R E S E A R C H
Open Access
On some new matrix transformations
Rahmet Sava¸s
1*and Ekrem Sava¸s
2*Correspondence:
[email protected] 1Department of Mathematics, Medeniyet University, Istanbul, Turkey
Full list of author information is available at the end of the article
Abstract
In this paper, we characterize some matrix classes (ω(p,s),Vσλ), (ωp(s),Vσλ) and (ωp(s),Vσλ)regunder appropriate conditions.
1 Introduction
Letwdenote the set of all real and complex sequencesx= (xk). Byl∞andc, we denote the Banach spaces of bounded and convergent sequencesx= (xk) normed byx=supk|xk|, respectively. A linear functionalLonl∞is said to be a Banach limit [] if it has the following properties:
() L(x)≥ifn≥(i.e.,xn≥for alln),
() L(e) = , wheree= (, , . . .),
() L(Dx) =L(x), where the shift operatorDis defined byD(xn) ={xn+}.
LetBbe the set of all Banach limits onl∞. A sequencex∈∞is said to be almost con-vergent if all Banach limits ofxcoincide. Letcˆdenote the space of the almost convergent sequences. Lorentz [] has shown that
ˆ
c=
x∈l∞:lim
m dm,n(x) exists, uniformly inn
,
where
dm,n(x) =
xn+xn++xn++· · ·+xn+m
m+ .
The study of regular, conservative, coercive and multiplicative matrices is important in the theory of summability. In [], King used the concept of the almost convergence of a sequence introduced by Lorentz to define more general classes of matrices than those of regular and conservative ones.
In [], Schaefer defined the concepts ofσ-conservative,σ-regular andσ-coercive ma-trices and characterized the matrix classes (c,Vσ), (c,Vσ)reg and (l∞,Vσ), whereVσ
de-notes the set of all bound sequences, all of whose invariant means (orσ-means) are equal. In [], Mursaleen characterized the classes (c(p),Vσ), (c(p),Vσ)reg and (l∞(p),Vσ) of
ma-trices, which generalized the results due to Schaefer []. In [], Mohiuddine and Aiyup defined the spaceω(p,s) and obtained necessary and sufficient conditions to characterize the matrices of classes (ω(p,s),Vσ), (ωp(s),Vσ) and (ωp(s),Vσ)reg.
Matrix transformations between sequence spaces have also been discussed by Savaş and Mursaleen [], Basarir and Savaş [], Mursaleen [, –], Vatan and Simsek [], Savaş [–], Vatan [] and many others.
In this paper we characterize the matrix classes from this space to the spaceVλ σ,i.e.,
we obtain necessary and sufficient conditions to characterize the matrices of classes (ω(p,s),Vλ
σ), (ωp(s),Vσλ) and (ωp(s),Vσλ)reg.
2 Preliminaries
Letσbe a one-to-one mapping from the setNof natural numbers into itself. A continuous linear functionalϕonl∞is said to be an invariant mean orσ-mean if and only if
(i) ϕ(x)≥when the sequencex= (xk)hasxk≥for allk;
(ii) ϕ(e) = ;
(iii) ϕ(x) =ϕ(xσ(k))for allx∈l∞.
LetVσ denote the set of bounded sequences all of whoseσ-means are equal. We say
that a sequencex= (xk) isσ-convergent if and only ifx∈Vσ. Forσ(n) =n+ , the setVσ
is reduced to the setcˆof almost convergent sequences [, ]. Ifx= (xn), writeTx= (xσ(n)). It is easy to show that
Vσ =
x∈l∞:lim
m tmn(x) =L, uniformly inn;L=σ-limx
,
where
tmn(x) =
m+ m
j=
Tjxn
andσm(n) denotes themth iterate ofσ atn.
Ifpkis real and positive, we define (see Maddox [])
c(p) =
x: lim
k→∞|xk| pk=
and
c(p) =x: lim
k→∞|xk–l|
pk= for somel
.
The classes (ω(p,s),Vσ), (ωp(s),Vσ) and (ωp(s),Vσ)reg have been defined by Mohiuddine and Aiyup [] and, forp= (pk) withpk> , the spaceω(p,s) is defined fors≥ by
ω(p,s) =
x:
n
n
k=
k–s|xk–l|pk→,n→ ∞for somel,s≥
,
wheres= (sk) is an arbitrary sequence withsk= ( k= , , . . .). Ifpk=p, for eachk, we haveω(p,s) =ωp(s).
The sequence space
ω(p) =
x:
n
n
k=
|xk–l|pk→,n→ ∞
We further define the following.
Letλ= (λm) be a non-decreasing sequence of positive numbers tending to∞such that λm+≤λm+ , λ= .
A sequencex= (xk) of real numbers is said to be (σ,λ)-convergent to a numberLif and only ifx∈Vλ
σ, where
Vλ σ =
x∈l∞:m→∞lim tmn(x) =L, uniformly inn;L= (σ,λ)-limx
,
tmn(x) = λm
i∈Im
xσi(n),
andIm= [m–λm+ ,m]. Note thatc⊂Vσλ⊂l∞. Forσ(n) =n+ ,Vσλreduces to the space
ˆ
Vλof almostλ-convergent sequences []; and if we takeσ(n) =n+ andλm=m, then
Vλ
σ reduces toˆc(see []). Further, if we takeλm=m, thenVσλreduces toVσ.
IfEis a subset ofω, then we writeE+for a generalized Köthe-Toeplitz dual ofE;i.e.,
E+=
a: k
akxkconverges for everyx∈E
.
If <pk≤, thenω+(p) =M, where
M=
a: ∞
r=
max
r
r·N–pk |a k|
<∞for some integerN>
,
and max is the maximum taken over r≤k< r+(see Theorem , []).
IfXis a topological linear space, we denote byX∗the continuous dual ofX;i.e., the set of all continuous linear functionals onX. Obviously,
ω(p,s)∗=
a: ∞
r=
max
r
r·N–
pkak
sk
<∞for some integerN>
.
3 Main results
LetXandYbe two nonempty subsets of the spacewof complex sequences. LetA= (ank) (n,k= , , . . .) be an infinite matrix of complex numbers. We writeAx= (An(x)) ifAn(x) :=
kankxkconverges for eachn. (Throughout,
kwill denote summation overkfromk= tok=∞.) Ifx= (xk)∈Ximplies thatAx= (An(x))∈Y, we say thatAdefines a (matrix) transformation fromXtoY and we denote it byA:X→Y. By (X,Y) we mean the class of matricesAsuch thatA:X→Y.
We now characterize the matrices in the class (c(p),Vσλ(p)). We write
tm,n(Ax) =
k
a(n,k,m)xk,
where
a(n,k,m) = λm
i∈Im
Theorem . Let <pk≤,then A∈(ω(p,s),Vσλ)if and only if (i) there exists an integerB> such that for everyn
Dn=sup m
∞
r=
max
r
r·B–
pka(n,k,m)
sk <∞;
(ii) α(k)={ank}∞n=∈Vσλfor eachk; (iii) α={kank}∞n=∈Vσλ.
In this case theσ-limof Ax is(limx)[u–kuk] +kukxkfor every x∈w(p,s),where
u=σ-lima and uk=σ-lima(k),k= , , . . . .
Proof Suppose thatA∈(ω(p,s),Vλ
σ). Defineek= (, , . . . , , , . . .) having in thekth
coor-dinate sequence. Sinceeandekare inω(p,s), necessity of (ii) and (iii) is clear. We know that
ka(n,k,m)xk converges for eachm,nandx∈ω(p,s). Therefore (a(n,k,m))k∈ω+(p,s) and
∞
r=
max
r
r·B–
pka(n,k,m)
sk <∞
for eachm,n(see []). Furthermore, if fmn(x) =tmn(Ax), then{fmn}m is a sequence of continuous linear functionals onω(p,s) such thatlimmtmn(Ax) exists. Therefore, by using the Banach-Steinhaus theorem, the necessity of (i) follows immediately.
Conversely, suppose that the conditions (i), (ii) and (iii) hold andx∈ω(p,s). We know that (a(n,k,m))k anduk are inω+(p,s) and that the series
ka(n,k,m)xk and
kukxk converge for eachm,n. Write
c(n,k,m) =a(n,k,m) –uk.
Then
k
a(n,k,m)xk=
k
ukxk+l
k
c(n,k,m) + k
c(n,k,m)(xk–l)
by (ii) for some integerk> , we have
lim
m
k≤k
c(n,k,m)(xk–l) = , uniformly inn,
wherelis the limit ofxforx∈ω(p,s). Since
sup
m,n
r
max
r
r·B–
pkc(n,k,m)≤Dn,
lim
m
k≤k
a(n,k,m) –uk
sk
sk(xk–l)= ,
uniformly inn, whence
lim
n
k
a(n,k,m)xk–l·u+
k
Theorem . Let≤pk<∞,then A∈(ωp(s),Vσλ)if and only if (i) for everyn
M(A) =sup
m
r
rp
r
a(n,skk,m)q
q
<∞,
wherep–+q–= ;
(ii) α(k)∈Vσλfor eachk; (iii) α∈Vλ
σ.
Proof Assume that the conditions are satisfied and letx∈ωp(s). Then tmn(Ax)≤
∞
r=
r
a(n,k,m)skxk
sk
≤
∞
r=
r
a(n,skk,m)q
q
·
r
|xk|p
p
,
and hencetmn(Ax) is absolutely and uniformly convergent for eachm,n. Note that (i) and (ii) imply that
∞
r=
pr
r
|skuk|
q
≤M(A) <∞,
so that by Hölder’s inequality,kukxk<∞. Now as in the converse part of Theorem ., it follows thatA∈(ωp(s),Vσλ).
Conversely, suppose thatA∈(ωp(s),Vσλ). Sinceekandeare inωp(s), the necessity of (ii) and (iii) is clear. For the necessity of (i), suppose that
tmn(Ax) =
k
a(n,k,m)xk
exits for eachnwheneverx∈ωp(s). Then, for eachnandr≥, write
fnr(x) =
r
a(n,k,m)xk.
Then{fnr}mis a sequence of continuous linear functionals onωp(s). Since fnr(x)≤
r
a(n,skk,m)q
q
·
r
|sk·xk|p
p
≤pr r
a(n,skk,m)q
q
· x,
it follows that for eachn,
lim
j j
r=
fmr(x) =tmn(Ax)
is in the dual spaceωp∗. Hence there exists aKmnsuch that
For eachnand any integerj> , definex∈ωp(s) as in [] (Theorem , p.), we get j
r=
pr
r
a(n,skk,m)q
q
≤Kmn.
Hence, for eachn,
∞
r=
pr
r
a(n,skk,m)q
q
≤Kmn<∞. (.)
Sincetmn(Ax) is absolutely convergent, we get
tmn(Ax)≤ ∞
r=
pr
r
a(n,skk,m)q
q
x,
so that
Kmn≤ ∞
r=
rp
r
a(n,skk,m)q
q
. (.)
By virtue of (.) and (.),
Kmn= ∞
r=
pr
r
a(n,skk,m)q
q
.
Finally, by (Theorem , [], p.) for everyn, the existence oflimmtmn(Ax) onωp(s) implies that
sup
m
Kmn=sup m
∞
r=
pr
r
a(n,skk,m)q
q
<∞,
which is (i).
Theorem . Let <pk<∞,then A∈(ωp(s),Vσ)regif and only if conditions(i), (ii)with σ-lim= and(iii)withσ-lim= +of Theorem.hold.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors completed the paper together. Both authors read and approved the final manuscript.
Author details
1Department of Mathematics, Medeniyet University, Istanbul, Turkey.2Department of Mathematics, Istanbul Ticaret University, Uskudar, Istanbul, Turkey.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
We would like to thank the referees for their valuable comments.
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doi:10.1186/1029-242X-2013-254
Cite this article as:Sava¸s and Sava¸s:On some new matrix transformations.Journal of Inequalities and Applications2013