Rahmet Savaş

Ekrem Savaş

Journal of Inequalities and Applications

Springer - Publisher Connector

Savas¸ and Savas¸ Journal of Inequalities and Applications 2013, 2013:254
http://www.journaloﬁnequalitiesandapplications.com/content/2013/1/254
RESEARCH Open Access
On some newmatrix transformations
Rahmet Savas¸1* and Ekrem Savas¸2
*Correspondence:
rahmet.savas@medeniyet.edu.tr
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),Vλσ ), (ωp(s),V
λ
σ ) and
(ωp(s),Vλσ )reg under appropriate conditions.
1 Introduction
Let w denote the set of all real and complex sequences x = (xk). By l∞ and c, we denote the
Banach spaces of bounded and convergent sequences x = (xk) normed by ‖x‖ = supk|xk|,
respectively. A linear functional L on l∞ is said to be a Banach limit [] if it has the following
properties:
() L(x)≥  if n≥  (i.e., xn ≥  for all n),
() L(e) = , where e = (, , . . .),
() L(Dx) = L(x), where the shift operator D is deﬁned by D(xn) = {xn+}.
Let B be the set of all Banach limits on l∞. A sequence x ∈ ∞ is said to be almost con-
vergent if all Banach limits of x coincide. Let cˆ denote the space of the almost convergent
sequences. Lorentz [] has shown that
cˆ =
{
x ∈ l∞ : limm dm,n(x) exists, uniformly in n
}
,
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 of σ -conservative, σ -regular and σ -coercive ma-
trices and characterized the matrix classes (c,Vσ ), (c,Vσ )reg and (l∞,Vσ ), where Vσ de-
notes the set of all bound sequences, all of whose invariant means (or σ -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 space ω(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.
© 2013 Savas¸ and Savas¸; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Savas¸ and Savas¸ Journal of Inequalities and Applications 2013, 2013:254 Page 2 of 7
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In this paper we characterize the matrix classes from this space to the space V λσ , i.e.,
we obtain necessary and suﬃcient conditions to characterize the matrices of classes
(ω(p, s),V λσ ), (ωp(s),V λσ ) and (ωp(s),V λσ )reg.
2 Preliminaries
Let σ be a one-to-onemapping from the setN of natural numbers into itself. A continuous
linear functional ϕ on l∞ is said to be an invariant mean or σ -mean if and only if
(i) ϕ(x)≥  when the sequence x = (xk) has xk ≥  for all k;
(ii) ϕ(e) = ;
(iii) ϕ(x) = ϕ(xσ (k)) for all x ∈ l∞.
Let Vσ denote the set of bounded sequences all of whose σ -means are equal. We say
that a sequence x = (xk) is σ -convergent if and only if x ∈ Vσ . For σ (n) = n + , the set Vσ
is reduced to the set cˆ of almost convergent sequences [, ].
If x = (xn), write Tx = (xσ (n)). It is easy to show that
Vσ =
{
x ∈ l∞ : limm tmn(x) = L, uniformly in n;L = σ -limx
}
,
where
tmn(x) =

m + 
m∑
j=
Tjxn
and σm(n) denotes themth iterate of σ at n.
If pk is real and positive, we deﬁne (see Maddox [])
c(p) =
{
x : lim
k→∞
|xk|pk = 
}
and
c(p) =
{
x : lim
k→∞
|xk – l|pk =  for some l
}
.
The classes (ω(p, s),Vσ ), (ωp(s),Vσ ) and (ωp(s),Vσ )reg have been deﬁned by Mohiuddine
and Aiyup [] and, for p = (pk) with pk > , the space ω(p, s) is deﬁned for s≥  by
ω(p, s) =
{
x : n
n∑
k=
k–s|xk – l|pk → ,n→ ∞ for some l, s≥ 
}
,
where s = (sk) is an arbitrary sequence with sk =  (k = , , . . .). If pk = p, for each k, we
have ω(p, s) = ωp(s).
The sequence space
ω(p) =
{
x : n
n∑
k=
|xk – l|pk → ,n→ ∞
}
for some l, which has been investigated by Maddox is the special case of ω(p, s) which
corresponds to s = . Obviously ω(p)⊂ ω(p, s).
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We further deﬁne the following.
Let λ = (λm) be a non-decreasing sequence of positive numbers tending to ∞ such that
λm+ ≤ λm + , λ = .
A sequence x = (xk) of real numbers is said to be (σ ,λ)-convergent to a number L if and
only if x ∈ V λσ , where
V λσ =
{
x ∈ l∞ : limm→∞ tmn(x) = L, uniformly in n;L = (σ ,λ)-limx
}
,
tmn(x) =

λm
∑
i∈Im
xσ i(n),
and Im = [m – λm + ,m]. Note that c⊂ V λσ ⊂ l∞. For σ (n) = n + ,V λσ reduces to the space
Vˆλ of almost λ-convergent sequences []; and if we take σ (n) = n +  and λm =m, then
V λσ reduces to cˆ (see []). Further, if we take λm =m, then V λσ reduces to Vσ .
If E is a subset of ω, then we write E+ for a generalized Köthe-Toeplitz dual of E; i.e.,
E+ =
{
a :
∑
k
akxk converges for every x ∈ E
}
.
If  < pk ≤ , then ω+(p) =M, where
M =
{
a :
∞∑
r=
max
r
{(
r ·N–) pk |ak|} <∞ for some integer N > 
}
,
and max is the maximum taken over r ≤ k < r+ (see Theorem , []).
If X is a topological linear space, we denote by X∗ the continuous dual of X; i.e., the set
of all continuous linear functionals on X. Obviously,
[
ω(p, s)
]∗ =
{
a :
∞∑
r=
max
r
{(
r ·N–) pk ∣∣∣∣aksk
∣∣∣∣
}
<∞ for some integer N > 
}
.
3 Main results
Let X and Y be two nonempty subsets of the space w of complex sequences. Let A = (ank)
(n,k = , , . . .) be an inﬁnite matrix of complex numbers. We write Ax = (An(x)) if An(x) :=∑
k ankxk converges for each n. (Throughout,
∑
k will denote summation over k from k = 
to k = ∞.) If x = (xk) ∈ X implies that Ax = (An(x)) ∈ Y , we say that A deﬁnes a (matrix)
transformation from X to Y and we denote it by A : X → Y . By (X,Y ) we mean the class
of matrices A such that A : X → Y .
We now characterize the matrices in the class (c(p),V λσ (p)). We write
tm,n(Ax) =
∑
k
a(n,k,m)xk ,
where
a(n,k,m) = 
λm
∑
i∈Im
aσ i(n),k .
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Theorem . Let  < pk ≤ , then A ∈ (ω(p, s),V λσ ) if and only if
(i) there exists an integer B >  such that for every n
Dn = sup
m
∞∑
r=
max
r
(
r · B–) pk ∣∣∣∣a(n,k,m)sk
∣∣∣∣ <∞;
(ii) α(k) = {ank}∞n= ∈ V λσ for each k;
(iii) α = {∑k ank}∞n= ∈ V λσ .
In this case the σ -lim of Ax is (limx)[u –
∑
k uk] +
∑
k ukxk for every x ∈ w(p, s), where
u = σ -lima and uk = σ -lima(k), k = , , . . . .
Proof Suppose thatA ∈ (ω(p, s),V λσ ). Deﬁne ek = (, , . . . , , , . . .) having  in the kth coor-
dinate sequence. Since e and ek are inω(p, s), necessity of (ii) and (iii) is clear.We know that∑
k a(n,k,m)xk converges for each m, n and x ∈ ω(p, s). Therefore (a(n,k,m))k ∈ ω+(p, s)
and
∞∑
r=
max
r
(
r · B–) pk ∣∣∣∣a(n,k,m)sk
∣∣∣∣ <∞
for each m, n (see []). Furthermore, if fmn(x) = tmn(Ax), then {fmn}m is a sequence of
continuous linear functionals on ω(p, s) such that limm tmn(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 and x ∈ ω(p, s). We know
that (a(n,k,m))k and uk are in ω+(p, s) and that the series
∑
k a(n,k,m)xk and
∑
k ukxk
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 integer k > , we have
lim
m
∑
k≤k
c(n,k,m)(xk – l) = , uniformly in n,
where l is the limit of x for x ∈ ω(p, s). Since
sup
m,n
∑
r
max
r
(
r · B–) pk ∣∣c(n,k,m)∣∣≤ Dn,
lim
m
∑
k≤k
∣∣∣∣a(n,k,m) – uksk
∣∣∣∣∣∣sk(xk – l)∣∣ = ,
uniformly in n, whence
lim
n
∑
k
a(n,k,m)xk – l · u +
∑
k
uk(xk – l). 
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Theorem . Let ≤ pk <∞, then A ∈ (ωp(s),V λσ ) if and only if
(i) for every n
M(A) = sup
m
∑
r

r
p
(∑
r
∣∣∣∣a(n,k,m)sk
∣∣∣∣
q) q
<∞,
where p– + q– = ;
(ii) α(k) ∈ V λσ for each k;
(iii) α ∈ V λσ .
Proof Assume that the conditions are satisﬁed and let x ∈ ωp(s). Then
∣∣tmn(Ax)∣∣≤ ∞∑
r=
∑
r
∣∣∣∣a(n,k,m)skxksk
∣∣∣∣≤
∞∑
r=
(∑
r
∣∣∣∣a(n,k,m)sk
∣∣∣∣
q) q
·
(∑
r
|xk|p
) 
p
,
and hence tmn(Ax) is absolutely and uniformly convergent for eachm, n. Note that (i) and
(ii) imply that
∞∑
r=

r
p
(∑
r
|skuk|
) 
q
≤M(A) <∞,
so that by Hölder’s inequality,
∑
k ukxk <∞. Now as in the converse part of Theorem .,
it follows that A ∈ (ωp(s),V λσ ).
Conversely, suppose that A ∈ (ωp(s),V λσ ). Since ek and e are in ωp(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 each n whenever x ∈ ωp(s). Then, for each n and r ≥ , write
fnr(x) =
∑
r
a(n,k,m)xk .
Then {fnr}m is a sequence of continuous linear functionals on ωp(s). Since
∣∣fnr(x)∣∣≤
(∑
r
∣∣∣∣a(n,k,m)sk
∣∣∣∣
q) q
·
(∑
r
|sk · xk|p
) 
p
≤  rp
(∑
r
∣∣∣∣a(n,k,m)sk
∣∣∣∣
q) q
· ‖x‖,
it follows that for each n,
lim
j
j∑
r=
fmr(x) = tmn(Ax)
is in the dual space ω∗p . Hence there exists a Kmn such that
∣∣∣∣a(n,k,m)sk
∣∣∣∣≤ Kmn‖x‖. (.)
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For each n and any integer j > , deﬁne x ∈ ωp(s) as in [] (Theorem , p.), we get
j∑
r=

r
p
(∑
r
∣∣∣∣a(n,k,m)sk
∣∣∣∣
q) q
≤ Kmn.
Hence, for each n,
∞∑
r=

r
p
(∑
r
∣∣∣∣a(n,k,m)sk
∣∣∣∣
q) q
≤ Kmn <∞. (.)
Since tmn(Ax) is absolutely convergent, we get
∣∣tmn(Ax)∣∣≤ ∞∑
r=

r
p
(∑
r
∣∣∣∣a(n,k,m)sk
∣∣∣∣
q) q
‖x‖,
so that
Kmn ≤
∞∑
r=

r
p
(∑
r
∣∣∣∣a(n,k,m)sk
∣∣∣∣
q) q
. (.)
By virtue of (.) and (.),
Kmn =
∞∑
r=

r
p
(∑
r
∣∣∣∣a(n,k,m)sk
∣∣∣∣
q) q
.
Finally, by (Theorem , [], p.) for every n, the existence of limm tmn(Ax) on ωp(s)
implies that
sup
m
Kmn = sup
m
∞∑
r=

r
p
(∑
r
∣∣∣∣a(n,k,m)sk
∣∣∣∣
q) q
<∞,
which is (i). 
Theorem . Let  < pk < ∞, then A ∈ (ωp(s),Vσ )reg if and only if conditions (i), (ii) with
σ -lim =  and (iii) with σ -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.
Received: 21 November 2012 Accepted: 18 April 2013 Published: 20 May 2013
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doi:10.1186/1029-242X-2013-254
Cite this article as: Savas¸ and Savas¸: On some newmatrix transformations. Journal of Inequalities and Applications 2013
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