Characterization of certain matrix classes involving
generalized difference summability spaces
Hemen Dutta
Abstract. The main aim of this article is to generalize the notion of
almost convergent, Cesàro summable and lacunary summable spaces by
using a generalized difference operator defined associating a sequence of
non-zero scalars and characterize some matrix classes involving these sequence spaces. In this article we also introduce the idea of difference
infinite matrices. It is expected that these investigations will generalize
several notions associated with thus constructed spaces as well as of matrix
transformations.
M.S.C. 2000: 46B20, 46E30, 40C05.
Key words: Almost convergent sequences, Cesàro summable sequences, lacunary
summable sequences, matrix transformations, difference sequence spaces.
1
Introduction
Let w denote the space of all scalar sequences and any subspace of w is called a
sequence space. Let `∞ , c and c0 be the spaces of bounded, convergent and null
sequences x = (xk ) with complex terms, respectively, normed by
kxk∞ = sup |xk |
k
The notion of difference sequence spaces was introduced by Kizmaz [10]. The
notion was further generalized by Et and Colak [4] by introducing the spaces `∞ (∆s ),
c(∆s ) and c0 (∆s ). Another type of generalization of the difference sequence spaces
is due to Tripathy and Esi [15], who studied the spaces `∞ (∆r ), c(∆r ) and c0 (∆r ).
Tripathy, Esi and Tripathy [16] generalized the above notions and unified these as
follows:
Let r, s be non-negative integers, then for Z a given sequence space we have
Z(∆sr ) = {x = (xk ) ∈ w : (∆sr xk ) ∈ Z},
where ∆sr x = (∆sr xk ) = (∆s−1
xk − ∆s−1
xk+r ) and ∆0r xk = xk for all k ∈ N , which
r
r
is equivalent to the following binomial representation:
µ ¶
n
X
s
xk+ri
∆sr xk =
(−1)i
i
i=0
Applied Sciences, Vol.11, 2009, pp.
60-67.
c Balkan Society of Geometers, Geometry Balkan Press 2009.
°
Characterization of certain matrix classes
61
Let r, s be non-negative integers and v = (vk ) be a sequence of non-zero scalars.
Then for Z, a given sequence space we define the following sequence spaces:
Z(∆s(vr) ) = {x = (xk ) ∈ w : (∆s(vr) xk ) ∈ Z}, for Z = `∞ , c and c0
s−1
0
where(∆s(vr) xk ) = (∆s−1
(vr) xk − ∆(vr) xk−r ) and ∆(vr) xk = vk xk for all k ∈N , which is
equivalent to the following binomial representation:
∆s(vr) xk
µ ¶
n
X
i s
=
(−1)
vk−ri xk−ri .
i
i=0
In this expansion it is important to note that we take vk−ri = 0 and xk−ri = 0 for
non-positive values of k − ri.
In the next section we shall show that these spaces can be made BK-spaces under
the norm
kxk = sup |∆s(vr) xk |
k
For s=1 and vk =1 for all k ∈N , we get the spaces `∞ (∆r ), c(∆r ) and c0 (∆r ).
For r=1 and vk =1 for all k ∈N , we get the spaces `∞ (∆s ), c(∆s ) and c0 (∆s ). For
r = s = 1 and vk =1 for all k ∈ N , we get the spaces `∞ (∆), c(∆) and c0 (∆).
Let E and F be two sequence spaces and A = (ank ) be an infinite matrix of
real or complex numbers ank , where n, k ∈ N .Then we say that A defines a matrix
mapping from E into F , and denote it by writing A : E −→ F if for every sequence
x = (xk ) ∈ E the sequence Ax = {(Ax)n }, the A-transform of x, is in F , where
(1.1)
(Ax)n =
∞
X
ank xk , (n ∈ N )
k=1
We denote be (E, F ) the class of all matrices A such that A : E −→ F . Thus
A ∈ (E, F ) if and only if the series on the right hand side of (1.1) converges for each
n ∈ N and every x ∈ E, and we have Ax = {(Ax)n }n∈N ∈ F for all x ∈ E. A
sequence x is said to be A-summable to l if Ax converges to l which is called the
A-limit of x. Further we write (E, F, P ) to the subset of (E, F ) which preserves the
limit or sum.
Interest in general matrix transformation theory was, to some extent, stimulated
by special results in summability theory which were obtain by Cesàro, Borel and
others, at the turn of the 20th century. It was however the celebrated German mathematician O. Toeplitz who, in 1911, brought the methods of linear space theory to
bear on problems connected with matrix transformations on sequence spaces. Toeplitz
characterized all those infinite matrices A = (ank ), n, k ∈ N , which map the space of
convergent sequences into itself, leaving the limit of each convergent sequence invariant (See for instance Maddox [12]).
A linear functional L on `∞ is said to be a Banach limit (see Banach [1]) if it has
the properties:
(i) L(x) ≥ 0 if x ≥ 0,
(ii) L(e) = 1, where e = (1, 1, 1, ...),
(iii) L(Dx) = L(x), where D is the shift operator defined by (Dxn ) = (xn+1 ).
62
Hemen Dutta
Let B be the set of all Banach limits on `∞ . A sequence x is said to be almost
convergent to a number l if L(x) = l for all L in B. Let ĉ denote the set of all almost
convergent sequences. Lorentz [11] proved that
m
1 X
xn+i exists uniformly in n}.
m→∞ m + 1
i=0
ĉ = {x : lim
By a lacunary sequence θ = (kq ); q = 1, 2, 3, ..., where k0 = 0, we mean an
increasing sequence of non-negative integers with hq = (kq − kq−1 ) → ∞ as q → ∞.
We denote Iq = (kq−1 , kq ].
Let Cθ denote the space of all lacunary summable sequences. Then
1 X
xk exists}.
r→∞ hr
Cθ = {x : lim
k∈Ir
Let σ1 denote the space of all Cesàro summable sequences. Then
n
1X
xi exists}.
n→∞ n
i=1
σ1 = {x : lim
Let r, s be non negative integers and v = (vk ) be a sequence of non-zero scalars.
Then we define the following spaces:
p−1
1X s
∆(vr) xn+i exists uniformly in n},
p→∞ p
i=0
ĉ(∆s(vr) ) = {x : lim
n
1X s
∆(vr) xi exists},
n→∞ n
i=1
σ1 (∆s(vr) ) = {x : lim
1 X s
∆(vr) xk exists}.
q→∞ hq
Cθ (∆s(vr) ) = {x : lim
k∈Iq
ĉ(∆s(vr) ),
We call the spaces
and Cθ (∆s(vr) ) as the spaces of ∆s(vr) -almost
s
convergent, ∆(vr) -Cesàro summable and ∆s(vr) -lacunary summable sequences respectively. For s = 0 and vk = 1 for all k ∈ N we get the spaces of almost convergent([1],
[11]), Cesàro summable and lacunary summable ([6], [8]) sequences respectively.
2
σ1 (∆s(vr) )
Main Results
In this section we show that the spaces Z(∆s(vr) ), for Z = `∞ , c and c0 are BK-spaces.
We also characterize the matrix classes (c, ĉ(∆s(vr) )), (c, σ1 (∆s(vr) )), (c, Cθ (∆s(vr) )),
(c, ĉ(∆s(vr) ), P ), (c, σ1 (∆s(vr) ), P ) and (c, Cθ (∆s(vr) ), P ).
Theorem 2.1. The spaces Z(∆s(vr) ), for Z = `∞ , c and c0 are BK-spaces under the
norm defined by
kxk = sup |∆s(vr) xk |
.
k
Characterization of certain matrix classes
63
Proof. First we show that the spaces Z(∆s(vr) ) are normed linear spaces normed by
k.k.
For x = θ, we have kxk = 0. Conversely, let kxk = 0. Then using definition of
norm, we have
sup |∆s(vr) xk | = 0
k
It follows that
∆s(vr) xk = 0 for all k ≥ 1.
Let k = 1, then ∆s(vr) x1 =
s
P
(−1)i
¡s¢
i=0
v
v1−ri x1−ri = 0 and so v1 x1 = 0, by putting
v1−ri = 0 and x1−ri = 0 for i = 1, ..., s. Hence x1 = 0, since (λk ) is a sequence of
non-zero scalars. Similarly taking k = 2, ..., rs, we have x2 = . . . = xrs = 0. Next
s
¡ ¢
P
let k = rs + 1, then ∆s(vr) xrs+1 =
(−1)i vs v1+rs−ri x1+rs−ri = 0. Since x1 = x2 =
i=0
. . . = xrs = 0, we must have vrs+1 xrs+1 = 0 and thus xrs+1 = 0. Proceeding in this
way we can conclude that xk = 0 for all k ≥ 1. Hence x = θ.
Again it is easy to show that kx + yk ≤ kxk + kyk and for any scalar α, kαxk =
|α|kxk.
Thus Z(∆s(vr) ) is a normed linear space normed by k.k.
Now we show that Z(∆s(vr) ) is a Banach space under the norm k.k.
Let (xi ) be a Cauchy sequence in Z(∆s(vr) ), where xi = (xik ) = (xi1 , xi2 , ...) for each
i ≥ 1. Then for a given ε > 0, there exists a positive integer n0 such that
kxi − xj k = sup |∆s(vr) (xik − xjk )| < ε for all i, j ≥ n0
k
It follows that
|∆s(vr) (xik − xjk )| < ε for all i, j ≥ n0 and for all k ≥ 1.
This implies that (∆s(vr) xik ) is a Cauchy sequence in C for all k ≥ 1 and so it is
convergent in C for all k ≥ 1.
Let lim ∆s(vr) xik = yk , say for each k ≥ 1. Considering k = 1, 2, ..., rs, ..., we can
i→∞
easily conclude that lim xik = xk , say exists for each k ≥ 1.
i→∞
Now we can have
lim |∆s(vr) (xik − xjk )| < ε for all i ≥ n0 and k ≥ 1
j→∞
Hence
sup |∆s(vr) (xik − xk )| < ε for all i ≥ n0 .
k
i
This implies that (x − x) ∈ Z(∆s(vr) ). Since Z(∆s(vr) ) is a linear space, x = xi − (xi −
x) ∈ Z(∆s(vr) ).
Hence Z(∆s(vr) ) are complete.
From the above proof we can easily conclude that kxi −xk → 0 implies |xik −xk | → 0
as i → ∞, for each k ≥ 1.
Hence we can conclude that Z(∆s(vr) ) are BK-spaces.
64
Hemen Dutta
This completes the proof.
¤
Remark. One may find it interesting to see the above proof by taking particular
values of r and s in the difference operator ∆s(vr) . For example, let us take r = 3 and
s = 2, then ∆2(v3) xk = vk xk - 2vk−3 xk−3 + vk−6 xk−6 . For k = 2, we have ∆2(v3) x2 =
v2 x2 - 2v−1 x−1 + v−4 x−4 = v2 x2 . Also for k = 7, we have ∆2(v3) x7 = v7 x7 - 2v4 x4 +
v1 x1 etc.
Theorem 2.2. A ∈ (c, ĉ(∆s(vr) )) if and only if
∞
P
(i) sup{
k=1
1
p|
n+p−1
P
j=n
∆s(vr) ajk | : p ∈ N } < ∞, n = 1, 2, ...
(ii) there exists αk , k = 1, 2, ..., such that
n+p−1
P
lim p1
∆s(vr) ajk = αk , uniformly in n and
p→∞
j=n
(iii) there exists α such that
n+p−1
∞
P P
∆s(vr) ajk = α, uniformly in n,
lim p1
p→∞
j=n k=1
where ∆s(vr) ajk =
s
P
(−1)i
i=0
¡ s¢
i vk aj−ri,k and we take aj−ri,k = 0 for non-positive
values of j-ri. (e.g., ∆2(3) v1 a11 = v1 a11 −2v1 a−2,1 +v1 a−5,1 = a11 , ∆2(3) v1 a71 = v1 a71 −
2v1 a41 + v1 a11 etc.)
Proof. Suppose A ∈ (c, ĉ(∆s(vr) )). Fix n and put Spn (x) =
∆s(vr) (Ax)j
=
∆s−1
(vr) (Ax)j
−
∆s−1
(vr) (Ax)j−r
=
1
p
n+p−1
P
∞
X
j=n
∆s(vr) (Ax)j , where
∆s(vr) ajk xk .
k=1
∆s(vr) (Ax)j , j
We write Bj (x) =
= 1, 2, .... Then we observe that (Bj ) is a sequence
of bounded linear operators on c and so Spn , p = 1, 2, ... is a bounded linear operator on
c such that lim Spn (x) exists uniformly in n. Now (i) follows from an application of
p→∞
uniform boundedness principle. Let e = (1, 1, ...) and ek = (0, 0, ..., 1, 0, ...), k = 1, 2, ...
where 1 is in the k th position. Since e and ek , k = 1, 2, ... are convergent sequences,
lim Spn (ek ) and lim Spn (e) must exists uniformly in n. Hence (ii) and (iii) must
p→∞
p→∞
hold.
For the converse part let (xk ) converges to l and the conditions (i), (ii) and (iii)
hold. We write
(2.1)
∞ n+p−1
∞ n+p−1
∞ n+p−1
1X X
1X X
1 X X
∆s(vr) ajk
∆s(vr) ajk xk =
∆s(vr) ajk (xk − l) + l
p
p
p
j=n
j=n
j=n
k=1
k=1
k=1
Since (xk ) converges to l, using (i) and (ii) it follows that the first term on the right
∞
P
of (2.1) tends to
αk (xk − l) as p → ∞. Again since (iii) holds, the second term
k=1
on the right of (2.1) tends to lα as p → ∞.
Characterization of certain matrix classes
∞ n+p−1
P
P
1
∆s(vr) ajk xk
p
p→∞ k=1 j=n
Therefore lim
65
exists uniformly in n.
This completes the proof.
¤
Proof of the following two Theorems follow by applying similar arguments as
applied to prove Theorem 2.2.
Theorem 2.3. A ∈ (c, σ1 (∆s(vr) )) if and only if
∞
P
(i) sup{
k=1
1
p|
p
P
j=1
∆s(vr) ajk | : p ∈ N } < ∞,
(ii) there exists αk , k = 1, 2, ..., such that
p
P
lim p1
∆s(vr) ajk = αk , and
p→∞
j=1
(iii) there exists α such that
p P
∞
P
lim p1
∆s(vr) ajk = α.
p→∞
j=1 k=1
Theorem 2.4. A ∈ (c, Cθ (∆s(vr) )) if and only if
∞
P
(i) sup{
k=1
1
hq |
P
j∈Iq
∆s(vr) ajk | : q ∈ N } < ∞,
(ii) there exists
P αks, k = 1, 2, ..., such that
∆(vr) ajk = αk , and
lim h1q
q→∞
j∈Iq
(iii) there exists α such that
∞
P P
lim h1q
∆s(vr) ajk = α.
q→∞
j∈Iq k=1
Theorem 2.5. A ∈ (c, ĉ(∆s(vr) ), P ) if and only if
∞
P
(i) sup{
k=1
1
p→∞ p
(ii) lim
1
p→∞ p
(iii) lim
1
p|
n+p−1
P
j=n
n+p−1
P
j=n
∆s(vr) ajk | : p ∈ N } < ∞, n = 1, 2, ...
∆s(vr) ajk = 0, uniformly in n, k = 1, 2, ...
n+p−1
∞
P P
j=n k=1
∆s(vr) ajk = 1 uniformly in n.
Proof. Suppose A ∈ (c, ĉ(∆s(vr) ), P ). Then the condition (i) holds by Theorem 2.2.
Also (ii) and (iii) hold since we must have ĉ(∆s(vr) ) − lim Aek = 0, k = 1, 2, ... and
ĉ(∆s(vr) ) − lim Ae = 1 respectively.
For converse part suppose the conditions hold and (xk ) converges to l. Then by
Theorem 2.2, we have A ∈ (c, ĉ(∆s(vr) )) and using (2.1) we must have
∞ n+p−1
1X X
∆s(vr) ajk xk = l.
p→∞ p
j=n
lim
k=1
66
Hemen Dutta
This completes the proof.
¤
Proof of the following two Theorems follow by applying similar arguments as
applied to prove Theorem 2.5.
Theorem 2.6. A ∈ (c, σ1 (∆s(vr) ), P ) if and only if
∞
P
(i) sup{
k=1
1
p|
p
P
j=1
∆s(vr) ajk | : p ∈ N } < ∞,
p
P
1
∆s ajk
p
p→∞ j=1 (vr)
(ii) lim
= 0, k = 1, 2, ...
p P
∞
P
1
∆s ajk
p
p→∞ j=1 k=1 (vr)
(iii) lim
= 1.
Theorem 2.7. A ∈ (c, Cθ (∆s(vr) ), P ) if and only if
∞
P
(i) sup{
k=1
1
hq |
P
1
q→∞ hq j∈I
q
(ii) lim
P
j∈Iq
∆s(vr) ajk | : q ∈ N } < ∞,
∆s(vr) ajk = 0, k = 1, 2, ...
∞
P P
1
∆s ajk
h
q→∞ q j∈I k=1 (vr)
q
(iii) lim
= 1.
Remark. By taking s = 0, and vk = 1 for all k ∈ N , in Theorem 2.2 and Theorem
2.5, we get the matrix classes (c, ĉ), (c, ĉ, P ) studied by King [9].
3
Conclusions
In the present paper we present a generalization of the notion of almost convergent, Cesàro summable and lacunary summable spaces by using the generalized
difference operator ∆s(vr) and study the spaces of these difference sequences for BKspaces. We also characterize the matrix classes involving these sequence spaces and
the space c. Further we characterize the matrix classes which preserves the limit. One
may find it interesting to see ([14], [2], [7]) for some results on matrices and functional
analytic studies on spaces.
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Authors’ addresses:
Hemen Dutta
Department of Mathematics,
A.D.P. College, Nagaon-782002, Assam, India.
E-mail: hemen− dutta08@rediffmail.com