R E S E A R C H
Open Access
Two new sequence spaces generated by the
composition of
mth order generalized
difference matrix and lambda matrix
Mustafa Cemil Bi¸sgin
*and Abdulcabbar Sönmez
*Correspondence: [email protected] Department of Mathematics, Faculty of Science, Erciyes University, Talas Street, Melikgazi/Kayseri, 38039, Turkey
Abstract
In this work, we introduce two sequence spacescλ
0(Gm) andcλ(Gm) generated by the
composition ofmth order generalized difference matrix and lambda matrix and define an isomorphism between new sequence spaces and classical sequence spaces. Afterward, we investigate inclusion relations and obtain the Schauder basis of those spaces. Furthermore, we determine their
α-,
β
- andγ
-duals. Lastly, we characterize some matrix classes related to those spaces.MSC: 40C05; 40H05; 46B45
Keywords: matrix transformations; matrix domain; Schauder basis;
α
-,β- and
γ
-duals; matrix classes1 Introduction
The family of all real (or complex) valued sequences is denoted byw.wis a vector space under point-wise addition and scalar multiplication. Any vector subspace ofwis called a sequence space. In the literature, the classical sequence spaces are symbolized with∞, c,candp which are called all bounded, null, convergent and absolutelyp-summable
sequence spaces, respectively, where ≤p<∞.
A sequence space X with a linear topology is called aK-space provided each of the maps
pi:X→Cdefined bypi(x) =xiis continuous for alli∈N. It is assumed thatwis always
en-dowed with its locally convex topology generated by the sequence{pn}∞n=of semi-norms onw, where pn(x) =|xn|,n= , , , . . . . AK-spaceXis called anFK-space provided X
is a complete linear metric space. AnFK-space whose topology is normable is called a
BK-space [].
The classical sequence spaces ∞,c andcareBK-spaces with their usualsup-norm defined byx∞=supk∈N|xk|andpis aBK-space with itsp-normdefined by
xp=
∞
k= |xk|p
p ,
wherep∈[,∞) [].
Given an infinite matrixA= (ank) withank∈C, for alln,k∈Nand a sequencex∈w, the A-transform ofxis defined by
(Ax)n= ∞
k=
ankxk (.)
and is assumed to be convergent for alln∈N[]. For using simple notations here and in what follows, the summation without limits runs from to∞. Ifx∈Ximplies that
Ax∈Y, then we say thatAdefines a matrix mapping fromX intoY and denote it by
A:X−→Y. By using the notation (X:Y), we mean the class of all infinite matricesAsuch thatA:X−→Y.
For an arbitrary sequence spaceX,XAis called the domain of an infinite matrixAand
is defined by
XA=
x= (xk)∈w:Ax∈X
, (.)
which is also a sequence space. Bybsandcswe denote the spaces of all bounded and con-vergent series, and define them by means of the matrix domain of the summation matrix
S= (snk) such thatbs= (∞)Sandcs=cS, respectively, whereS= (snk) is defined by
snk=
⎧ ⎨ ⎩
, ≤k≤n, , k>n,
which is a triangle matrix too. A matrixAis called a triangle ifank= fork>nandann=
for alln,k∈N. Moreover, a triangle matrixAuniquely has an inverseA–=Bwhich is also a triangle matrix.
Unless stated otherwise, any term with negative subscript is assumed to be zero. The theory of matrix transformation was prompted by summability theory which was obtained by Cesàro, Riesz and others. The Cesàro mean of order one and the Riesz mean according to the sequencep= (pn) are defined by using the matricesC= (cnk) andRp= (rpnk) such
that
cnk=
⎧ ⎨ ⎩
n+, ≤k≤n,
, k>n, and r
p nk=
⎧ ⎨ ⎩
pk
Pn, ≤k≤n, , k>n,
respectively, wherep> ,pn≥ (n≥) andPn=
n k=pk.
Moreover, the theory of matrix transformation has been continued until nowadays. Many authors have constructed new sequence spaces by using matrix domain of infinite matrices. For example, (∞)NqandcNq in [],arpandar∞in [],arandarcin [],Z(u,v;p)
in [], Xp andX∞ in [],epr ander∞ in [],r∞t ,rtandrct in [],rpt in [],eranderc in
[],˜cand˜cin []. Also, many authors introduced new sequence spaces by using espe-cially difference matrices. For instance,c(),c() and∞() in [],c(),c() and
∞() in [],ar
() andarc() in [],c(m),c(m) and∞(m) in [],c(p),c(p) and∞(p) in [],c(u:),c(u:) and∞(u:) in [],c(u,,p),c(u,,p) and
m
∞(p) in [],rq(p,Bm) in [],ˆ∞,ˆc,ˆcandˆpin [],c(B),c(B),∞(B) andp(B) in
[],c((m)),c((m)) and∞((m)) in [],(um)Xin [].
In this work, we introduce two sequence spacescλ
(Gm) and cλ(Gm) generated by the composition ofmth order generalized difference matrix and lambda matrix and define an isomorphism between new sequence spaces and classical sequence spaces. Afterward, we investigate inclusion relations and obtain the Schauder basis of those spaces. Furthermore, we determine theirα-,β- andγ-duals. Lastly, we characterize some matrix classes related to those spaces.
2 Two new sequence spaces
In this section, we give some historical information and define the sequence spacescλ
(Gm) andcλ(Gm) generated by the composition ofmth order generalized difference matrix and
lambda matrix. Moreover, we speak of some inclusion relations.
The idea of using the notion ofλ-convergent was first motivated by Mursaleen and No-man in []. They defined the sequence spacescλ
,λ∞ andcλ by means of the Lambda
matrix= (λnk) such that
cλ
=
x= (xk)∈w:n→∞lim
λn n
k=
(λk–λk–)xk=
,
cλ=
x= (xk)∈w:n→∞lim
λn n
k=
(λk–λk–)xkexists
and
λ∞=
x= (xk)∈w:sup n∈N
λn n
k=
(λk–λk–)xk
<∞
,
whereλ= (λk) consists of positive reals such that
<λ<λ<· · · and lim
k→∞λk=∞
and the lambda matrix= (λnk) is defined by
λnk=
⎧ ⎨ ⎩
λk–λk–
λn , ≤k≤n,
, k>n
for allk,n∈N. Here, we would like to touch on a point, if we takeλn=n+ andλn=Pn, for
alln∈N, we obtain the Cesàro mean of order one and the Riesz mean matrix, respectively. So, the= (λnk) matrix generalizes theC= (cnk) andRp= (rnkp ) matrices.
Also, they improved their work by constructing the spacescλ
() andcλ() in []. The sequence spacescλ
() andcλ() are defined by
cλ() =
x= (xk)∈w:n→∞lim
λn n
k=
(λk–λk–)(xk–xk–) =
and
cλ() =
x= (xk)∈w:n→∞lim
λn n
k=
(λk–λk–)(xk–xk–) exists
,
whereis a difference matrix.
Afterward, Sönmez and Başar defined the sequence spacescλ
(B) andcλ(B) in [] and improved Mursaleen and Noman’s work as follows:
cλ
(B) =
x= (xk)∈w:n→∞lim
λn n
k=
(λk–λk–)(bxk+bxk–) =
and
cλ(B) =
x= (xk)∈w:n→∞lim
λn n
k=
(λk–λk–)(bxk+bxk–) exists
,
whereB=B(b,b) is called a double band (generalized difference) matrix defined by
bnk=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
b, k=n,
b, k=n– ,
, otherwise.
Letrandsbe non-zero real numbers, then themth order generalized difference matrix
Gm(r,s) = (gnkm(r,s)) is defined by
gnkm(r,s) =
⎧ ⎨ ⎩
m–
n–k
rm–n+k–sn–k, max{,n–m+ } ≤k≤n,
, otherwise
for all n,k ∈N and m∈N ={, , , . . .}. We want to recall that G(r,s) =B(b,b),
G(r,s) =B(b
,b,b), G(r,s) =B(b,b,b,b), G(r,s) =B(b,b,b,b,b), . . . where
B(b,b),B(b,b,b),B(b,b,b,b),B(b,b,b,b,b), . . . are double band (that is, the generalized difference matrix), triple band, quadruple band, quinary band, . . . matrix, re-spectively. Moreover, Gm(, –) =m,G(, –) =,G(, –) =. So, our results ob-tained from the matrix domain of the mth order generalized difference matrixGm are
more general and more extensive than the results on the matrix domain of B(b,b),
B(b,b,b),B(b,b,b,b),B(b,b,b,b,b), . . . ,m,and.
By considering the definition ofmth order generalized difference matrixGm, we define
the sequence spacescλ
(Gm) andcλ(Gm) as follows:
cλ
Gm=
x= (xk)∈w:n→∞lim
λn n
k=
(λk–λk–)
m–
ϑ=
m–
ϑ
rm–ϑ–sϑx k–ϑ=
and
cλGm=
x= (xk)∈w:n→∞lim
λn n
k=
(λk–λk–)
m–
ϑ=
m–
ϑ
rm–ϑ–sϑxk–ϑexists
If we recall the notation of (.), the sequence spacescλ
(Gm) andcλ(Gm) can be redefined by the matrix domain ofGmas follows:
cλ
Gm=cλ
Gm and cλ
Gm=cλ
Gm. (.)
Also, by constructing a triangle matrixTmλ= (tmλ nk) so that
tmλ nk = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ λn m– ϑ= m– ϑ
rm–ϑ–sϑ(λ
k+ϑ–λk+ϑ–), k<n–m+ , λn m– ϑ= m– ϑ–
rm–ϑsϑ–(λ
n–m+ϑ+–λn–m+ϑ), k=n–m+ ,
λn m– ϑ= m– ϑ–
rm–ϑ+sϑ–(λ
n–m+ϑ+–λn–m+ϑ), k=n–m+ ,
.. .
rm–(λn––λn–)+(m–)rm–s(λn–λn–)
λn , k=n– ,
rm–(λn–λn–)
λn , k=n,
, k>n
for alln,k∈Nandm∈N, we rearrange the sequence spacescλ(Gm) andcλ(Gm) by means of theTmλ= (tmλ
nk) matrix as follows:
cλGm= (c)Tmλ and cλ
Gm=cTmλ. (.)
So, for a given arbitrary sequencex= (xk), theTmλ-transform ofxis denoted by
yk=
Tmλxk=
λk k
j=
(λj–λj–)
m–
ϑ=
m–
ϑ
rm–ϑ–sϑxj–ϑ (.)
for allk∈N, or, by using another representation, we can rewrite the sequencey= (yk) as
follows:
yk=
λk k–m+
j=
m–
ϑ=
m–
ϑ
rm–ϑ–sϑ(λj+ϑ–λj+ϑ–)xj+· · ·+
rm–(λk–λk–)
λk
xk (.)
for allk∈N.
Theorem . The sequence spaces cλ
(Gm)and cλ(Gm)are BK -spaces according to their
norms defined by
xcλ
(Gm)=xcλ(Gm)=T
mλ
xn∞=sup
n∈N
Tmλxn.
Proof It is known thatcandcareBK-spaces with theirsup-norm[]. Also (.) holds andTmλ= (tmλ
nk) is a triangle matrix. If we consider these three facts and Theorem ..
of Wilansky [], we conclude thatcλ
(Gm) andcλ(Gm) areBK-spaces. This step completes
the proof.
Theorem . The sequence spaces cλ
(Gm)and cλ(Gm)are linearly isomorphic to the
Proof To avoid the repetition of similar statements, we give the proof of the theorem for only the sequence spacecλ(Gm). For the proof, the existence of a linear bijection between
the spacescλ(Gm) andcshould be shown. Let us define a transformationLsuch thatL: cλ(Gm)−→c,L(x) =Tmλx. Then it is clear that for allx∈cλ(Gm),L(x) =Tmλx∈c. Also, it
is trivial thatLis a linear transformation andx= wheneverL(x) = . On account of this,
Lis injective.
Furthermore, for a given sequencey= (yk)∈c, we define the sequencex= (xk) as follows:
xk=
rm–
k
j=
m+k–j–
m–
–s
r
k–jj
i=j–
(–)j–i λi
λj–λj–
yi
for allk∈N. Then, for everyn∈N, we obtain
m–
ϑ=
m–
ϑ
rm–ϑ–sϑxk–ϑ= k
i=k–
(–)k–i λi
λk–λk–
yi.
If we consider the equality above, for alln∈N, we conclude that
Tmλxn=
λn n
k=
(λk–λk–)
m–
ϑ=
m–
ϑ
rm–ϑ–sϑxk–ϑ
=
λn n
k=
(λk–λk–)
k
i=k–
(–)k–i λi
λk–λk–
yi
=
λn n
k=
k
i=k–
(–)k–iλiyi
=yn.
So,Tmλx=yand sincey∈c, we bring to a conclusion thatTmλx∈c. Hence, we conclude
thatx∈cλ(Gm) andL(x) =y. ThusLis surjective.
Furthermore, we have for everyx∈cλ(Gm) that
L(x)∞=Tmλx∞=xcλ(Gm).
So,Lis norm preserving. As a consequence,Lis a linear bijection. This step shows that the spacescλ(Gm) andcare linearly isomorphic, namelycλ(Gm)∼=c.
Lemma .[] The inclusions c⊂cλand c⊂cλhold.
Theorem . The inclusion cλ
(Gm)⊂cλ(Gm)strictly holds.
Proof It is well known that every null sequence is also convergent. So, the inclusion
cλ
(Gm)⊂cλ(Gm) holds. Now we define a sequencex= (xk) such that
xk=
rm–
k
j=
m+j–
m–
–s
r
for allk∈N. Then we obtain by (.) that
Tmλx n=
λn n
k=
(λk–λk–) =
for alln∈N, which gives us thatTmλx=e, wheree= (, , . . .). ThenTmλx=e∈c\c
, namelyx∈cλ(Gm)\cλ
(Gm). This shows that the inclusioncλ(Gm)⊂cλ(Gm) strictly holds.
This step completes the proof.
Theorem . If r+s= ,the inclusion c⊂cλ
(Gm)is strict.
Proof It is clear thatx,x,x, . . . ,mx∈c
wheneverx∈c. Assume thatr+s= and
x∈c. ThenGmx=rm–mxand because ofmx∈c
,Gmx∈c. If we consider this fact and Lemma ., we deduce thatGmx∈cλ
. This shows thatx∈cλ(Gm). As a consequence,
c⊂cλ
(Gm) holds. Now we define a sequencey= (yk) such thatyk=lnkfor allk∈Nand k>m. Then it is obvious thatGmy∈cbuty∈/ c. Because ofc⊂cλ, we conclude that
Gmy∈cλ
and therebyy∈cλ(Gm). This shows thatc⊂cλ(Gm) strictly holds if r+s= .
This step completes the proof.
If we combine Theorem . and Theorem ., we give the following result.
Corollary . If r+s= ,the inclusions c⊂cλ(Gm)and c⊂cλ(Gm)are strict.
Now we define two sequencesx= (xk) andy= (yk) such thatxk=kk+andyk=
√
k+mfor allk∈Nandm∈N. Then we can see thatx∈∞andGmx∈c⊂cλ, namelyx∈cλ(Gm). Also, it is clear thaty∈cλ
(Gm)\∞. These two facts give us the following corollary.
Corollary . The spaces∞ and cλ
(Gm)overlap but the space∞ does not include the
space cλ
(Gm).
Now we give the following lemma which is needed in the next theorem.
Lemma .[] A∈(∞:c)⇔limn→∞
k|ank|= .
Theorem . Let a sequence z= (zk)be as follows:
zk=
λk–λk–
m–
ϑ=
m–
ϑ
rm–ϑ–sϑ(λk+ϑ–λk+ϑ–)
for all k∈N.Then the inclusion∞⊂cλ(Gm)is strict if and only if z∈cλ.
Proof We assume that the inclusion∞⊂cλ
(Gm) holds. Then it is obvious that for every
x∈∞,x∈cλ(Gm), namelyTmλ∈c. ThusTmλ∈(∞:c). If we consider the last result and Lemma ., we deduce that
lim
n→∞
k
Also, by using the definition of the matrixTmλ= (tmλ
nk), we obtain by the equality above
that
lim
n→∞
λn n–m+
k= m– ϑ=
m–
ϑ
rm–ϑ–sϑ(λk+ϑ–λk+ϑ–)
= , (.) lim n→∞ λn m– ϑ=
m–
ϑ–
rm–ϑsϑ–(λn–m+ϑ+–λn–m+ϑ)
= , (.) .. . (.) lim n→∞
rm–(λn––λn–) + (m– )rm–s(λn–λn–)
λn
= , (.+m)
lim
n→∞r
m–λn–λn–
λn
= . (.+m)
By considering equalities (.), (.), . . . , (.+m) and (.+m), we conclude that
lim
n→∞
λn–
λn
= , lim
n→∞
λn–
λn
= , . . . , lim
n→∞
λn–m+
λn
= .
For alln≥m– , we write the equality as follows:
λn n–m+
k= m– ϑ=
m–
ϑ
rm–ϑ–sϑ(λk+ϑ–λk+ϑ–)
=λn–m+
λn
λn–m+
n–m+
k=
(λk–λk–)zk
. (.+m)
By combininglimn→∞λn–m+
λn = , (.) and (.+m), we deduce that
lim
n→∞
λn–m+
n–m+
k=
(λk–λk–)zk= . (.+m)
This means thatz∈cλ
.
On the contrary, assume thatz∈cλ
. Then we have (.+m). Also, for alln≥m– , we write the inequality as follows:
≤r
m–λ
n–m++ (m– )rm–s(λn–m+–λ) +· · ·+sm–(λn–λm–)
λn = λn n–m+
k=
m–
ϑ=
m–
ϑ
rm–ϑ–sϑ(λ
k+ϑ–λk+ϑ–)
≤
λn n–m+
k= m– ϑ=
m–
ϑ
rm–ϑ–sϑ(λk+ϑ–λk+ϑ–)
=
λn n–m+
k=
(λk–λk–)zk
≤
λn–m+
n–m+
k=
By combining the last inequality and (.+m), we conclude that
lim
n→∞ rm–λ
n–m++ (m– )rm–s(λn–m+–λ) +· · ·+sm–(λn–λm–)
λn
= .
Specially, if we take (r= ,s= – andm= ), (r= ,s= – andm= ), . . . then we ob-tainlimn→∞λn–λn–
λn = ,limn→∞
λn––λn–
λn = , . . . , respectively. These equalities show that (.+m), (.+m), . . . , (.) and (.) hold, respectively. If we take into account the last result and Lemma ., we conclude thatTmλ∈(
∞:c). Thus, the inclusion∞⊂cλ(Gm)
holds and is strict by Corollary .. This step completes the proof.
3 The Schauder basis and
α
-,β
- andγ
-dualsIn the present section, we give the Schauder basis and determineα-,β- andγ-duals of the sequence spacescλ
(Gm) andcλ(Gm).
Let (X,xX) be a normed space. A set{xk:xk∈X,k∈N}is called a Schauder basis for Xif for everyx∈Xthere exist unique scalarsμk,k∈N, such thatx=
kμkxk;i.e.,
x–
n
k=
μkxk
X
−→
asn→ ∞.
Note that the Hamel basis is free from topology, whereas the Schauder basis involves convergence and hence topology (see []).
For example, lete(k)be a sequence with in thekth place and zeros elsewhere, and let
e= (, , , . . .). Then the sequence (e(k)) is a Schauder basis forc
. Moreover,{e,e,e, . . .} is a Schauder basis forc.
Due to the transformation,Ldefined in the proof of Theorem . is an isomorphism; the inverse image of (e(k)) is a Schauder basis forcλ
(Gm). Now we give the following results.
Theorem . Letσk={Tmλx}kfor all k∈N.For every fixed k∈N,we define the sequences h= (hn)and hm(kλ)(r,s) ={hmn(λk)(r,s)}n∈Nsuch that
hn=
rm–
n
k=
m+k–
m–
–s
r
k
,
hmλ n(k)(r,s) =
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
rm–(–sr)n–k[
(m+n–k– m– )λk
r(λk–λk–) +
(m+n–k– m– )λk
s(λk+–λk) ], k<n,
λk
rm–(λ
k–λk–), k=n,
, k>n.
Then
(a) The sequence{hmλ
(k)(r,s)}k∈Nis a Schauder basis for the spacec
λ
(Gm),and every
x∈cλ
(Gm)has a unique representation of the form
x=
k
(b) The sequence{h,hmλ
()(r,s),h
mλ
()(r,s), . . .}is a Schauder basis for the spacec
λ(Gm),and
everyx∈cλ(Gm)has a unique representation of the form
x=lh+
k
[σk–l]hm(kλ)(r,s),
wherel=limk→∞σk.
If we consider the results of Theorem . and Theorem ., we give the following result.
Corollary . The sequence spaces cλ
(Gm)and cλ(Gm)are separable.
Given arbitrary sequence spacesXandY, the setM(X,Y) defined by
M(X,Y) =y= (yk)∈w:xy= (xkyk)∈Yfor allx= (xk)∈X
(.)
is called the multiplier space ofXandY. For a sequence spaceZwithY⊂Z⊂X, one can easily observe thatM(X,Y)⊂M(Z,Y) andM(X,Y)⊂M(X,Z) hold, in turn.
By using the sequence spaces,csandbsand notation (.), theα-,β- andγ-duals of a sequence spaceXare defined by
Xα=M(X,
), Xβ=M(X,cs) and Xγ=M(X,bs),
respectively.
Now we give some properties which are needed in the next lemma
sup
K∈F
n
k∈K ank
p<∞, (.)
sup
n∈N
k
|ank|<∞, (.)
lim
n→∞ank=αk for eachk∈N, (.)
lim
n→∞
k
ank=α, (.)
whereFis the collection of all finite subsets ofNandp∈[,∞).
Lemma .[] Let A= (ank)be an infinite matrix,then the following hold: (i) A= (ank)∈(c:) = (c:)⇔(.)holds withp= ;
(ii) A= (ank)∈(c:c)⇔(.)and(.)hold;
(iii) A= (ank)∈(c:c)⇔(.),(.)and(.)hold; (iv) A= (ank)∈(c:∞) = (c:∞)⇔(.)holds;
(v) A= (ank)∈(c:p) = (c:p)⇔(.)holds with≤p<∞;
Theorem . Define the set umλ
(r,s)by
umλ(r,s) =
a= (an)∈w:sup K∈F
n
k∈K
dmnkλ<∞
,
where the matrix Dmλ= (dmλ
nk)is defined by means of the sequence a= (an)by
dnkmλ(r,s) =
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
rm–(–sr)n–k[
(m+n–k– m– )λk
r(λk–λk–) +
(m+n–k– m– )λk
s(λk+–λk) ]an, k<n,
λn
rm–(λn–λ
n–)an, k=n,
, k>n.
Then{cλ
(Gm)}α={cλ(Gm)}α=umλ(r,s).
Proof For givena= (an)∈w, by taking into account the sequencex= (xn) that is defined
in the proof of Theorem ., we obtain
anxn=
rm–
n
k=
m+n–k–
m–
–s
r
n–kk
i=k–
(–)k–i λi
λk–λk–
anyi=Dmnλ(y)
for alln∈N. If we consider the equality above, we conclude thatax= (anxn)∈whenever
x= (xk)∈cλ(Gm) orcλ(Gm) if and only ifDmλy∈whenevery= (yk)∈corc. This means thata= (an)∈ {cλ(Gm)}α={cλ(Gm)}αif and only ifDmλ∈(c:) = (c:). If we consider this and Lemma .(i), we write
a= (an)∈
cλGmα=cλGmα ⇔ sup
K∈F
n
k∈K
dnkmλ<∞
and conclude{cλ
(Gm)}α={cλ(Gm)}α=umλ(r,s). This step completes the proof.
Theorem . Given the sets umλ
(r,s),umλ(r,s),umλ(r,s)and umλ(r,s)as follows:
umλ(r,s) =
a= (ak)∈w: ∞
j=k
m+n–j–
m–
–s
r
n–j
ajexists for all k∈N
,
umλ(r,s) =
a= (ak)∈w:sup n∈N
n–
k=
bmkλ(n)<∞
,
umλ(r,s) =
a= (ak)∈w:sup n∈N
λn rm–(λ
n–λn–)
an
<∞
and
umλ(r,s) =
a= (ak)∈w:
k
rm–
k
j=
m+j–
m–
–s
r
j
akconverges
where
bmkλ(n) =λk
ak rm–(λ
k–λk–)
+
rm–
n
j=k+
–s
r
n–j m+n–j–
m–
r(λk–λk–) +
m+n–j–
m–
s(λk+–λk)
aj
, k<n.
Then {cλ
(Gm)}β=um
λ
(r,s)∩um
λ
(r,s)∩um
λ
(r,s)and{cλ(Gm)}β =um
λ
(r,s)∩um
λ
(r,s)∩
umλ(r,s).
Proof Givena= (ak)∈w, by considering the sequencex= (xk) that is defined in the proof
of Theorem ., we obtain
n
k=
akxk= n
k=
rm–
k
j=
m+k–j–
m–
–s
r
k–jj
i=j–
(–)j–i λi
λj–λj–
yi ak = n– k= λk ak rm–(λ
k–λk–)
+
rm–
n
j=k+
–s
r
n–j m+n–j–
m–
r(λk–λk–) +
m+n–j–
m–
s(λk+–λk)
aj
yk
+ λn
rm–(λ
n–λn–)
anyn
=
n–
k=
bmkλ(n)yk+
λn rm–(λ
n–λn–)
anyn
=Vnmλ(y)
∀n∈N, where the matrixVmλ= (vmλ
nk) is defined as follows:
vmλ nk(r,s) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
bmλ
k (n), k<n, λn
rm–(λn–λ
n–)an, k=n,
, k>n
for alln,k∈N. Thenax= (akxk)∈cswheneverx= (xk)∈cλ(Gm) if and only ifVmλy∈c whenevery= (yk)∈c. This shows thata= (ak)∈ {cλ(Gm)}βif and only ifVmλ∈(c:c). If we consider this and Lemma .(ii), we obtain
∞
j=k
m+n–j–
m–
–s
r
n–j
ajexists ∀k∈N, (.)
sup
n∈N n–
k=
and
sup
n∈N
λn rm–(λ
n–λn–)
an
<∞. (.)
These results show that{cλ
(Gm)}β=umλ(r,s)∩umλ(r,s)∩umλ(r,s).
By using a similar way, we obtaina= (ak)∈ {cλ(Gm)}βif and only ifVmλ∈(c:c). If we
consider this and Lemma .(iii), we conclude that (.), (.) and (.) hold. Moreover, one can easily see that
rm–
n
k=
k
j=
m+j–
m–
–s
r
j ak=
n–
k=
bmkλ(n) + λn
rm–(λ
n–λn–)
an=
k vmnkλ.
As a consequence, we derive from (.) that
rm–
k
j=
m+j–
m–
–s
r
j ak
∈cs.
Since condition (.) is weaker, it can be omitted. Therefore we conclude that{cλ(Gm)}β=umλ
(r,s)∩um
λ
(r,s)∩um
λ
(r,s). This step
com-pletes the proof.
Theorem . {cλ
(Gm)}γ ={cλ(Gm)}γ =umλ(r,s)∩umλ(r,s).
Proof It can be proved by combining the proof method of Theorem . and Lemma .(iv).
4 Matrix transformations
In the present section, we determine some matrix classes related to the sequence spaces
cλ
(Gm) andcλ(Gm). Let us begin with two lemmas which are needed in the proof of theo-rems.
Lemma .[] Any matrix map between BK -spaces is continuous.
Lemma .[] Let X,Y be any two sequence spaces,A be an infinite matrix and U be a triangle matrix.Then A∈(X:YU)⇔UA∈(X:Y).
For simplicity of notation, in what follows, we use the following equalities.
bmnkλ(i) =λk
ank rm–(λ
k–λk–)
+
rm–
i
j=k+
–s
r
n–j m+n–j–
m–
r(λk–λk–) +
m+n–j–
m–
s(λk+–λk)
anj
, k<i
and
bmλ nk =λk
ank rm–(λ
k–λk–)
+
rm–
∞
j=k+
–s
r
n–j m+n–j–
m–
r(λk–λk–) +
m+n–j–
m–
s(λk+–λk)
anj
for alln,k,i∈Nprovided the convergence of the series. Also, unless stated otherwise, we assume throughout Section that the sequencey= (yk) is connected with the sequence x= (xk) as follows:
xk=
rm–
k
j=
m+k–j–
m–
–s
r
k–jj
ϕ=j–
(–)j–ϕ λϕ
λj–λj–
yϕ
for allk∈N.
Theorem . Given an infinite matrix A= (ank)of complex numbers,the following state-ments hold.
() Let≤p<∞.ThenA∈(cλ(Gm) :
p)if and only if
sup
K∈F
n
k∈K bmλ
nk
p<∞, (.)
sup
i∈N i–
k=
bmλ
nk(i)<∞ (n∈N), (.)
rm–
k
j=
m+j–
m–
–s
r
j ank
∞
k=
∈cs (n∈N), (.)
lim
k→∞
λk rm–(λ
k–λk–)
ank=an (n∈N), (.)
(an)∈p. (.)
() A∈(cλ(Gm) :
∞)if and only if (.)and(.)hold,and
sup
n∈N
k
bmλ
nk<∞, (.)
(an)∈∞. (.)
Proof For a given sequencex= (xk)∈cλ(Gm), we assume that conditions (.)-(.) hold.
Then, by remembering Theorem ., we deduce that{ank}k∈N∈ {cλ(Gm)}β for alln∈N.
Therefore theA-transform ofxexists. Moreover, it is trivial thaty∈c, namely∃l∈C limk→∞|yk–l|= . Furthermore, if we consider Lemmas . and ., we conclude that the
matrixBmλ∈(c:
p), where ≤p<∞.
Now, we consider the following equality:
i
k=
ankxk= i–
k=
bmλ nk(i)yk+
λi rm–(λ
i–λi–)
aniyi (n,i∈N). (.)
Then Bmλyexists and the series
kbmnkλyk converges for alln∈N. Moreover, we derive
from (.) that the series∞j=k(–sr)n–j[(m+mn––j–)
r(λk–λk–) +
(m+mn––j–)
s(λk+–λk)]anjconverges for alln,k∈N; and thereforelimi→∞bmλ
obtain by (.) that
k
ankxk=
k
bmnkλyk+lan (.)
for alln∈N. Then we write the equality above as follows:
An(x) =Bmnλ(y) +lan (.)
for alln∈N. Also, we know (Bmλy)
n∈panda= (an)∈p. Then we haveBmλyp<∞ andanp<∞. By takingp-norm (.) side by side, we obtain that
Axp≤B
mλ y
p+|l|anp<∞.
ThereforeAx∈pand soA∈(cλ(Gm) :p).
On the contrary, assume that A∈(cλ(Gm) :
p), where ≤p<∞. This leads us to
{ank}k∈N ∈ {cλ(Gm)}β for alln∈N. Then, if we consider Theorem ., conditions (.)
and (.) hold.
We know thatcλ(Gm) and
pareBK-spaces. If we combine this fact and Lemma ., we
conclude that there is a constantM> such that
Axp≤Mxcλ(Gm) (.)
holds for allx∈cλ(Gm). Let us define a sequencez= (z
k) such thatz=
k∈Khm(kλ)(r,s) for every fixedk∈N, where the sequencehmλ
(k)(r,s) ={hmn(kλ)(r,s)}n∈NandK∈F. We know from Theorem . thatz∈cλ(Gm) andTmλ(hmλ
(k)(r,s)) =e(k),∀k∈N. Then we obtain
zcλ(Gm)=Tmλ(z)
∞=
k∈K
Tmλh(mkλ)(r,s)
∞
=
k∈K e(k)
∞ =
and
An(z) =
k∈K An
hm(kλ)(r,s)=
k∈K
j
anjhmj(kλ)(r,s) =
k∈K bmnkλ
for alln∈N. Since inequality (.) holds for everyx∈cλ(Gm), the inequality is satisfied
also forz∈cλ(Gm). Then we have
n
k∈K bmnkλ
pp ≤M
for allK∈F. Therefore (.) holds. If we consider this and Lemma .(v), we conclude that Bmλ= (bmλ
HenceAxandBmλyexist. So one can easily see that the series
kankxkand
kbmnkλykare
convergent for alln∈N. Thus, we conclude that
lim
i→∞ i–
k=
bmnkλ(i)yk=
k
bmnkλyk (.)
or alln∈N. As a consequence, if we pass to the limit in (.) asi→ ∞, we obtain
lim
i→∞
λi rm–(λ
i–λi–)
aniyiexists
for alln∈N. Because ofy= (yk)∈c\c, this leads us
lim
i→∞
λi rm–(λ
i–λi–)
aniexists
for alln∈N. Therefore, (.) holds. If we takel=limk→∞yk, also relation (.) holds.
Because ofAx,Bmλy∈
p, we concludea= (an)∈p. The last result is the necessity of
(.). This step completes the proof of part ().
If we take Lemma .(iv) instead of Lemma .(v), the second part of theorem can be
proved similarly.
Moreover, from (.) we derive
lim
i→∞ i–
k=
bmnkλ(i)=
k
bmnkλ≤sup
n∈N
k
bmnkλ. (.)
If we combine (.) and (.), we conclude that
lim
i→∞ i–
k=
bmnkλ(i)
exists for eachn∈N. So, condition (.) implies condition (.).
Theorem . Given an infinite matrix A= (ank)of complex numbers,the following state-ments hold.
() Let≤p<∞.ThenA∈(cλ
(Gm) :p)if and only if (.)and(.)hold,and ∞
j=k
–s
r
n–j m+n–j–
m–
r(λk–λk–) +
m+n–j–
m–
s(λk+–λk)
anjexists for alln,k∈N, (.)
λk rm–(λ
k–λk–)
ank
∞
k=
∈∞ for alln∈N. (.)
() A∈(cλ
(Gm) :∞)if and only if (.),(.)and(.)hold.
Proof From Lemma .(iv) and (v), we know (c:p) = (c:p) and (c:∞) = (c:∞).
Theorem . A∈(cλ(Gm) :c)if and only if (.), (.)and(.)hold,and the conditions
lim
n→∞an=a, (.)
lim
n→∞b mλ
nk =αk, ∀k∈N, (.)
lim
n→∞
k bmλ
nk =α (.)
hold.
Proof Given arbitraryx∈cλ(Gm), we assume that conditions (.), (.), (.), (.),
(.) and (.) hold for an infinite matrixA= (ank). We consider Theorem ., and
con-dition (.) implies concon-dition (.). Then we conclude that{ank}k∈N∈ {cλ(Gm)}β for all n∈N, and soAxexists. From (.) and (.) we have
k
j=
|αj|=n→∞lim k
j=
bmnjλ≤sup
n∈N
j
bmnjλ<∞
for allk∈N. This leads us to (αk)∈, and therefore the series
kαk(yk–l) converges,
where limk→∞yk=l and soy∈c. If we combine Lemma .(iii) with conditions (.),
(.) and (.), we deduce thatBmλ= (bmλ
nk)∈(c:c). Also from condition (.) we have
An(x) =
k
ankxk=
k
bmnkλyk+lan.
With a basic calculation, we obtain
k
ankxk=
k
bmnkλ(yk–l) +l
k
bmnkλ+lan (.)
for alln∈N. If we pass to the limit in (.), we write
lim
n→∞An(x) =
k
αk(yk–l) +l(α+a).
This shows thatAx∈cand soA∈(cλ(Gm) :c).
On the contrary, we assume that A∈(cλ(Gm) :c). Since every convergent sequence
is also bounded, we deduce that A∈(cλ(Gm) :
∞). If we consider this fact and
Theo-rem ., we conclude that conditions (.), (.) and (.) hold. Let us take the sequences
hmλ
(k)(r,s) ={hm
λ
n(k)(r,s)}n∈N∈c
λ(Gm) andz= khm
λ
(k)(r,s) defined in Theorem . and the proof of Theorem ., respectively. Then it is clear thatAhmλ
(k)(r,s) ={bm
λ
nk}n∈N∈cfor
ev-eryk∈N. Hence condition (.) holds. Moreover, from Theorem . we know that the
transformationL:cλ(Gm)−→c,L(x) =Tmλxis continuous. So, we write
Tnmλ(z) =
k
Tnmλh(mkλ)(r,s)=
k
for alln∈N, where
δnk=
⎧ ⎨ ⎩
, k=n,
, k=n.
This leads us toTmλz=e∈cand soz∈cλ(Gm).
It is well known thatcis aBK-space. If we combine Theorem . and Lemma ., we conclude that the matrix transformationA:cλ(Gm)−→cis continuous. Therefore the
equality
An(z) =
k An
hm(kλ)(r,s)=
k bmnkλ
holds for alln∈N. This shows that (.) holds.
By considering conditions (.), (.), (.) and Lemma .(iii), we deduce thatBmλ=
(bmλ
nk)∈(c:c). Hence, (.), (.) and the last result give us that condition (.) holds for
allx∈cλ(Gm) andy∈c. Finally, if we considerAx,Bmλy∈cand (.), we conclude that
condition (.) holds. This step completes the proof.
Theorem . A∈(cλ(Gm) :c
)if and only if(.), (.), (.)and the following conditions
hold:
lim
n→∞an= , (.)
lim
n→∞b mλ
nk = , ∀k∈N, (.)
lim
n→∞
k bmλ
nk = . (.)
Proof In Theorem ., if we take Lemma .(vi) instead of Lemma .(iii), the present
theorem can be proved by using a similar way.
Theorem . A∈(cλ
(Gm) :c)if and only if (.), (.), (.)and(.)hold.
Proof If we combine Lemma .(ii) Theorem . and Theorem .(), the present theorem
can be proved by using a similar way.
Theorem . A∈(cλ
(Gm) :c)if and only if (.), (.), (.), (.), (.)and(.)
hold.
Proof If we combine Lemma .(vii), Theorem . and Theorem ., the present theorem
can be proved by using a similar way.
Now, by using Lemma ., we give one more result.
Corollary . Given an infinite matrix A= (ank)of complex numbers,we define a matrix E= (enk)as follows:
enk=
λn n
j=
(λj–λj–)
m–
ϑ=
m–
ϑ
for all n,k∈N.Then A belongs to matrix classes(c:cλ(Gm)), (c:cλ(Gm)), (p:cλ(Gm)), (c:cλ(Gm)), (c:cλ(Gm))and(p:cλ(Gm))if and only if E belongs to matrix classes(c:c), (c:c), (p:c), (c:c), (c:c)and(p:c).
Finally, we put a period to our work by mentioning as of now that the sequence space
f(Gm) of almost convergent sequences derived by the domain ofmth order generalized
difference matrix will be defined and studied analogously in the next paper.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors read and approved the final manuscript.
Acknowledgements
We would like to express our thanks to the anonymous reviewers for their valuable comments.
Received: 10 January 2014 Accepted: 4 July 2014 Published:23 Jul 2014 References
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