ∈{cs,fs,bs}, and θ∈{f,c,fs,ℓ∞}, where c(p), c0(p), and ℓ∞(p) denote the space of Maddox convergent, null and bounded sequence spaces defined by Maddox [12].Lemma 1 (see [13]).
The setfs has no Schauder basis.


## 2. The Sequence Spacesafr, af0r, and afsr Derived by the Domain of the Matrix Ar

In the present section, we introduce the sequence spacesafr, af0r, and afsr as the set of all sequences such that Ar-transforms of them are in the spaces f, f0, and fs, respectively. Further, this section is devoted to examination of the basic topological properties of the sets afr, af0r, and afsr. Recently, Aydın and Başar [14] studied the sequence spaces acr and a0r:(6)acr={x=(xk)∈ω:limn→∞1n+1∑k=0n‍(1+rk)xkexists},a0r={x=(xk)∈ω:limn→∞1n+1∑k=0n‍(1+rk)xk=0},
where Ar denotes the matrix Ar=(ankr) defined by
(7)ankr={1+rkn+1(0⩽k⩽n),0(k>n).Now we introduce the sequence spacesafr, afr, and afsr as the sets of all sequences such that their Ar-transforms are in the spaces f, f0, and fs, respectively; that is,
(8)afr={∑i=0k‍x=(xk)∈ω:00000∃α∈ℂ∋limm→∞∑k=0m‍1m+1∑i=0k‍1k+1(1+ri)xn+i00000000000000000000000=αuniformlyinn∑i=0k‍},af0r={∑i=0k‍x=(xk)∈ω:00000limm→∞∑k=0m‍1m+1∑i=0k‍1+rik+1xn+i=000000000000000000uniformlyinn∑i=0k‍},afsr={∑i=0j‍x=(xk)∈ω:00000∃β∈ℂ∋limm→∞∑k=0m‍∑j=0n+k‍∑i=0j‍1+rji+1xn+j000000000000000=βuniformlyinn∑i=0j‍}.
We can redefine the spaces afsr, afr, and af0r by the notation of (2):
(9)af0r=(f0)Ar,afr=fAr,afsr=(fs)Ar.
It is known by Başar [15] that the method is regular for 0<r<1. We assume unless stated otherwise that 0<r<1.Define the sequencey=(yk), which will be frequently used, as the Ar-transform of a sequence x=(xk); that is,
(10)yk(r)=∑i=0k‍1+rik+1xi(k∈ℕ).Theorem 2.
The spacesafr and afsr have no Schauder basis.Proof.
Since it is known that the matrix domainμA of a normed sequence space μ has a basis if and only if μ has a basis whenever A=(ank) is a triangle [16, Remark 2.4] and the space f has no Schauder basis by [7, Corollary 3.3], we have that afr has no Schauder basis. Since the set fs has no basis in Lemma 1, afsr has no Schauder basis.Theorem 3.
The following statements hold.(i)
The setsafrandaf0rare linear spaces with the coordinatewise addition and scalar multiplication which areBK-spaces with the norm(11)∥x∥afr=supm|∑k=0m‍1m+1∑i=0k‍1+rik+1xi+n|.(ii)
The setafsris a linear space with the coordinatewise addition and scalar multiplication which is aBK-space with the norm(12)∥x∥afsr=supm|∑k=0m‍1m+1∑j=0k+n‍∑i=0j‍1+rij+1xi|.Proof.
Since the second part can be similarly proved, we only focus on the first part. Since the sequence spacesf and f0 endowed with the norm ∥·∥∞ are BK-spaces (see [1, Example 7.3.2(b)]) and the matrix Ar=(ankr) is normal, Theorem 4.3.2 of Wilansky [17, p.61] gives the fact that the spaces afr and af0r are BK-spaces with the norm in (11).Now, we may give the following theorem concerning the isomorphism between our spaces and the setsf, f0, and fs.Theorem 4.
The sequence spacesafr, af0r, and afsr are linearly isomorphic to the sequence spaces f, f0, and fs, respectively; that is, afr≅f, af0r≅f0, and afsr≅fs.Proof.
To prove the fact thatafr≅f, we should show the existence of a linear bijection between the spaces afr and f. Consider the transformation T defined with the notation of (2) from afr to f by x↦y=Tx=Arx. The linearity of T is clear. Further, it is clear that x=θ whenever Tx=θ, and hence, T is injective.
Lety=(yk)∈afr, and define the sequence x=(xk(r)) by
(13)xk=11+rk[(k+1)yk-kyk-1]foreachk∈ℕ,
whence
(14)fAr-limx=limm→∞∑k=0m‍1m+1×∑i=0k‍(1+ri)xi+n1+kuniformlyinn=limm→∞∑k=0m‍1m+1×∑i=0k(1+ri)[1/(1+ri)(yi+n(k+1)-yi+n-1k)]1+k000000000000000000000000000000000uniformlyinn=limm→∞1m+1∑k=0m‍yk+nuniformlyinn=f-limy
which implies that x∈afr. As a result, T is surjective. Hence, T is a linear bijection which implies that the spaces afr and f are linearly isomorphic, as desired. Similarly, the isomorphisms af0r≅f0 and afsr≅fs can be proved.Theorem 5.
The inclusionf⊂afr strictly holds.Proof.
Letx=(xk)∈c. Since c⊂f, x∈f. Because Ar is regular for 0<r<1, Arx∈c. Therefore, since limArx=f-limArx, we see that x∈afr. So we have that the inclusion f⊂afr holds. Further, consider the sequence t=(tk(r)) defined by tk(r)=(2k+1)/(1+rk)(-1)k∀k∈ℕ. Then, since Art=(-1)n∈f, x∈afr. One can easily see that t∉f. Thus, t∈afr∖f, and this completes the proof.Theorem 6.
The sequence spacesafr and ℓ∞ overlap, but neither of them contains the other.Proof.
Let us consider the sequenceu=(uk(r)) defined by uk(r)=1/(1+rk) for all ℕ. Then, since Aru=e∈f, u∈afr. It is clear that u∈ℓ∞. This means that the sequence spaces afr and ℓ∞ are not disjoint. Now, we show that the sequence space afr and ℓ∞ do not include each other. Let us consider the sequence t=(tk(r)) defined as in proof of Theorem 5 above and z=(zk(r)) = (0,…,0,1/(1+r101),…,1/(1+r110),0,…,0,1/(1+r211),…,1/(1+r231),0,…,0,…) where the blocks of 0’s are increasing by factors of 100 and the blocks of 1/(1+rk)’s are increasing by factors of 10. Then, since Art=(-1)n∈f, t∈afr, but t∉ℓ∞. Therefore, t∈afr∖ℓ∞. Also, the sequence z∉afr since Arz=(0,…,0,1,…,1,0,…,0,1,…,1,0,…,0,…)∉f where the blocks of 0’s are increasing by factors of 100 and the blocks of 1’s are increasing by factors of 10, but z is bounded. This means that z∈ℓ∞∖afr. Hence, the sequence spaces afr and ℓ∞ overlap, but neither of them contains the other. This completes the proof.Theorem 7.
Let the spacesaf0r, acr, and afr be given. Then,  (i)
af0r⊂afrstrictly hold;(ii)
acr⊂afrstrictly hold.Proof.
(i) Letx=(xk)∈af0r which means that Arx∈f0. Since f0⊂f, Arx∈f. This implies that x∈afr. Thus, we have af0r⊂afr.
Now, we show that this inclusion is strict. Let us consider the sequenceu=(uk(r)) defined as in proof of Theorem 6 for all k∈ℕ. Consider the following:
(15)fAr-limu=limm→∞∑k=0m‍1m+1∑i=0k‍1+rik+1ui+n=limm→∞∑k=0m‍1m+1=e=(1,1,…)
which means that u∈afr∖af0r; that is to say, the inclusion is strict.(ii)
Letx=(xk)∈acr which means that Arx∈c. Since c⊂f, Arx∈f. This implies that x∈afr. Thus, we have acr⊂afr. Furthermore, let us consider the sequence t={tk(r)} defined as in proof of Theorem 5 for all k∈ℕ. Then, since Art=(-1)n∈f∖c, t∈afr∖acr. This completes the proof.


## 3. Certain Matrix Mappings on the Setsafr, afsr and Some Duals

In this section, we will characterize some matrix transformations between the spaces ofAr almost convergent sequence and almost convergent series in addition to paranormed and classical sequence spaces after giving β- and γ-duals of the spaces afsr and  afr. We start with the definition of the beta- and gamma-duals.Ifx and y are sequences and X and Y are subsets of ω, then we write x·y=(xkyk)k=0∞, x-1*Y={a∈ω:a·x∈Y} and
(16)M(X,Y)=⋂x∈X‍x-1*Y={a:a·x∈Y∀x∈X}
for the multiplier space of X and Y. One can easily observe for a sequence space Z with Y⊂Z and Z⊂X that inclusions M(X,Y)⊂M(X,Z) and M(X,Y)⊂M(Z,Y) hold, respectively. The α-,β-, and γ-duals of a sequence space, which are, respectively, denoted by Xα,Xβ, and Xγ, are defined by
(17)Xα=M(X,ℓ1),Xβ=M(X,cs),Xγ=M(X,bs).
It is obvious that Xα⊂Xβ⊂Xγ. Also, it can easily be seen that the inclusions Xα⊂Yα, Xβ⊂Yβ, and Xγ⊂Yγ hold whenever Y⊂X.Lemma 8 (see [18]).
A
=
(
a
n
k
)
∈
(
f
:
ℓ
∞
) if and only if
(18)supn∑k‍|ank|<∞.Lemma 9 (see [18]).
A
=
(
a
n
k
)
∈
(
f
:
c
) if and only if (18) holds and there are α,αk∈ℂ such that
(19)limn→∞ank=αk∀k∈ℕ,(20)limn→∞∑k‍ank=α,(21)limn→∞∑k‍|Δ(ank-αk)|=0.Theorem 10.
Define the setst1r and t2r by
(22)t1r={a=(ak)∈ω:∑k‍|Δ(ak1+rk)(k+1)|<∞},t2r={a=(ak)∈ω:supk|ak(k+1)1+rk|<∞},
where Δ(ak/(1+rk))=ak/(1+rk)-ak+1/(1+rk+1) for all k∈ℕ. Then (afr)γ=t1r∩t2r.Proof.
Take any sequencea=(ak)∈ω, and consider the following equality:
(23)∑k=0n‍akxk=∑k=0n‍ak[∑i=k-1k‍(-1)k-ji+11+riyi]=∑k=0n-1‍Δ(ak1+rk)(k+1)yk00000+n+11+rnanyn=(Ty)n,
where T={tnkr} is
(24)tnkr={Δ(ak1+rk)(k+1)(0⩽k⩽n-1)n+11+rnan(k=n)0(k>n)
for all k,n∈ℕ. Thus, we deduce from (23) that ax=(akxk)∈bs whenever x=(xk)∈afr if and only if Ty∈ℓ∞ whenever y=(yk)∈f where T={tnkr} is defined in (24). Therefore, with the help of Lemma 8,  (afr)γ=t1r∩t2r.Theorem 11.
Theβ-dual of the space afr is the intersection of the sets
(25)t3r={a=(ak)∈ω:limn→∞∑k‍|Δ(tnkr-αk)|=0},t4r={a=(ak)∈ω:(k+11+rkak)∈cs},
where limn→∞tnkr=αk for all k∈ℕ. Then, (afr)β=t3r∩t4r.Proof.
Let us take any sequencea∈ω. By (23), ax=(akxk)∈cs whenever x=(xk)∈afr if and only if Ty∈c whenever y=(yk)∈f. It is obvious that the columns of that matrix T in c where T={tnkr} defined in (24), we derive the consequence by Lemma 9 that (afr)β=t3r∩t4r.Theorem 12.
Theγ-dual of the space afsr is the intersection of the sets
(26)c1r={a=(ak)∈ω:∑k‍|Δ[Δ(ak1+rk)(k+1)00000000000000000000000+ak1+rk(k+1)]|<∞∑k‍},c2r={a=(ak)∈ω:(ak(k+1)1+rk)∈c0}.
In other words, we have (afsr)γ=c1r∩c2r.Proof.
We obtain from (23) that ax=(akxk)∈bs whenever x=(xk)∈afsr if and only if Ty∈ℓ∞ whenever y=(yk)∈fs, where T={tnkr} is defined in (24). Therefore, by Lemma 19(viii), (afsr)γ=c1r∩c2r.Theorem 13.
Define the setc3r by
(27)c3r={a=(ak)∈ω:limn→∞∑k‍|Δ2(tnkr)|exists}.
Then, (afsr)β=c1r∩c2r∩c3r.Proof.
This may be obtained in the same way as mentioned in the proof of Theorem12 with Lemma 19(viii) instead of Lemma 19(vii). So we omit details.For the sake of brevity, the following notations will be used:(28)a(n,k,m)=1m+1∑i=0m‍an+i,k,a(n,k)=∑i=0n‍aik,a^nk=Δ(ank1+rk)(k+1)=(ank1+rk-an,k+11+rk+1)(k+1),Δank=an,k-an,k+1,a~nk=∑j=0n‍(1+rj)ejkn+1,
for all k,n∈ℕ. Assume that the infinite matrices A=(ank) and B=(bnk) map the sequences x=(xk) and y=(yk) which are connected with relation (10) to the sequences u=(un) and v=(vn), respectively; that is,
(29)un=(Ax)n=∑k‍ankxk∀n∈ℕ,(30)vn=(By)n=∑k‍bnkyk∀n∈ℕ.
One can easily conclude here that the method A is directly applied to the terms of the sequence x=(xk), while the method B is applied to the Ar-transform of the sequence x=(xk). So the methods A and B are essentially different.Now, suppose that the matrix productBAr exists which is a much weaker assumption than the conditions on the matrix B belonging to any matrix class, in general. It is not difficult to see that the sequence in (30) reduces to the sequence in (29) under the application of formal summation by parts. This leads us to the fact that BAr exists and is equal to A and (BAr)x=B(Arx) 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) which are connected with the relation
(31)a^nk=bnk=Δ(ank1+rk)(k+1)orank=(1+rk)∑j=k∞‍bnj1+j00000000000000000000∀k,n∈ℕ.Note that the methodsA and B are not necessarily equivalent since the order of summation may not be reversed. We now give the following fundamental theorem connected with the matrix mappings on/into the almost convergent spaces afr and afsr.Theorem 14.
Suppose that the entries of the infinite matricesA=(ank) and B=(bnk) are connected with relation (31) for all k,n∈ℕ, and let λ be any given sequence space. Then, A∈(afr:λ) if and only if
(32)B∈(f:λ),{n+11+rkank}k∈ℕ∈c0.Proof.
Suppose thatA=(ank) and B=(bnk) are connected with the relation (31), and let λ be any given sequence space, and keep in mind that the spaces afr and f are norm isomorphic.
LetA∈(afr:λ), and take any sequence x∈afr, and keep in mind that y=Arx. Then, (ank)k∈ℕ∈(afr)β; that is, (32) holds for all n∈ℕ and BAr exists which implies that (bnk)k∈ℕ∈ℓ1=fβ for each n∈ℕ. Thus, By exists for all y∈f, and thus, we have m→∞ in the equality
(33)∑k=0m‍bnkyk=∑k=0m‍∑j=km‍(1+rk)bnj1+jxk
for all m,n∈ℕ, and we have (31) By=Ax which means that B∈(f:λ). On the other hand, assume that (32) holds and B∈(f:λ). Then, we have (bnk)k∈ℕ∈ℓ1 for all n∈ℕ which gives together with (υnk)k∈ℕ∈(afr)β for each n∈ℕ that Ax exists. Then, we obtain from the equality
(34)∑k=0m‍ankxk=∑k=0m-1‍Δ(ank1+rk)(k+1)yk+m+11+rmanmym=∑k=0m‍bnkyk
for all m,n∈ℕ, as m→∞, that Ax=By, and this shows that A∈(afr:λ).Theorem 15.
Suppose that the entries of the infinite matricesE=(enk) and F=(fnk) are connected with the relation
(35)fnk=a~nk
for all m,n∈ℕ and λ is any given sequence space. Then, E∈(λ:afr) if and only if F∈(λ:f).Proof.
Letx=(xk)∈λ, and consider the following equality:
(36)∑j=0n‍∑k=0m‍1+rjn+1ejkxk=∑k=0m‍fnkxk
for all k,m,n∈ℕ, which yields as m→∞ that Ex∈afr whenever x∈λ if and only if Fx∈f whenever x∈λ. This step completes the proof.Theorem 16.
Letλ be any given sequence space, and the matrices A=(ank) and B=(bnk) are connected with the relation (31). Then, A∈(afsr:λ) if and only if B∈(fs:λ) and (ank)k∈ℕ∈(afsr)β for all n∈ℕ.Proof.
The proof is based on the proof of Theorem14.Theorem 17.
Letλ be any given sequence space, and the elements of the infinite matrices E=(enk) and F=(fnk) are connected with relation (35). Then, E=(enk)∈(λ:afsr) if and only if F∈(λ:fs).Proof.
The proof is based on the proof of Theorem15.By Theorems14, 15, 16, and 17, we have quite a few outcomes depending on the choice of the space λ to characterize certain matrix mappings. Hence, by the help of these theorems, the necessary and sufficient conditions for the classes (afr:λ), (λ:afr), (afsr:λ) and (λ:afsr) may be derived by replacing the entries of A and B by those ofB=A(Ar)-1, and F=ArE, respectively, where the necessary and sufficient conditions on the matrices E and F are read from the concerning results in the existing literatureLemma 18.
LetA=(ank) be an infinite matrix. Then, the following statements hold:(i)
A∈(c0(p):f)   if and only if(37)∃N>1∋supm∈ℕ∑k‍|a(n,k,m)|N-1/pk<∞,∀n∈ℕ,∃αk∈ℂ∀k∈ℕ∋limm→∞a(n,k,m)=αkuniformlyinn;(ii)
A∈(c(p):f)   if and only if (37) and(38)∃α∈ℂ∋limm→∞∑k‍a(n,k,m)=αuniformlyinn;(iii)
A∈(ℓ∞(p):f)   if and only if (37) and(39)∃N>1∋limm→∞∑k‍|a(n,k,m)-αk|N1/pk=0000000000000000000000000uniformlyinn.Lemma 19.
LetA=(ank) be an infinite matrix. Then, the following statements hold:(i)
(Duran, [19])A∈(ℓ∞:f)if and only if (18) holds and(40)f-limank=αkexistsforeachfixedk,(41)limm→∞∑k‍|a(n,k,m)-αk|=0uniformlyinn;(ii)
(King, [20])A∈(c:f)if and only if (18), (40) hold and(42)f-lim∑k‍ank=α;(iii)
(Başar and Çolak, [21])A∈(cs:f)if and only if (40) holds and(43)supn∈ℕ∑k‍|Δank|<∞;(iv)
(Başar and Çolak, [21])A∈(bs:f)if and only if (40), (43) hold and(44)limkank=0existsforeachfixedn,(45)limq→∞∑k‍1q+1∑i=0q‍|Δ[a(n+i,k)-αk]|=0uniformlyinn;(v)
(Duran, [19])A∈(f:f)if and only if (18), (40), and (42) hold and(46)limm→∞∑k‍|Δ[a(n,k,m)-αk]|=0uniformlyinn;(vi)
(Başar, [22])A∈(fs:f)if and only if (40), (44), (46), and (45) hold;(vii)
(Öztürk, [23])A∈(fs:c)if and only if (19), (43), and (44) hold and(47)limn→∞∑k‍|Δ2ank|=α;(viii)
A∈(fs:ℓ∞)if and only if (43) and (44) hold;(ix)
(Başar and Solak, [24])A∈(bs:fs)if and only if (44), (45) hold and(48)supn∈ℕ∑k‍|Δa(n,k)|<∞,f-lima(n,k)=αkexistsforeachfixedk;(x)
(Başar, [22])A∈(fs:fs)if and only if (45), (48) hold and(49)limq→∞∑k‍1q+1∑i=0q‍|Δ2[a(n+i,k)-αk]|=0uniformlyinn;(xi)
(Başar and Çolak, [21])A∈(cs:fs)if and only if (48) holds;(xii)
(Başar, [25])A∈(f:cs)if and only if(50)supn∈ℕ∑k‍|a(n,k)|<∞,(51)∑n‍ank=αkexistsforeachfixedk,(52)∑n‍∑k‍ank=α,(53)limm→∞∑k‍|Δ[a(n,k)-αk]|=0.Now we give our main results which are related to matrix mappings on/into the spaces of almost convergent seriesafsr and sequences afr.Corollary 20.
LetA=(ank) be an infinite matrix. Then, the following statements hold.(i)
A∈(afsr:f)if and only if{ank}k∈ℕ∈(afr)βfor alln∈ℕand (40), (44) hold witha^nkinstead ofank, (46) holds witha^(n,k,m)instead ofa(n,k,m), and (45) holds witha^(n,k)instead ofa(n,k).(ii)
A∈(afsr:c)if and only if{ank}k∈ℕ∈(afr)βfor alln∈ℕand (19), (43), (44), and (47) hold witha^nkinstead ofank.(iii)
A∈(afsr:ℓ∞)if and only if{ank}k∈ℕ∈(afr)βfor alln∈ℕand (43) and (44) hold witha^nkinstead ofank.(iv)
A∈(afsr:fs)if and only if{ank}k∈ℕ∈(afsr)βfor alln∈ℕand (45), (48), and (49) hold witha^(n,k)instead ofa(n,k).(v)
A∈(cs:afsr)if and only if (48) holds witha~(n,k)instead ofa(n,k).(vi)
A∈(bs:afsr)if and only if (44) holds witha~nkinstead ofankand (45), (48) hold witha~(n,k)instead ofa(n,k).(vii)
A∈(fs:afsr)if and only if (45), (48), and (49) hold witha~(n,k)instead ofa(n,k).Corollary 21.
LetA=(ank) be an infinite matrix. Then, the following statements hold.(i)
A∈(c(p):afsr)if and only if (37)and (38) hold witha~(n,k,m)instead ofa(n,k,m).(ii)
A∈(c0(p):afsr)if and only if (37) holds witha~(n,k,m)instead ofa(n,k,m).(iii)
A∈(ℓ∞(p):afsr)if and only if (37) and (39) hold witha~(n,k,m)instead ofa(n,k,m).Corollary 22.
LetA=(ank) be an infinite matrix. Then, the following statements hold.(i)
A∈(afr:ℓ∞)if and only if{ank}k∈ℕ∈(afr)βfor alln∈ℕand (18) holds witha^nkinstead ofank.(ii)
A∈(afr:c)if and only if{ank}k∈ℕ∈(afr)βfor alln∈ℕand (18), (19), (20), and (21) hold witha^nkinstead ofank.(iii)
A∈(afr:cs)if and only if{ank}k∈ℕ∈(afr)βfor alln∈ℕand (50),(53) hold witha^(n,k)instead ofa(n,k)and (51),(52) hold witha^nkinstead ofank.Corollary 23.
LetA=(ank) be an infinite matrix. Then, the following statements hold.(i)
A∈(ℓ∞:afr)if and only if  (18), (40) hold witha~nkinstead ofankand (41) holds witha~(n,k,m)instead ofa(n,k,m).(ii)
A∈(f:afr)if and only if (18), (40), and (46) hold witha~(n,k,m)instead ofa(n,k,m)and (42) holds witha~nkinstead ofank.(iii)
A∈(c:afr)if and only if (18), (40), and (42) hold witha~nkinstead ofank.(iv)
A∈(bs:afr)if and only if (40), (43), and (44) hold witha~nkinstead ofankand (45) holds witha~(n,k)instead ofa(n,k).(v)
A∈(fs:afr)if and only if (40), (44) hold witha~nkinstead ofank, (46) holds witha~(n,k,m)instead ofa(n,k,m), and (45) holds witha~(n,k)instead ofa(n,k).(vi)
A∈(cs:afr)if and only if (40) and (43) hold witha~nkinstead ofank.Remark 24.
Characterization of the classes(afr:f∞), (f∞:afr), (afsr:f∞), and (f∞:afsr) is redundant since the spaces of almost bounded sequences f∞ and ℓ∞ are equal.



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