Matrix Transformations on Some Difference Sequence Spaces

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 6 (May. - Jun. 2013), PP 01-04
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Matrix Transformations on Some Difference Sequence Spaces
Z. U. Siddiqui, A. Kiltho
Department of Mathematics and Statistics, University of Maiduguri, Nigeria
Abstract: The sequence spaces 𝑙∞(𝑢, 𝑣, ∆), 𝑐0(𝑢, 𝑣, ∆) and 𝑐(𝑢, 𝑣, ∆) were recently introduced. The matrix
classes (𝑐 𝑢, 𝑣, ∆ : 𝑐) and (𝑐 𝑢, 𝑣, ∆ : 𝑙∞) were characterized. The object of this paper is to further determine
the necessary and sufficient conditions on an infinite matrix to characterize the matrix classes (𝑐 𝑢, 𝑣, ∆ ∶ 𝑏𝑠)
and (𝑐 𝑢, 𝑣, ∆ ∶ 𝑙 𝑝). It is observed that the later characterizations are additions to the existing ones.
Keywords- Difference operators, Duals, Generalized weighted mean, Matrix transformations
I. Introduction
The sequence spaces 𝑙∞(∆), 𝑐0(∆) and 𝑐(∆) were first introduced by Kizmaz [6] in 1981. Similar to the
sequence spaces 𝑙∞(𝑝), 𝑐0(𝑝) and 𝑐(𝑝) for 𝑝 𝑘 > 1 of Maddox [7] and Simons [10], the ∆- sequence spaces
above were extended to ∆𝑙∞(𝑝), ∆𝑐0(𝑝) and ∆𝑐(𝑝) by Ahmad and Mursaleen [1] in … The concept of
difference operators has been discussed and used by Polat and Başar [8] and by Altay and Başar [2], both in
2007.
The idea of generalized weighted mean was applied by Altay and Başar [3], in 2006. This concept
depends on the idea of 𝐺 𝑢, 𝑣 - transforms which has been used by Polat, et al [10] and by Basarir and Kara [4].
We shall need the following sequence spaces:
𝜔 = {𝑥 = 𝑥 𝑘 ∶ x is any sequence }
𝑐 = {𝑥 = 𝑥 𝑘 ∈ 𝜔: 𝑥 𝑘 converges, i.e. lim 𝑘→∞ 𝑥 𝑘 exists }
𝑐0 = 𝑥 = 𝑥 𝑘 ∈ 𝜔: limk→∞ 𝑥 𝑘 = 0 , the set of all null sequences
𝑙∞ = 𝑚 = 𝑥 = 𝑥 𝑘 ∈ 𝜔: 𝑥 ∞ = 𝑠𝑢𝑝𝑛 𝑥 𝑘 < ∞
𝑙1 = 𝑙 = 𝑥 = 𝑥 𝑘 ∈ 𝜔: 𝑥 1 = 𝑥 𝑘 < ∞∞
𝑘=0
𝑙 𝑝 = 𝑥 = 𝑥 𝑘 ∈ 𝜔: 𝑥 𝑝 = 𝑥 𝑘
𝑝
< ∞; 1 ≤ 𝑝 < ∞
𝜙 = {𝑥 = 𝑥 𝑘 ∈ 𝜔: ∃ 𝑁 ∈ ℕ 𝑕such that ∀ 𝑘 ≥ 𝑁, 𝑥 𝑘 = 0}, the set of finitely nonzero sequences
𝑏𝑠 = 𝑥 = 𝑥 𝑘 ∈ 𝜔: 𝑥 𝑏𝑠 = 𝑠𝑢𝑝𝑛 𝑥 𝑘
𝑛
𝑘=0 < ∞ , the set of all sequences with bounded partial sums
𝑋 𝛽
= {𝑎 = 𝑎 𝑘 ∈ 𝜔: 𝑎 𝑘 𝑥 𝑘 ∈ 𝑐∞
𝑘=0 , ∀ 𝑥 ∈ 𝑋}
Note that 𝑥 = (𝑥 𝑘 ) is used throughout for the convention (𝑥 𝑘 ) = (𝑥 𝑘 ) 𝑘=0
∞
. We take 𝑒 = (1, 1, 1, … ) and 𝑒 𝑘
for
the sequence whose only nonzero term is 1 in the 𝑘th place for each 𝑘 ∈ ℕ, where ℕ = {0, 1, 2, 3, … }. Any
vector subspace of 𝜔 is called a sequence subspace. A sequence space 𝑋 is FK if it is a complete linear metric
space with continuous coordinates 𝑃𝑛 ∶ 𝑋 → ℂ, defined by 𝑃𝑛 𝑥 = 𝑥 𝑛 ∀ 𝑥 = (𝑥 𝑘 ) ∈ 𝑋 with 𝑛 ∈ ℕ. A normed
FK space is BK-space or Banach space with continuous coordinates. An FK space has AK- property if 𝑥[𝑚]
→ 𝑥
in 𝑋, where 𝑥[𝑚]
= 𝑥 𝑘 𝑒 𝑘𝑛
𝑘=0 is the mth
- section of 𝑥. If 𝜑 is dense in 𝑋 then it has an AD- property (see Boos
[5]). A matrix domain of a sequence space 𝑋, is defined as 𝑋𝐴 = 𝑥 = (𝑥 𝑘 ∈ 𝜔 ∶ 𝐴𝑥 ∈ 𝑋 }.
Let 𝒰 be the set of all sequences 𝑢 = (𝑢 𝑘 ) with 𝑢 𝑘 ≠ 0 ∀ 𝑘 ∈ ℕ, and for 𝑢 ∈ 𝒰 let
1
𝑢
=
1
𝑢 𝑘
. Then
for 𝑢, 𝑣 ∈ 𝒰 define the matrix 𝐺 𝑢, 𝑣 = (𝑔 𝑛𝑘 ) by
𝑔 𝑛𝑘 =
𝑢 𝑛 𝑣 𝑘, for 0 ≤ 𝑘 ≤ 𝑛,
0, for 𝑘 > 𝑛 ∀ 𝑘, 𝑛 ∈ ℕ
This matrix is called the generalized weighted mean. The sequence 𝑦 = (𝑦 𝑘) in the sequence spaces
𝜆 𝑢, 𝑣, Δ = {𝑥 = (𝑥 𝑘 ) ∈ 𝜔 ∶ 𝑦 = 𝑢 𝑘 𝑣𝑖∆𝑥𝑖 ∈ 𝑋},𝑘
𝑖=0 𝜆 ∈ {𝑙∞, 𝑐, 𝑐0} (1)
is the 𝐺 𝑢, 𝑣, ∆ −transform of a given sequence 𝑥 = (𝑥 𝑘 ). It is defined by
𝑦 = 𝑢 𝑘 𝑣𝑖∆𝑥𝑖
𝑘
𝑖=0
= 𝑢 𝑘 ∇𝑣𝑖 𝑥𝑖
𝑘
𝑖=0
where,
∇𝑣𝑖 = 𝑣𝑖 − 𝑣𝑖+1 and ∆𝑥 = ∆𝑥𝑖 = 𝑥𝑖 − 𝑥𝑖−1,
and taking all negative subscripts to be naught. The spaces (1) were defined in [9]. If 𝑋 is any normed sequence
space the matrix domain 𝑋 𝐺(𝑢,𝑣,∆) is the generalized weighted mean difference sequence space [9]. Our object is
to characterize the matrix classes 𝑐 𝑢, 𝑣, ∆ ∶ 𝑙 𝑝 and (𝑐 𝑢, 𝑣, ∆ ∶ 𝑏𝑠). However, matrix class characterizations
are done with help of 𝛽 −duals, and so we need the following

Lemma 1.1 [9]: Let 𝑢, 𝑣, ∈ 𝒰, 𝑎 = 𝑎 𝑘 ∈ 𝜔 and the matrix 𝐷 = (𝑑 𝑛𝑘 ) by
𝑑 𝑛𝑘 =
1
𝑢 𝑛 𝑣 𝑘
−
1
𝑢 𝑛 𝑣 𝑘+1
𝑎 𝑘 ; 0 ≤ 𝑘 < 𝑛 ,
1
𝑢 𝑛 𝑣 𝑛
𝑎 𝑛 ; 𝑘 = 𝑛
0; 𝑘 > 𝑛
and let 𝑑1, 𝑑2, 𝑑3, 𝑑4 and 𝑑5 be the sets
𝑑1 = {𝑎 = 𝑎 𝑘 ∈ 𝜔 ∶ 𝑠𝑢𝑝𝑛 𝑑 𝑛𝑘𝑘∈𝒦 < ∞};𝑛
𝑑2 = {𝑎 = 𝑎 𝑘 ∈ 𝜔 ∶ 𝑠𝑢𝑝𝑛 𝑑 𝑛𝑘 < ∞};𝑛
𝑑3 = {𝑎 = 𝑎 𝑘 ∈ 𝜔 ∶ 𝑙𝑖𝑚 𝑛→∞ 𝑑 𝑛𝑘 exists for each 𝑛 ∈ ℕ}
Then, [𝑐0 𝑢, 𝑣, ∆ ] 𝛽
= 𝑑1 ∩ 𝑑2 ∩ 𝑑3.
II. Methodology
If A is an infinite matrix with complex entries 𝑎 𝑛𝑘 (𝑛, 𝑘 ∈ ℕ), then 𝐴 = (𝑎 𝑛𝑘 ) is used for 𝐴 =
(𝑎 𝑛𝑘 ) 𝑛,𝑘=0
∞
and 𝐴 𝑛 is the sequence in the nth
row of A, or 𝐴 𝑛 = (𝑎 𝑛𝑘 ) 𝑘=0
∞
for every 𝑛 ∈ ℕ. The A- transform of a
sequence x is defined as
𝐴𝑥 = (𝐴 𝑛 (𝑥)) 𝑛=0
∞
= lim 𝑛→∞ 𝑎 𝑛𝑘 𝑥 𝑘
∞
𝑘=0 (𝑛 ∈ ℕ)
provided the series on the right converges for each n and for all 𝑥 ∈ 𝑋. The pair (𝑋, 𝑌) is referred to as a matrix
class, so that
𝐴 ∈ (𝑋, 𝑌) ⟺
𝐴 𝑛 ∈ 𝑋 𝛽
∀ 𝑛 ∈ ℕ
and
𝐴𝑥 ∈ 𝑌 ∀ 𝑥 ∈ 𝑋, in the norm of 𝑌
(2)
In this paper we shall take 𝑋 = 𝑐(𝑢, 𝑣, ∆) and 𝑌 ∈ 𝑙 𝑝, 𝑏𝑠 . We shall need the following lemma for the proof of
Theorems 3.1 and 3.2 as our main results in section 3:
Lemma 2.1 [9]: The sequence spaces 𝜆 𝑢, 𝑣, Δ for 𝜆 ∈ {𝑙∞, 𝑐, 𝑐0} are complete normed linear spaces with the
norm 𝑥 𝜆 𝑢,𝑣,Δ = sup 𝑘 𝑢 𝑘 ∆𝑥𝑖
𝑘
𝑖=0 = 𝑦 𝜆. They are also BK spaces with both AK- and AD- properties.
Further, let 𝑦 ∈ 𝑐0 and define 𝑥 = 𝑥 𝑘 by
𝑥 𝑘 =
1
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
𝑦𝑖 +
1
𝑢 𝑘 𝑣 𝑘
𝑦 𝑘; 𝑘 ∈ ℕ𝑘−1
𝑖=0
then 𝑥 ∈ 𝑐0 𝑢, 𝑣, ∆ .
An infinite matrix A maps a BK space 𝑋 continuously into the space 𝑏𝑠 if and only if the sequence the
sequence of functional {𝑓𝑛 } defined by
𝑓𝑛 𝑥 = 𝑎 𝑛𝑘 𝑥 𝑘, 𝑛 = 1, 2, 3, …∞
𝑘=1
𝑚
𝑛=1
is bounded in the dual space of 𝑋.
III. Main Results
Theorem 3.1. 𝐴 ∈ 𝑐 𝑢, 𝑣, ∆ ∶ 𝑙 𝑝 for 𝑝 > 1, if and only if
(i) 𝑠𝑢𝑝𝑛
1
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
𝑎 𝑛𝑘 +
1
𝑢 𝑘 𝑣 𝑘
𝑎 𝑛𝑘
𝑘−1
𝑖=1𝑘∈𝒦
𝑝
< ∞,
(ii) 𝑙𝑖𝑚 𝑛→∞
1
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
𝑎 𝑛𝑘 +
1
𝑢 𝑘 𝑣 𝑘
𝑎 𝑛𝑘
𝑘−1
𝑖=1 = 𝑎 𝑘 , exists
(iii) 𝑙𝑖𝑚 𝑛→∞
1
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
𝑎 𝑛𝑘 +
1
𝑢 𝑘 𝑣 𝑘
𝑎 𝑛𝑘
𝑘−1
𝑖=1 = 𝑎𝑛
𝑘=0 , exists
Proof: Since 𝑐 𝑢, 𝑣, ∆ and 𝑙 𝑝 are BK spaces, we suppose that (i), (ii) and (iii) hold and take 𝑥 = (𝑥 𝑘 ) ∈
𝑐 𝑢, 𝑣, ∆ . Then by (2) and Lemma 1.1, 𝐴 𝑛 ∈ [𝑐 𝑢, 𝑣, ∆ ] 𝛽
for all 𝑛 ∈ ℕ, which implies the existence of the A-
transform of 𝑥, or 𝐴𝑥 exists for each 𝑛. It is also clear that the associated sequence 𝑦 = (𝑦 𝑘) is in 𝑐 and hence
𝑦 ∈ 𝑐0. Again, since 𝑐 𝑢, 𝑣, ∆ has AK (Lemma 2.1) and contains 𝜙, by the mth
partial sum of the series
𝑎 𝑛𝑘 𝑥 𝑘
∞
𝑘=0 we have
𝑎 𝑛𝑘 𝑥 𝑘 =
1
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
+
1
𝑢 𝑘 𝑣 𝑘
𝑘−1
𝑖=1 𝑎 𝑛𝑘 𝑦 𝑘,𝑚
𝑘=0
𝑚
𝑘=0
which becomes
𝑎 𝑛𝑘 𝑥 𝑘 =
1
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
𝑎 𝑛𝑘 +
1
𝑢 𝑘 𝑣 𝑘
𝑎 𝑛𝑘
𝑘−1
𝑖=1 𝑦 𝑘,∞
𝑘=0
∞
𝑘=0 for 𝑝 > 1,

⟹ 𝐴𝑥 𝑙 𝑝
≤ sup 𝑛
1
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
𝑎 𝑛𝑘 𝑦 𝑘 +
1
𝑢 𝑘 𝑣 𝑘
𝑎 𝑛𝑘 𝑦 𝑘
𝑝
𝑘−1
𝑘=0
1/𝑝
𝑘
≤ 𝑦 𝑘 𝑙 𝑝
𝑠𝑢𝑝𝑛
1
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
𝑎 𝑛𝑘
𝑝
𝑘−1
𝑘=0
1/𝑝
+
𝑎 𝑛𝑘
𝑢 𝑘 𝑣 𝑘
𝑝
𝑘−1
𝑘=0
1/𝑝
𝑘 < ∞
⟹ 𝐴𝑥 ∈ 𝑙 𝑝 and hence 𝐴 ∈ 𝑐 𝑢, 𝑣, ∆ ∶ 𝑙 𝑝 .
Conversely, let 𝐴 ∈ 𝑐 𝑢, 𝑣, ∆ ∶ 𝑙 𝑝 , 1 < 𝑝 < ∞. Then again by (2) and Lemma 1.1, 𝐴 𝑛 ∈ [𝑐 𝑢, 𝑣, ∆ ] 𝛽
for all
𝑛 ∈ ℕ implying (ii) and (iii) for all 𝑥 ∈ 𝑐 𝑢, 𝑣, ∆ and 𝑦 ∈ 𝑙 𝑝 . To prove (i), let the continuous linear functional
𝑓𝑛 (𝑛 ∈ ℕ) be defined on (𝑐 𝑢, 𝑣, ∆ )∗
, the continuous dual of 𝑐 𝑢, 𝑣, ∆ . Since the series 𝑎 𝑛𝑘 𝑥 𝑘
∞
𝑘=0 converges
for each 𝑥 and for each 𝑛, then 𝑓𝐴 𝑛
∈ (𝑐 𝑢, 𝑣, ∆ )∗
; where
𝑓𝐴 𝑛
𝑥 = 𝑎 𝑛𝑘 𝑥 𝑘
∞
𝑘=0 ∀ 𝑥 ∈ 𝑐 𝑢, 𝑣, ∆ .
⟹ 𝑓𝐴 𝑛
= 𝐴 𝑛 𝑙 𝑝
= 𝑎 𝑛𝑘
𝑝∞
𝑘=0
1
𝑝 < ∞, for all 𝑛 ∈ ℕ,
with 𝐴 𝑛 ∈ [𝑐 𝑢, 𝑣, ∆ ] 𝛽
. This means that the functional defined by the rows of A on 𝑐 𝑢, 𝑣, ∆ are pointwise
bounded, and by the Banach-Steinhaus theorem these functional are uniformly bounded. Hence there exists a
constant 𝑀 > 0, such that 𝑓𝐴 𝑛
≤ 𝑀, ∀ 𝑛 ∈ ℕ, yielding (i).
Theorem 3.2: 𝐴 ∈ 𝑐 𝑢, 𝑣, ∆ ∶ 𝑏𝑠 if and only if conditions (ii) and (iii) of Theorem 3.1 hold, and
(iv) 𝑠𝑢𝑝 𝑚
1
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
𝑎 𝑛𝑘 +
1
𝑢 𝑘 𝑣 𝑘
𝑎 𝑛𝑘
𝑘−1
𝑖=1 < ∞𝑚
𝑛=1𝑘 .
Proof. Suppose𝐴 ∈ 𝑐 𝑢, 𝑣, ∆ ∶ 𝑏𝑠 . Then 𝐴 𝑛 ∈ [𝑐 𝑢, 𝑣, ∆ ] 𝛽
for all 𝑛 ∈ ℕ. Since 𝑒 𝑘 = 𝛿 𝑛𝑘 , where 𝛿 𝑛𝑘 = 1
(𝑛 = 𝑘) and = 0 (𝑛 ≠ 𝑘), belongs to 𝑐 𝑢, 𝑣, ∆ , the necessity of (ii) holds. Similarly by taking 𝑥 = 𝑒 =
(1, 1, 1, … ) ∈ 𝑐 𝑢, 𝑣, ∆ we get (iii). We prove the necessity of (i) as follows:
Suppose 𝐴 ∈ 𝑐 𝑢, 𝑣, ∆ ∶ 𝑏𝑠 . Then it implies
𝐴 𝑟(𝑥) < ∞𝑚
𝑛=1 , 𝑚 = 1, 2, 3, …,
where,
𝐴 𝑟 𝑥 = 𝑎 𝑟𝑘𝑘 (
𝑦 𝑘
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
+
𝑦 𝑘
𝑢 𝑘 𝑣 𝑘
)𝑘−1
𝑖=0
converges for each 𝑟 whenever 𝑥 ∈ 𝑐 𝑢, 𝑣, ∆ , which follows by the Banach-Steinhaus theorem that
𝑠𝑢𝑝 𝑘 𝑎 𝑛𝑘 < ∞, each 𝑟. Hence 𝐴 𝑟 defines an element of [𝑐 𝑢, 𝑣, ∆ ]∗
for each 𝑟.
Now define
𝑞 𝑚 𝑥 = 𝐴 𝑟(𝑥) , 𝑟 = 1,2,3, …𝑚
𝑛=1
𝑞 𝑚 is subadditive. Moreover, 𝐴 𝑟 is a bounded linear functional on 𝑐 𝑢, 𝑣, ∆ implies each 𝑞 𝑚 is a sequence of
continuous seminorms on 𝑐 𝑢, 𝑣, ∆ such that
𝑠𝑢𝑝 𝑚 𝑞 𝑚 𝑥 = 𝐴 𝑟(𝑥) < ∞∞
𝑟=1 for each 𝑥 ∈ 𝑐 𝑢, 𝑣, ∆ .
Thus there exists a constant 𝑀 > 0 such that
𝐴 𝑟(𝑥) ≤ 𝑀 𝑥 𝑐 𝑢,𝑣,∆
∞
𝑟=1
which implies (i).
Sufficiency: Suppose (i) – (iii) of the theorem hold. Then 𝐴 𝑛 ∈ [𝑐 𝑢, 𝑣, ∆ ] 𝛽
. If 𝑥 ∈ 𝑐 𝑢, 𝑣, ∆ , it
suffices to show that 𝐴 𝑛 (𝑥) ∈ 𝑏𝑠 in the norm of the sequence space 𝑏𝑠.
Now, 𝑎 𝑛𝑘 𝑥 𝑘 =
1
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
+
1
𝑢 𝑘 𝑣 𝑘
𝑘−1
𝑖=1 𝑎 𝑛𝑘 𝑦 𝑘
𝑛
𝑘=0
𝑛
𝑘=0

≤ 𝑠𝑢𝑝𝑛
1
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
𝑎 𝑛𝑘 +
𝑎 𝑛𝑘
𝑢 𝑘 𝑣 𝑘
𝑦 𝑘
𝑘−1
𝑖=0
𝑛
𝑘=0 by (i)
≤ 𝑦 𝑘 𝑠𝑢𝑝𝑛
1
𝑢 𝑘
1
𝑣𝑖
−
1
𝑣𝑖+1
+
1
𝑢 𝑘 𝑣 𝑘
𝑘−1
𝑖=1 𝑎 𝑛𝑘 < ∞∞
𝑘=0 , as 𝑛 → ∞.
This implies 𝐴 𝑛 (𝑥) ∈ 𝑏𝑠 or 𝐴 ∈ 𝑐 𝑢, 𝑣, ∆ ∶ 𝑏𝑠 . □
Concluding Remarks
The generalization obtained here still admit improvement in the sense that the conditions obtained here
may further be simplified resulting in less restrictions on the involved matrices.
References
[1] Ahmad, Z. U., and Mursaleen, Kőthe-Teoplitz Duals of Some New Sequence Spaces and Their Matrix Transformations, Pub. Dé
L’institut Mathématique Nouvelle série tome 42 (56), 1987, p. 57-61
[2] Altay, B., and F. Başar, Some Paranormed Sequence Spaces of Non-absolute Type Derived by Weighted Mean, J. math. Anal. Appl.
319, (2006), p. 494-508
[3] Altay B., and F. Başar, The Fine Spectrum and the Matrix Domain of the Difference Operator ∆ on the Sequence Space lp, 0 < 𝑝 <
1), Comm. Math. Anal. Vol. 2 (2), 2007, p. 1-11
[4] Başarir, M., and E. E. Kara, On Some Difference Sequence Spaces of Weighted Means and Compact Operators, Ann. Funct. Anal.
2, 2011, p. 114-129
[5] Boos, J., Classical and Modern Methods in Summability, Oxford Sci. Pub., Oxford, 2000
[6] Kizmaz, H., On Certain Sequence Spaces, Canad. Math. Bull. Vol. 24 (2), 1981, p. 169-176
[7] Maddox, I. J., Spaces of Strongly Summable Sequences, Quart. J. Math. Oxford,18 (2), 1967, p. 345-355
[8] Polat, H., and F. Başar, Some Euler Spaces of Difference Sequences of Order m∗
, Acta Mathematica Scienta, 2007, 27B (2), p. 254-
266
[9] Polat, H., Vatan K. and Necip S., Difference Sequence Spaces Derived by Generalized Weighted Mean, App. Math. Lett. 24
(5), 2011, p. 608-614
[10] Simons, S., The Sequences Spaces,l(pv) and m(pv), Proc. London Math. Soc. 15 (3), 1965, p. 422-436

Matrix Transformations on Some Difference Sequence Spaces

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