# On the Domain of the TriangleA(λ) on the Spaces of Null, Convergent, and Bounded Sequences

Authors: Naim L. Braha; Feyzi Başar
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduce the spaces ofA(λ)-null, A(λ)-convergent, and A(λ)-bounded sequences. We examine some topological properties of the spaces and give some inclusion relations concerning these sequence spaces. Furthermore, we compute α-, β-, and γ-duals of these spaces. Finally, we characterize some classes of matrix transformations from the spaces of A(λ)-bounded and A(λ)-convergent sequences to the spaces of bounded, almost convergent, almost null, and convergent sequences and present a Steinhaus type theorem.


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## Body

## 1. Introduction

Byω, we denote the space of all complex sequences. If x∈ω, then we simply write x=(xk) instead of x=(xk)k=0∞. Also, we will use the conventions that e=(1,1,…), and e(n) is the sequence whose only nonzero term is 1 in the nth place for each n∈ℕ, where ℕ={0,1,2,…}. Any vector subspace of ω is called a sequence space. We will write ℓ∞, c and c0 for the sequence spaces of all bounded, convergent, and null sequences, respectively. Further, by ℓp with 1≤p<∞, we denote the sequence space of all p-absolutely convergent series, that is, ℓp={x=(xk)∈ω:∑k|xk|p‍<∞}. For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. Moreover, we write bs and cs for the spaces of all bounded and convergent series, respectively. A sequence space μ is called an FK-space if it is a complete linear metric space with continuous coordinates pn:μ→ℂ, where ℂ denotes the complex field and pn(x)=xn for all x=(xn)∈μ and every n∈ℕ. A normed FK-space is called a BK-space, that is, a BK-space is a Banach space with continuous coordinates. The sequence spaces c0 and c are BK-spaces with the usual sup-norm given by ∥x∥∞=supn∈ℕ|xn|. Also, the space ℓp is a BK-space with the usual norm ∥·∥p defined by
(1)∥x∥p=(∑n‍|xn|p)1/p,
where 1≤p<∞. A sequence (yn) in a normed space X is called a Schauder basis for X if for every x∈X there is a unique sequence (αn) of scalars such that x=∑nαnyn, that is,
(2)limn→∞∥x-(α0y0+α1y1+α2y2+⋯+αnyn)∥=0.
The alpha-, beta-, and gamma-duals μα, μβ, and μγ of a sequence space μ are, respectively, defined by
(3)μα={a=(ak)∈ω:ax=(akxk)∈ℓ1∀x=(xk)∈μ},μβ={a=(ak)∈ω:ax=(akxk)∈cs∀x=(xk)∈μ},μγ={a=(ak)∈ω:ax=(akxk)∈bs∀x=(xk)∈μ}.IfA is an infinite matrix with complex entries ank, where k,n∈ℕ, then we write A=(ank) instead of A=(ank)n,k=0∞. Also, we write An for the sequence in the nth row of the matrix A, that is, An=(ank)k=0∞ for every n∈ℕ. Further, if x=(xk)∈ω then we define the A-transform of x as the sequence Ax={(Ax)n}n=0∞, where
(4)(Ax)n=∑k‍ankxk
provided the series on the right hand side of (4) convergent for each n∈ℕ.Furthermore, the sequencex is said to be A-summable to l∈ℂ if Ax converges to l which is called the A-limit of x. In addition, let μ and ν be sequence spaces. Then, we say that A defines a matrix mapping from μ into ν if for every sequence x∈μ the A-transform of x exists and is in ν. Moreover, we write (μ:ν) for the class of all infinite matrices that map μ into ν. Thus, A∈(μ:ν) if and only if An∈μβ for all n∈ℕ and Ax∈ν for all x∈μ. The matrix domain μA of an infinite matrix A in a sequence space μ is defined by
(5)μA={x∈ω:Ax∈μ}
which is a sequence space. The approach constructing a new sequence space by means of the matrix domain of a triangle matrix was employed by several authors, see for instance [1–4]. In this paper, we introduce the spaces of A(λ)-null, A(λ)-convergent, and A(λ)-bounded sequences which generalize the results given in [2]. Further, we define some related BK-spaces and construct their bases. Moreover, we establish some inclusion relations concerning those spaces and determine their alpha-, beta-, and gamma-duals. Finally, we characterize some classes of matrix transformations on these sequence spaces.


## 2. Notion ofA(λ)-Null, A(λ)-Convergent, and A(λ)-Bounded Sequences

Letλ=(λk) be a strictly increasing sequence of positive real numbers tending to infinity, as k→∞ and λn+1≥2λn for each n∈ℕ. From this last relation, it follows that Δ2λn≥0. The first and second differences are defined as follows: Δλk=λk-λk-1 and Δ2λk=Δ(Δλk)=λk-2λk-1+λk-2 for all k∈ℕ, where λ-1=λ-2=0.Letx=(xk) be a sequence of complex numbers, such that x-1=x-2=0. We say that the sequence x=(xk) is A(λ)-strongly convergent to a number l if
(6)limn→∞1λn-λn-1∑k=0n‍|(λk-2λk-1+λk-2)(xk-l)|=0.
This generalizes the concept of Λ-strong convergence (see [5]).Lemma 1 (see [5]).
A sequence
x
=
(
x
n
)
of complex numbers
λ
-strongly converges to a number
l
if and only if
x
=
(
x
n
)
converges to
l
in the ordinary sense and
(7)
lim
n
→
∞
1
λ
n
-
λ
n
-
1
∑
k
=
0
n
‍
λ
k
-
1
|
x
k
-
x
k
-
1
|
=
0
.Let us define the sequencey=(yn) by the A(λ)-transform of a sequence x=(xk), that is,
(8)yn=(Aλx)n=1λn-λn-1∑k=0n‍(λk-2λk-1+λk-2)xk
for all n∈ℕ. Throughout the text, we suppose that the terms of the sequences x=(xk) and y=(yk)   are connected with the relation (8).Lemma 2 (see [5]).
If a sequence
(
y
n
)
converges to
l
in the ordinary sense and condition (7) of Lemma 1 holds, then the sequence
x
=
(
x
n
)
of complex numbers
A
(
λ
)
-strongly converges to
l.Remark 3 (see [5]).
From above results, we can conclude the following. The sequencex=(xn) of complex numbers A(λ)-strongly converges to l if and only if the following relation holds:
(9)limn→∞1λn-λn-1∑k=0n‍(λk-2λk-1+λk-2)(xk-l)=0.Now, we define the infinite matrixA(λ)={ank(λ)}n,k=0∞ by
(10)ank(λ)={λk-2λk-1+λk-2λn-λn-1,0≤k≤n;0,k>n
for all n,k∈ℕ. Then, A(λ)-transform of a sequence x∈ω is the sequence Aλx={(Aλx)n}n=0∞, where (Aλx)n is given by the relation (8) for every n∈ℕ. Thus, the sequence x is A(λ)-convergent if and only if x is A(λ)-summable. Further, if x is A(λ)-convergent then the A(λ)-limit of x exists and coincides with the ordinary limit of x, that is, to say that the method A(λ) is regular.


## 3. The Spaces ofA(λ)-Null, A(λ)-Convergent, and A(λ)-Bounded Sequences

We introduce the classesAλ(c0), Aλ(c), and Aλ(ℓ∞) of all A(λ)-null, A(λ)-convergent, and A(λ)-bounded sequences of complex numbers, that is,
(11)Aλ(c0)={x=(xk)∈ω:limn→∞(Aλx)n=0},Aλ(c)={x=(xk)∈ω:∃l∈ℂ∋limn→∞(Aλx)n=l},Aλ(ℓ∞)={x=(xk)∈ω:supn∈ℕ|(Aλx)n|<∞}.
Obviously, Aλ(c0), Aλ(c), and Aλ(ℓ∞) are the linear spaces with respect to the usual operations coordinatewise addition and scalar multiplication of sequences. Here and after, by X we denote any of the spaces c0, c, and ℓ∞. It is not hard to see that the quantity
(12)∥x∥Aλ(X)∶=supn∈ℕ1λn-λn-1∑k=0n‍|(λk-2λk-1+λk-2)xk|
is finite for every x=(xk)∈Aλ(X), and ∥·∥Aλ(X) is a norm on Aλ(X).Denote by∥·∥bv the usual bv-norm, that is, to say that
(13)∥x∥bv∶=∑k‍|xk-xk-1|.
With the notation of (5), we can redefine the spaces Aλ(c0),Aλ(c), and Aλ(ℓ∞) as follows:
(14)Aλ(c0)=(c0)A(λ),Aλ(c)=(c)A(λ),Aλ(ℓ∞)=(ℓ∞)A(λ).Theorem 4.
The sequence spaces
A
λ
(
c
0
)
,
A
λ
(
c
),
andAλ(ℓ∞)areBK-spaces with the norm given by(15)∥x∥Aλ(X)=∥Aλx∥∞=supn∈ℕ|(Aλx)n|.Proof.
This follows from Theorem  4.3.12 given in [6] and the relations (14).Theorem 5.
The sequence spaces
A
λ
(
c
0
)
,
A
λ
(
c
),
andAλ(ℓ∞)are norm isomorphic to the spacesc0,c,
andℓ∞, respectively.Proof.
Since the matrixA(λ) is triangle, it has unique inverse which is also triangle matrix (see [6, 1.4.8]). Therefore, the linear operator, defined by T:Aλ(X)→X,Tx=Aλx for all x∈Aλ(X), is bijective and norm preserving by relation (15).As a consequence of Theorems4 and 5, we get the following result.Corollary 6.
Define the sequence
e
(
n
)
(
λ
)
∈
A
λ
(
c
0
)
for every fixed
n
∈
ℕ
by
(16)
e
k
(
n
)
(
λ
)
=
{
(
-
1
)
k
-
n
λ
n
-
λ
n
-
1
λ
k
-
2
λ
k
-
1
+
λ
k
-
2
,
n
≤
k
≤
n
+
1
,
0
,
o
t
h
e
r
w
i
s
e
,
where
k
∈
ℕ
. Then, one has the following.
(1)
The sequence
{
e
k
(
0
)
(
λ
)
,
e
k
(
1
)
(
λ
)
,
e
k
(
2
)
(
λ
)
,
…
}
is a Schauder basis for the space
A
λ
(
c
0
),
and everyx∈Aλ(c0)has a unique representation:x=∑n‍(Aλx)nek(n)(λ).
(2)
The sequence
{
e
,
e
k
(
0
)
(
λ
)
,
e
k
(
1
)
(
λ
)
,
e
k
(
2
)
(
λ
)
,
…
}
is a Schauder basis for the space
A
λ
(
c
),
and everyx∈Aλ(c)has a unique representation:x=le+∑n‍[(Aλx)n-l]ek(n)(λ), wherel=limn→∞(Aλx)n.


## 4. Some Inclusion Relations Related to the New Spaces

In this section, we give some inclusion relations concerning the spacesAλ(c0), Aλ(c), and Aλ(ℓ∞).Theorem 7.
The inclusions
A
λ
(
c
0
)
⊂
A
λ
(
c
)
⊂
A
λ
(
ℓ
∞
)
strictly hold.Proof.
Let us suppose thatx=(xn)∈Aλ(c0), then it follows that x=(xn)∈Aλ(c) and x=(xn)∈Aλ(ℓ∞). In what follows we show that these inclusions are strict. The first inclusion follows from the fact that every sequence, which converges in ordinary sense, converges in A(λ)-sense to the same limit. To prove the strictness of the inclusion Aλ(c)⊂Aλ(ℓ∞), define the sequence x=(xk) by
(17)xk=(-1)kλk-λk-2λk-2λk-1+λk-2
for all k∈ℕ. Then, it follows that
(18)(Aλx)n=1λn-λn-1∑k=0n(λk-2λk-1+λk-2)xk=1λn-λn-1∑k=0n‍(-1)k(λk-λk-2)=(-1)n.
Therefore, it is trivial that x=(xk)∈Aλ(ℓ∞)∖Aλ(c).Theorem 8.
The equality
A
λ
(
c
0
)
∩
c
=
c
0
holds.Proof.
First, we prove thatAλ(c0)∩c⊂c0. If a sequence x=(xn) converges in the ordinary sense to l then it follows that xn→l converges in A(λ)-sense, too. This gives the first inclusion. The converse inclusion follows from Lemma 1, in [5].In what follows we describe some properties of the sequence(λn) in the space ℓ∞.Theorem 9.
For the sequence
(
λ
n
)
which is given in Section2, the following relations are satisfied:
(i)
(
Δ
λ
n
-
1
/
(
Δ
λ
n
-
Δ
λ
n
-
1
)
)
n
=
1
∞
∉
ℓ
∞  if and only ifliminfn→∞((Δλn+1-Δλn)/Δλn)=0;
(ii)
(
Δ
λ
n
-
1
/
(
Δ
λ
n
-
Δ
λ
n
-
1
)
)
n
=
1
∞
∈
ℓ
∞  if and only ifliminfn→∞((Δλn+1-Δλn)/Δλn)>0.Proof.
(i) Let us start with the expression(19)sn(x)=xn-(Aλx)n=1λn-λn-1∑k=0n‍(λk-2λk-1+λk-2)(xn-xk).
After some calculations, we get
(20)sn(x)=Δλn-1Δλn[sn(x)+(Aλx)n+(Aλx)n-1]=Δλn-1Δλn-Δλn-1[(Aλx)n-(Aλx)n-1].
On the other hand, from the definition of the sequence(λn) we have
(21)λn≥2λn-1⟹Δ2λn≥0⟹Δλn≥Δλn-1⟹ΔλnΔλn-1≥1.
From the last relation, we have following two possibilities:(a)
liminfn→∞((Δλn-Δλn-1)/Δλn-1)>0 or(b)
liminfn→∞((Δλn-Δλn-1)/Δλn-1)=0.
Part (a) is satisfied if and only if(Δλn-1/(Δλn-Δλn-1))n=1∞ is bounded. Part (b) is satisfied if and only if (Δλn-1/(Δλn-Δλn-1))n=1∞ is unbounded.Lemma 10.
The inclusions
c
0
⊂
A
λ
(
c
0
)
and
c
⊂
A
λ
(
c
)
hold. Those spaces coincide if and only if
s
(
x
)
∈
c
0
for every
x
∈
A
λ
(
c
0
),
respectively,Aλ(c),
wheres(x)={sn(x)}n=0∞.Lemma 11.
The inclusion
ℓ
∞
⊂
A
λ
(
ℓ
∞
)
holds. Those spaces coincide if and only if
s
(
x
)
∈
ℓ
∞  for everyx∈Aλ(ℓ∞).Theorem 12.
The inclusions
c
0
⊂
A
λ
(
c
0
)
,
c
⊂
A
λ
(
c
)
and
ℓ
∞
⊂
A
λ
(
ℓ
∞
)
strictly hold if and only if
(22)
liminf
n
→
∞
Δ
λ
n
+
1
-
Δ
λ
n
Δ
λ
n
>
0
.Proof.
Let us suppose thatℓ∞⊂Aλ(ℓ∞) is strict. Then, from Lemma 11, it follows that there exists a sequence x=(xn)∈Aλ(ℓ∞) such that s(x)={sn(x)}n=0∞∉ℓ∞. Since x=(xn)∈Aλ(ℓ∞), we have Aλx∈ℓ∞ which leads us to the fact that {xn-sn(x)}∈ℓ∞. On the other hand, from relation (20), it follows that (Δλn-1/(Δλn-Δλn-1))∉ℓ∞. The last relation is equivalent to
(23)liminfn→∞Δλn+1-ΔλnΔλn=0
by part (i) of Theorem 9. In a similar way we can conclude that the inclusions c0⊂Aλ(c0), c⊂Aλ(c) are strict. In what follows we prove the sufficiency. Let
(24)liminfn→∞Δλn+1-ΔλnΔλn=0.
Then, from, part (i) of Theorem 9, it follows that (Δλn-1/(Δλn-Δλn-1))∉ℓ∞ and ((Δλn-1+Δλn)/(Δλn-Δλn-1))∉ℓ∞. Let us define the sequence x=(xn) by
(25)xn=(-1)nΔλn-1+ΔλnΔλn-Δλn-1
for all n∈ℕ. Then, we get the following estimation:
(26)|(Aλx)n|=1λn-λn-1|∑k=0n(-1)kΔλk-1+ΔλkΔλk-Δλk-1×(λk-2λk-1+λk-2)∑k=0n‍(-1)k|=1λn-λn-1|∑k=0n(-1)k(Δλk-1+Δλk)|=1λn-λn-1|(-1)nΔλn|=1.
Hence, Aλx∈ℓ∞ which means that x∈Aλ(ℓ∞)∖ℓ∞. If liminfn→∞((Δλn+1-Δλn)/Δλn)=0. Then, there exists a subsequence (nr) such that
(27)limr→∞Δλnr+1-ΔλnrΔλnr=0.
Now, let us define the sequence x=(xn) by
(28)xn={1,n=nr,-Δλk-1-Δλk-2Δλk-Δλk-1,n=nr,0,otherwise,
for all n∈ℕ. It follows from (28) that x∉c. On the other hand,
(29)(Aλx)n={Δλk-Δλk-1Δλk,n=nr,0,otherwise.
Now, from the relations (27) and (29), we derive that x=(xn)∈Aλ(c0)⊂Aλ(c). This completes the proof.As an immediate result of Theorem12, we have the following.Corollary 13.
The equalities
c
0
=
A
λ
(
c
0
)
,
c
=
A
λ
(
c
),
andℓ∞=Aλ(ℓ∞)are satisfied if and only if(30)liminfn→∞Δλn+1-ΔλnΔλn>0.Proposition 14.
The following statements hold.(i)
AlthoughcandAλ(c0)overlap, the spaceAλ(c0)does not include the spacec.(ii)
Althoughℓ∞andAλ(c)overlap, the spaceAλ(c)does not include the spaceℓ∞.Proposition 15.
Ifliminfn→∞((Δλn+1-Δλn)/Δλn)=0, then the following statements hold.(i)
Neither of the spacescandAλ(c0)includes the other.(ii)
Neither of the spacesAλ(c0)andℓ∞includes the other.(iii)
Neither of the spacesAλ(c)andℓ∞includes the other.


## 5. Theα-, β-, and γ-Duals of the Spaces Aλ(c0), Aλ(c), and Aλ(ℓ∞)

In this section, we determine the alpha-, beta-, and gamma-duals of the spacesAλ(c0), Aλ(c), and Aλ(ℓ∞).We need the following lemma due to Stieglitz and Tietz [3] in proving Theorem 17.Lemma 16.
A
=
(
a
n
k
)
∈
(
c
0
:
ℓ
1
)
=
(
c
:
ℓ
1
)
if and only if
(31)
sup
K
,
N
∈
ℱ
|
∑
n
∈
N
‍
∑
k
∈
K
‍
a
n
k
|
<
∞
.
Here and after, by ℱ one denotes the collection of all finite subsets of ℕ.Theorem 17.
The
α
-dual of the spaces
A
λ
(
c
0
)
,
A
λ
(
c
),
andAλ(ℓ∞)is the set(32)a1(λ)={a=(an)∈ω:∑n‍λn-λn-1λn-2λn-1+λn-2|an|<∞}.Proof.
Define the matrixB=(bnk) with the aid of a sequence a=(an) as follows:
(33)bnk={(-1)n-kλk-λk-1λn-2λn-1+λn-2an,n-1≤k≤n,0,0≤k≤n-1ork>n.
Then, x=(xn)∈Aλ(c0), we have from Theorem 5(34)anxn=an∑k=n-1n(-1)n-k‍λk-λk-1λn-2λn-1+λn-2yk=(By)n
for all n∈ℕ. From the relation (34), it follows that ax=(anxn)∈ℓ1 whenever x∈Aλ(c0) if and only if By∈ℓ1 whenever y=(yk)∈c0, that is, a∈{Aλ(c0)}α if and only if B∈(c0:ℓ1). By Lemma 16, this is possible if and only if
(35)supK,N∈ℱ|∑n∈N∑k∈Kbnk|<∞.
Now, from definition of the sets K,N and the matrix B=(bnk), it follows that (35) holds if and only if
(36)∑n‍λn-λn-1λn-2λn-1+λn-2|an|<∞
which gives that {Aλ(c0)}α=a1(λ).
In a similar way, one can show thata1(λ) is the α-dual of the spaces Aλ(c) and Aλ(ℓ∞). So, we omit the details.Theorem 18.
Define the sets
A
,
B
,
C,andDas follows:(37)A={∑k|Δ[akλk-2λk-1+λk-2(λk-λk-1)]∑k‍|a=(ak)∈ω:∑k|Δ[akλk-2λk-1+λk-2(λk-λk-1)]∑k‍|},B={a=(ak)∈ω:supk∈ℕ|ak(λk-λk-1)λk-2λk-1+λk-2|},C={a=(ak)∈ω:limk→∞ak(λk-λk-1)λk-2λk-1+λk-2exists},D={a=(ak)∈ω:limk→∞ak(λk-λk-1)λk-2λk-1+λk-2=0}.
Then, one has {Aλ(c0)}β=A∩B, {Aλ(c)}β=A∩C and {Aλ(ℓ∞)}β=A∩D.Proof.
Since the proof is similar for the spacesAλ(c0) and Aλ(ℓ∞), we consider only the space Aλ(c). Let u=(uk)∈ω. Then, taking into account the relation (8) between the sequences x=(xk) and y=(yk), we obtain that
(38)∑k=0n‍ukxk=∑k=0n‍uk∑j=k-1k(-1)k-jλj-λj-1λk-2λk-1+λk-2yj=∑k=0n-1‍(λk-λk-1)Δ(ukλk-2λk-1+λk-2)yk+(λn-λn-1)unynλn-2λn-1+λn-2,=(By)n,∀n∈ℕ,
where
(39)Δ(ukλk-2λk-1+λk-2)=ukλk-2λk-1+λk-2-uk+1λk+1-2λk+λk-1,
and the matrix B=(bnk) is defined by
(40)bnk={(λk-λk-1)×Δ(ukλk-2λk-1+λk-2),0≤k≤n-1,(λn-λn-1)unλn-2λn-1+λn-2,k=n,0,k>n
for all k,n∈ℕ. Therefore, one can easily see from (38) that ux=(ukxk)∈cs with x=(xk)∈Aλ(c) if and only if By∈c with y=(yk)∈c, where B=(bnk) is defined by (40). That is, to say that u=(uk)∈{Aλ(c)}β if and only if B is a matrix satisfying the conditions of Kojima-Schur's theorem (cf. Başar [7, Theorem 3.3.3, page 35]). This leads to the fact that {Aλ(c)}β=A∩C.Theorem 19.
The
γ
-dual of the spaces
A
λ
(
c
0
)
,
A
λ
(
c
),andAλ(ℓ∞)is the setA∩B.Proof.
This is similar to the proof of Theorem18. So, we omit the details.


## 6. Some Matrix Transformation Related to Sequence SpacesAλ(c0), Aλ(c), and Aλ(ℓ∞)

In this section, we characterize the matrix transformations from the spacesAλ(ℓ∞) and Aλ(c) into the spaces ℓ∞, f, f0, c, and c0 of bounded, almost convergent, almost null, convergent, and null sequences, respectively. We write throughout for brevity that
(41)Δank=ank-an,k+1,a~nk=(λk-λk-1)Δ(ankλk-2λk-1+λk-2),a(n,k)=∑‍j=0najk,cnk=∑‍j=0n1j+1ajk,dnk=san-1,k+rank,enk=tan-2,k+san-1,k+rank,a(n,k,m)=1m+1∑‍j=0man+j,k
for all k,m,n∈ℕ, and we use these abbreviations with other letters, where r,s,t∈ℝ∖{0}.Theorem 20.
A
=
(
a
n
k
)
∈
(
A
λ
(
X
)
:
ℓ
∞
)
if and only if
(42)
(
λ
k
-
λ
k
-
1
λ
k
-
2
λ
k
-
1
+
λ
k
-
2
a
n
k
)
k
∈
ℕ
∈
c
0
f
o
r
e
a
c
h
f
i
x
e
d
n
∈
ℕ
(43)
sup
n
∈
ℕ
∑
k
‍
|
a
~
n
k
|
<
∞
.Proof.
Suppose that the conditions (42) and (43) hold, and take any x=(xk)∈Aλ(X). Then, the sequence (ank)k∈ℕ∈{Aλ(X)}β for all n∈ℕ, and this implies the existence of the A-transform of x.
Let us now consider the following equality derived by using the relation (8) from the mth partial sum of the series ∑k‍ankxk:
(44)∑k=0m‍ankxk=∑k=0m-1‍a~nkyk+λm-λm-1λm-2λm-1+λm-2anmym
for all m,n∈ℕ. Therefore, we obtain from (44) with (42), as m→∞, that
(45)∑k‍ankxk=∑k‍a~nkyk,∀n∈ℕ.
Now, by taking the sup-norm in (45), we derive that
(46)supn∈ℕ|(Ax)n|≤∥y∥∞supn∈ℕ∑k‍|a~nk|<∞
which shows the sufficiency of the conditions (42) and (43).
Conversely, suppose thatA=(ank)∈(Aλ(X):ℓ∞). Then, since (ank)k∈ℕ∈{Aλ(X)}β for all n∈ℕ by the hypothesis, the necessity of (42) is trivial and (45) holds. Consider the continuous linear functionals fn defined on Aλ(X) by the sequences an=(ank)k∈ℕ as
(47)fn(x)=∑kankxk,∀n∈ℕ.
Since Aλ(ℓ∞)≅ℓ∞, Aλ(c)≅c and Aλ(c0)≅c0, it should follow with (45) that ∥fn∥=∥a~n∥∞. This just says that the functionals defined by the rows of A on Aλ(X) are pointwise bounded. Hence, by Banach-Steinhaus theorem, fn's are uniformly bounded which gives that there exists a constant K>0 such that ∥fn∥≤K for all n∈ℕ. It therefore follows that ∑k‍|a~nk|=∥fn∥≤K holds for all n∈ℕ which shows the necessity of the condition (43).
This step completes the proof.Prior to characterizing the class of infinite matrices from the spaceAλ(ℓ∞) into the space of almost convergent sequences, we give a short knowledge on the concept of almost convergence. The shift operatorP is defined on ω by Pn(x)=xn+1 for all n∈ℕ. A Banach limit L is defined on ℓ∞, as a non-negative linear functional, such that L(Px)=L(x) and L(e)=1. A sequence x=(xk)∈ℓ∞ is said to be almost convergent to the generalized limit α if all Banach limits of x coincide and are equal to α [8] and is denoted by f-limxk=α. Let Pi be the composition of P with itself i times and write for a sequence x=(xk)(48)tmn(x)=1m+1∑i=0m‍Pni(x),∀m,n∈ℕ.
Lorentz [8] proved that f-limxk=α if and only if limm→∞tmn(x)=α, uniformly in n. It is well known that a convergent sequence is almost convergent such that its ordinary and generalized limits are equal. By f0 and f, we denote the spaces of almost null and almost convergent sequences, respectively. Now, we can give the lemma characterizing the almost coercive matrices.Lemma 21 (see [9, Theorem 1]).
A
=
(
a
n
k
)
∈
(
ℓ
∞
:
f
)
if and only if
(49)
sup
n
∈
ℕ
∑
k
|
a
n
k
|
<
∞
;
(50)
∃
α
k
∈
ℂ
∋
f
-
lim
a
n
k
=
α
k
f
o
r
e
a
c
h
f
i
x
e
d
k
∈
ℕ
;
(51)
lim
m
→
∞
∑
k
‍
|
a
(
n
,
k
,
m
)
-
α
k
|
=
0
u
n
i
f
o
r
m
l
y
i
n
n
.Theorem 22.
A
=
(
a
n
k
)
∈
(
A
λ
(
ℓ
∞
)
:
f
) if and only if the conditions (42) and (43) hold, and
(52)∃αk∈ℂ∋f-lima~nk=a~kforeachfixedk∈ℕ;(53)limm→∞∑k‍|a~(n,k,m)-a~k|=0uniformlyinn.Proof.
LetA∈(Aλ(ℓ∞):f). Then, since f⊂ℓ∞, the necessity of (42) and (43) is immediately obtained from Theorem 20. To prove the necessity of (52), consider the sequence e(n)(λ)={ek(n)(λ)}n∈ℕ∈Aλ(ℓ∞), defined by (16) for every fixed k∈ℕ. Since Ax exists and is in f for every x∈Aλ(ℓ∞), one can easily see that Ae(n)(λ)=(a~nk)n∈ℕ∈f for all k∈ℕ, that is, the condition (52) is necessary.
Define the matrixB=(bnk) by bnk=a~nk for all k,n∈ℕ. Then, we derive from the equality (45) that Ax=By. Since A=(ank)∈(Aλ(ℓ∞):f) by the hypothesis, we have B∈(ℓ∞:f). Therefore, the matrix B satisfies the condition (51) of Lemma 21 which is equivalent to the condition (53).
Conversely, suppose that the matrixA satisfies the conditions (42), (43), (52), and (53), and x∈Aλ(ℓ∞). Reconsider the equality Ax=By obtained from (45) with bnk instead of a~nk. Then, the conditions (49), (50), and (51) are satisfied by the matrix B. Hence, B is almost coercive by Lemma 21 and this gives by passing to f-limit in (45) that Ax∈f, that is, A∈(Aλ(ℓ∞):f), as desired.
This concludes the proof.As a direct consequence of Theorem22, we have the following.Corollary 23.
A
=
(
a
n
k
)
∈
(
A
λ
(
ℓ
∞
)
:
f
0
)
if and only if the conditions (42) and (43) hold, and (52) and (53) hold with
a
~
k
=
0
for all
k
∈
ℕ
.Theorem 24.
A
=
(
a
n
k
)
∈
(
A
λ
(
ℓ
∞
)
:
c
)
if and only if the condition (42) holds, and the conditions
(54)
∑
k
‍
|
a
~
n
k
|
c
o
n
v
e
r
g
e
u
n
i
f
o
r
m
l
y
i
n
n
∈
ℕ
;
(55)
∃
α
k
∈
ℂ
∋
lim
n
→
∞
a
n
k
=
α
k
f
o
r
e
a
c
h
f
i
x
e
d
k
∈
ℕ
.Corollary 25.
A
=
(
a
n
k
)
∈
(
A
λ
(
ℓ
∞
)
:
c
0
)  if and only if the conditions (42) and (54) hold, and (55) also holds with αk=0 for all k∈ℕ.Now, we give the following lemma due to King [10] characterizing the class of almost conservative matrices.Lemma 26.
A
=
(
a
n
k
)
∈
(
c
:
f
)
if and only if (49) and (50) hold, and
(56)
∃
α
∈
ℂ
∋
f
-
lim
∑
k
a
n
k
=
α
.Theorem 27.
A
=
(
a
n
k
)
∈
(
A
λ
(
c
)
:
f
)
if and only if the conditions (42), (43), and (52) hold, and the condition
(57)
∃
α
∈
ℂ
∋
f
-
lim
∑
k
‍
a
~
n
k
=
a
~
.
also holds.Proof.
This is obtained by a similar way used in proving Theorem22 with Lemma 26 instead of Lemma 21. So, to avoid the repetition of the similar statements we omit the details.Corollary 28.
A
=
(
a
n
k
)
∈
(
A
λ
(
c
)
:
f
)
ρ
if and only if the conditions (42) and (43) hold, and the conditions (52) and (57) also hold with
α
k
=
0
for all
k
∈
ℕ
and
a
~
=
ρ
, respectively; where by
(
A
λ
(
c
)
:
f
)
ρ
, we denote the class of infinite matrices
A
such that
f
-
lim
A
x
=
ρ
[
A
(
λ
)
-
lim
x
]
for all
x
∈
A
λ
(
c
)
.Now, we give the following Steinhaus type theorem.Theorem 29.
The classes
(
A
λ
(
ℓ
∞
)
:
f
)
and
(
A
λ
(
c
)
:
f
)
ρ
are disjoint, where
ρ
∈
ℝ
∖
{
0
}
.Proof.
Suppose that the classes(Aλ(ℓ∞):f) and (Aλ(c):f)ρ are not disjoint. Then, there is at least one A=(ank) in the set (Aλ(ℓ∞):f)∩(Aλ(c):f)ρ. Therefore, one can derive by combining (53) and (52) with αk=0 for all k∈ℕ that
(58)limm→∞∑k‍|a~(n,k,m)|=0uniformlyinn
which is contrary to the condition (57) with a~=ρ≠0. This completes the proof.Lemma 30 (see [11, Lemma 5.3]).
Let
μ
,
ν
be any two sequence spaces,
A
an infinite matrix, and
B
a triangle matrix. Then,
A
∈
(
μ
:
ν
B
)
if and only if
B
A
∈
(
μ
:
ν
)
.It is trivial that Lemma30 has several consequences. Indeed, combining Lemma 30 with Theorems 20, 22, 24, and 27 and Corollaries 23, 25, and 28 by choosing B as one of the special matrices C1,Er,Rt,Δ,Δ(1),Ar, or S, one can easily obtain the following results.Corollary 31.
Let
A
=
(
a
n
k
)
be an infinite matrix over the complex field. Then, the following statements hold.
(i)
E
=
(
e
n
k
)
∈
(
A
λ
(
X
)
:
b
v
∞
)
if and only if (42) and (43) hold with
e
n
k
-
e
n
-
1
,
k
instead of
a
n
k
, where
b
v
∞
denotes the space of allsequences x=(xk)such that(xk-xk-1)∈ℓ∞and was introduced by Başar and Altay [11].
(ii)
E
=
(
e
n
k
)
∈
(
A
λ
(
X
)
:
e
∞
r
)
if and only if (42) and (43) hold with
∑
j
=
0
n
(
n
j
)
(
1
-
r
)
n
-
j
r
j
e
j
k
instead of
a
n
k
, where
e
∞
r
denotes the space of all sequences
x
=
(
x
k
)
such that
(
∑
j
=
0
n
(
n
j
)
(
1
-
r
)
n
-
j
r
j
e
k
j
x
j
)
∈
ℓ
∞
and was introduced by Altayet al. [12].
(iii)
E
=
(
e
n
k
)
∈
(
A
λ
(
X
)
:
X
∞
)
if and only if (42) and (43) hold with
e
(
n
,
k
)
/
(
n
+
1
)
instead of
a
n
k
, where
X
∞
denotes the space of all sequences
x
=
(
x
k
)
such that
(
∑
j
=
0
n
‍
x
j
/
(
n
+
1
)
)
∈
ℓ
∞
and was introduced by Ng and Lee [13].
(iv)
E
=
(
e
n
k
)
∈
(
A
λ
(
X
)
:
r
∞
t
)
if and only if (42) and (43) hold with
∑
j
=
0
n
‍
t
j
e
j
k
/
T
n
instead of
a
n
k
, where
r
∞
t
denotes the space of all sequences
x
=
(
x
k
)
such that
(
∑
j
=
0
n
‍
t
j
x
j
/
T
n
)
∈
ℓ
∞
and was introduced by Altay and Başar [14].
(v)
E
=
(
e
n
k
)
∈
(
A
λ
(
X
)
:
b
s
)
if and only if (42) and (43) hold with
e
(
n
,
k
)
instead of
a
n
k
.Corollary 32.
Let
A
=
(
a
n
k
)
be an infinite matrix over the complex field. Then, the following statements hold.(i)
E=(enk)∈(Aλ(ℓ∞):c(Δ))if and only if (42), (54), and (55) hold withenk-en+1,kinstead ofank, wherec(Δ)denotes the space of all sequencesx=(xk)such that(xk-xk+1)∈cand was introduced by Kızmaz [15].(ii)
E=(enk)∈(Aλ(ℓ∞):ecr)if and only if (42), (54), and (55) hold with∑j=0n(nj)(1-r)n-jrjejkinstead ofank, whereecrdenotes the space of all sequencesx=(xk)such thatErx∈cand was introduced by Altay and Başar [16].(iii)
E=(enk)∈(Aλ(ℓ∞):c~)if and only if (42), (54), and (55) hold withe(n,k)/(n+1)instead ofank, wherec~denotes the space of all sequencesx=(xk)such thatC1x∈cand was introduced by Şengönül and Başar [17].(iv)
E=(enk)∈(Aλ(ℓ∞):rct)if and only if (42), (54), and (55) hold with∑j=0n‍tjejk/Tninstead ofank, whererctdenotes the space of all sequencesx=(xk)such thatRtx∈cand was introduced by Altay and Başar [18].(v)
E=(enk)∈(Aλ(ℓ∞):cs)if and only if (42), (54) and (55) hold withe(n,k)instead ofank.Corollary 33.
Let
A
=
(
a
n
k
)
be an infinite matrix over the complex field. Then, the following statements hold.
(i)
A
=
(
a
n
k
)
∈
(
A
λ
(
ℓ
∞
)
:
f
^
)
if and only if (42), (43), (52), and (53) hold with
d
n
k
instead of
a
n
k
, where
f
^
denotes the space of all sequences
x
=
(
x
k
)
such that
B
(
r
,
s
)
x
∈
f
and was introduced by Başar and Kirişçi [19].
(ii)
A
=
(
a
n
k
)
∈
(
A
λ
(
ℓ
∞
)
:
f
^
0
)
if and only if (42) and (43) hold, and (52) and (53) hold with
α
k
=
0
for all
k
∈
ℕ
and
d
n
k
instead of
a
n
k
, where
f
^
0
denotes the space of all sequences
x
=
(
x
k
)
such that
B
(
r
,
s
)
x
∈
f
0
and was introduced by Başar and Kirişçi [19].
(iii)
A
=
(
a
n
k
)
∈
(
A
λ
(
c
)
:
f
^
)
if and only if (42), (43), (52), and (53) hold with
d
n
k
instead of
a
n
k
.
(iv)
A
=
(
a
n
k
)
∈
(
A
λ
(
c
)
:
f
^
0
)
if and only if (42) and (43) hold, and (52) and (53) also hold with
α
k
=
0
for all
k
∈
ℕ
and
α
=
1
, respectively, hold with
α
k
=
0
for all
k
∈
ℕ
and
d
n
k
instead of
a
n
k
.
(v)
A
=
(
a
n
k
)
∈
(
A
λ
(
c
)
:
f
^
)
ρ
if and only if the conditions of Corollary28 hold with
d
n
k
instead of
a
n
k
.Corollary 34.
LetA=(ank) be an infinite matrix over the complex field. Then, the following statements hold. (i)
A=(ank)∈(Aλ(ℓ∞):f~) if and only if (42), (43), (52), and (53) hold with cnk instead of ank, where f~ denotes the space of all sequences x=(xk) such that C1x∈f and was introduced by Kayaduman and Şengönül [20].(ii)
A=(ank)∈(Aλ(ℓ∞):f~0) if and only if (42) and (43) hold, and (52) and (53) hold with αk=0 for all k∈ℕ and cnk instead of ank, where f~0 denotes the space of all sequences x=(xk) such that C1x∈f0 and was introduced by Kayaduman and Şengönül [20].(iii)
A=(ank)∈(Aλ(c):f~) if and only if (42), (43), (52), and (53) hold with cnk instead of ank.(iv)
A=(ank)∈(Aλ(c):f~0) if and only if (42) and (43) hold, and (52) and (53) also hold with αk=0 for all k∈ℕ and α=1, respectively, with cnk instead of ank.(v)
A=(ank)∈(Aλ(c):f~)ρ if and only if the conditions of Corollary 28 hold with cnk instead of ank.Corollary 35.
LetA=(ank) be an infinite matrix over the complex field. Then, the following statements hold.(i)
A=(ank)∈(Aλ(ℓ∞):f(B)) if and only if (42), (43), (52), and (53) hold with enk instead of ank, where f(B) denotes the space of all sequences x=(xk) such that B(r,s,t)x∈f and was introduced by Sönmez [21].(ii)
A=(ank)∈(Aλ(ℓ∞):f0(B)) if and only if (42) and (43) hold, and (52) and (53) hold with αk=0 for all k∈ℕ and enk instead of ank, where f0(B) denotes the space of all sequences x=(xk) such that B(r,s,t)x∈f0 and was introduced by Sönmez [21].(iii)
A=(ank)∈(Aλ(c):f(B)) if and only if the conditions (42), (43), (52), and (53) hold with enk instead of ank.(iv)
A=(ank)∈(Aλ(c):f0(B)) if and only if the conditions (42) and (43) hold, and the conditions (52) and (53) also hold with αk=0 for all k∈ℕ and α=1, respectively, hold with αk=0 for all k∈ℕ and enk instead of ank.(v)
A=(ank)∈(Aλ(c):f(B))ρ if and only if the conditions of Corollary 28 hold with enk instead of ank.


## 7. Conclusion

Mursaleen and Noman [2, 22, 23] have studied the domains ℓ∞λ, cλ, c0λ, and ℓpλ of the matrix Λ in the classical sequence spaces ℓ∞, c, c0, and ℓp, respectively. Malkowsky and Rakočević [24] characterized some classes of matrix transformations and investigated related compact operators involving the spaces of Λ-null, Λ-convergent, and Λ-bounded sequences. Quite recently, Sönmez and Başar [25] have introduced the spaces c0λ(B) and cλ(B) of generalized difference sequences which generalize the paper due to Mursaleen and Noman [26]. Mursaleen and Noman [26] have derived some inclusion relations and determined the alpha-, beta-, and gamma-duals of those spaces and constructed their Schauder bases. Finally, Sönmez and Başar [25] have characterized some matrix classes from the spaces c0λ(B) and cλ(B) to the spaces ℓp, c0, and c. In the present paper, we emphasize the domains Aλ(c0), Aλ(c), and Aλ(ℓ∞) of the matrix A(λ) in the classical sequence spaces c0, c, and ℓ∞. Our results are more general and comprehensive than the corresponding results of Mursaleen and Noman [2, 22, 23] derived with the matrix Λ. We should note that, as a natural continuation of the present paper, one can study the domains Aλ(ℓp) and Aλ(bvp) of the matrix A(λ) in the classical sequence space ℓp and in the sequence space bvp with 0<p<1 and 1≤p<∞, where bvp denotes the space of all sequences x=(xk) such that (xk-xk-1)∈ℓp and introduced in the case 1≤p<∞ by Başar and Altay [11] and in the case 0<p<1 by Altay and Başar [27].



---

*Source: 476363-2013-05-28.xml*