Matrix Transformations on Paranormed Sequence Spaces Related To De La Vallée-Pousin Mean

International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 7 || September. 2016 || PP-04-07
www.ijmsi.org 4 | Page
Matrix Transformations on Paranormed Sequence Spaces
Related To De La Vallée-Pousin Mean
Zakawat U. Siddiqui and Ado Balili
Department of Mathematics and Statistics, University of Maiduguri, Maiduguri, Nigeria
ABSTRACT: In this paper, we determine the necessary and sufficient conditions to characterize the matrices
which transform paranormed sequence spaces into the spaces 𝑉𝜎 (𝜆) and 𝑉𝜎
∞
(𝜆) , where 𝑉𝜎 (𝜆) denotes the
space of all (𝜎, 𝜆)-convergent sequences and 𝑉𝜎
∞
(𝜆) denotes the space of all (𝜎, 𝜆)-bounded sequences defined
using the concept of de la Vallée-Pousin mean.
Keywords: de la Vall𝑒e-Pousin Mean, 𝜎-convergence, Invariant Mean, Matrix Transformations
Mathematics Subject Classification: 40A05, 40C05, 40D05
I. INTRODUCTION
We shall denote the space of real valued sequences by 𝜔.Any vector subspace of 𝜔 is called a sequence space. if
𝑥 𝜖 𝜔 , then we write 𝑥 = 𝑥 𝑘 instead of 𝑥 = (𝑥 𝑘 ) 𝑘=0
∞
. We denote the spaces of all finite, bounded, convergent
and null sequences by , 𝑙∞, 𝑐, and 𝑐0, respectively. Further, we shall use the conventions that 𝑒 = 1, 1, 1,
…and 𝑒(𝑘) as the sequence whose only non zero term is 1 in the kth place for each 𝑘 𝜖 ℕ,
A sequence space X with linear topology is called a K-space if each of the maps 𝑝𝑖 : 𝑋 → ℂ: defined by 𝑝𝑖 𝑥 =
𝑥𝑖 is continuous for all 𝑖 ∈ ℕ. A K-space is called an FK-space if X is complete linear metric space; a BK-space
is a normed FK-space.
A linear topological space X over the real field ℝ is said to be a paranormed space if there is a sub additive
function 𝑝 ∶ 𝑋 → ℝ such that 𝑝 𝜃 = 0, 𝑝 𝑥 = −𝑥 , and scalar multiplication is continuous, i. e. 𝛼 𝑛 − 𝛼 →
0 and 𝑝 𝑥 𝑛 → 0 imply 𝑝 𝛼 𝑛 𝑥 𝑛 − 𝛼𝑥 → 0 𝑎𝑠 𝑛 → ∞ for all 𝑥 ∈ 𝑋 and 𝛼 ∈ ℝ, where 𝜃 is the zero vector in
the linear space X.
Assume here and after that 𝑥 = 𝑥 𝑘 be a sequence such that 𝑥 𝑘 ≠ 0, ∀ 𝑘 𝜖 ℕ and (𝑝 𝑘 ) be the bounded sequence
of strictly positive real numbers with 𝑠𝑢𝑝𝑝 𝑘 = 𝐻 𝑎𝑛𝑑 𝑀 = max⁡(1 , 𝐻).then , the sequence spaces
𝑐0 𝑝 = { 𝑥 = 𝑥 𝑘 ∈ 𝜔 ∶ lim 𝑘→∞ |𝑥 𝑘 | 𝑝 𝑘 = 0}
𝑐 𝑝 = { 𝑥 = 𝑥 𝑘 ∈ 𝜔 : lim
𝑘→∞
|𝑥 𝑘 − 𝑙| 𝑝 𝑘 = 0, 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑙 ∈ ℂ}
ℓ∞ 𝑝 = { 𝑥 = 𝑥 𝑘 ∈ 𝜔 ∶ sup 𝑘 |𝑥 𝑘 | 𝑝 𝑘 < ∞ } and
ℓ 𝑝 = { 𝑥 = 𝑥 𝑘 ∈ 𝜔 ∶ |𝑥 𝑘
∞
𝑘 | 𝑝 𝑘 < ∞ }
were defined and studied by Maddox [1] and Simons [2].
If 𝑝 𝑘 = 𝑝 , ∀ 𝑘 ∈ ℕ for some constant 𝑝 > 0, then these spaces reduce to 𝑐0 , 𝑐, ℓ∞, 𝑎𝑛𝑑 ℓ 𝑝, respectively. Note
that 𝑐0 𝑝 is a linear metric space paranormed by 𝑕 𝑥 = sup 𝑘 |𝑥 𝑘 |
𝑝 𝑘
𝑀 . ℓ∞ 𝑝 , 𝑐 𝑝 fail to be linear metric
space because the continuity of multiplication does not hold for them. These two spaces turn out to be linear
metric spaces if and only if inf 𝑝 𝑘 > 0. ℓ 𝑝 is linear metric space paranormed by 𝑤 𝑥 = ( |𝑥 𝑘𝑘 | 𝑝 𝑘 )
1
𝑀 .
These sequence spaces are complete paranormed spaces in their respective paranorm if and only if inf 𝑝 𝑘 > 0.
However, these are not normed spaces, in general. (see Aydin and Basar [3] and Karakaya et al, [4]).
The above sequence spaces were further generalized. Bulut and Çakar [5] defined the sequence space
ℓ 𝑝, 𝑠 = { 𝑥 = 𝑥 𝑘 𝜖 𝜔 : 𝑘−𝑠
𝑘 |𝑥 𝑘 | 𝑝 𝑘 < ∞, 𝑠 ≥ 0},
which generalized the sequence space ℓ 𝑝 . They showed that ℓ 𝑝, 𝑠 isa linear sequence space paranormed by
𝑔 𝑥 = ( 𝑘−𝑠
|𝑥 𝑘𝑘 | 𝑝 𝑘 )
1
𝑀 .
Basarir [6] generalized the other sequence spaces as follows:
ℓ∞ 𝑝, 𝑠 = 𝑥 = 𝑥 𝑘 ∈ 𝜔 ∶ sup 𝑘 𝑘−𝑠
𝑥 𝑘 | 𝑝 𝑘 < ∞
𝑐0 𝑝, 𝑠 = 𝑥 = 𝑥 𝑘 ∈ 𝜔 ∶ 𝑘−𝑠
𝑥 𝑘 | 𝑝 𝑘 → 0, 𝑘 → ∞
𝑐 𝑝, 𝑠 = { 𝑥 = 𝑥 𝑘 ∈ 𝜔 : 𝑘−𝑠
𝑥 𝑘 − 𝑙| 𝑝 𝑘 → 0, 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑙, 𝑘 → ∞
It is easy to see that 𝑐0 𝑝, 𝑠 is paranormed by 𝑞 𝑥 = sup 𝑘(𝑘−𝑠
|𝑥 𝑘 | 𝑝 𝑘 )
1
𝑀 . Also ℓ∞ 𝑝, 𝑠 and 𝑐 𝑝, 𝑠 are
paranormed by 𝑞 𝑥 iff 𝑖𝑛𝑓𝑝 𝑘 > 0. All the spaces defined above are complete in their topologies.
Let X and Y be two sequence spaces and 𝐴 = (𝑎 𝑛𝑘 ) 𝑛,𝑘=1
∞
be an infinite matrix of real or complex numbers. We
denote 𝐴𝑥 = 𝐴 𝑛 𝑥 , 𝐴 𝑛 𝑥 = 𝑎 𝑛𝑘𝑘 𝑥 𝑘 provided that the series on the right converges for each n. If
𝑥 = 𝑥 𝑘 ∈ 𝑋, implies that 𝐴𝑥 ∈ 𝑌, then we say that A defines a matrix transformation from X into Y, and by
(X, Y) we denote the class of such matrices.

Matrix Transformations On Paranormed Sequence Spaces Related To De La Vallée-Pousin Mean
Let 𝜎 be a one to one mapping from the set ℕ of natural numbers into itself. A continuous linear functional 𝜑
on the space ℓ∞ is said to be an invariant mean or 𝜎-mean if and only if
( i) 𝜑 𝑥 ≥ 0 if 𝑥 ≥ 0, 𝑖. 𝑒 𝑥 𝑘 ≥ 0, for all 𝑘 ,
(ii) 𝜑 𝑒 = 1, where 𝑒 = 1, 1, 1, … ,
(iii) 𝜑 𝑥 = 𝜑 𝑥 𝜎 𝑘 , for all 𝑥 ∈ ℓ∞.
Though out this we consider the mapping 𝜎 which has no finite orbit, that is 𝜎 𝑝
𝑘 ≠ 𝑘 for all integer 𝑘 ≥
0 𝑎𝑛𝑑 𝑝 ≥ 1, where 𝜎 𝑝
𝑘 denotes the p-th iterate of 𝜎 at k. Note that a 𝜎 −mean extends the limit functional
on the space c in the sense that 𝜎 𝑥 = 𝑙𝑖𝑚𝑥 for all 𝑥 ∈ ℂ, (see Mursaleen [7]). Consequently, 𝑐 ⊂ 𝑉𝜎 , the set of
bounded sequence all of whose 𝜎 −means are equal.
We say that a sequence 𝑥 = 𝑥 𝑘 is 𝜎 −convergent iff 𝑥 ∈ 𝑉𝜎 . Using this concept, Schaefer [8] defined and
characterized 𝜎 −convervative and 𝜎 −coercive matrices. If 𝜎 is translation, then 𝑉𝜎 is reduced to 𝑓 of almost
convergent sequences (Lorentz [9]). As an application of almost convergence, Mohiuddine [10] established
some approximation theorems for sequences of positive linear operators through this concept. The idea of
𝜎 −convergence for double sequences was introduced in Çakar et al [11].
II. BASIC DEFINITIONS
Definition 2.1 (de la Vallée-Poussin mean): Let 𝜆 = 𝜆 𝑚 be a non –decreasing sequence of positive numbers
tending to ∞ such that 𝜆 𝑚+1 ≤ 𝜆 𝑚 + 1, 𝜆1 = 0, then
𝜌 𝑚 𝑥 =
1
𝜆 𝑚
𝑥𝑗𝑗∈𝐼 𝑚
is called the generalized de la Vallée-Poussin mean, where 𝐼 𝑚 = [𝑚 − 𝜆 𝑚 + 1, 𝑚 ].
Definition 2.2 (Mursaleen et al [12]) A sequence 𝑥 = 𝑥 𝑘 of real numbers is said to be 𝜎, 𝜆 −convergent to a
number L iff
lim 𝑚→∞
1
𝜆 𝑚
𝑥 𝜎 𝑗 (𝑛)𝑗 ∈𝐼 𝑚
= 𝐿
uniformly in n, and 𝑉𝜎 (𝜆) denotes the set of all such sequences i.e
𝑉𝜎 𝜆 = { 𝑥 ∈ ℓ∞ ∶ lim 𝑚→∞
1
𝜆 𝑚
𝑥 𝜎 𝑗 (𝑛)𝑗∈𝐼 𝑚
= 𝐿, uniformly in n}
Note that a convergent sequence is 𝜎, 𝜆 −convergent but converse need not hold.
2.1 Remark
(i) If 𝜎 𝑛 = 𝑛 + 1, then 𝑉𝜎 (𝜆) is reduced to the 𝑓𝜆 (see Mursaleen et al [13])
(ii) If 𝜆 𝑚 = 𝑚, then 𝑉𝜎 (𝜆) is reduced to the space 𝑉𝜎
(iii) If 𝜎 𝑛 = 𝑛 + 1 and 𝜆 𝑚 = 𝑚, then 𝑉𝜎 (𝜆) is reduced to the space 𝑓, (almost convergent sequences)
(iv) 𝑐 ⊂ 𝑉𝜎 𝜆 ⊂ ℓ∞.
Definition 2.3 (Mohiuddine [13]) A sequence 𝑥 = (𝑥 𝑘 ) of real numbers is said to be 𝜎, 𝜆 −bounded if and
only if 𝑠𝑢𝑝 𝑚,𝑛 |
1
𝜆 𝑚
| < ∞, and we denote by 𝑉𝜎
∞
(𝜆), the set of all such sequences i. e
𝑉𝜎
∞
(𝜆) = 𝑥 ∈ ℓ∞ ∶ sup 𝑚,𝑛 |𝑡 𝑚𝑛 𝑥 < ∞ },
where, 𝑡 𝑚𝑛 𝑥 =
1
𝜆 𝑚
2.2 Remark
𝑐 ⊂ 𝑉𝜎 (𝜆) ⊂ 𝑉𝜎
∞
(𝜆) ⊂ ℓ∞.
III. SOME KNOWN RESULTS
The following results play vital role in our main results
Lemma 3.1 (Theorem 1; Bulut and Çakar [5]):
(i) If 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞, 𝑎𝑛𝑑 𝑝 𝑘
−1
+ 𝑞 𝑘
−1
= 1,for k∈ ℕ, then
ℓ†
𝑝, 𝑠 = { 𝑎 = 𝑎 𝑘 : 𝑘 𝑠 𝑞 𝑘−1∞
𝑘=1 𝑁
−
𝑞 𝑘
𝑝 𝑘 |𝑎 𝑘 | 𝑞 𝑘 < ∞, 𝑠 > 0, 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑁 > 1}
(ii) If 0 < 𝑚 = 𝑖𝑛𝑓𝑘 𝑝 𝑘 ≤ 𝑝 𝑘 ≤ 1, for each k =1,2,3, …, then ℓ†
𝑝, 𝑠 = 𝑚(𝑝, 𝑠), where
𝑚 𝑝, 𝑠 = { 𝑎 = 𝑎 𝑘 : sup 𝑘 𝑘−𝑠
|𝑎 𝑘 | 𝑝 𝑘 < ∞, 𝑠 ≥ 1
Lemma 3.2 (Theorem3; Bulut and Çakar [5]): ( i ) If 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞, for every 𝑘 ∈ ℕ, then
𝐴 ∈ (ℓ 𝑝, 𝑠 , ℓ∞) if and only if there exists an integer N> 1, such that
sup 𝑛 |𝑎 𝑛𝑘
∞
𝑘=1 | 𝑞 𝑘 𝑁−𝑞 𝑘 𝑘 𝑠(𝑞 𝑘−1)
< ∞, (3.1)
(ii) If 0 < 𝑚 = 𝑖𝑛𝑓𝑘 𝑝 𝑘 ≤ 𝑝 𝑘 ≤ 1, for each 𝑘 ∈ ℕ, then 𝐴 ∈ (ℓ 𝑝, 𝑠 , ℓ∞) if and only if
sup 𝑛𝑘 |𝑎 𝑛𝑘 | 𝑝 𝑘 𝑘 𝑠
= 𝑀 < ∞. (3.2)
Lemma 3.3 (Theorem 4; Bulut and Çakar [5]):
(i) Let 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞ for every k. Then 𝐴 ∈ (ℓ 𝑝, 𝑠 , 𝑐) if and only if (3.1) holds together with

𝑎 𝑛𝑘 → 𝛼 𝑘, (𝑛 → ∞, 𝑘 𝑓𝑖𝑥𝑒𝑑 ) (3.3)
( ii ) If 0 < 𝑚 = 𝑖𝑛𝑓𝑘 𝑝 𝑘 ≤ 𝑝 𝑘 ≤ 1, for some 𝑘 ∈ ℕ.Then 𝐴 ∈ (ℓ 𝑝, 𝑠 , 𝑐), if and only if condition (3.2) and
(3.3)
Lemma 3.4 (Theorem 2.1 Mohiuddine [13]): The spaces 𝑉𝜎 (𝜆) and 𝑉𝜎
∞
(𝜆) are BK spaces with the norm
𝑥 = sup 𝑚,𝑛≥0 |𝑡 𝑚𝑛 𝑥 |. (3.4)
Lemma 3.5 (Theorem 3.1 Mohiuddine [13] ): Let 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞, for every 𝑘 ∈ ℕ.Then
𝐴 ∈ (ℓ 𝑝 , 𝑉𝜎
∞
𝜆 ) if and only if there exists an integer 𝑁 > 1 such that
sup 𝑚,𝑛 |
1
𝜆 𝑚
𝑘 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗 ∈𝐼 𝑚
| 𝑞 𝑘 𝑁−𝑞 𝑘 < ∞, (3.5)
Lemma 3.6 (Theorem 3.2 Mohiuddine [13]):
(i) 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞, for every 𝑘 ∈ ℕ.Then 𝐴 ∈ (ℓ 𝑝 , 𝑉𝜎 𝜆 ) if and only if ( i ) condition (3.5)
( ii ) lim 𝑚→∞
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗 ∈𝐼 𝑚
= 𝛼 𝑘 uniformly in n, for every 𝑘 ∈ ℕ
IV. MAIN RESULTS
We shall prove the following results.
Theorem 4.1 Let 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞ for every 𝑘 ∈ ℕ. Then 𝐴 ∈ (ℓ 𝑝, 𝑠 , 𝑉𝜎
∞
𝜆 ) if and only if there
exists an integer 𝐷 > 1, such that
sup 𝑚,𝑛 |
1
𝜆 𝑚
𝑘 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
| 𝑞 𝑘 𝐷−𝑞 𝑘 𝑘 𝑠(𝑞 𝑘−1)
< ∞. (4.1)
Proof. Sufficiency. Let (4.1) hold and that 𝑥 ∈ ℓ(𝑝, 𝑠).Using the inequality
𝑎 𝑏 ≤ 𝐷(|𝑎| 𝑞
𝐷−𝑞
+ 𝑏| 𝑝
for 𝐷 > 0 and 𝑎, 𝑏 complex numbers (𝑝−1
+ 𝑞−1
= 1) (Maddox [14] )
We have
𝑡 𝑚𝑛 𝐴𝑥 = |
1
𝜆 𝑚
𝑘 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗∈𝐼 𝑚
𝑥 𝑘 |
⟹ 𝐷[|
1
𝜆 𝑚
𝑘 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
| 𝑞 𝑘 𝐷−𝑞 𝑘 . 𝑘 𝑠(𝑞 𝑘−1)
+ 𝑘−𝑠
𝑥 𝑘 | 𝑝 𝑘 ,
where 𝑝 𝑘
−1
+ 𝑞 𝑘
−1
= 1.
Taking the supremum over m, n on both sides and using (4.1), we get
𝐴𝑥 ∈ 𝑉𝜎
∞
𝜆 for 𝑥 ∈ ℓ(𝑝, 𝑠). i. e 𝐴 ∈ (ℓ 𝑝, 𝑠 , 𝑉𝜎
∞
𝜆 ).
Necessity. Let 𝐴 ∈ ℓ 𝑝, 𝑠 , 𝑉𝜎
∞
𝜆 . We put 𝑞 𝑛 𝑥 = sup 𝑘 𝑘−𝑠
𝑡 𝑚𝑛 𝐴𝑥 .
It is easy to see that for 𝑛 ≥ 0, 𝑞 𝑛 is a continuous semi norm on ℓ(𝑝, 𝑠) and (𝑞 𝑛 ) is point wise bounded on
ℓ(𝑝, 𝑠). Suppose that (4.1) is not true. Then there exists 𝑥 ∈ ℓ(𝑝, 𝑠) with sup 𝑛 𝑞 𝑛 𝑥 = ∞. By the principle of
condensation of singularities (Yosida [15]), the set 𝑥 ∈ ℓ 𝑝, 𝑠 : sup 𝑛 𝑞 𝑛 𝑥 = ∞ is of second category in
ℓ(𝑝, 𝑠) and hence non empty. Thus there exists 𝑥 ∈ ℓ(𝑝, 𝑠) with sup 𝑛 𝑞 𝑛 𝑥 = ∞.But this contradict the fact
that (𝑞 𝑛 ) is pointwise bounded on ℓ(𝑝, 𝑠). Now by the Banach-Steinhaus theorem, there is a constant M such
that
𝑞 𝑛 𝑥 ≤ 𝑀𝑔(𝑥) (4.2)
Now define a sequence 𝑥 = (𝑥 𝑘 ) by
𝑥 𝑘 = 𝛿
𝑀
𝑝 𝑘 (𝑠𝑔𝑛
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
)|
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
| 𝑞 𝑘−1
𝑉−1
𝐷
−
𝑞 𝑘
𝑝 𝑘 . 𝑘 𝑠 𝑞 𝑘−1
1 ≤ 𝑘 ≤ 𝑘0
0, 𝑘 > 𝑘0
where 0 < 𝛿 < 1 and 𝑉 = |
1
𝜆 𝑚
𝑘0
𝑘=1 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗 ∈𝐼 𝑚
| 𝑞 𝑘 𝐷−𝑞 𝑘 . 𝑘 𝑠(𝑞 𝑘−1)
.
Then it is easy to see that 𝑥 ∈ ℓ(𝑝, 𝑠) and 𝑔(𝑥) ≤ 𝛿.
Applying this sequence to (4.2) we get the condition (4.1)
This completes the proof of the theorem.
Theorem 4.2 Let 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞ for every 𝑘 ∈ ℕ. Then 𝐴 ∈ (ℓ 𝑝, 𝑠 , 𝑉𝜎 𝜆 ) if and only if
(i) Condition (4.1) of Theorem 4.1 holds
(ii) lim 𝑚
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗∈𝐼 𝑚
= 𝛼 𝑘 , uniformly in n for every 𝑘 ∈ ℕ.
Proof. Sufficiency. Let ( i ) and ( ii ) hold and 𝑥 ∈ ℓ(𝑝, 𝑠). For 𝑗 ≥ 1
|
1
𝜆 𝑚
𝑗
𝑘=1
𝑎 𝜎 𝑗 𝑛 ,𝑘
𝑗 ∈𝐼 𝑚
| 𝑞 𝑘 𝐷−𝑞 𝑘 . 𝑘 𝑠(𝑞 𝑘−1)
≤ sup
𝑚
|
1
𝜆 𝑚
𝑘
𝑎 𝜎 𝑗 𝑛 ,𝑘
𝑗 ∈𝐼 𝑚
| 𝑞 𝑘 𝐷−𝑞 𝑘 . 𝑘 𝑠(𝑞 𝑘−1)
< ∞
for every n. Therefore
|𝛼 𝑘 | 𝑞 𝑘
𝑘 𝐷−𝑞 𝑘 . 𝑘−𝑠 𝑞 𝑘−1
= lim𝑗 lim 𝑘 |
1
𝜆 𝑚
𝑗
𝑘=1 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
| 𝑞 𝑘 . 𝐷−𝑞 𝑘 . 𝑘−𝑠 𝑞 𝑘−1

≤ sup 𝑘 |
1
𝜆 𝑚
𝑘 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗∈𝐼 𝑚
| 𝑞 𝑘 𝐷−𝑞 𝑘 . 𝑘 𝑠(𝑞 𝑘−1)
< ∞,
where 𝑝 𝑘
−1
+ 𝑞 𝑘
−1
= 1. Consequently reasoning as in the proof of the sufficiency of Theorem 4.1, the series
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘 𝑥 𝑘𝑗∈𝐼 𝑚𝑘 and 𝛼 𝑘𝑘 𝑥 𝑘 converge for every 𝑛, 𝑚, 𝑗 𝑎𝑛𝑑 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ ℓ(𝑝, 𝑠). For a given
𝜀 > 0 𝑎𝑛𝑑 𝑥 ∈ ℓ(𝑝, 𝑠), choose 𝑘0 such that
( 𝑘−𝑠∞
𝑘=𝑘0+1 |𝑥 𝑘 | 𝑝 𝑘 )
1
𝑀 𝜀, (4.3)
where 𝑀 = sup 𝑘 𝑝 𝑘 condition ( ii ) implies that there exists 𝑚0 such that
[
1
𝜆 𝑚
𝑚0
𝑘=1 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
− 𝛼 𝑘 | <
𝜀
2
for all 𝑚 ≥ 𝑚0 and uniformly in n. Now, since
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘 𝑥 𝑘𝑗∈𝐼 𝑚𝑘 and 𝛼 𝑘𝑘 𝑥 𝑘 converge (absolutely) uniformly in m, n and for every 𝑥 ∈ ℓ(𝑝, 𝑠), we have
[
1
𝜆 𝑚
∞
𝑘=𝑘0+1 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
− 𝛼 𝑘 𝑥 𝑘 converges uniformly in m, n. Hence by conditions (i) and (ii)
[
1
𝜆 𝑚
∞
− 𝛼 𝑘 | <
𝜀
2
for 𝑚 ≥ 𝑚0 and uniformly in n. Therefore
[
1
𝜆 𝑚
∞
− 𝛼 𝑘 𝑥 𝑘 → ∞, (𝑚 → ∞) , uniformly in n, i. e
lim 𝑚
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘 𝑥 𝑘𝑗∈𝐼 𝑚𝑘 = 𝛼 𝑘𝑘 𝑥 𝑘 (4.4)
Necessity. Let 𝐴 ∈ (ℓ 𝑝, 𝑠 , 𝑉𝜎 𝜆 ). Since 𝑉𝜎 𝜆 ⊂ 𝑉𝜎
∞
𝜆 . Condition (i) follows by Theorem 4.1. Since 𝑒(𝑘)
=
(0, 0, 0, … , 1 𝑘𝑡𝑕 place , 0, 0, … ) ∈ ℓ(𝑝, 𝑠) and condition (ii) follows immediately by (4.4).
This completes the proof of the theorem.
V. CONCLUSION
The notion of invariant mean and de la Vallée-Poussin mean plays very active role in the recent research on
matrix transformations. With the help of these two notions, the concept of (𝜎, 𝜆)-convergent sequences denoted
by 𝑉𝜎 𝜆 and (𝜎, 𝜆)-bounded sequences denoted by 𝑉𝜎
∞
𝜆 and also (𝜎, 𝜃)-convergent sequence, (𝑉𝜎 𝜃 ) and
(𝜎, 𝜃)-bounded sequence, (𝑉𝜎
∞
𝜃 ) sequences and many others have evolved. Related to these sequence spaces,
many matrix classes have been characterized. As we have characterized the matrix classes (ℓ 𝑝, 𝑠 , 𝑉𝜎 𝜆 ) and
(ℓ 𝑝, 𝑠 , 𝑉𝜎
∞
𝜆 ) in our main results here, some other characterizations may also evolve.
V1. ACKNOWLEDGEMENT
The authors are thankful to Mr. Sighat U. Siddiqui for his logistic support in the communication process of the
paper, which resolved some technical problems.
REFERENCES
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