Gen. Math. Notes, Vol. 13, No. 1, November 2012, pp.21-31
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2012
www.i-csrs.org
Available free online at http://www.geman.in
On Regular Cesàro Double Sequence Spaces
Yurdal Sever
Malatya Fen Lisesi, Malatya \ Türkiye
E-mail: yurdalsever@hotmail.com
(Received: 20-10-12 / Accepted: 14-11-12)
Abstract
In this study, we define the double sequence space Cesr and examine some
properties of this sequence space. Furthermore, we determine the β(r)-dual of
the space Cesr .
Keywords: Double sequence space, seminormed sequence space, β-dual.
1
Introduction
By Ω, we denote the set of all real or complex valued double sequences which
is a vector space with coordinatewise addition and scalar multiplication. Any
vector subspace of Ω is called as a double sequence space. The space Mu of
all bounded double sequences is defined by
Mu = x = (xmn ) ∈ Ω : kxk∞ = sup |xmn | < ∞
m,n∈N
which is a Banach space with the norm k·k∞ ; where N = {1, 2, 3, . . .}. Consider
the sequence x = (xmn ) ∈ Ω. If for every ε > 0 there exists n0 = n0 (ε) ∈ N
and ` ∈ C such that
|xmn − `| < ε
for all m, n > n0 then we call that the double sequence x is convergent in
the Pringsheim’s sense to the limit ` and write P − lim xmn = `; where C
denotes the complex field. By Cp , we denote the space of all convergent double sequences in the Pringsheim’s sense. It is well-known that there are such
sequences in the space Cp but not in the space Mu . So, we may mention the
22
Yurdal Sever
space Cbp of the double sequences which are both convergent in the Pringsheim’s sense and bounded, i.e., Cbp = Cp ∩ Mu . By Cbp0 , we denote the space
of the double sequences which are both convergent to zero in the Pringsheim’s
sense and bounded. A sequence in the space Cp is said to be regularly convergent if it is a single convergent sequence with respect to each index and denote
the set of all such sequences by Cr .
Let λ be the space of double sequences, converging with respect to
Psome
linear convergence rule υ − lim : λ → C. P
The sum of a doubleP
seriesP i,j xij
n
with respect to this rule is defined by υ − ij xij = υ − limm,n m
i=1
j=1 xij .
Let λ, µ be two spaces of double sequences, converging with respect to the
linear convergence rules υ1 − lim and υ2 − lim, respectively, and A = (amnkl )
also be a four dimensional matrix of real or complex numbers. Define the set


!


X
(υ2 )
amnkl xkl
exists and Ax ∈ λ . (1)
λA = (xkl ) ∈ Ω : Ax = υ2 −


k,l
m,n∈N
Then, we say, with the notation of (1), that A maps the space λ into the space
(υ )
µ if µ ⊂ λA 2 and denote the set of all four dimensional matrices, mapping
the space λ into the space µ, by (λ : µ). It is trivial that for any matrix
A ∈ (λ : µ), (amnkl )k,l∈N is in the β(υ2 )-dual λβ(υ2 ) of the space λ for all
m, n ∈ N. An infinite matrix A is said to be Cυ -conservative if Cυ ⊂ (Cv )A . The
characterizations of some four dimensional matrix transformations between
double sequence spaces have been given by Robison [16], Hamilton [8] and
Zeltser [22].
Lemma 1.1 ([16, 8, 22]) A = (amnkl ) ∈ (Cr , Cr ) if and only if
X
|amnkl | < ∞,
sup
m,n
k,l
r- lim amnkl = akl exists (k, l ∈ N),
m,n
r- lim
m,n
r- lim
m,n
X
(2)
X
amnkl = v exists,
(3)
(4)
k,l
amnkl0 = ul0 and r- lim
m,n
k
X
amnk0 l = vk0 (k0 , l0 ∈ N).
(5)
l
The arithmetic (or Cesáro ) mean smn of a double sequence x = (xmn ) is
defined by
m
smn =
n
1 XX
xjk ,
mn j=1 k=1
(m, n ∈ N).
On Regular Cesàro Double Sequence Spaces
23
We say that x = (xmn ) is regularly (C, 1, 1) summable or regularly Cesáro
summable to some number ` if
r − lim smn = `,
where (C, 1, 1) = (cmnkl ) is a four dimensional matrix defined by
1
, (1 ≤ k ≤ m and 1 ≤ l ≤ n)
mn
cmnkl =
0 , (otherwise)
(6)
(The letter ” C ” comes from the name ” Cesáro ”.)
We shall write throughout for simplicity in notation for all m, n, k, l ∈ N
that
410 amn = amn − am+1,n ,
401 amn = amn − am,n+1 ,
411 amn = 401 (410 amn ) = 410 (401 amn ).
Now, we may summarize the knowledge given in some document on the
double sequence spaces. Móricz [10] proved that the double sequence space
Cp is complete under the pseudonorm kxkP = limN →∞ supk,l>N |xkl | and the
sets Cbp and Cbp0 are Banach spaces under the norm k · k∞ . Gökhan and Çolak
[5, 6, 7] extended these space to the paranormed double sequence spaces, determined their duals and gave some inclusion relations. The summability of
double sequences defining by the product of two complex single sequences, Jardas and Sarapa [9] proved the Silverman-Toeplitz and Steinhaus type theorems
for three dimensional matrices. Boos, Leiger and Zeller [3] defined the concept
of V-SM-method by the application domain of a matrix sequence A = (A(υ) )
of infinite matrices and gave the consistency theory for such type methods
and introduce the notions of e, be and c convergence for double sequences. By
using the gliding hump method, Zeltser [19] recently characterized the classes
of four dimensional matrix mappings from λ into µ; where λ, µ ∈ {Ce , Cbe }.
Also employing the same arguments, Zeltser [20] gave the theorems determining the necessary and sufficient conditions for Ce -SM and Cbe -SM-methods to
be concervative and coercive. Zeltser [21] considered the dual pairs hE, E β(v) i
of double sequence spaces E and E β(v) , where E β(v) denotes the β-dual of E
with respect to v-convergence of double sequences for v ∈ {p, bp, r} and introduced two oscillating properties for a double sequence space E. Also, Zeltser
[22] emphasized two types of summability methods of double sequences defined by four dimensional matrices which preserve the regular convergence and
the Cc -convergence of double sequences and extends some well-known facts of
summability to four dimensional matrices. By using the definitions of limit
inferior, limit superior and the core of a double sequence with the notion of
the regularity of four dimensional matrices, Patterson [14] proved an invariant
24
Yurdal Sever
core theorem. Also, Patterson [15] determined the sufficient conditions on a
four dimensional matrix in order to be stronger than the convergence in the
Pringsheim’s sense and derives some results concerning with the summability
of double sequences. Mursaleen and Edely [11] recently introduced the statistical convergence and Cauchy for double sequences and gave the relation between statistical convergent and strongly Cesàro summable double sequences.
Mursaleen [12] and Mursaleen and Edely [13] defined the almost strong regularity of matrices for double sequences and apply these matrices to establish a
core theorem and introduced the M -core for double sequences and determined
those four dimensional matrices transforming every bounded double sequence
x = (xjk ) into one whose core is a subset of the M -core of x. Quite recently,
Altay and Başar [1] defined some spaces of double sequences. Çakan and Altay
[4] investigated statistical core for double sequences and studied an inequality related to the statistical and P-cores of bounded double sequences. Başar
and Sever [2] examined some properties of the space Lq . Subramanian and
Mishra [17, 18] defined some new double sequence spaces and examined their
properties.
In this study, we define the double sequence space Cesr and examine some
properties of this sequence space. Furthermore, we determine the β(r)-dual of
the space Cesr .
2
Regular Cesàro Double Sequence Spaces
In this section, we introduce the set Cesr consisting of double sequences whose
Cesáro transforms are convergent in the regular sense. That is to say that
!
)
(
m,n
1 X
xjk ∈ Cr = (Cr )(C,1,1) .
Cesr = (xjk ) ∈ Ω :
mn j,k=1
We show that Cesr is a Banach space and is isomorphic to the space Cr .
Theorem 2.1 The set Cesr becomes a linear space with the coordinatewise
addition and scalar multiplication of double sequences and Cesr is a Banach
space with the norm k · kCesr defined by
m,n
1 X
kxkCesr = sup xjk (7)
m,n mn
j,k=1
and is linearly isomorphic to the space Cr .
Proof. The first part of the theorem is a routine verification and so we
omit it.
25
On Regular Cesàro Double Sequence Spaces
Now, we may show that Cesr is a Banach space with norm defined
by
n
o∞(7).
(l)
l
l
Let (x )l∈N be any Chauchy sequence in the space Cesr , where x = xmn
m,n=1
for every fixed l ∈ N. Then, for a given ε > 0 there exists a positive integer
n0 (ε) such that
kxl − xr kCesr
m,n
1 X
= sup xljk − xrjk < ε
mn
m,n
j,k=1
for all l, r > n0 (ε) which yields for every m, n that
m,n
m,n
1 X
X
1
xljk −
xrjk < ε.
mn
mn j,k=1 j,k=1
This means that
1
mn
Pm,n
l
j,k=1 xjk
is a Cauchy sequence with complex terms
l∈N
for every fixed m, n ∈ N. Since C is complete, it converges, say
m,n
m,n
1 X l
1 X
lim
xjk =
xjk .
l→∞ mn
mn
j,k=1
j,k=1
Using these infinitely many limits, we define the sequence
(8)
1
mn
Pm,n
j,k=1 xjk
.
m,n∈N
It is seen by (8) that
m,n
m,n
1 X
X
1
l
lim xjk −
xjk l→∞ mn
mn j,k=1 j,k=1
= 0.
(9)
Cr
Now we show that x = (xjk ) in Cesr . Let m, n, p, q ∈ N. Since
m,n
p,q
m,n
m,n
1 X
1 X
1 X
1 X l xjk −
xjk ≤ xjk −
xjk mn
mn
pq
mn
j,k=1
j,k=1
j,k=1
j,k=1
m,n
p,q
1 X
X
1
+
xljk −
xljk mn
pq
j,k=1
j,k=1
p,q
p,q
1 X
X
1
l
+
xjk −
xjk pq
pq
j,k=1
≤ 3ε
j,k=1
26
Yurdal Sever
and for fixed n0 and m, p ∈ N,
p,n0
m,n0
1 m,n
1 m,n
X0
X0
1 X
1 X l xjk −
xjk ≤ xjk −
x mn0
mn0
pn0 j,k=1 mn0 j,k=1 jk j,k=1
j,k=1
p,n0
0
1 m,n
X
X
1
xljk −
xljk +
mn0
pn
0
j,k=1
j,k=1
p,n0
p,n0
1 X
1 X
l
+
xjk −
xjk pn0
pn0
j,k=1
j,k=1
≤ 3ε
similarly, for fixed m0 , we can show that the sequence
n
1
m0 n
Pm0 ,n
j,k=1
xjk
o
is
n∈N
convergent. This shows that x ∈ Cesr .
To prove the fact Cesr and Cr linearly isomorphic, we should define a
lineer bijection between the spaces Cesr and Cr . Consider the transformation
T defined, from Cesr to Cr by x 7→ T x.
T : Cesr → Cr
m
x 7→ T x =
n
1 XX
xjk
mn j=1 k=1
!
= (smn ) = s
(i) Let x = (xjk ), y = (yjk ) ∈ Cesr and α ∈ C. Then, T is linear since
!
m
n
1 XX
αxjk + yjk
T (αx + y) =
mn j=1 k=1
!
!
m
n
m
n
1 XX
1 XX
= α
xjk +
yjk
mn j=1 k=1
mn j=1 k=1
= αT x + T y.
(ii) The equation,

x11
1
 2 (x11 + x21 )

..

.
Tx = 
P
 1 m
 m j=1 xj1
..
.
1
(x11
2
+ x12 )
1
(x11 + x12 + x21 + x22 )
4
..
.
P
m P2
1
j=1
k=1 xjk
2m
..
.
1
(x11
3

+ x12 + x13 )
...
1
(x11 + x12 + x13 + x21 + x22 + x23 ) . . . 
6
..
.. 
.
. 
=0
P
P

m
3
1
... 
j=1
k=1 xjk
3m
..
..
.
.
leads to the fact that

x11 = 0 x12 = 0 x13 = 0 . . . 

x21 = 0 x22 = 0 x23 = 0 . . .
=⇒ x = 0
..
..
..
.. 

.
.
.
.
27
On Regular Cesàro Double Sequence Spaces
and this means that T is a bijection.
(iii) Let us take s ∈ Cr and define the sequence x = (xjk ) via s by
xjk = jksjk − (j − 1)ksj−1,k − j(k − 1)sj,k−1 + (j − 1)(k − 1)sj−1,k−1 ; (j, k ∈ N),
where s0,0 = 0, s0,1 = 0 and s1,0P= 0.PSince s ∈ Cr , there exists L ∈ C such
m
n
1
that r − limmn smn = L and mn
j=1
k=1 xjk = smn ,
m X
n
1 X
r- lim xjk − L = 0.
m,n mn
j=1 k=1
Therefore, x ∈ Cesr . So that, T is surjective.
The conditions (i)-(iii) are satisfied, so T is a linear isomorphism between
the linear spaces Cesr and Cr . This step concludes the proof.
Theorem 2.2 The space Cr is subset of the space Cesr .
Proof. Let x = (xkl ) ∈ Cr . Consider the matrix (C, 1, 1) = (cmnkl ), defined by
(6). Then (C, 1, 1) transform of x is
{(C, 1, 1)x}mn =
=
∞ X
∞
X
cmnkl xkl
k=1 l=1
n
m X
X
1
mn
xkl for all m, n ∈ N.
k=1 l=1
Since the matrix (C, 1, 1) is in the class (Cr , Cr ), the sequence (C, 1, 1)x is in
Cr , hence x ∈ Cesr . This shows that the inclusion Cr ⊂ Cesr holds.
Let us define the sequence x = (xmn ) by
1 , (m = n),
xmn =
0 , (m 6= n).
Since the (C, 1, 1) transform of x is s = (smn ) =
min{m,n}
mn
and r −
m,n∈N
lim smn = 0, x = (xmn ) is in Cesr but not in Cr . This shows that the inclusion
Cr ⊂ Cesr is strict.
3
Dual of the Space Cesr
In this section, we shall determine the β(r)-dual of the space Cesr . Although
the β-dual of the spaces of single sequences are unique, the β-duals of the
28
Yurdal Sever
double sequence spaces may be more than one with respect to υ-convergence.
The β(υ)-dual of a double sequence space λ, denoted by λβ(υ) , is defined by
(
)
X
λβ(υ) = (aij ) ∈ Ω : υ −
aij xij exists for all (xij ) ∈ λ .
i,j
It is easy to see for any two spaces λ and µ of double sequences that µβ(υ) ⊂
λβ(υ) whenever λ ⊂ µ.
Now, we can give the β-dual of the space Cesr with respect to the rconvergence using the technique used in [1].
Theorem 3.1 The β(r)-dual of the space Cesr is the set
n
X
Υr−r = a ∈ Ω :
|kl411 akl | < ∞, (kl∆10 akl )k , (kl∆01 akl )l ∈ `1 ,
k,l
o
(klakl ) ∈ Mu , (akl ) ∈ CS r ,
where l1 and CS r denote the space of absolutely summable single sequences and
the space of double sequences consisting of all double series whose sequence of
partial sums are in the space Cr , respectively.
Proof. Suppose that x = (xkl ) ∈ Cesr . Let us define the sequence s = (smn )
as
n
m
1 XX
xkl for all m, n ∈ N.
smn =
mn k=1 l=1
Then, s is in the space Cr , by Teorem 2.1. Let us determine the neccessary
and sufficient condition in order to the series
∞ X
∞
X
akl xkl
(10)
k=1 l=1
is to be r-convergent for a sequence a = (akl ) ∈ Ω. We obtain m, nth partial
sums of the series in (10) that
Pm Pn
zmn =
l=1 akl xkl
Pk=1
m−1 Pn−1
=
k=1
l=1 skl (kl411 akl ) P
P
m−1
+ k=1 skn (kn410 akn ) + n−1
l=1 sml (ml401 aml )
+smn (mnamn )
(11)
for all m, n ∈ N. (11) may be rewritten by the matrix representation as follows:
zmn =
m X
n
X
k=1 l=1
bmnkl skl = (Bs)mn
(12)
On Regular Cesàro Double Sequence Spaces
29
for all m, n ∈ N, where B = (bmnkl ) is the four dimensional matrix defined by
bmnkl

kl411 akl




 kn410 akn
ml401 aml
=


mnamn



0
,
,
,
,
,
k ≤ m − 1 and l ≤ n − 1
k ≤ m − 1 and l = n
k = m and l ≤ n − 1
k = m and l = n
otherwise
(13)
We therefore read from the equality (11) that ax = (akl xkl ) ∈ CS r whenever
x = (xkl ) ∈ Cesr if and only if z = (zkl ) ∈ Cr whenever s = (skl ) ∈ Cr which
leads to the fact that B = (bmnkl ), defined by (13), is in the class (Cr , Cr ).
Thus we see from Lemma 1.1 for the matrix B, defined by (13), the conditions
(2)-(5) hold. We drive that
X
|kl411 akl | < ∞,
(14)
k,l
sup
n
sup
m
X
|kn410 akn | < ∞
(15)
|ml401 aml | < ∞,
(16)
k
X
l
sup |mnamn | < ∞
(17)
m,n
and
∞ X
∞
X
akl exists.
(18)
k=1 l=1
β(r)
This shows that Cesr
= Υr−r which completes the proof.
Acknowledgements
The author would like to express his pleasure to referee for his/her careful
reading and making some useful comments which improved the presentation
of the paper.
30
Yurdal Sever
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