On matrix operators on the series space $|\bar{N}_p^θ|_k$
Recently, the space $|\bar{N}_p^θ|_k$ has been generated from the set of $k$-absolutely convergent series $\ell_k$ as the set of series summable by the absolute weighted method. In the paper, we investigate some properties of this space, such as $\beta$ -duality and the relationship with \ell k and...
Saved in:
| Date: | 2017 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2017
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1800 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507665061380096 |
|---|---|
| author | Mohapatra, R. N. Sarigol, M. A. Мохапатра, Р. Н. Сарігол, М. А. |
| author_facet | Mohapatra, R. N. Sarigol, M. A. Мохапатра, Р. Н. Сарігол, М. А. |
| author_sort | Mohapatra, R. N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:27:02Z |
| description | Recently, the space $|\bar{N}_p^θ|_k$
has been generated from the set of $k$-absolutely convergent series $\ell_k$ as the set of series
summable by the absolute weighted method. In the paper, we investigate some properties of this space, such as $\beta$ -duality
and the relationship with \ell k and then show that each element in the classes
$\Bigl(|\bar{N}_p|,\;|\bar{N}_p^θ|_k\Bigr)$
and
$\Bigl(|\bar{N}_p^θ|_k,\;|\bar{N}_q|\Bigr)$
of infinite matrices corresponds to a continuous linear operator and also characterizes these classes. Hence, in the special case, we
deduce some well-known results of Sarıg¨ol, Bosanquet, Orhan, and Sunouchi. |
| first_indexed | 2026-03-24T02:12:55Z |
| format | Article |
| fulltext |
UDC 517.9
R. N. Mohapatra (Univ. Central Florida, USA),
M. A. Sarıgöl (Univ. Pamukkale, Denizli, Turkey)
ON MATRIX OPERATORS ON THE SERIES SPACE | \bfitN \bfittheta
\bfitp | \bfitk
ПРО МАТРИЧНI ОПЕРАТОРИ В ПРОСТОРАХ РЯДIВ
\bigm| \bigm| \bigm| \bfitN \bfittheta
\bfitp
\bigm| \bigm| \bigm|
\bfitk
Recently, the space
\bigm| \bigm| \bigm| N\theta
p
\bigm| \bigm| \bigm|
k
has been generated from the set of k-absolutely convergent series \ell k as the set of series
summable by the absolute weighted method. In the paper, we investigate some properties of this space, such as \beta -duality
and the relationship with \ell k and then show that each element in the classes
\Bigl( \bigm| \bigm| Np
\bigm| \bigm| , \bigm| \bigm| \bigm| N\theta
q
\bigm| \bigm| \bigm|
k
\Bigr)
and
\Bigl( \bigm| \bigm| \bigm| N\theta
p
\bigm| \bigm| \bigm|
k
,
\bigm| \bigm| Nq
\bigm| \bigm| \Bigr) of infinite
matrices corresponds to a continuous linear operator and also characterizes these classes. Hence, in the special case, we
deduce some well-known results of Sarıgöl, Bosanquet, Orhan, and Sunouchi.
Нещодавно простiр
\bigm| \bigm| \bigm| N\theta
p
\bigm| \bigm| \bigm|
k
було згенеровано з множини \ell k всiх k-абсолютно збiжних рядiв, як множину рядiв
сумовних за абсолютним ваговим методом. В роботi дослiджено деякi властивостi цього простору такi, як \beta -
дуальнiсть i зв’язок з \ell k i, крiм того, доведено, що кожний елемент класiв
\Bigl( \bigm| \bigm| Np
\bigm| \bigm| , \bigm| \bigm| \bigm| N\theta
q
\bigm| \bigm| \bigm|
k
\Bigr)
та
\Bigl( \bigm| \bigm| \bigm| N\theta
p
\bigm| \bigm| \bigm|
k
,
\bigm| \bigm| Nq
\bigm| \bigm| \Bigr)
нескiнченних матриць вiдповiдає неперервному лiнiйному оператору i також харахтеризує цi класи. Таким чином,
у частинному випадку, нами виведено загальновiдомi результати Сарiголя, Босанке, Орхана та Сунучi.
1. Introduction. Let \omega be the set of all complex sequences and \ell k be the set of k-absolutely
convergent series. Let A = (anv) be an arbitrary infinite matrix of complex numbers and (\theta n) be a
positive sequence. By A(x) = (An (x)) , we denote the A-transform of the sequence x = (xv) , i.e.,
An (x) =
\infty \sum
v=0
anvxv,
provided that the series are convergent for v, n \geq 0. For (U, V ) , we write the set of all infinite matrices
A which map a sequence space U into a sequence V, and also the sets UA = \{ x \in \omega : A(x) \in U\}
and
U\beta =
\Biggl\{
\psi = (\psi n) :
\infty \sum
n=0
\psi nxn is convergent for all x \in U
\Biggr\}
are said to be the domain of a matrix A in U and the \beta dual of U, respectively.
Now let \Sigma av be a given infinite series with n th partial sum sn. Then the series \Sigma av is said to
be summable | A, \theta | k, k \geq 1, if [16]
\infty \sum
n=1
\theta k - 1
n | An (s) - An - 1 (s) | k<\infty .
In the case A =
\bigl(
N, pn
\bigr)
and \theta n = Pn/pn, the summability | A, \theta | k is reduced to the summability
methods | N, pn, \theta n| k and | N, pn| k, [3, 19], respectively. Also | A, \theta | k = | C,\alpha | k for A = (C,\alpha ) and
\theta n = n, in Flett’s notation [5]. By a weighted mean matrix A = (anv), we mean that
anv =
\left\{ pv/Pn, 0 \leq v \leq n,
0, v > n,
(1.1)
c\bigcirc R. N. MOHAPATRA, M. A. SARIGÖL, 2017
1524 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
ON MATRIX OPERATORS ON THE SERIES SPACE | N\theta
p| k 1525
where (pn) is a positive sequence with Pn = p0 + p1 + . . .+ pn \rightarrow \infty as n\rightarrow \infty , P - 1 = p - 1 = 0.
Throughout the paper, (qv) denotes a positive sequence with Qn = q0+q1+. . .+qn \rightarrow \infty as n\rightarrow \infty ,
and k\ast also denotes the conjugate of k > 1, i.e., 1/k + 1/k\ast = 1, 1/k\ast = 0 for k = 1. The series
space | N \theta
p| k has been defined in [14] as the set of all series summable by the summability method
| N, pn, \theta n| k. Say \mu k = | N \theta
p| k and \lambda k = | N \theta
q| k, for brevity. Then the series \Sigma av is | N, pn, \theta n| k
summable if and only if the sequence a = (av) \in \mu k. On the other hand, if we take the matrix A as
in (1.1), then, we can write
An(s) =
1
Pn
n\sum
v=0
pvsv =
1
Pn
n\sum
v=0
(Pn - Pv - 1) av, P - 1 = 0,
which implies
A0(s) = a0, An(s) - An - 1(s) =
pn
PnPn - 1
n\sum
v=0
Pv - 1av for n \geq 1.
We define the sequence space \mu k by
\mu k =
\left\{ a = (an) \in \omega :
\infty \sum
n=1
\theta k - 1
n
\bigm| \bigm| \bigm| \bigm| \bigm| pn
PnPn - 1
n\sum
v=1
Pv - 1av
\bigm| \bigm| \bigm| \bigm| \bigm|
k
<\infty
\right\} , 1 \leq k <\infty .
One can restate the space \mu k as the domain of the matrix T = (tnv) in the space \ell k of k-absolutely
convergent series, i.e., \mu k = (lk)T , where
tnv =
\left\{
1 n = 0, v = 0,
\theta
1/k\ast
n pnPv - 1
PnPn - 1
, 1 \leq v \leq n,
0, v > n or n \geq 1, v = 0,
(1.2)
for all n, v \in \BbbN = \{ 0, 1, 2, . . .\} . Besides, it is well known that lk is the BK -space (i.e., Banach
space with continuos coordinates) with respect to its natural norm | | x| | lk =
\Bigl( \sum \infty
v=0
| xv| k
\Bigr) 1/k
for
k \geq 1. Hence, since the matrix T is triangle and \mu k = (lk)T , it is immediate by Theorem 4.3.2 of
Wilansky [22, p. 63] that \mu k is also BK -space with respect to the norm | | x| | \mu k
= | | T (x)| | lk , k \geq 1.
We refer the reader to [13] for the case \theta n = Pn/pn and pn = 1, and also to [1] for full knowledge
on the normed sequence spaces and domain of triangle matrices in normed or paranormed sequence
spaces and the matrix transformations and summability theory.
The problems of absolute summability factors and comparision of these methods goes to old
rather and uptill now were widely examined by many authors, (see, for example, [3 – 6, 8 – 20]).
Now, by different standpoint we note that most of these results correspond to the special matrices
I,W \in (\mu 1, \lambda k) and (\mu k, \lambda 1) , where I is an infinite identity matrix and the matrix W = (wnv)
defined by wnv = \varepsilon n for v = n, zero otherwise. More recently, the above mentioned space \mu k has
been derived by the matrix T from the space \ell k and triangle matrix operators defined on that space
have been investigated in [14]. Note that although in the most cases the new space generated by the
summability matrix from \ell k is the expansion or the contraction of the original space, it can be seen
in some cases that these spaces overlap.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
1526 R. N. MOHAPATRA, M. A. SARIGÖL
2. Main results. In this paper we compute \beta -dual of \mu k and exhibit some inclusion relations
between the spaces \mu k and \ell k. We also show that each element in the classes (\mu 1, \lambda k) and (\mu k, \lambda 1)
of infinite matrices corresponds to a continous linear operator and characterize these operators, which
includes some known results of Sarıgöl [14], Bosanquet [4], Orhan and Sarıgöl [12] and Sunouchi
[20] as a special case. More precisely, we prove the following theorems.
Theorem 2.1. Let 1 < k <\infty and (\theta n) be a positive sequence. Then
\mu \beta 1 =
\biggl\{
\psi : \mathrm{s}\mathrm{u}\mathrm{p}
v
\biggl\{ \bigm| \bigm| \bigm| \bigm| 1pv (Pv\psi v - Pv - 1\psi v+1)
\bigm| \bigm| \bigm| \bigm| + Pv
pv
| \psi v|
\biggr\}
<\infty
\biggr\}
and
\mu \beta k =
\Biggl\{
\psi :
\infty \sum
v=1
1
\theta v
\bigm| \bigm| \bigm| \bigm| 1pv (Pv\psi v - Pv - 1\psi v+1)
\bigm| \bigm| \bigm| \bigm| k\ast <\infty , \mathrm{s}\mathrm{u}\mathrm{p}
v
| Lv| <\infty
\Biggr\}
, (2.1)
where
Lv = \theta - 1/k\ast
v
Pv
pv
\psi v.
Theorem 2.2. Let (\theta n) be a positive sequence. Then
(i) \ell k \subset \mu k holds for 1 < k <\infty if and only if
\mathrm{s}\mathrm{u}\mathrm{p}
m
\Biggl\{
m\sum
v=1
P k\ast
v - 1
\Biggr\} 1/k\ast \Biggl\{ \infty \sum
n=m
\theta k - 1
n
\biggl(
pn
PnPn - 1
\biggr) k
\Biggr\} 1/k
<\infty ; (2.2)
(ii) \mu k \subset \ell k holds for 1 < k <\infty if and only if
\mathrm{s}\mathrm{u}\mathrm{p}
m
\theta - 1/k\ast
m
Pm
pm
<\infty ; (2.3)
(iii) \mu k \subset \ell 1 holds for 1 \leq k <\infty if and only if
\infty \sum
v=1
1
\theta v
\biggl(
Pv
pv
\biggr) k\ast
<\infty ;
(iv) \ell 1 \subset \mu k holds for 1 \leq k <\infty if and only if
\mathrm{s}\mathrm{u}\mathrm{p}
v
Pv - 1
\infty \sum
n=v
\theta k - 1
n
\biggl(
pn
PnPn - 1
\biggr) k
<\infty .
Note that, \mu k = \ell k for \theta npn = O(Pn) and Pn = O(pn), and also if npn\theta n = O(Pn), then (2.2)
holds but not (2.3) and so the inclusion \mu k \subset \ell k holds strictly. Besides, \ell 1 \subset \mu 1 for all sequence
(pn) , since
\infty \sum
n=v
pn
PnPn - 1
=
1
Pv - 1
.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
ON MATRIX OPERATORS ON THE SERIES SPACE | N\theta
p| k 1527
Theorem 2.3. Let 1 \leq k <\infty . Assume that A = (anv) is an arbitrary infinite matrix and (\theta n)
is a positive sequence. Then (\mu 1, \lambda k) \subset B (\mu 1, \lambda k) , i.e., there exists a continuous linear operator
LA such that LA(x) = A(x), and A \in (\mu 1, \lambda k) if and only if, for n = 0, 1, . . . ,
\mathrm{s}\mathrm{u}\mathrm{p}
v
\biggl\{ \bigm| \bigm| \bigm| \bigm| 1pv (Pvanv - Pv - 1an,v+1)
\bigm| \bigm| \bigm| \bigm| + Pv
pv
| anv|
\biggr\}
<\infty , (2.4)
\mathrm{s}\mathrm{u}\mathrm{p}
j
\infty \sum
n=1
\bigm| \bigm| \bigm| \bigm| \bigm| \theta 1/k
\ast
n qn
QnQn - 1
n\sum
v=1
Qv - 1
pj
(Pjavj - Pj - 1av,,j+1)
\bigm| \bigm| \bigm| \bigm| \bigm|
k
<\infty . (2.5)
Theorem 2.4. Let 1 < k <\infty . Assume that A = (anv) is an arbitrary infinite matrix and (\theta n)
is a positive sequence. Then, (\mu k, \lambda 1) \subset B (\mu k, \lambda 1) , i.e., there exists a bouned linear operator LA
such that LA(x) = A(x), and A \in (\mu k, \lambda 1) if and only if
\mathrm{s}\mathrm{u}\mathrm{p}
v
\theta - 1/k\ast
v
Pv
pv
| anv| <\infty , n = 0, 1, . . . , (2.6)
\infty \sum
v=1
1
\theta v
\bigm| \bigm| \bigm| \bigm| 1pv (Pvanv - Pv - 1an,v+1)
\bigm| \bigm| \bigm| \bigm| k\ast <\infty , n = 0, 1, . . . , (2.7)
\infty \sum
j=0
1
\theta j
\Biggl( \infty \sum
n=1
qn
QnQn - 1
\bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
v=1
Qv - 1
pj
(Pjavj - Pj - 1av,,j+1)
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggr) k\ast
<\infty . (2.8)
3. Needed lemmas. We need the following lemmas for the proof of our theorems.
Lemma 3.1 [18]. Let 1 < k <\infty . Then, A \in (\ell k, \ell 1) if and only if
\infty \sum
v=0
\Biggl( \infty \sum
n=0
| anv|
\Biggr) k\ast
<\infty ,
Lemma 3.2 [7]. Let 1 \leq k <\infty . Then A \in (\ell 1, \ell k) if and only if
\mathrm{s}\mathrm{u}\mathrm{p}
v
\infty \sum
n=0
| anv| k <\infty .
Lemma 3.3 [21]. Let 1 < k <\infty . Then
(i) A \in (\ell 1, c) if and only if
\mathrm{l}\mathrm{i}\mathrm{m}
n
anv exists for v \geq 0 and \mathrm{s}\mathrm{u}\mathrm{p}
n,v
| anv| <\infty ;
(ii) A \in (\ell k, c) if and only if
holds for v \geq 0 and \mathrm{s}\mathrm{u}\mathrm{p}
n
\infty \sum
v=0
| anv| k
\ast
<\infty .
Lemma 3.4 [2]. Let 1 < k \leq s <\infty . Then A \in (\ell k, \ell k) if and only if
\mathrm{s}\mathrm{u}\mathrm{p}
m
\Biggl(
m\sum
v=0
ck
\ast
v
\Biggr) 1/k\ast \Biggl( \infty \sum
n=m
akv
\Biggr) 1/k
<\infty ,
where A = (anv) is a factorable matrix of nonnegative numbers, i.e., anv = ancv for 0 \leq v \leq n,
and zero otherwise.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
1528 R. N. MOHAPATRA, M. A. SARIGÖL
4. Proofs of theorems.
Proof of Theorem 2.1. Let \psi \in \mu \beta k . Then, the series
\sum \infty
v=0
\psi vxv is convergent for every
x \in \mu k, and also x \in \mu k if and only if R \in \ell k, where R = (Rn)n\in N is defined by
R0 = x0, Rn =
\theta
1/k\ast
n pn
PnPn - 1
n\sum
v=1
Pv - 1xv for n \geq 1. (4.1)
Furher, it can be written from Abel’s partial summation and (4.1) that
m\sum
v=0
\psi vxv = \psi 0R0 +
m - 1\sum
v=1
(Pv\psi v - Pv - 1\psi v+1)
Rv
pv
+
Pm
pm
\psi mRm =
= \psi 0R0 +
m - 1\sum
v=1
(Pv\psi v - Pv - 1\psi v+1)
\theta
- 1/k\ast
v
pv
Rv +
Pm
pm
\psi m\theta
- 1/k\ast
m Rm =
\infty \sum
v=0
wmvRv,
where
wmv =
\left\{
\psi 0, v = 0,
(Pv\psi v - Pv - 1\psi v+1)
\theta
- 1/k\ast
v
pv
, 1 \leq v \leq m - 1,
Pm
pm
\psi m\theta
- 1/k\ast
m , v = m,
0, v > m.
Now, \psi \in \mu \beta k \leftrightarrow W \in (\ell k, c) . Therefore, it follows from Lemma 3.3 that
\mathrm{s}\mathrm{u}\mathrm{p}
m
\Biggl\{
m - 1\sum
v=1
1
\theta v
\bigm| \bigm| \bigm| \bigm| 1pv (Pv\psi v - Pv - 1\psi v+1)
\bigm| \bigm| \bigm| \bigm| k\ast + \bigm| \bigm| \bigm| \bigm| Pm
pm
\psi m\theta
- 1/k\ast
m
\bigm| \bigm| \bigm| \bigm| k\ast
\Biggr\}
<\infty
if and only if (2.2) is satisfied.
Theorem 2.1 is proved.
Proof of Theorem 2.2. (i) Let us define the matrix T by (1.2). Then, it is easily seen that lk \subset \mu k
if and only if T \in (lk, lk) . Hence, applying Lemma 3.4 with the matrix T, it follows that T \in (lk, lk)
if and only if (2.2) holds, which completes the proof.
(ii) Take x \in \mu k, then y = T (x) \in lk and so \mu k \subset lk states that if y \in lk then T - 1(y) \in lk, or,
equivalently, T - 1 \in (lk, lk) . Now, if we say T - 1 = S = (snv), then we have
snv =
\left\{
1, n = 0, v = 0,
- dn - 1
Pn - 1
, v = n - 1, n \geq 2,
dn
Pn - 1
, v = n, n \geq 1,
0 v > n or n \geq 1, v = 0,
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
ON MATRIX OPERATORS ON THE SERIES SPACE | N\theta
p| k 1529
where
dn =
Pn - 1Pn
\theta
1/k\ast
n pn
for n \geq 1.
Now, if y \in lk, thenS0(y) = y0,
Sn(y) =
n\sum
v=1
snvyv =
1
Pn - 1
( - dn - 1yn - 1 + dnyn) for n \geq 1
and so, since Pn - 1 < Pn for all n \geq 1,
| | S(y)| | lk \leq
\Biggl\{
| y0| k +
\infty \sum
n=1
\Biggl( \bigm| \bigm| \bigm| \bigm| dn - 1
Pn - 1
yn - 1
\bigm| \bigm| \bigm| \bigm| k + \bigm| \bigm| \bigm| \bigm| dnPn - 1
yn
\bigm| \bigm| \bigm| \bigm| k
\Biggr) \Biggr\} 1/k
=
= O(1) \mathrm{s}\mathrm{u}\mathrm{p}
n
dn
Pn - 1
| | y| | lk <\infty ,
which shows that (2.3) is sufficient for T - 1 \in (lk, lk) . Conversely, if T - 1 \in (lk, lk) , then S :
lk \rightarrow lk is a bounded linear operator since lk is a BK -space. Hence, there exists some constant M
such that
| | S(x)| | lk \leq M | | x| | lk for all x \in lk. (4.2)
Applying (4.2) with xv = 1 for v = n and otherwise, we get, for all v \geq 0,
| svv| \leq
\Biggl( \infty \sum
n=0
| snv| k
\Biggr) 1/k
\leq M,
which implies (2.3).
Theorem 2.2 is proved.
The other parts can be easily proved by Lemmas 3.2 and 3.3.
Proof of Theorem 2.3. It is clear from the definition of matrix transformation that LA is a linear
operator, and since \mu k is a BK -space, it is also continuous by Theorem 4.2.8 of Wilanky [22, p. 57].
For the second part, A \in (\mu 1, \lambda k) if and only if (anj)
\infty
j=0 \in \mu \beta 1 and A(x) = (An(x)) \in \lambda k. But, by
Theorem 2.1, (anj)
\infty
j=0 \in \mu \beta 1 if and only if (2.8) holds. Further, since (xn) \in \mu 1 \leftrightarrow (Rn) \in \ell 1 by
(4.1) with k = 1, then
Pm
pm
anmRm \rightarrow 0 as m\rightarrow \infty , for each n. By the inversion of (4.1), we get
m\sum
v=0
anvxv =
m - 1\sum
v=0
(Pvanv - Pv - 1an,v+1)
Rv
pv
+
Pm
pm
anmRm
which implies
An (x) =
\infty \sum
v=0
(Pvanv - Pv - 1an,v+1)
Rv
pv
.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
1530 R. N. MOHAPATRA, M. A. SARIGÖL
On the other hand, t\ast \in \ell k if and only if A(x) = (An(x)) \in \lambda k, whenever t\ast 0 = A0(x) and, for
n \geq 1,
t\ast n =
\theta
1/k\ast
n qn
QnQn - 1
n\sum
v=1
Qv - 1Av(x) =
=
\infty \sum
j=0
n\sum
v=1
\theta
1/k\ast
n qnQv - 1
QnQn - 1
(Pjavi - Pj - 1av,j+1)
Rj
pj
=
\infty \sum
j=0
cnjRj ,
where, for j = 0, 1, 2, . . . ,
cnj =
\left\{
(Pjanj - Pj - 1an,j+1)
1
pj
, n = 0,\sum n
v=1
\theta
1/k\ast
n qnQv - 1
QnQn - 1
(Pjavj - Pj - 1av,j+1)
1
pj
, n \geq 1.
Now, A \in (\mu 1, \lambda k) \leftrightarrow C \in (\ell 1, \ell k) , i.e., equivalently, \mathrm{s}\mathrm{u}\mathrm{p}j
\sum \infty
n=0
| cnj | k < \infty , by Lemma 3.2.
Thus, it follows from the definition of the matrix C that
\mathrm{s}\mathrm{u}\mathrm{p}
j
\infty \sum
n=0
| cnj | k = \mathrm{s}\mathrm{u}\mathrm{p}
j
\Biggl\{ \bigm| \bigm| \bigm| \bigm| (Pja0j - Pj - 1a0,j+1)
1
pj
\bigm| \bigm| \bigm| \bigm| k +
+
\infty \sum
n=1
\theta k - 1
\bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
v=1
qnQv - 1
QnQn - 1
(Pjavi - Pj - 1av,j+1)
1
pj
\bigm| \bigm| \bigm| \bigm| \bigm|
k
\right\} <\infty ,
if and only if (2.9) is satisfied by (2.4).
Theorem 2.3 is proved.
Proof of Theorem 2.4. The first part is as in the proof of Theorem 2.3. For the second part,
A \in (\mu k, \lambda 1) if and only if (anj)
\infty
j=0 \in \mu \beta k and A(x) = (An(x)) \in \lambda 1 for every x \in \mu k. But, it
follows from Lemma 3.3 that (anj)
\infty
j=0 \in \mu \beta k if and only if (2.6) and (2.7) hold. Also, by (4.1), since
x \in \mu k \leftrightarrow R \in \ell k , then
Pm
pm
anm\theta
- 1/k\ast
m Rm \rightarrow 0 as m \rightarrow \infty for each n, by (2.6). By Abel’s partial
summation and (4.1), we get
m\sum
v=0
anvxv = an0R0 +
m - 1\sum
v=1
(Pvanv - Pv - 1an,v+1)
\theta
- 1/k\ast
v
pv
Rv +
Pm
pm
\theta - 1/k\ast
m anmRm,
and so
An (x) =
\infty \sum
v=0
(Pvanv - Pv - 1an,v+1)
\theta
- 1/k\ast
v
pv
Rv.
On the other hand, t\ast \in \ell 1 if and only if A(x) = (An(x)) \in \lambda 1 whenever
t\ast 0 =
\infty \sum
v=0
(Pva0v - Pv - 1a0,v+1)
\theta
- 1/k\ast
v
pv
Rv
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
ON MATRIX OPERATORS ON THE SERIES SPACE | N\theta
p| k 1531
and, for n \geq 1,
t\ast n =
n\sum
v=1
qnQv - 1
QnQn - 1
Av(x) =
=
\infty \sum
j=1
\Biggl(
n\sum
v=1
qnQv - 1
QnQn - 1
(Pjavi - Pj - 1av,j+1)
\theta
- 1/k\ast
j
pj
\Biggr)
Rj =
\infty \sum
j=0
dnjRj ,
where
dnj =
\left\{
(Pja0j - Pj - 1a0,j+1)
\theta
- 1/k\ast
j
pj
, n = 0, j \geq 0,
\sum n
v=1
qnQv - 1
QnQn - 1
(Pjavi - Pj - 1av,j+1)
\theta
- 1/k\ast
j
pj
, n \geq 1, j \geq 0.
Now, A \in (\mu k, \lambda 1) \leftrightarrow D \in (\ell k, \ell 1) . Applying Lemma 3.1 to the matrix D gives
\infty \sum
j=0
\Biggl( \infty \sum
n=0
| dnj |
\Biggr) k\ast
=
\infty \sum
j=0
\Biggl(
| d0j | +
\infty \sum
n=1
| dnj |
\Biggr) k\ast
<\infty
which holds if and only if condition (2.8) is satisfied by (2.7).
Theorem 2.4 is proved.
5. Applications. Theorem 2.3 and 2.4 have several consequences depending on the choice of an
infinite matrix A, the sequences \theta = (\theta n) , p = (pn) and q = (qn) , For example, if we take A =W
(resp. \varepsilon v = 1) then A \in (\mu 1, \lambda k) gives us summability factors of the form that if \Sigma av is summable
\mu 1, then \Sigma \varepsilon vav is summable \lambda k (resp. \mu 1 \subset \lambda k).
If A is chosen any triangular matrix in Theorem 2.3, then, it is obvious that (2.4) holds, so it can
be omitted, and also (2.5) is reduced to
\mathrm{s}\mathrm{u}\mathrm{p}
i
1
pkj
\infty \sum
n=j
\theta k - 1
n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
v=j
qnQv - 1
QnQn - 1
(Pjavj - Pj - 1av,,j+1)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
k
<\infty , (5.1)
equivalently,
\mathrm{s}\mathrm{u}\mathrm{p}
j
\left\{
\bigm| \bigm| \bigm| \bigm| \bigm| \theta
1/k\ast
j qjPjajj
Qjpj
\bigm| \bigm| \bigm| \bigm| \bigm|
k
+
\infty \sum
n=j+1
\theta k - 1
n | \Gamma (n, j)| k
\right\} <\infty , (5.2)
where
\Gamma (n, j) =
qnQv - 1
QnQn - 1
\biggl[
Pj
pj
(avj - av,,j+1) + av,,j+1
\biggr]
.
Further, if (5.1) is satisfied, then A : \mu 1 \rightarrow \lambda k is a continuous linear mapping and so there exists a
number M such that
| | A(x)| | \lambda k
\leq M | | x| | \mu 1 for all x \in \mu 1. (5.3)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
1532 R. N. MOHAPATRA, M. A. SARIGÖL
Taking any v \geq 0, we apply (5.3) with xv+1 = 1, xm = 0,m \not = v+1. Hence, it can be obtained that
v = 0, 1, . . .
\infty \sum
n=v+1
\theta k - 1
n
\bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
m=v+1
qnQm - 1
QnQn - 1
am,v+1
\bigm| \bigm| \bigm| \bigm| \bigm|
k
\leq Mk (5.4)
(see, also, [14). Thus, it is easily seen from (5.2) that (5.1) implies (5.4),
\mathrm{s}\mathrm{u}\mathrm{p}
j
\theta
1/k\ast
j qjPj
Qjpj
| ajj | <\infty (5.5)
and
\mathrm{s}\mathrm{u}\mathrm{p}
j
\biggl(
Pj
pj
\biggr) k \infty \sum
n=j+1
\theta k - 1
n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
v=j
qnQv - 1
QnQn - 1
(avj - av,j+1)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
k
<\infty . (5.6)
Conversely, by considering (5.2), it is deduced from (5.4), (5.5) and (5.6) that (5.1) is satisfied.
So (5.1) is equivalent to (5.4), (5.5) and (5.6), which gives the following result of [14].
Corollary 5.1. Let A be any triangle matrix and (\theta n) be a positive sequence.Then, A \in (\mu 1, \lambda k) ,
1 \leq k <\infty , if and only if (5.4), (5.5) and (5.6) are satisfied.
It is well known that the case A = I and k = 1 of this result was given by Bosanquet [4] and
Sunouchi [20], and also the case A = I, \theta n = n and k \geq 1 was established by Orhan and Sarıgöl
[12].
Further, if we take A as triangular matrix in Theorem 2.4, then (2.6) and (2.7) hold directly, and
(2.8) reduces to
\infty \sum
j=0
1
pk
\ast
j \theta j
\left( \infty \sum
n=j
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
v=j
qnQv - 1
QnQn - 1
(Pjavj - Pj - 1av,,j+1)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) k\ast
<\infty . (5.7)
So we get the following main result of [14].
Corollary 5.2. Let 1 < k < \infty and 1/k + 1/k\ast = 1. Let A be a triangle matrix and (\theta n) be a
positive sequence. Then A \in (\mu k, \lambda 1) if and only if (5.7) is satisfied.
References
1. Başar F. Summability theory and Its applications. – Istanbul: Bentham Sci. Publ., 2012.
2. Bennett G. Some elemantery inequalities // Quart. J. Math. (Oxford). – 1987. – 38. – P. 401 – 425.
3. Bor H., Thorpe B. On some absolute summability methods // Analysis. – 1987. – 7, № 2. – P. 145 – 152.
4. Bosanquet L. S. Review on G. Sunouchi’s paper // Notes Fourier Anal., 18, absolute summability of a series with
constant terms // Math. Rev. – 1950. – 654. – 11.
5. Flett T. M. On an extension of absolute summability and some theorems of Littlewood and Paley // Proc. London
Math. Soc. – 1957. – 7. – P. 113 – 141.
6. Kuttner B. Some remarks on summability factors // Indian J. Pure and Appl. Math. – 1985. – 16, № 9. – P. 1017 – 1027.
7. Maddox I. J. Elements of functinal analysis. – London; New York: Cambridge Univ. Press, 1970.
8. Mazhar S. M. On the absolute summability factors of infinite series // Tohoku Math. J. – 1971. – 23. – P. 433 – 451.
9. Mehdi M. R. Summability factors for generalized absolute summability I // Proc. London Math. Soc. – 1960. – 3,
№ 10. – P. 180 – 199.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
ON MATRIX OPERATORS ON THE SERIES SPACE | N\theta
p| k 1533
10. McFadden L. Absolute Nörlund summability // Duke Math. J. – 1942. – 9. – P. 168 – 207.
11. Mohapatra R. N., Das G. Summability factors of lower-semi matrix transformations // Monatsh. Math. – 1975. – 79. –
S. 307 – 315.
12. Orhan C., Sarıgöl M. A. On absolute weighted mean summability // Rocky Mountain J. Math. – 1993. – 23, № 3. –
P. 1091 – 1097.
13. Sarıgöl M. A. Necessary and sufficient conditions for the equivalence of the summability methods | N, pn| k and
| C, 1| k // Indian J. Pure and Appl. Math. – 1991. – 22, № 6. – P. 483 – 489.
14. Sarıgöl M. A. Matrix transformatins on fields of absolute weighted mean summability // Stud. Sci. Math. Hung. –
2011. – 48, № 3. – P. 331 – 341.
15. Sarıgöl M. A., Bor H. Characterization of absolute summability factors // J. Math. Anal. and Appl. – 1995. – 195. –
P. 537 – 545.
16. Sarıgöl M. A. On local properties of factored Fourier series // Appl. Math. and Comput. – 2010. – 216. – P. 3386 – 3390.
17. Orhan C., Sarıgöl M. A. Characterization of summaility factors for Riesz methods // Kuwait J. Sci. Eng. – 1994. –
21. – P. 1 – 7.
18. Orhan C., Sarıgöl M. A. Extension of Mazhar’s theorem on summability factors // Kuwait J. Sci. – 2015. – 42, № 3. –
P. 28 – 35.
19. Sulaiman W. T. On summability factors of infinite series // Proc. Amer. Math. Soc. – 1992. – 115. – P. 313 – 317.
20. Sunouchi G. Notes on Fourier analysis, 18, absolute summability of a series with constant terms // Tohoku Math. J. –
1949. – 1. – P. 57 – 65.
21. Stieglitz M., Tietz H. Matrixtransformationen von Folgenraumen Eine Ergebnisüberischt // Math Z. – 1977. – 154. –
S. 1 – 16.
22. Wilansky A. Summability through functional analysis // North-Holland Math. Stud., 1984. – 85.
Received 24.09.09
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
|
| id | umjimathkievua-article-1800 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:12:55Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/44/bbe92ed2d3b3ebf2f4892838419d3c44.pdf |
| spelling | umjimathkievua-article-18002019-12-05T09:27:02Z On matrix operators on the series space $|\bar{N}_p^θ|_k$ Про матричнi оператори в просторах рядiв $|\bar{N}_p^θ|_k$ Mohapatra, R. N. Sarigol, M. A. Мохапатра, Р. Н. Сарігол, М. А. Recently, the space $|\bar{N}_p^θ|_k$ has been generated from the set of $k$-absolutely convergent series $\ell_k$ as the set of series summable by the absolute weighted method. In the paper, we investigate some properties of this space, such as $\beta$ -duality and the relationship with \ell k and then show that each element in the classes $\Bigl(|\bar{N}_p|,\;|\bar{N}_p^θ|_k\Bigr)$ and $\Bigl(|\bar{N}_p^θ|_k,\;|\bar{N}_q|\Bigr)$ of infinite matrices corresponds to a continuous linear operator and also characterizes these classes. Hence, in the special case, we deduce some well-known results of Sarıg¨ol, Bosanquet, Orhan, and Sunouchi. Нещодавно простiр $|\bar{N}_p^θ|_k$ було згенеровано з множини $\ell_k$ всiх $k$-абсолютно збiжних рядiв, як множину рядiв сумовних за абсолютним ваговим методом. В роботi дослiджено деякi властивостi цього простору такi, як $\beta$ - дуальнiсть i зв’язок з $\ell k$ i, крiм того, доведено, що кожний елемент класiв $\Bigl(|\bar{N}_p|,\;|\bar{N}_p^θ|_k\Bigr)$ та $\Bigl(|\bar{N}_p^θ|_k,\;|\bar{N}_q|\Bigr)$ нескiнченних матриць вiдповiдає неперервному лiнiйному оператору i також харахтеризує цi класи. Таким чином, у частинному випадку, нами виведено загальновiдомi результати Сарiголя, Босанке, Орхана та Сунучi. Institute of Mathematics, NAS of Ukraine 2017-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1800 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 11 (2017); 1524-1533 Український математичний журнал; Том 69 № 11 (2017); 1524-1533 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1800/782 Copyright (c) 2017 Mohapatra R. N.; Sarigol M. A. |
| spellingShingle | Mohapatra, R. N. Sarigol, M. A. Мохапатра, Р. Н. Сарігол, М. А. On matrix operators on the series space $|\bar{N}_p^θ|_k$ |
| title | On matrix operators on the series space $|\bar{N}_p^θ|_k$ |
| title_alt | Про матричнi оператори в просторах рядiв $|\bar{N}_p^θ|_k$ |
| title_full | On matrix operators on the series space $|\bar{N}_p^θ|_k$ |
| title_fullStr | On matrix operators on the series space $|\bar{N}_p^θ|_k$ |
| title_full_unstemmed | On matrix operators on the series space $|\bar{N}_p^θ|_k$ |
| title_short | On matrix operators on the series space $|\bar{N}_p^θ|_k$ |
| title_sort | on matrix operators on the series space $|\bar{n}_p^θ|_k$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1800 |
| work_keys_str_mv | AT mohapatrarn onmatrixoperatorsontheseriesspacebarnpthk AT sarigolma onmatrixoperatorsontheseriesspacebarnpthk AT mohapatrarn onmatrixoperatorsontheseriesspacebarnpthk AT sarígolma onmatrixoperatorsontheseriesspacebarnpthk AT mohapatrarn promatričnioperatorivprostorahrâdivbarnpthk AT sarigolma promatričnioperatorivprostorahrâdivbarnpthk AT mohapatrarn promatričnioperatorivprostorahrâdivbarnpthk AT sarígolma promatričnioperatorivprostorahrâdivbarnpthk |