Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 949282, 10 pages
http://dx.doi.org/10.1155/2013/949282
Research Article
Domain of the Double Sequential Band Matrix 𝐵(̃𝑟, 𝑠̃) in
the Sequence Space ℓ(𝑝)∗
Havva Nergiz and Feyzi BaGar
Department of Mathematics, Faculty of Arts and Sciences, Fatih University, The Hadımköy Campus, Büyükçekmece,
34500 Istanbul, Turkey
Correspondence should be addressed to Feyzi Başar; feyzibasar@gmail.com
Received 17 October 2012; Accepted 29 January 2013
Academic Editor: Ferhan M. Atici
Copyright © 2013 H. Nergiz and F. Başar. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
The sequence space ℓ(𝑝) was introduced by Maddox (1967). Quite recently, the domain of the generalized difference matrix 𝐵(𝑟, 𝑠)
̃ 𝑝)
in the sequence space ℓ𝑝 has been investigated by Kirişçi and Başar (2010). In the present paper, the sequence space ℓ(𝐵,
of nonabsolute type has been studied which is the domain of the generalized difference matrix 𝐵(̃𝑟, 𝑠̃) in the sequence space
̃ 𝑝) have been determined, and the Schauder basis has
ℓ(𝑝). Furthermore, the alpha-, beta-, and gamma-duals of the space ℓ(𝐵,
̃
been given. The classes of matrix transformations from the space ℓ(𝐵, 𝑝) to the spaces ℓ∞ , c and c0 have been characterized.
̃ 𝑝) to the Euler, Riesz, difference,
Additionally, the characterizations of some other matrix transformations from the space ℓ(𝐵,
and so forth sequence spaces have been obtained by means of a given lemma. The last section of the paper has been devoted to
conclusion.
1. Preliminaries, Background, and Notation
By 𝑤, we denote the space of all real valued sequences. Any
vector subspace of 𝑤 is called a sequence space. We write ℓ∞ ,
𝑐, and 𝑐0 for the spaces of all bounded, convergent, and null
sequences, respectively. Also by 𝑏𝑠, 𝑐𝑠, ℓ1 , and ℓ𝑝 , we denote
the spaces of all bounded, convergent, absolutely convergent
and 𝑝-absolutely convergent series, respectively, where 1 <
𝑝 < ∞.
A linear topological space 𝑋 over the real field R is said
to be a paranormed space if there is a subadditive function
𝑔 : 𝑋 → R such that 𝑔(𝜃) = 0, 𝑔(𝑥) = 𝑔(−𝑥) and
scalar multiplication is continuous; that is, |𝛼𝑛 − 𝛼| → 0
and 𝑔(𝑥𝑛 − 𝑥) → 0 imply 𝑔(𝛼𝑛 𝑥𝑛 − 𝛼𝑥) → 0 for all 𝛼’s
in R and all 𝑥’s in 𝑋, where 𝜃 is the zero vector in the linear
space 𝑋.
Assume here and after that (𝑝𝑘 ) is a bounded sequence
of strictly positive real numbers with sup 𝑝𝑘 = 𝐻 and
𝑀 = max{1, 𝐻}. Then, the linear spaces ℓ(𝑝) were defined
by Maddox [1] (see also Simons [2] and Nakano [3])
as follows:
󵄨 󵄨𝑝
ℓ (𝑝) = {𝑥 = (𝑥𝑘 ) ∈ 𝑤 : ∑󵄨󵄨󵄨𝑥𝑘 󵄨󵄨󵄨 𝑘 < ∞} ,
𝑘
(1)
(0 < 𝑝𝑘 ≤ 𝐻 < ∞)
which is the complete space paranormed by
1/𝑀
󵄨 󵄨𝑝
𝑔 (𝑥) = (∑󵄨󵄨󵄨𝑥𝑘 󵄨󵄨󵄨 𝑘 )
.
(2)
𝑘
For simplicity in notation, here and in what follows, the
summation without limits runs from 0 to ∞. We assume
−1
throughout that 𝑝𝑘−1 + (𝑝𝑘󸀠 ) = 1 and denote the collection
of all finite subsets of N = {0, 1, 2, . . .} by F and use the
convention that any term with negative subscript is equal to
naught.
Let 𝜆, 𝜇 be any two sequence spaces and let 𝐴 =
(𝑎𝑛𝑘 ) be an infinite matrix of real or complex numbers 𝑎𝑛𝑘 ,
2
Abstract and Applied Analysis
where 𝑛, 𝑘 ∈ N. Then, we say that 𝐴 defines a matrix mapping
from 𝜆 into 𝜇, and we denote it by writing 𝐴 : 𝜆 → 𝜇; if for
every sequence 𝑥 = (𝑥𝑘 ) ∈ 𝜆 the sequence 𝐴𝑥 = {(𝐴𝑥)𝑛 }, the
𝐴-transform of 𝑥, is in 𝜇, where
(𝐴𝑥)𝑛 = ∑𝑎𝑛𝑘 𝑥𝑘 ,
for each 𝑛 ∈ N.
𝑘
(3)
By (𝜆 : 𝜇), we denote the class of all matrices 𝐴 such that
𝐴 : 𝜆 → 𝜇. Thus, 𝐴 ∈ (𝜆 : 𝜇) if and only if the series on the
right side of (3) converges for each 𝑛 ∈ N and every 𝑥 ∈ 𝜆,
and we have 𝐴𝑥 = {(𝐴𝑥)𝑛 }𝑛∈N ∈ 𝜇 for all 𝑥 ∈ 𝜆. A sequence
𝑥 is said to be 𝐴-summable to 𝛼 if 𝐴𝑥 converges to 𝛼 which
is called the 𝐴-limit of 𝑥.
The shift operator 𝑃 is defined on 𝜔 by (𝑃𝑥)𝑛 = 𝑥𝑛+1 for
all 𝑛 ∈ N. A Banach limit 𝐿 is defined on ℓ∞ , as a nonnegative
linear functional, such that 𝐿(𝑃𝑥) = 𝐿(𝑥) and 𝐿(𝑒) = 1, where
𝑒 = (1, 1, 1, . . .). A sequence 𝑥 = (𝑥𝑘 ) ∈ ℓ∞ is said to be almost
convergent to the generalized limit 𝑙 if all Banach limits of 𝑥
are 𝑙 and is denoted by 𝑓 − lim 𝑥𝑘 = 𝑙. Lorentz [4] proved that
𝑓 − lim 𝑥𝑘 = 𝑙
1 𝑚
∑ 𝑥𝑘+𝑛 = 𝑙 uniformly in 𝑛.
𝑚→∞𝑚 + 1
𝑘=0
iff lim
(4)
It is well known that a convergent sequence is almost
convergent such that its ordinary and generalized limits are
equal. By 𝑓, we denote the space of all almost convergent
sequences; that is,
𝑓 :=
𝑚
𝑥𝑛+𝑘
= 𝑙 uniformly in 𝑛}.
𝑚
+1
𝑘=0
(5)
{𝑥 = (𝑥𝑘 ) ∈ 𝜔 : ∃𝑙 ∈ C ∋ lim ∑
𝑚→∞
Define the double sequential band matrix 𝐵(̃𝑟, 𝑠̃) =
{𝑏𝑛𝑘 (𝑟𝑘 , 𝑠𝑘 )} by
𝑟,
{
{𝑘
𝑏𝑛𝑘 (𝑟𝑘 , 𝑠𝑘 ) = {𝑠𝑘 ,
{
{0,
𝑘 = 𝑛,
𝑘 = 𝑛 − 1,
otherwise
(6)
for all 𝑘, 𝑛 ∈ N, where 𝑟̃ = (𝑟𝑘 ) and 𝑠̃ = (𝑠𝑘 ) are
the convergent sequences. We should note that the double
sequential band matrices were firstly used by Srivastava and
Kumar [5, 6], Panigrahi and Srivastava [7], and Akhmedov
and El-Shabrawy [8].
The main purpose of this paper, which is a continuation
of Kirişçi and Başar [9], is to introduce the sequence space
̃ 𝑝) of nonabsolute type consisting of all sequences whose
ℓ(𝐵,
𝐵(̃𝑟, 𝑠̃)-transforms are in the space ℓ(𝑝). Furthermore, the
basis is constructed and the alpha-, beta-, and gamma-duals
̃ 𝑝). Moreover, the matrix
are computed for the space ℓ(𝐵,
̃ 𝑝) to some sequence
transformations from the space ℓ(𝐵,
spaces are characterized. Finally, we note open problems and
further suggestions.
It is clear that Δ(1) can be obtained as a special case of
𝐵(̃𝑟, 𝑠̃) for 𝑟̃ = 𝑒 and 𝑠̃ = −𝑒 and it is also trivial that 𝐵(̃𝑟, 𝑠̃)
is reduced in the special case 𝑟̃ = 𝑟𝑒 and 𝑠̃ = 𝑠𝑒 to the
generalized difference matrix 𝐵(𝑟, 𝑠). So, the results related to
the matrix domain of the matrix 𝐵(̃𝑟, 𝑠̃) are more general and
more comprehensive than the corresponding consequences
of the matrix domains of Δ(1) and 𝐵(𝑟, 𝑠).
The rest of this paper is organized as follows. In Section 2,
̃ 𝑝) is defined and proved that
the linear sequence space ℓ(𝐵,
it is a complete paranormed space with a Schauder basis.
Section 3 is devoted to the determination of alpha-, beta-,
̃ 𝑝). In Section 4, the
and gamma-duals of the space ℓ(𝐵,
̃ 𝑝) : 𝑓), (ℓ(𝐵,
̃ 𝑝) : 𝑐),
̃ 𝑝) : ℓ∞ ), (ℓ(𝐵,
classes (ℓ(𝐵,
̃ 𝑝) : 𝑐0 ) of infinite matrices are characterized.
and (ℓ(𝐵,
Additionally, the characterizations of some other classes of
̃ 𝑝) to the Euler,
matrix transformations from the space ℓ(𝐵,
Riesz, difference, and so forth sequence spaces are obtained
by means of a given lemma. In the final section of the paper,
open problems and further suggestions are noted.
̃ of
2. The Sequence Space ℓ(𝐵,𝑝)
Nonabsolute Type
In this section, we introduce the complete paranormed linear
̃ 𝑝).
sequence space ℓ(𝐵,
The matrix domain 𝜆 𝐴 of an infinite matrix 𝐴 in a
sequence space 𝜆 is defined by
𝜆 𝐴 = {𝑥 = (𝑥𝑘 ) ∈ 𝜔 : 𝐴𝑥 ∈ 𝜆} .
(7)
Choudhary and Mishra [10] defined the sequence space ℓ(𝑝)
which consists of all sequences such that 𝑆-transforms of
them are in the space ℓ(𝑝), where 𝑆 = (𝑠𝑛𝑘 ) is defined by
1,
𝑠𝑛𝑘 = {
0,
0 ≤ 𝑘 ≤ 𝑛,
𝑘 > 𝑛,
(8)
for all 𝑘, 𝑛 ∈ N. Başar and Altay [11] have recently examined
the space 𝑏𝑠(𝑝) which is formerly defined by Başar in [12]
as the set of all series whose sequences of partial sums are
in ℓ∞ (𝑝). More recently, Aydın and Başar [13] have studied
the space 𝑎𝑟 (𝑢, 𝑝) which is the domain of the matrix 𝐴𝑟 in
the sequence space ℓ(𝑝), where the matrix 𝐴𝑟 = {𝑎𝑛𝑘 (𝑟)} is
defined by
𝑘
{ 1 + 𝑟 𝑢 , 0 ≤ 𝑘 ≤ 𝑛,
𝑎𝑛𝑘 (𝑟) = { 𝑛 + 1 𝑘
𝑘 > 𝑛,
{0,
(9)
for all 𝑘, 𝑛 ∈ N, (𝑢𝑘 ) such that 𝑢𝑘 ≠ 0 for all 𝑘 ∈ N and 0 < 𝑟 <
1. Altay and Başar [14] have studied the sequence space 𝑟𝑡 (𝑝)
which is derived from the sequence space ℓ(𝑝) of Maddox
by the Riesz means 𝑅𝑡 . With the notation of (7), the spaces
ℓ(𝑝), 𝑏𝑠(𝑝), 𝑎𝑟 (𝑢, 𝑝), and 𝑟𝑡 (𝑝) can be redefined by
ℓ (𝑝) = [ℓ (𝑝)]𝑆 ,
𝑎𝑟 (𝑢, 𝑝) = [ℓ (𝑝)]𝐴𝑟 ,
𝑏𝑠 (𝑝) = [ℓ∞ (𝑝)]𝑆 ,
𝑟𝑡 (𝑝) = [ℓ (𝑝)]𝑅𝑡 .
(10)
Abstract and Applied Analysis
3
Following Choudhary and Mishra [10], Başar and Altay
[11], Altay and Başar [14–17], and Aydın and Başar [13, 18],
̃ 𝑝) as the set of all
we introduce the sequence space ℓ(𝐵,
sequences whose 𝐵(̃𝑟, 𝑠̃)-transforms are in the space ℓ(𝑝); that
is
̃ 𝑝) := {(𝑥𝑘 ) ∈ 𝑤 : ∑󵄨󵄨󵄨𝑠𝑘−1 𝑥𝑘−1 + 𝑟𝑘 𝑥𝑘 󵄨󵄨󵄨𝑝𝑘 < ∞} ,
ℓ (𝐵,
󵄨
󵄨
(11)
(0 < 𝑝𝑘 ≤ 𝐻 < ∞) .
It is trivial that in the case 𝑝𝑘 = 𝑝 for all 𝑘 ∈ N, the sequence
̃ 𝑝) is reduced to the sequence space ℓ̃𝑝 which is
space ℓ(𝐵,
introduced by Kirişçi and Başar [9]. With the notation of (7),
̃ 𝑝) as follows:
we can redefine the space ℓ(𝐵,
̃ 𝑝) := [ℓ (𝑝)]
ℓ (𝐵,
𝐵(̃𝑟,̃𝑠) .
(12)
Define the sequence 𝑦 = (𝑦𝑘 ), which will be frequently used,
as the 𝐵(̃𝑟, 𝑠̃)-transform of a sequence 𝑥 = (𝑥𝑘 ); that is,
∀𝑘 ∈ N.
(13)
̃ 𝑝) are linearly isomorphic by
Since the spaces ℓ(𝑝) and ℓ(𝐵,
̃ 𝑝) if
Corollary 4, one can easily observe that 𝑥 = (𝑥𝑘 ) ∈ ℓ(𝐵,
and only if 𝑦 = (𝑦𝑘 ) ∈ ℓ(𝑝), where the sequences 𝑥 = (𝑥𝑘 )
and 𝑦 = (𝑦𝑘 ) are connected with the relation (13).
Now, we may begin with the following theorem which is
essential in the text.
̃ 𝑝) is a complete linear metric space paraTheorem 1. ℓ(𝐵,
normed by the paranorm
1/𝑀
󵄨
󵄨𝑝
ℎ (𝑥) = (∑󵄨󵄨󵄨𝑠𝑘−1 𝑥𝑘−1 + 𝑟𝑘 𝑥𝑘 󵄨󵄨󵄨 𝑘 )
ℎ (𝜆 𝑛 𝑥(𝑛) − 𝜆𝑥) ≤ ℎ [(𝜆 𝑛 − 𝜆) (𝑥(𝑛) − 𝑥)]
+ ℎ [𝜆 (𝑥(𝑛) − 𝑥)]
𝑘
𝑦𝑘 = {𝐵 (̃𝑟, 𝑠̃) 𝑥}𝑘 = 𝑟𝑘 𝑥𝑘 + 𝑠𝑘−1 𝑥𝑘−1 ,
Let (𝜆 𝑛 ) be a sequence of scalars with 𝜆 𝑛 → 𝜆, as 𝑛 →
∞
̃ 𝑝)
∞, and let (𝑥(𝑛) )𝑛=0 be a sequence of elements 𝑥(𝑛) ∈ ℓ(𝐵,
with ℎ(𝑥(𝑛) − 𝑥) → 0, as 𝑛 → ∞. We observe that
.
(14)
𝑘
̃ 𝑝) is linear with
Proof. It is easy to see that the space ℓ(𝐵,
respect to the coordinate-wise addition and scalar multiplication. Therefore, we first show that it is a paranormed space
with the paranorm ℎ defined by (14).
It is clear that ℎ(𝜃) = 0 where 𝜃 = (0, 0, 0, . . .) and ℎ(𝑥) =
̃ 𝑝).
ℎ(−𝑥) for all 𝑥 ∈ ℓ(𝐵,
̃
Let 𝑥, 𝑦 ∈ ℓ(𝐵, 𝑝); then by Minkowski’s inequality we have
+ ℎ [(𝜆 𝑛 − 𝜆) 𝑥] .
It follows from 𝜆 𝑛 → 𝜆 (𝑛 → ∞) that |𝜆 𝑛 − 𝜆| < 1 for all
sufficiently large 𝑛; hence
lim ℎ [(𝜆 𝑛 − 𝜆) (𝑥(𝑛) − 𝑥)] ≤ lim ℎ (𝑥(𝑛) − 𝑥) = 0. (17)
𝑛→∞
𝑛→∞
Furthermore, we have
lim ℎ [𝜆 (𝑥(𝑛) − 𝑥)] ≤ max {1, |𝜆|𝑀} lim ℎ (𝑥(𝑛) − 𝑥) = 0.
𝑛󳨀→∞
𝑛󳨀→∞
(18)
Also, we have
󵄨
󵄨
lim ℎ [(𝜆 𝑛 − 𝜆) 𝑥] ≤ lim 󵄨󵄨󵄨𝜆 𝑛 − 𝜆󵄨󵄨󵄨 ℎ (𝑥) = 0.
𝑛󳨀→∞
𝑛󳨀→∞
𝑘
1/𝑀
󵄨
󵄨𝑝 /𝑀 𝑀
= {∑[󵄨󵄨󵄨𝑠𝑘−1 (𝑥𝑘−1 +𝑦𝑘−1)+𝑟𝑘 (𝑥𝑘 +𝑦𝑘)󵄨󵄨󵄨 𝑘 ] }
𝑘
1/𝑀
󵄨
󵄨𝑝
≤ (∑󵄨󵄨󵄨𝑠𝑘−1 𝑥𝑘−1 + 𝑟𝑘 𝑥𝑘 󵄨󵄨󵄨 𝑘 )
󵄨󵄨 ̃ 𝑛
̃ 𝑛 ) − (𝐵𝑥
̃ 𝑚 ) 󵄨󵄨󵄨󵄨 ≤ [∑󵄨󵄨󵄨󵄨(𝐵𝑥
̃ 𝑚 ) 󵄨󵄨󵄨󵄨𝑝𝑘 ]
󵄨󵄨(𝐵𝑥 ) − (𝐵𝑥
󵄨
󵄨
𝑘
𝑘󵄨
𝑘
𝑘󵄨
1/𝑀
𝑘
(20)
= ℎ (𝑥𝑛 − 𝑥𝑚 ) < 𝜀
̃ 0 )𝑘 , (𝐵𝑥
̃ 1 )𝑘 , (𝐵𝑥
̃ 2 )𝑘 , . . .} is a
for every 𝑛, 𝑚 > 𝑛0 (𝜀), {(𝐵𝑥
Cauchy sequence of real numbers for every fixed 𝑘 ∈ N. Since
̃ 𝑛 )𝑘 → (𝐵𝑥)
̃ 𝑘 as 𝑛 → ∞.
R is complete, it converges, say (𝐵𝑥
̃ 0 , (𝐵𝑥)
̃ 1 , (𝐵𝑥)
̃ 2 , . . . we
Using these infinitely many limits (𝐵𝑥)
̃ 0 , (𝐵𝑥)
̃ 1 , (𝐵𝑥)
̃ 2 , . . .}. For each 𝐾 ∈ N
define the sequence {(𝐵𝑥)
and 𝑛, 𝑚 > 𝑛0 (𝜀)
1/𝑀
𝐾
󵄨 ̃ 𝑛
̃ 𝑚 ) 󵄨󵄨󵄨󵄨𝑝𝑘 ]
[ ∑ 󵄨󵄨󵄨󵄨(𝐵𝑥
)𝑘 − (𝐵𝑥
𝑘󵄨
𝑘
(19)
Then, we obtain from (16), (17), (18), and (19) that ℎ(𝜆 𝑛 𝑥(𝑛) −
𝜆𝑥) → 0, as 𝑛 → ∞. This shows that ℎ is a paranorm on
̃ 𝑝).
ℓ(𝐵,
Furthermore, if ℎ(𝑥) = 0, then (∑𝑘 |𝑠𝑘−1 𝑥𝑘−1 +
𝑟𝑘 𝑥𝑘 |𝑝𝑘 )1/𝑀 = 0. Therefore |𝑠𝑘−1 𝑥𝑘−1 + 𝑟𝑘 𝑥𝑘 |𝑝𝑘 = 0 for each
𝑘 ∈ N. If we put 𝑘 = 0, since 𝑠−1 = 0 and 𝑟0 ≠ 0, we have
𝑥0 = 0. For 𝑘 = 1, since 𝑥0 = 0 we have 𝑥1 = 0. Continuing
in this way, we obtain 𝑥𝑘 = 0 for all 𝑘 ∈ N. That is, 𝑥 = 𝜃. This
shows that ℎ is a total paranorm.
̃ 𝑝) is complete. Let {𝑥𝑛 } be any
Now, we show that ℓ(𝐵,
̃
Cauchy sequence in ℓ(𝐵, 𝑝) where 𝑥𝑛 = {𝑥0(𝑛) , 𝑥1(𝑛) ,𝑥2(𝑛) , . . .}.
Here and after, for short we write 𝐵̃ instead of 𝐵(̃𝑟, 𝑠̃). Then
for a given 𝜀 > 0, there exists a positive integer 𝑛0 (𝜀) such that
ℎ(𝑥𝑛 − 𝑥𝑚 ) < 𝜀 for all 𝑛, 𝑚 > 𝑛0 (𝜀). Since for each fixed 𝑘 ∈ N
1/𝑀
󵄨
󵄨𝑝
ℎ (𝑥 + 𝑦) = [∑󵄨󵄨󵄨𝑠𝑘−1 (𝑥𝑘−1 + 𝑦𝑘−1 ) + 𝑟𝑘 (𝑥𝑘 + 𝑦𝑘 )󵄨󵄨󵄨 𝑘 ]
(16)
≤ ℎ (𝑥𝑛 − 𝑥𝑚 ) < 𝜀.
(21)
𝑘=0
1/𝑀
󵄨
󵄨𝑝
+ (∑󵄨󵄨󵄨𝑠𝑘−1 𝑦𝑘−1 + 𝑟𝑘 𝑦𝑘 󵄨󵄨󵄨 𝑘 )
By letting 𝑚, 𝐾 → ∞, we have for 𝑛 > 𝑛0 (𝜀) that
𝑘
= ℎ (𝑥) + ℎ (𝑦) .
󵄨 ̃ 𝑛
̃ 󵄨󵄨󵄨󵄨𝑝𝑘 ]
)𝑘 − (𝐵𝑥)
ℎ (𝑥 − 𝑥) = [∑󵄨󵄨󵄨󵄨(𝐵𝑥
𝑘󵄨
𝑛
(15)
𝑘
1/𝑀
< 𝜀.
(22)
4
Abstract and Applied Analysis
̃ 𝑝). Since ℓ(𝐵,
̃ 𝑝) is a linear space,
This shows us 𝑥𝑛 − 𝑥 ∈ ℓ(𝐵,
̃
we conclude that 𝑥 ∈ ℓ(𝐵, 𝑝); It follows that 𝑥𝑛 → 𝑥, as 𝑛 →
̃ 𝑝), thus we have shown that ℓ(𝐵,
̃ 𝑝) is complete.
∞ in ℓ(𝐵,
Therefore, one can easily check that the absolute property
̃ 𝑝); that is, 𝑔1 (𝑥) ≠ 𝑔1 (|𝑥|),
does not hold on the space ℓ(𝐵,
̃ 𝑝) is the sequence space
where |𝑥| = (|𝑥𝑘 |). This says that ℓ(𝐵,
of nonabsolute type.
̃ 𝑝) is stronger than coorTheorem 2. Convergence in ℓ(𝐵,
dinate-wise convergence.
Proof. First we show that ℎ(𝑥𝑛 − 𝑥) → 0, as 𝑛 → ∞ implies
𝑥𝑘𝑛 → 𝑥𝑘 ; as 𝑛 → ∞ for every 𝑘 ∈ N. We fix 𝑘, then we have
󵄨
󵄨𝑝𝑘
lim 󵄨󵄨𝑠 𝑥(𝑛) + 𝑟𝑘 𝑥𝑘(𝑛) − 𝑠𝑘−1 𝑥𝑘−1 − 𝑟𝑘 𝑥𝑘 󵄨󵄨󵄨󵄨
𝑛 → ∞󵄨󵄨 𝑘−1 𝑘−1
𝑘
𝑛→∞
𝑀
̃ 𝑝) of nonabsolute type
Corollary 4. The sequence space ℓ(𝐵,
is linearly paranorm isomorphic to the space ℓ(𝑝), where 0 <
𝑝𝑘 ≤ 𝐻 < ∞ for all 𝑘 ∈ N.
̃ 𝑝) has AK.
Theorem 5. The space ℓ(𝐵,
̃ 𝑝), we put
Proof. For each 𝑥 = (𝑥𝑘 ) ∈ ℓ(𝐵,
𝑚
𝑥⟨𝑚⟩ = ∑ 𝑥𝑘 𝑒(𝑘) ,
(26)
̃ 𝑝) be given. Then, there is 𝑁 = 𝑁(𝜀) ∈
Let 𝜀 > 0 and 𝑥 ∈ ℓ(𝐵,
N such that
∞
󵄨
󵄨𝑝
∑ 󵄨󵄨󵄨𝑠𝑘−1 𝑥𝑘−1 + 𝑟𝑘 𝑥𝑘 󵄨󵄨󵄨 𝑘 < 𝜀𝑀.
= 0.
(27)
𝑘=𝑁
Hence, we have for 𝑘 = 0 that
󵄨
󵄨
lim 󵄨󵄨𝑠 𝑥(𝑛) + 𝑟0 𝑥0(𝑛) − 𝑠−1 𝑥−1 − 𝑟0 𝑥0 󵄨󵄨󵄨󵄨 = 0,
𝑛 → ∞ 󵄨󵄨 −1 −1
Then we have for all 𝑚 ≥ 𝑁,
(24)
𝑚
ℎ (𝑥 − 𝑥⟨𝑚⟩ ) = ℎ (𝑥 − ∑ 𝑥𝑘 𝑒(𝑘) )
𝑘=1
which gives the fact that |𝑥0(𝑛) − 𝑥0 | → 0, as 𝑛 → ∞.
Similarly, for each 𝑘 ∈ N, we have |𝑥𝑘(𝑛) − 𝑥𝑘 | → 0, as
𝑛 → ∞.
A sequence space 𝜆 with a linear topology is called a
𝐾-space provided each of the maps 𝑝𝑖 : 𝜆 → C defined by
𝑝𝑖 (𝑥) = 𝑥𝑖 is continuous for all 𝑖 ∈ N, where C denotes the
complex field. A 𝐾-space 𝜆 is called an 𝐹𝐾-space provided 𝜆 is
complete linear metric space. An 𝐹𝐾-space whose topology
is normable is called a 𝐵𝐾-space. Given a 𝐵𝐾-space 𝜆 ⊃ 𝜙,
we denote the 𝑛th section of a sequence 𝑥 = (𝑥𝑘 ) ∈ 𝜆 by
𝑥[𝑛] := ∑𝑛𝑘=0 𝑥𝑘 𝑒(𝑘) , and we say that 𝑥 = (𝑥𝑘 ) has the property
𝐴𝐾 if lim𝑛 → ∞ ‖ 𝑥 − 𝑥[𝑛] ‖𝜆 = 0. If 𝐴𝐾 property holds for
every 𝑥 ∈ 𝜆, then we say that the space 𝜆 is called 𝐴𝐾-space
(cf. [19]). Now, we may give the following.
Theorem 3. (ℓ𝑝 )𝐵̃ is the linear space under the coordinatewise
addition and scalar multiplication which is the 𝐵𝐾-space with
the norm
󵄨
󵄨𝑝
‖𝑥‖ := (∑󵄨󵄨󵄨𝑠𝑘−1 𝑥𝑘−1 + 𝑟𝑘 𝑥𝑘 󵄨󵄨󵄨 )
∀𝑚 ∈ {1, 2, . . .} .
𝑘=0
󵄨
󵄨𝑝𝑘
(𝑛)
≤ lim ∑󵄨󵄨󵄨󵄨𝑠𝑘−1 𝑥𝑘−1
+ 𝑟𝑘 𝑥𝑘(𝑛) − 𝑠𝑘−1 𝑥𝑘−1 − 𝑟𝑘 𝑥𝑘 󵄨󵄨󵄨󵄨 (23)
𝑛→∞
= lim [ℎ (𝑥𝑛 − 𝑥)]
Let us suppose that 1 < 𝑝𝑘 ≤ 𝑠𝑘 for all 𝑘 ∈ N. Then, it
is known that ℓ(𝑝) ⊂ ℓ(𝑠) which leads us to the immediate
̃ 𝑝) ⊂ ℓ(𝐵,
̃ 𝑠).
consequence that ℓ(𝐵,
With the notation of (13), define the transformation 𝑇
̃ 𝑝) to ℓ(𝑝) by 𝑥 󳨃→ 𝑦 = 𝑇𝑥. Since 𝑇 is linear and
from ℓ(𝐵,
bijection, we have the following.
1/𝑀
∞
󵄨
󵄨𝑝
= ( ∑ 󵄨󵄨󵄨𝑠𝑘−1 𝑥𝑘−1 + 𝑟𝑘 𝑥𝑘 󵄨󵄨󵄨 𝑘 )
(28)
𝑘=𝑚+1
∞
󵄨
󵄨𝑝
≤ ( ∑ 󵄨󵄨󵄨𝑠𝑘−1 𝑥𝑘−1 + 𝑟𝑘 𝑥𝑘 󵄨󵄨󵄨 𝑘 )
1/𝑀
< 𝜀.
𝑘=𝑁
This shows that 𝑥 = ∑𝑘 𝑥𝑘 𝑒(𝑘) .
Now we have to show that this representation is unique.
We assume that 𝑥 = ∑𝑘 𝜆 𝑘 𝑒(𝑘) . Then for each 𝑘,
󵄨
󵄨𝑝 1/𝑀
(󵄨󵄨󵄨𝑠𝑘−1 𝜆 𝑘−1 + 𝑟𝑘 𝜆 𝑘 − 𝑠𝑘−1 𝑥𝑘−1 − 𝑟𝑘 𝑥𝑘 󵄨󵄨󵄨 𝑘 )
1/𝑀
󵄨
󵄨𝑝
≤ (∑󵄨󵄨󵄨𝑠𝑘−1 𝜆 𝑘−1 + 𝑟𝑘 𝜆 𝑘 − 𝑠𝑘−1 𝑥𝑘−1 − 𝑟𝑘 𝑥𝑘 󵄨󵄨󵄨 𝑘 )
𝑘
= ℎ (𝑥 − 𝑥) = 0.
(29)
1/𝑝
,
where 1 ≤ 𝑝 < ∞.
𝑘
(25)
Proof. Because the first part of the theorem is a routine
verification, we omit the detail. Since ℓ𝑝 is the 𝐵𝐾-space with
respect to its usual norm (see [20, pages 217-218]) and 𝐵(̃𝑟, 𝑠̃)
is a normal matrix, Theorem 4.3.2 of Wilansky [21, page 61]
gives the fact that (ℓ𝑝 )𝐵̃ is the 𝐵𝐾-space, where 1 ≤ 𝑝 <
∞.
Hence, 𝑠𝑘−1 𝜆 𝑘−1 + 𝑟𝑘 𝜆 𝑘 = 𝑠𝑘−1 𝑥𝑘−1 + 𝑟𝑘 𝑥𝑘 for each 𝑘.
For 𝑘 = 0, 𝑟0 𝜆 0 = 𝑟0 𝑥0 . Since 𝑟0 ≠ 0, we have 𝜆 0 = 𝑥0 .
For 𝑘 = 1, 𝑠0 𝜆 0 + 𝑟1 𝜆 1 = 𝑠0 𝑥0 + 𝑟1 𝑥1 . Since 𝑟1 ≠ 0, we also
have 𝜆 1 = 𝑥1 .
Continuing in this way, we obtain 𝜆 𝑘 = 𝑥𝑘 for each 𝑘.
Therefore, the representation is unique.
We firstly define the concept of the Schauder basis for a
paranormed sequence space and next give the basis of the
̃ 𝑝).
sequence space ℓ(𝐵,
Abstract and Applied Analysis
5
Let (𝑋, 𝑔) be a paranormed space. A sequence (𝑏𝑘 ) of the
elements of 𝑋 is called a basis for 𝑋 if and only if, for each
𝑥 ∈ 𝑋, there exists a unique sequence (𝛼𝑘 ) of scalars such
that
(i) Let 0 < 𝑝𝑘 ≤ 1 for all 𝑘 ∈ N. Then, 𝐴 ∈ (ℓ(𝑝) : ℓ∞ ) if
and only if
󵄨 󵄨𝑝
sup 󵄨󵄨󵄨𝑎𝑛𝑘 󵄨󵄨󵄨 𝑘 < ∞.
𝑛,𝑘∈N
(34)
𝑛
lim 𝑔 (𝑥 − ∑ 𝛼𝑘 𝑏𝑘 ) = 0.
𝑛→∞
(30)
𝑘=0
The series ∑𝑘 𝛼𝑘 𝑏𝑘 which has the sum 𝑥 is then called the
expansion of 𝑥 with respect to (𝑏𝑛 ) and written as 𝑥 = ∑𝑘 𝛼𝑘 𝑏𝑘 .
Since it is known that the matrix domain 𝜆 𝐴 of a sequence
space 𝜆 has a basis if and only if 𝜆 has a basis whenever
𝐴 = (𝑎𝑛𝑘 ) is a triangle (cf. [22, Remark 2.4]), we have the
following.
̃ 𝑘 for all
Corollary 6. Let 0 < 𝑝𝑘 ≤ 𝐻 < ∞ and 𝛼𝑘 = (𝐵𝑥)
𝑘 ∈ N. Define the sequence 𝑏(𝑘) = {𝑏𝑛(𝑘) }𝑛∈N of the elements of
̃ 𝑝) by
the space ℓ(𝐵,
(−1)𝑛−𝑘 𝑛−1
𝑠𝑗
𝑗=𝑘
𝑟𝑗
{
{
𝑏𝑛(𝑘) := { 𝑟𝑛
{
{0,
∏
,
0 ≤ 𝑘 ≤ 𝑛,
(31)
𝑘 > 𝑛,
for every fixed 𝑘 ∈ N. Then, the sequence {𝑏(𝑘) }k∈N given by (31)
̃ 𝑝) and any 𝑥 ∈ ℓ(𝐵,
̃ 𝑝) has a unique
is a basis for the space ℓ(𝐵,
(𝑘)
representation of the form 𝑥 := ∑𝑘 𝛼𝑘 𝑏 .
3. The Alpha-, Beta-, and Gamma-Duals of
̃
the Space ℓ(𝐵,𝑝)
In this section, we state and prove the theorems determining
the alpha-, beta-, and gamma-duals of the sequence space
̃ 𝑝) of nonabsolute type.
ℓ(𝐵,
For the sequence spaces 𝜆 and 𝜇, the set 𝑆(𝜆, 𝜇) defined
by
𝑆 (𝜆, 𝜇)
:= {𝑧 = (𝑧𝑘 ) ∈ 𝜔 : 𝑥𝑧 = (𝑥𝑘 𝑧𝑘 ) ∈ 𝜇 ∀𝑥 = (𝑥𝑘 ) ∈ 𝜆}
(32)
is called the multiplier space of the spaces 𝜆 and 𝜇. With the
notation of (32), the alpha-, beta-, and gamma-duals of a
sequence space 𝜆, which are, respectively, denoted by 𝜆𝛼 , 𝜆𝛽 ,
and 𝜆𝛾 , are defined by
𝜆𝛼 := 𝑆 (𝜆, ℓ1 ) ,
𝜆𝛽 := 𝑆 (𝜆, 𝑐𝑠) ,
𝜆𝛾 := 𝑆 (𝜆, 𝑏𝑠) .
(33)
Since the case 0 < 𝑝𝑘 ≤ 1 may be established in similar
way to the proof of the case 1 < 𝑝𝑘 ≤ 𝐻 < ∞, we omit the
detail of that case and give the proof only for the case 1 < 𝑝𝑘 ≤
𝐻 < ∞ in Theorems 10–12 below.
We begin with quoting three lemmas which are needed in
proving Theorems 10–12.
Lemma 7 ([23, (i) and (ii) of Theorem 1]). Let 𝐴 = (𝑎𝑛𝑘 ) be
an infinite matrix. Then, the following statements hold.
(ii) Let 1 < 𝑝𝑘 ≤ 𝐻 < ∞ for all 𝑘 ∈ N. Then, 𝐴 ∈ (ℓ(𝑝) :
ℓ∞ ) if and only if there exists an integer 𝑀 > 1 such
that
󵄨
󵄨𝑝󸀠
sup∑󵄨󵄨󵄨󵄨𝑎𝑛𝑘 𝑀−1 󵄨󵄨󵄨󵄨 𝑘 < ∞.
𝑛∈N 𝑘
(35)
Lemma 8 ([23, Corollary for Theorem 1]). Let 0 < 𝑝𝑘 ≤ 𝐻 <
∞ for all 𝑘 ∈ N. Then, 𝐴 = (𝑎𝑛𝑘 ) ∈ (ℓ(𝑝) : 𝑐) if and only if
(34) and (35) hold, and
lim 𝑎
𝑛 → ∞ 𝑛𝑘
= 𝛽𝑘 ,
∀𝑘 ∈ N.
(36)
Lemma 9 ([24, Theorem 5.1.0]). Let 𝐴 = (𝑎𝑛𝑘 ) be an infinite
matrix. Then, the following statements hold
(i) Let 0 < 𝑝𝑘 ≤ 1 for all 𝑘 ∈ N. Then, 𝐴 ∈ (ℓ(𝑝) : ℓ1 ) if
and only if
󵄨󵄨
󵄨󵄨𝑝𝑘
󵄨󵄨
󵄨󵄨
sup sup󵄨󵄨󵄨 ∑ 𝑎𝑛𝑘 󵄨󵄨󵄨 < ∞.
󵄨󵄨
𝑁∈F 𝑘∈N 󵄨󵄨󵄨𝑛∈𝑁
󵄨
(37)
(ii) Let 1 < 𝑝𝑘 ≤ 𝐻 < ∞ for all 𝑘 ∈ N. Then, 𝐴 ∈ (ℓ(𝑝) :
ℓ1 ) if and only if there exists an integer 𝑀 > 1 such that
󵄨󵄨
󵄨󵄨𝑝𝑘󸀠
󵄨󵄨
󵄨󵄨
−1
sup ∑󵄨󵄨󵄨 ∑ 𝑎𝑛𝑘 𝑀 󵄨󵄨󵄨 < ∞.
󵄨󵄨
𝑁∈F 𝑘 󵄨󵄨󵄨𝑛∈𝑁
󵄨
(38)
Theorem 10. Define the sets 𝑆1 (𝑝) and 𝑆2 (𝑝) by
{
𝑆1 (𝑝) = ⋃ {𝑎 = (𝑎𝑘 ) ∈ 𝜔 :
𝑀>1
{
󵄨󵄨
󵄨󵄨𝑝𝑘󸀠
󵄨󵄨 (−1)𝑛−𝑘 𝑛−1 𝑠𝑗
󵄨
}
−1 󵄨󵄨
󵄨
sup ∑󵄨󵄨󵄨 ∑
∏ 𝑎𝑛 𝑀 󵄨󵄨󵄨 < ∞} ,
𝑟𝑛 𝑗=𝑘 𝑟𝑗
󵄨󵄨
𝑁∈F 𝑘 󵄨󵄨𝑛∈𝑁
󵄨
󵄨
}
󵄨󵄨
󵄨󵄨𝑝𝑘
{
}
󵄨󵄨󵄨 (−1)𝑛−𝑘 𝑛−1 𝑠𝑗 󵄨󵄨󵄨
𝑆2 (𝑝) = {𝑎 = (𝑎𝑘 ) ∈ 𝜔 : sup sup󵄨󵄨󵄨 ∑
∏ 𝑎𝑛 󵄨󵄨󵄨 < ∞}.
𝑟 󵄨
N∈F 𝑘∈N 󵄨󵄨𝑛∈𝑁 𝑟𝑛
𝑗=𝑘 𝑗 󵄨󵄨
󵄨
{
}
(39)
Then,
̃ 𝑝)}𝛼 = {𝑆1 (𝑝) , 1 < 𝑝𝑘 ≤ 𝐻 < ∞, ∀𝑘 ∈ N, (40)
{ℓ (𝐵,
𝑆2 (𝑝) , 0 < 𝑝𝑘 ≤ 1, ∀𝑘 ∈ N.
Proof. Let us take any 𝑎 = (𝑎𝑛 ) ∈ 𝜔. By using (13) we obtain
that
(−1)𝑛−𝑘 𝑛−1 𝑠𝑗
∏ 𝑦
𝑟𝑛 𝑗=𝑘 𝑟𝑗 𝑘
𝑘=0
𝑛
𝑥𝑛 = ∑
(41)
6
Abstract and Applied Analysis
holds for all 𝑛 ∈ N which leads us to
(−1)𝑛−𝑘 𝑛−1 𝑠𝑗
∏ 𝑎 𝑦 = (𝐶𝑦)𝑛 ,
𝑟𝑛 𝑗=𝑘 𝑟𝑗 𝑛 𝑘
𝑘=0
𝑛
𝑎𝑛 𝑥𝑛 = ∑
𝐷𝑦 ∈ 𝑐 whenever 𝑦 = (𝑦𝑘 ) ∈ ℓ(𝑝). Therefore, we derive from
(35) and (36) that
(𝑛 ∈ N) ,
(42)
where 𝐶 = (𝑐𝑛𝑘 ) is defined by
(−1)𝑛−𝑘 𝑛−1
𝑠𝑗
𝑗=𝑘
𝑟𝑗
{
{
𝑐𝑛𝑘 = { 𝑟𝑛
{
{0,
∏
𝑎𝑛 ,
0 ≤ 𝑘 ≤ 𝑛,
∞
for all 𝑘, 𝑛 ∈ N. Thus, we observe by combining (42) with the
condition (37) of Part (i) of Lemma 9 that 𝑎𝑥 = (𝑎𝑛 𝑥𝑛 ) ∈ ℓ1
̃ 𝑝) if and only if 𝐶𝑦 ∈ ℓ1 whenever
whenever 𝑥 = (𝑥𝑘 ) ∈ ℓ(𝐵,
̃ 𝑝)}𝛼 = 𝑆1 (𝑝).
𝑦 = (𝑦𝑘 ) ∈ ℓ(𝑝). That means {ℓ(𝐵,
Theorem 11. Define the sets 𝑆3 (𝑝), 𝑆4 (𝑝), and 𝑆5 (𝑝) by
𝑆3 (𝑝) =
󵄨󵄨 𝑛
󵄨󵄨𝑝𝑘󸀠
󵄨
{
}
󵄨󵄨󵄨 (−1)𝑖−𝑘 𝑖−1 𝑠𝑗
−1 󵄨󵄨
⋃ {𝑎 = (𝑎𝑘 ) ∈ 𝜔 : sup∑󵄨󵄨󵄨∑
∏ 𝑎𝑖 𝑀 󵄨󵄨󵄨 < ∞},
𝑟𝑖 𝑗=𝑘 𝑟𝑗
󵄨󵄨
𝑛∈N 𝑘 󵄨󵄨𝑖=𝑘
𝑀>1
󵄨
󵄨
{
}
∞
{
}
(−1)𝑖−𝑘 𝑖−1 𝑠𝑗
𝑆4 (𝑝) = {𝑎 = (𝑎𝑘 ) ∈ 𝜔 : ∑
∏ 𝑎𝑖 < ∞},
𝑟𝑖 𝑗=𝑘 𝑟𝑗
𝑖=𝑘
{
}
󵄨󵄨 𝑛
󵄨𝑝
󵄨󵄨 (−1)𝑖−𝑘 𝑖−1 𝑠𝑗 󵄨󵄨󵄨 𝑘
{
}
󵄨
󵄨
𝑆5 (𝑝) = {𝑎 = (𝑎𝑘 ) ∈ 𝜔 : sup 󵄨󵄨∑
∏ 𝑎𝑖 󵄨󵄨󵄨󵄨 < ∞} .
𝑟
𝑟
󵄨
󵄨
𝑛,𝑘∈N󵄨𝑖=𝑘
𝑖
𝑗=𝑘 𝑗 󵄨󵄨
󵄨
{
}
(44)
Then,
1 < 𝑝𝑘 ≤ 𝐻 < ∞ ∀𝑘 ∈ N,
0 < 𝑝𝑘 ≤ 1 ∀𝑘 ∈ N.
(45)
Proof. Take any 𝑎 = (𝑎𝑖 ) ∈ 𝜔 and consider the equation
obtained with (13) that
(−1)𝑖−𝑘 𝑖−1 𝑠𝑗 ]
∑𝑎𝑖 𝑥𝑖 = ∑ [ ∑
∏ 𝑦 𝑎
𝑟𝑖 𝑗=𝑘 𝑟𝑗 𝑘 𝑖
𝑖=0
𝑖=0 𝑘=0
[
]
𝑛
𝑛
𝑖
𝑛
𝑛
𝑖−𝑘 𝑖−1
(−1)
= ∑ [∑
𝑟𝑖
𝑘=0 𝑖=𝑘
[
s𝑗
∏ 𝑎𝑖 ] 𝑦𝑘
𝑟
𝑗=𝑘 𝑗
]
(46)
= (𝐷𝑦)𝑛 ,
𝑛
𝑑𝑛𝑘
𝑖−𝑘 𝑖−1 𝑠
𝑗
∏
𝑗=𝑘 𝑟𝑗
(48)
𝑠𝑗
∏ 𝑎𝑖 < ∞.
𝑟
𝑗=𝑘 𝑗
̃ 𝑝)}𝛽 = 𝑆3 (𝑝) ∩ 𝑆4 (𝑝).
This shows that {ℓ(𝐵,
Theorem 12.
̃ 𝑝)}𝛾 = {𝑆3 (𝑝) , 1 < 𝑝𝑘 ≤ 𝐻 < ∞, ∀𝑘 ∈ N, (49)
{ℓ (𝐵,
𝑆5 (𝑝) , 0 < 𝑝𝑘 ≤ 1, ∀𝑘 ∈ N.
Proof. From Lemma 7 and (46), we obtain that 𝑎𝑥 = (𝑎𝑖 𝑥𝑖 ) ∈
̃ 𝑝) if and only if 𝐷𝑦 ∈ ℓ∞
𝑏𝑠 whenever 𝑥 = (𝑥𝑖 ) ∈ ℓ(𝐵,
whenever 𝑦 = (𝑦𝑘 ) ∈ ℓ(𝑝), where 𝐷 = (𝑑𝑛𝑘 ) is defined by
̃ 𝑝)}𝛾 =
(47). Therefore, we obtain from (34) and (35) that {ℓ(𝐵,
̃ 𝑝)}𝛾 = 𝑆5 (𝑝) for 𝑝𝑘 ≤ 1.
𝑆3 (𝑝) for 1 < 𝑝𝑘 , {ℓ(𝐵,
4. Matrix Transformations on
̃
the Sequence Space ℓ(𝐵,𝑝)
In this section, we characterize some matrix transformations
̃ 𝑝). Theorem 13 gives the exact conditions of
on the space ℓ(𝐵,
the general case 0 < 𝑝𝑘 ≤ 𝐻 < ∞ by combining the cases
0 < 𝑝𝑘 ≤ 1 and 1 < 𝑝𝑘 ≤ 𝐻 < ∞. We consider only the case
1 < 𝑝𝑘 ≤ 𝐻 < ∞ and leave the case 0 < 𝑝𝑘 ≤ 1 to the reader
because it can be proved in similar way.
Theorem 13. Let 𝐴 = (𝑎𝑛𝑘 ) be an infinite matrix. Then, the
following statements hold.
̃ 𝑝) :
(i) Let 1 < 𝑝𝑘 ≤ 𝐻 < ∞ for all 𝑘 ∈ N. Then, 𝐴 ∈ (ℓ(𝐵,
ℓ∞ ) if and only if there exists an integer 𝑀 > 1 such
that
󵄨󵄨 𝑛
󵄨󵄨𝑝𝑘󸀠
󵄨󵄨 (−1)𝑖−𝑘 𝑖−1 𝑠𝑗
󵄨
−1 󵄨󵄨
󵄨
sup∑󵄨󵄨󵄨∑
∏ 𝑎𝑛𝑖 𝑀 󵄨󵄨󵄨 < ∞,
𝑟𝑖 𝑗=𝑘 𝑟𝑗
󵄨󵄨
𝑛∈N 𝑘 󵄨󵄨𝑖=𝑘
󵄨
󵄨
(50)
(−1)𝑖−𝑘 𝑖−1 𝑠𝑗
∏ 𝑎 < ∞.
𝑟𝑖 𝑗=𝑘 𝑟𝑗 𝑛𝑖
𝑖=𝑘
(51)
∞
∑
̃ 𝑝) : ℓ∞ )
(ii) Let 0 < 𝑝𝑘 ≤ 1 for all 𝑘 ∈ N. Then, 𝐴 ∈ (ℓ(𝐵,
if and only if the condition (51) holds, and
where 𝐷 = (𝑑𝑛𝑘 ) is defined by
(−1)
{
{∑
= {𝑖=𝑘 𝑟𝑖
{
{0,
𝑖−𝑘 𝑖−1
(−1)
𝑟𝑖
𝑖=𝑘
∑
(43)
𝑘>𝑛
̃ 𝑝)}𝛽 = {𝑆3 (𝑝) ∩ 𝑆4 (𝑝) ,
{ℓ (𝐵,
𝑆4 (𝑝) ∩ 𝑆5 (𝑝) ,
󵄨󵄨 𝑛
󵄨󵄨𝑝𝑘󸀠
󵄨󵄨 (−1)𝑖−𝑘 𝑖−1 𝑠𝑗
󵄨󵄨
−1
sup∑󵄨󵄨󵄨󵄨∑
∏ 𝑎𝑖 𝑀 󵄨󵄨󵄨󵄨 < ∞,
𝑟𝑖 𝑗=𝑘 𝑟𝑗
󵄨󵄨
𝑛∈N 𝑘 󵄨󵄨𝑖=𝑘
󵄨
󵄨
𝑎𝑖 ,
0 ≤ 𝑘 ≤ 𝑛,
(47)
𝑘>𝑛
for all 𝑘, 𝑛 ∈ N. Thus, we deduce from Lemma 8 with (46) that
̃ 𝑝) if and only if
𝑎𝑥 = (𝑎𝑖 𝑥𝑖 ) ∈ 𝑐𝑠 whenever 𝑥 = (𝑥𝑖 ) ∈ ℓ(𝐵,
󵄨󵄨 𝑛
󵄨𝑝
󵄨󵄨 (−1)𝑖−𝑘 𝑖−1 𝑠𝑗 󵄨󵄨󵄨 𝑘
󵄨
󵄨
sup 󵄨󵄨∑
∏ 𝑎 󵄨󵄨󵄨 < ∞.
𝑟𝑖 𝑗=𝑘 𝑟𝑗 𝑛𝑖 󵄨󵄨󵄨
𝑛,𝑘∈N󵄨󵄨𝑖=𝑘
󵄨
󵄨
(52)
Proof. Suppose that the conditions (50) and (51) hold, and
̃ 𝑝). In this situation, since {𝑎𝑛𝑘 }𝑘∈N ∈ {ℓ(𝐵,
̃ 𝑝)}𝛽 for
𝑥 ∈ ℓ(𝐵,
Abstract and Applied Analysis
7
every fixed 𝑛 ∈ N, the 𝐴-transform of 𝑥 exists. Consider the
following equality obtained by using the relation (13) that
(−1)𝑖−𝑘 𝑖−1 𝑠𝑗
∏ 𝑎 𝑦
𝑟𝑖 𝑗=𝑘 𝑟𝑗 𝑛𝑖 𝑘
𝑘=0 𝑖=𝑘
𝑚
̃ 𝑝) : 𝑓) if and only if 𝐹 ∈
for all 𝑘, 𝑛 ∈ N. Then, 𝐸 ∈ (ℓ(𝐵,
(ℓ(𝑝) : 𝑓) and
𝑚 𝑚
∑ 𝑎𝑛𝑘 𝑥𝑘 = ∑ ∑
𝑘=0
for all 𝑚, 𝑛 ∈ N. Taking into account the hypothesis we derive
from (53) as 𝑚 → ∞ that
(−1)𝑖−𝑘 𝑖−1 𝑠𝑗
∑𝑎𝑛𝑘 𝑥𝑘 = ∑∑
∏ 𝑎 𝑦,
𝑟𝑖 𝑗=𝑘 𝑟𝑗 𝑛𝑖 𝑘
𝑘
𝑘 𝑖=𝑘
∞
󵄨𝑝󸀠
󵄨
|𝑎𝑏| ≤ 𝑀 (󵄨󵄨󵄨󵄨𝑎𝑀−1 󵄨󵄨󵄨󵄨 + |𝑏|𝑝 ) ,
(55)
(𝑛)
𝑓𝑚𝑘
𝑚 (−1)𝑖 𝑖−1 𝑠𝑗
{
{∑
∏ 𝑒𝑛𝑖 ,
:= {𝑖=𝑘 𝑟𝑖 𝑗=𝑘 𝑟𝑗
{
{0,
𝑚
𝑚
𝑘 > 𝑚,
𝑘
𝑚
(−1)𝑘−𝑖 𝑘−1 𝑠𝑗 ]
∏ 𝑦
𝑟𝑖 𝑗=𝑖 𝑟𝑗 𝑖
]
𝑖 𝑖−1
𝑠𝑗
(61)
𝑚
(−1)
(𝑛)
= ∑ [∑
𝑦𝑘
∏ ] 𝑦𝑘 = ∑ 𝑓𝑚𝑘
𝑟
𝑟
𝑖 𝑗=𝑘 𝑗
𝑘=0 𝑖=𝑘
𝑘=0
[
]
󵄨󵄨 ∞
󵄨
󵄨󵄨 (−1)𝑖−𝑘 𝑖−1 𝑠𝑗 󵄨󵄨󵄨 󵄨 󵄨
󵄨
≤ sup∑ 󵄨󵄨󵄨∑
∏ 𝑎 󵄨󵄨󵄨 󵄨󵄨𝑦 󵄨󵄨
𝑟𝑖 𝑗=𝑘 𝑟𝑗 𝑛𝑖 󵄨󵄨󵄨 󵄨 𝑘 󵄨
𝑛∈N 𝑘 󵄨󵄨𝑖=𝑘
󵄨
󵄨
󵄨󵄨𝑝𝑘󸀠
󵄨󵄨 ∞ 𝑖−1
󵄨
󵄨󵄨 1
𝑠𝑗
󵄨 󵄨𝑝
−1 󵄨󵄨
󵄨
≤ sup∑𝑀 (󵄨󵄨󵄨∑ ∏ 𝑎𝑛𝑖 𝑀 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑦𝑘 󵄨󵄨󵄨 𝑘 )
󵄨󵄨
󵄨󵄨𝑖=𝑘 𝑟𝑖 𝑗=𝑘 𝑟𝑗
𝑛∈N 𝑘
󵄨
󵄨
󵄨󵄨 ∞ 𝑖−1
󵄨󵄨𝑝𝑘󸀠
󵄨󵄨 1
󵄨
𝑠𝑗
󵄨 󵄨𝑝
−1 󵄨󵄨
󵄨
≤ 𝑀 (sup∑󵄨󵄨󵄨∑ ∏ 𝑎𝑛𝑖 𝑀 󵄨󵄨󵄨 + ∑󵄨󵄨󵄨𝑦𝑘 󵄨󵄨󵄨 𝑘 ) < ∞.
𝑟
𝑟
󵄨
󵄨
𝑛∈N 𝑘 󵄨𝑖=𝑘 𝑖 𝑗=𝑘 𝑗
󵄨󵄨
𝑘
󵄨
(56)
̃ 𝑝) : ℓ∞ ) and 1 <
Conversely, suppose that 𝐴 ∈ (ℓ(𝐵,
𝑝𝑘 ≤ 𝐻 < ∞ for all 𝑘 ∈ N. Then 𝐴𝑥 exists for every
̃ 𝑝) and this implies that {𝑎𝑛𝑘 }𝑘∈N ∈ {ℓ(𝐵,
̃ 𝑝)}𝛽 for
𝑥 ∈ ℓ(𝐵,
all 𝑛 ∈ N. Now, the necessity of (51) is immediate. Besides,
we have from (54) that the matrix 𝐵 = (𝑏𝑛𝑘 ) defined by
𝑖−1
𝑖−𝑘
𝑏𝑛𝑘 = ∑∞
𝑖=𝑘 ((−1) /𝑟𝑖 )∏𝑗=𝑘 (𝑠𝑗 /𝑟𝑗 )𝑎𝑛𝑖 for all 𝑛, 𝑘 ∈ N, is in the
class (ℓ(𝑝) : ℓ∞ ). Then, 𝐵 satisfies the condition (35) which is
equivalent to (50).
This completes the proof.
Lemma 14 ([25, Theorem 1]). 𝐴 = (𝑎𝑛𝑘 ) ∈ (ℓ(𝑝) : 𝑓) if and
only if (34) and (35) hold, and
∃𝛼𝑘 ∈ C ∋ 𝑓 − lim 𝑎𝑛𝑘 = 𝛼𝑘
(60)
̃ 𝑝) : 𝑓) and take 𝑥 ∈ ℓ(𝐵,
̃ 𝑝).
Proof. Let 𝐸 = (𝑒𝑛𝑘 ) ∈ (ℓ(𝐵,
Then, we obtain the equality
∑ 𝑒𝑛𝑘 𝑥𝑘 = ∑ 𝑒𝑛𝑘 [∑
𝑘=0
𝑘=0
[𝑖=0
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
sup 󵄨󵄨󵄨∑𝑎𝑛𝑘 𝑥𝑘 󵄨󵄨󵄨
󵄨󵄨
𝑛∈N 󵄨󵄨󵄨 𝑘
󵄨
0 ≤ 𝑘 ≤ 𝑚,
for all 𝑚, 𝑘 ∈ N.
𝑚
where 𝑝 > 1 and 𝑝−1 + 𝑝󸀠−1 = 1, one can easily see that
(59)
(𝑛)
) with
for every fixed 𝑛 ∈ N, where 𝐹𝑛 = (𝑓𝑚𝑘
for each 𝑛 ∈ N. (54)
Now, by combining (54) with the following inequality (see
[23]) which holds for any 𝑀 > 0 and any 𝑎, 𝑏 ∈ C
for every fixed 𝑘 ∈ N. (57)
Theorem 15. Let the entries of the matrices 𝐸 = (𝑒𝑛𝑘 ) and 𝐹 =
(𝑓𝑛𝑘 ) be connected with the relation
𝑒𝑛𝑘 := 𝑠𝑘−1 𝑓𝑛,𝑘−1 + 𝑟𝑘 𝑓𝑛𝑘
𝐹𝑛 ∈ (ℓ (𝑝) : 𝑐)
(53)
(−1)𝑖 𝑖−1 𝑠𝑗
or 𝑓𝑛𝑘 := ∑
∏ 𝑒
𝑟𝑖 𝑗=𝑘 𝑟𝑗 𝑛𝑖
𝑖=𝑘
∞
(58)
for all 𝑚, 𝑛 ∈ N. Since 𝐸𝑥 exists, 𝐹𝑛 ∈ (ℓ(𝑝) : 𝑐). Letting
𝑚 → ∞ in the equality (61) we have 𝐸𝑥 = 𝐹𝑦. Since 𝐸𝑥 ∈ 𝑓,
then 𝐹𝑦 ∈ 𝑓. That is 𝐹 ∈ (ℓ(𝑝) : 𝑓).
Conversely, let 𝐹 ∈ (ℓ(𝑝) : 𝑓), and 𝐹𝑛 ∈ (ℓ(𝑝) : 𝑐), and
̃ 𝑝). Then, since (𝑓𝑛𝑘 )𝑘∈N ∈ {ℓ(𝑝)}𝛽 and 𝐹 ∈
take 𝑥 ∈ ℓ(𝐵,
̃ 𝑝)}𝛽 for all 𝑛 ∈ N. So,
(ℓ(𝑝) : 𝑓) we have (𝑒𝑛𝑘 )𝑘∈N ∈ {ℓ(𝐵,
𝐸𝑥 exists. Therefore we obtain from equality (61) as 𝑚 → ∞
̃ 𝑝) : 𝑓).
that 𝐸𝑥 = 𝐹𝑦, that is 𝐸 ∈ (ℓ(𝐵,
Theorem 16. Let 0 < 𝑝𝑘 ≤ 𝐻 < ∞ for all 𝑘 ∈ N. Then,
̃ 𝑝) : 𝑐) if and only if (50)–(52) hold and
𝐴 ∈ (ℓ(𝐵,
(−1)𝑘−𝑖 𝑘−1 𝑠𝑗
∏ 𝑎 = 𝛼𝑘 ,
𝑛→∞
𝑟𝑘 𝑗=𝑖 𝑟𝑗 𝑛𝑘
𝑘=𝑖
∞
lim ∑
for every fixed 𝑘 ∈ N.
(62)
̃ 𝑝) : 𝑐) and 1 < 𝑝𝑘 ≤ 𝐻 < ∞ for all
Proof. Let 𝐴 ∈ (ℓ(𝐵,
𝑘 ∈ N. Then, since the inclusion 𝑐 ⊂ ℓ∞ holds, the necessities
of (50) and (51) are immediately obtained from part (i) of
Theorem 13.
To prove the necessity of (62), consider the sequence 𝑏(𝑘)
̃ 𝑝) for every fixed
defined by (31) which is in the space ℓ(𝐵,
̃ 𝑝) exists
𝑘 ∈ N. Because the 𝐴-transform of every 𝑥 ∈ ℓ(𝐵,
and is in 𝑐 by the hypothesis,
{ ∞ (−1)𝑘−𝑖 𝑘−1 𝑠𝑗 }
𝐴𝑏(𝑘) = {∑
∈𝑐
∏ 𝑎
𝑟𝑘 𝑗=𝑖 𝑟𝑗 𝑛𝑘 }
𝑘=𝑖
}𝑛∈N
{
for every fixed 𝑘 ∈ N which shows the necessity of (62).
(63)
8
Abstract and Applied Analysis
Conversely suppose that conditions (50), (51), and (62)
̃ 𝑝). Then, 𝐴𝑥
hold, and take any 𝑥 = (𝑥𝑘 ) in the space ℓ(𝐵,
exists. We observe for all 𝑚, 𝑛 ∈ N that
󵄨󵄨 𝑚
󵄨󵄨𝑝𝑘󸀠
󵄨󵄨 (−1)𝑘−𝑖 𝑘−1 𝑠𝑗
󵄨󵄨
−1
∑ 󵄨󵄨󵄨󵄨∑
∏ 𝑎𝑛𝑘 𝑀 󵄨󵄨󵄨󵄨
𝑟𝑘 𝑗=𝑖 𝑟𝑗
󵄨
󵄨󵄨
𝑘=0󵄨󵄨𝑘=𝑖
󵄨
Corollary 19. Let 𝐴 = (𝑎𝑛𝑘 ) be an infinite matrix and define
the matrix 𝐶 = (𝑐𝑛𝑘 ) by
𝑚
󵄨󵄨 𝑚
󵄨󵄨𝑝𝑘󸀠
󵄨󵄨 (−1)𝑘−𝑖 𝑘−1 𝑠𝑗
󵄨󵄨
−1
≤ sup∑󵄨󵄨󵄨󵄨∑
∏ 𝑎𝑛𝑘 𝑀 󵄨󵄨󵄨󵄨 < ∞,
𝑟𝑘 𝑗=𝑖 𝑟𝑗
󵄨󵄨
𝑛∈N 𝑘 󵄨󵄨𝑘=𝑖
󵄨
󵄨
(64)
𝑚
󵄨󵄨 𝑚
󵄨󵄨𝑝𝑘󸀠
󵄨󵄨 (−1)𝑘−𝑖 𝑘−1 𝑠𝑗
󵄨󵄨
−1
lim ∑ 󵄨󵄨󵄨󵄨∑
∏ 𝑎𝑛𝑘 𝑀 󵄨󵄨󵄨󵄨
𝑚,𝑛󳨀→∞
𝑟𝑘 𝑗=𝑖 𝑟𝑗
󵄨
󵄨󵄨
𝑘=0󵄨󵄨𝑘=𝑖
󵄨
𝑚
󵄨󵄨 𝑚
󵄨󵄨𝑝𝑘󸀠
󵄨󵄨 (−1)𝑘−𝑖 𝑘−1 𝑠𝑗
󵄨󵄨
−1
≤ sup∑󵄨󵄨󵄨󵄨∑
∏ 𝑎𝑛𝑘 𝑀 󵄨󵄨󵄨󵄨 < ∞.
𝑟𝑘 𝑗=𝑖 𝑟𝑗
󵄨󵄨
𝑛∈N 𝑘 󵄨󵄨𝑘=𝑖
󵄨
󵄨
(65)
𝑚
󸀠
This shows that ∑𝑘 |𝛼𝑘 𝑀−1 |𝑝𝑘 < ∞ and so (𝛼𝑘 )𝑘∈N ∈
̃ 𝑝)}𝛽 which implies that the series ∑𝑘 𝛼𝑘 𝑥𝑘 converges for
{ℓ(𝐵,
̃ 𝑝).
every 𝑥 ∈ ℓ(𝐵,
Let us now consider the equality obtained from (54) with
𝑎𝑛𝑘 − 𝛼𝑘 instead of 𝑎𝑛𝑘
(−1)𝑘−𝑖 𝑘−1 𝑠𝑗
𝑖 𝑘=𝑖
𝑘
= ∑𝑐𝑛𝑖 𝑦𝑖 ,
𝑟𝑘
∏
𝑗=𝑖 𝑟𝑗
(𝑎𝑛𝑘 − 𝛼𝑘 ) 𝑦𝑖
(66)
∀𝑛 ∈ N,
𝑘
where 𝐶
=
(𝑐𝑛𝑖 ) defined by 𝑐𝑛𝑖
=
(𝑠
/𝑟
)(𝑎
−
𝛼
)
for
all
𝑛,
𝑖
∈
N.
∑𝑘=𝑖 ((−1)𝑘−𝑖 /𝑟𝑘 )∏𝑘−1
𝑛𝑘
𝑘
𝑗=𝑖 𝑗 𝑗
Therefore, we have at this stage from Lemma 8 that the
matrix 𝐶 belongs to the class (ℓ(𝑝) : 𝑐0 ) of infinite matrices.
Thus, we see by (66) that
lim ∑ (𝑎𝑛𝑘 − 𝛼𝑘 ) 𝑥𝑘 = 0.
𝑛→∞
𝑘
𝑛
𝑛
𝑐𝑛𝑘 = ∑ ( ) (1 − 𝑡)𝑛−𝑗 𝑡𝑗 𝑎𝑗𝑘 ,
𝑗
∀𝑛, 𝑘 ∈ N.
𝑗=0
which gives the fact that by letting 𝑚, 𝑛 → ∞ with (50) and
(62) that
∑ (𝑎𝑛𝑘 − 𝛼𝑘 ) 𝑥𝑘 = ∑ ∑
It is trivial that Lemma 18 has several consequences.
Indeed, combining Lemma 18 with Theorems 13, 15, and 16
and Corollary 17, one can derive the following results.
(67)
̃ 𝑝) and
Equation (67) means that 𝐴𝑥 ∈ 𝑐 whenever 𝑥 ∈ ℓ(𝐵,
this is what we wished to prove.
Therefore, we have the following
Corollary 17. Let 0 < 𝑝𝑘 ≤ 𝐻 < ∞ for all 𝑘 ∈ N. Then,
̃ 𝑝) : 𝑐0 ) if and only if (50)–(52) hold, and (62) also
𝐴 ∈ (ℓ(𝐵,
holds with 𝛼𝑘 = 0 for all 𝑘 ∈ N.
Now, we give the following lemma given by Başar and
Altay [26] which is useful for deriving the characterizations
of the certain matrix classes via Theorems 13, 15, and 16 and
Corollary 17.
Lemma 18 ([26, Lemma 5.3]). Let 𝜆, 𝜇 be any two sequence
spaces, let 𝐴 be an infinite matrix, and let 𝐵 also be a triangle
matrix. Then, 𝐴 ∈ (𝜆 : 𝜇𝐵 ) if and only if 𝐵𝐴 ∈ (𝜆 : 𝜇).
(68)
Then, the necessary and sufficient conditions in order to 𝐴
̃ 𝑝) : 𝑒𝑡 ), (ℓ(𝐵,
̃ 𝑝) : 𝑒𝑡 )
belongs to anyone of the classes (ℓ(𝐵,
∞
𝑐
̃ 𝑝) : 𝑒𝑡 ) are obtained from the respective ones in
and (ℓ(𝐵,
0
Theorems 13, 16 and Corollary 17 by replacing the entries of
𝑡
the matrix 𝐴 by those of the matrix 𝐶; where 0 < 𝑡 < 1, 𝑒∞
𝑡 𝑡
and 𝑒𝑐 , 𝑒0 , respectively, denote the spaces of all sequences whose
𝐸𝑡 -transforms are in the spaces ℓ∞ and 𝑐, 𝑐0 and are recently
studied by Altay et al. [27] and Altay and Başar [28], where 𝐸𝑡
denotes the Euler mean of order 𝑡.
Corollary 20. Let 𝐴 = (𝑎𝑛𝑘 ) be an infinite matrix and define
the matrix 𝐶 = (𝑐𝑛𝑘 ) by
𝑐𝑛𝑘 = 𝑠𝑎𝑛−1,𝑘 + 𝑟𝑎𝑛𝑘 ,
∀𝑛, 𝑘 ∈ N.
(69)
Then, the necessary and sufficient conditions in order to 𝐴
̂ is obtained from Theorem 15
̃ 𝑝) : 𝑓)
belongs to the class (ℓ(𝐵,
by replacing the entries of the matrix 𝐴 by those of the matrix
𝐶; where 𝑟, 𝑠 ∈ R \ {0} and 𝑓̂ denotes the space of all sequences
whose 𝐵(𝑟, 𝑠)-transforms are in the space 𝑓 and is recently
studied by Başar and Kirişçi [29].
Corollary 21. Let 𝐴 = (𝑎𝑛𝑘 ) be an infinite matrix and define
the matrix 𝐶 = (𝑐𝑛𝑘 ) by
𝑐𝑛𝑘 = 𝑡𝑎𝑛−2,𝑘 + 𝑠𝑎𝑛−1,𝑘 + 𝑟𝑎𝑛𝑘 ,
∀𝑛, 𝑘 ∈ N.
(70)
Then, the necessary and sufficient conditions in order to
̃ 𝑝) : 𝑓(𝐵)) is obtained from
𝐴 belongs to the class (ℓ(𝐵,
Theorem 15 by replacing the entries of the matrix 𝐴 by those of
the matrix 𝐶; where 𝑟, 𝑠, 𝑡 ∈ R \ {0} and 𝑓(𝐵) denotes the space
of all sequences whose 𝐵(𝑟, 𝑠, 𝑡)-transforms are in the space 𝑓
and is recently studied by Sönmez [30].
Corollary 22. Let 𝐴 = (𝑎𝑛𝑘 ) be an infinite matrix and define
the matrix 𝐶 = (𝑐𝑛𝑘 ) by
𝑐𝑛𝑘 =
1 𝑛
∑𝑎 ,
𝑛 + 1 𝑗=0 𝑗𝑘
∀𝑛, 𝑘 ∈ N.
(71)
Then, the necessary and sufficient conditions in order to 𝐴
̃ is obtained from Theorem 15
̃ 𝑝) : 𝑓)
belongs to the class (ℓ(𝐵,
by replacing the entries of the matrix 𝐴 by those of the matrix 𝐶,
where 𝑓̃ denotes the space of all sequences whose 𝐶1 -transforms
are in the space 𝑓 and is recently studied by Kayaduman and
Şengönül [31].
Abstract and Applied Analysis
9
Corollary 23. Let 𝐴 = (𝑎𝑛𝑘 ) be an infinite matrix and let 𝑡 =
(𝑡𝑘 ) be a sequence of positive numbers and define the matrix
𝐶 = (𝑐𝑛𝑘 ) by
𝑐𝑛𝑘 =
1 𝑛
∑𝑡 𝑎 ,
𝑇𝑛 𝑗=0 𝑗 𝑗𝑘
∀𝑛, 𝑘 ∈ N,
(72)
where 𝑇𝑛 = ∑𝑛𝑘=0 𝑡𝑘 for all 𝑛 ∈ N. Then, the necessary and
sufficient conditions in order to 𝐴 belongs to anyone of the
̃ 𝑝) : 𝑟𝑡 ) and (ℓ(𝐵,
̃ 𝑝) : 𝑟𝑡 )
̃ 𝑝) : 𝑟𝑡 ), (ℓ(𝐵,
classes (ℓ(𝐵,
∞
𝑐
0
are obtained from the respective ones in Theorems 13, 16 and
Corollary 17 by replacing the entries of the matrix 𝐴 by those
𝑡
, 𝑟𝑐𝑡 , and 𝑟0𝑡 are defined by Altay and
of the matrix 𝐶, where 𝑟∞
Başar in [32] as the spaces of all sequences whose 𝑅𝑡 -transforms
are, respectively, in the spaces ℓ∞ , 𝑐, and 𝑐0 , and are derived
𝑡
(𝑝), 𝑟𝑐𝑡 (𝑝) and 𝑟0𝑡 (𝑝) in the case
from the paranormed spaces 𝑟∞
𝑝𝑘 = 𝑝 for all 𝑘 ∈ N.
𝑡
Since the spaces 𝑟∞
, 𝑟𝑐𝑡 , and 𝑟0𝑡 reduce in the case 𝑡 = 𝑒 to
the Cesàro sequence spaces 𝑋∞ , 𝑐̃, and 𝑐̃0 of nonabsolute type,
respectively, Corollary 23 also includes the characterizations
̃ 𝑝) : 𝑋∞ ), (ℓ(𝐵,
̃ 𝑝) : 𝑐̃), and (ℓ(𝐵,
̃ 𝑝) : 𝑐̃0 ),
of the classes (ℓ(𝐵,
as a special case, where 𝑋∞ and 𝑐̃, 𝑐̃0 are the Cesàro spaces of
the sequences consisting of 𝐶1 -transforms are in the spaces
ℓ∞ and 𝑐, 𝑐0 and studied by Ng and Lee [33] and Şengönül and
Başar [34], respectively, where 𝐶1 denotes the Cesàro mean of
order 1.
Corollary 24. Let 𝐴 = (𝑎𝑛𝑘 ) be an infinite matrix and define
the matrix 𝐶 = (𝑐𝑛𝑘 ) by 𝑐𝑛𝑘 = 𝑎𝑛𝑘 − 𝑎𝑛+1,𝑘 for all 𝑛, 𝑘 ∈ N. Then,
the necessary and sufficient conditions in order to 𝐴 belongs
̃ 𝑝) : ℓ∞ (Δ)), (ℓ(𝐵,
̃ 𝑝) : 𝑐(Δ))
to anyone of the classes (ℓ(𝐵,
̃
and (ℓ(𝐵, 𝑝) : 𝑐0 (Δ)) are obtained from the respective ones in
Theorems 13 and 16 and Corollary 17 by replacing the entries
of the matrix 𝐴 by those of the matrix 𝐶, where ℓ∞ (Δ), 𝑐(Δ),
𝑐0 (Δ) denote the difference spaces of all bounded, convergent,
and null sequences and are introduced by Kızmaz [35].
Corollary 25. Let 𝐴 = (𝑎𝑛𝑘 ) be an infinite matrix and define
the matrix 𝐶 = (𝑐𝑛𝑘 ) by 𝑐𝑛𝑘 = ∑𝑛𝑗=0 𝑎𝑗𝑘 for all 𝑛, 𝑘 ∈ N. Then
the necessary and sufficient conditions in order to 𝐴 belongs to
̃ 𝑝) : 𝑏𝑠), (ℓ(𝐵,
̃ 𝑝) : 𝑐𝑠) and (ℓ(𝐵,
̃ 𝑝) :
anyone of the classes (ℓ(𝐵,
𝑐𝑠0 ) are obtained from the respective ones in Theorems 13, 16
and Corollary 17 by replacing the entries of the matrix 𝐴 by
those of the matrix 𝐶, where 𝑐𝑠0 denotes the set of those series
converging to zero.
5. Conclusion
The difference spaces ℓ∞ (Δ), 𝑐(Δ), and 𝑐0 (Δ) were introduced
by Kızmaz [35]. Since we essentially employ the infinite
matrices which is more different than Kızmaz and the other
authors following him, and use the technique of obtaining
a new sequence space by the matrix domain of a triangle
limitation method. Following this way, the domain of some
triangle matrices in the sequence space ℓ(𝑝) was recently
studied and were obtained certain topological and geometric
results by Altay and Başar [14, 16], Choudhary and Mishra
[10], Başar et al. [36], and Aydın and Başar [13]. Although
𝑏V(𝑒, 𝑝) = [ℓ(𝑝)]Δ is investigated, since 𝐵(1, −1) ≡ Δ, our
results are more general than those of Başar et al. [36]. Also
in case 𝑝𝑘 = 𝑝 for all 𝑘 ∈ N the results of the present study
are reduced to the corresponding results of the recent paper
of Kirişçi and Başar [9]. We should note that the difference
spaces Δ𝑐0 (𝑝), Δ𝑐(𝑝) and Δℓ∞ (𝑝) of Maddox’s spaces 𝑐0 (𝑝),
𝑐(𝑝), and ℓ∞ (𝑝) were studied by Ahmad and Mursaleen
[37]. Of course, a natural continuation of the present paper
is to study the sequence spaces [𝑐0 (𝑝)]𝐵(̃𝑟,̃𝑠) , [𝑐(𝑝)]𝐵(̃𝑟,̃𝑠) and
[ℓ∞ (𝑝)]𝐵(̃𝑟,̃𝑠) to generalize the main results of Ahmad and
Mursaleen [37] which fills up a gap in the existing literature.
It is clear that Δ(1) can be obtained as a special case of
𝐵(̃𝑟, 𝑠̃) for 𝑟̃ = 𝑒 and 𝑠̃ = −𝑒 and it is also trivial that 𝐵(̃𝑟, 𝑠̃)
is reduced in the special case 𝑟̃ = 𝑟𝑒 and 𝑠̃ = 𝑠𝑒 to the
generalized difference matrix 𝐵(𝑟, 𝑠). So, the results related to
the domain of the matrix 𝐵(̃𝑟, 𝑠̃) are much more general and
more comprehensive than the corresponding consequences
of the domain of the matrix 𝐵(𝑟, 𝑠). We should note from
now that the main results of the present paper are given as
an extended abstract without proof by Nergiz and Başar [38],
and our next paper will be devoted to some geometric and
̃ 𝑝).
topological properties of the space ℓ(𝐵,
Acknowledgments
The authors would like to thank Professor Bilâl Altay, Department of Mathematical Education, Faculty of Education,
İnönü University, 44280 Malatya, Turkey, for his careful
reading and constructive criticism of an earlier version of this
paper which improved the presentation and its readability.
The main results of this paper were presented in part at
the conference First International Conference on Analysis
and Applied Mathematics (ICAAM 2012) to be held October
18–21, 2012, in Gümüşhane, Turkey, at the University of
Gümüşhane.
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Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014