1 Introduction
Let X, Y be any subsets of ω, the set of all sequences of complex numbers, and \(A=(a_{nv})\) be an infinite matrix of complex numbers. By \(A(x)=(A_{n}(x))\), we indicate the A-transform of a sequence \(x= ( x_{v} ) \) if the series
are convergent for \(n\geq 0\). If \(Ax\in Y\), whenever \(x\in X\), then A, denoted by \(A:X\rightarrow Y\), is called a matrix transformation from X into Y, and we mean the class of all infinite matrices A such that \(A:X\rightarrow Y\) by \((X,Y)\). For \(c_{s}\), \(b_{s}\), and \(l_{p}\) (\(p\geq 1\)), we write the space of all convergent, bounded, p-absolutely convergent series, respectively. Further, the matrix domain of an infinite matrix A in a sequence space X is defined by
The α-, β-, and γ-duals of the space X are defined as follows:
A subspace X is called an FK space if it is a Frechet space, that is, a complete locally convex linear metric space, with continuous coordinates \(P_{n}:X\rightarrow C\) (\(n=1,2,\ldots \)), where \(P_{n}(x)=x_{n}\) for all \(x\in X\); an FK space whose metric is given by a norm is said to be a BK space. An FK space X including the set of all finite sequences is said to have AK if
for every sequence \(x\in X\), where \(e^{(v)}\) is a sequence whose only non-zero term is one in vth place for \(v\geq 0\). For example, it is well known that the Maddox space
is an FK space with AK with respect to its natural paranorm
where \(M=\max \{ 1,\sup_{n}p_{n} \} \); also it is even a BK space if \(p_{n}\geq 1\) for all n with respect to the norm
([19‐21, 29]).
$$ A_{n} ( x ) =\sum_{v=0}^{\infty }a_{nv}x_{v} $$
$$ X_{A}= \bigl\{ x= ( x_{n} ) \in \omega :A(x)\in X \bigr\} . $$
(1)
$$\begin{aligned}& X^{\alpha }= \bigl\{ \epsilon \in \omega :(\epsilon_{n}x_{n}) \in l_{1} \text{ for all } x\in X \bigr\} , \\ & X^{\beta }= \bigl\{ \epsilon \in \omega :(\epsilon_{n}x_{n}) \in c_{s} \text{ for all } x\in X \bigr\} , \\ & X^{\gamma }= \bigl\{ \epsilon \in \omega :(\epsilon_{n}x_{n}) \in b_{s} \text{ for all } x\in X \bigr\} . \end{aligned}$$
$$ \lim_{m\rightarrow \infty }x^{ [ m ] }=\lim_{m\rightarrow \infty }\sum _{v=0}^{m}x_{v}e^{(v)}=x $$
$$ l(p)= \Biggl\{ x=(x_{n}):\sum_{n=1}^{\infty } \vert x_{n} \vert ^{p_{n}}< \infty \Biggr\} $$
$$ g(x)= \Biggl( \sum_{n=0}^{\infty } \vert x_{n} \vert ^{p_{n}} \Biggr) ^{1/M}, $$
$$ \Vert x \Vert =\inf \Biggl\{ \delta >0:\sum_{n=0}^{\infty } \vert x_{n}/\delta \vert ^{p_{n}}\leq 1 \Biggr\} $$
Throughout this paper, we assume that \(0<\inf p_{n}\leq H<\infty \) and \(p_{n}^{\ast }\) is a conjugate of \(p_{n}\), i.e., \(1/p_{n}+1/p_{n} ^{\ast }=1\), \(p_{n}>1\), and \(1/p_{n}^{\ast }=0\) for \(p_{n}=1\).
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Let \(\sum a_{v}\) be a given infinite series with \(s_{n}\) as its nth partial sum, \(\phi = ( \phi_{n} ) \) be a sequence of positive real numbers and \(p= ( p_{n} ) \) be a bounded sequence of positive real numbers. The series \(\sum a_{v}\) is said to be summable \(\vert A,\phi_{n} \vert ( p ) \) if (see [10])
$$ \sum_{n=1}^{\infty } ( \phi_{n} ) ^{p_{n}-1} \bigl\vert A _{n} ( s ) -A_{n-1} ( s ) \bigr\vert ^{p_{n}}< \infty . $$
It should be noted that the summability \(\vert A,\phi_{n} \vert (p)\) includes some well-known summability methods for special cases of A, ϕ and \(p=(p_{n})\). For example, if we take \(A=E^{r}\) and \(p_{n}=k\) for all n, then it is reduced to the summability method \(\vert E,r \vert _{k}\) (see [12]) where Euler matrix \(E^{r}\) is defined by
for \(0< r<1\) and
$$ e_{nk}^{r}= \textstyle\begin{cases} {{{n}}\choose {{k}}}(1-r)^{n-k}r^{k}, &0\leq k\leq n , \\ 0,& k>n , \end{cases} $$
$$ e_{nk}^{1}= \textstyle\begin{cases} 0, &0\leq k< n , \\ 1, &k=n. \end{cases} $$
A large literature body, concerned with producing sequence spaces by means of matrix domain of a special limitation method and studying their algebraic, topological structure and matrix transformations, has recently grown. In this context, the sequence spaces \(\overline{l}(p)\), \(r_{p}^{t}\), \(l(u,v,p)\), and \(l(N^{t},p)\) were studied by Choudhary and Mishra [8], Altay and Başar [2, 3], Yeşilkayagil and Başar [37] by defining as the domains of the band, Riesz, the factorable, and Nörlund matrices in the \(l(p)\) (see also [1, 4‐6, 16‐18, 23‐28]).
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Also, some series spaces have been derived and examined by various absolute summability methods from a different point of view (see [13, 14, 32, 34]). In this paper, we generalize the space \(l(p)\) to the space \(\vert E_{\phi }^{r} \vert (p)\) derived by the absolute summability of Euler means and show that it is a paranormed space linearly isomorphic to \(l(p)\). Further, we determine α-, β-, and γ-duals of this space and construct its Schauder basis. Finally, we characterize certain matrix transformations on the space.
First, we remind some well-known lemmas which play important roles in our research.
2 Needed lemmas
Lemma 2.1
([11])
Let
\(p= ( p_{v} ) \)
and
\(q= ( q_{v} ) \)
be any two bounded sequences of strictly positive numbers.
(i)
If
\(p_{v}>1\)
for all
v, then
\(A\in ( l(p),l _{1} ) \)
if and only if there exists an integer
\(M>1\)
such that
$$ \sup \Biggl\{ \sum_{v=0}^{\infty } \biggl\vert \sum_{n \in K}a_{nv}M^{-1} \biggr\vert ^{p_{v}^{\ast }}:K\subset N \textit{ finite} \Biggr\} < \infty . $$
(2)
(ii)
If
\(p_{v}\leq 1\)
and
\(q_{v}\geq 1\)
for all
\(v\in N\), then
\(A\in ( l(p),l(q) ) \)
if and only if there exists some
M
such that
$$ \sup_{v}\sum_{n=0}^{\infty } \bigl\vert a_{nv}M^{-1/p_{v}} \bigr\vert ^{q_{n}}< \infty . $$
(iii)
If
\(p_{v}\leq 1\), then
\(A\in ( l(p),c ) \)
if and only if
and
\(A\in ( l(p),l_{\infty } ) \)
iff (b) holds.
$$ (\mathrm{a})\quad \lim_{n}a_{nv}\textit{ exists for each }v,\quad\quad (\mathrm{b})\quad \sup_{n,v} \vert a_{nv} \vert ^{p_{v}}< \infty , $$
(iv)
If
\(p_{v}>1\)
for all
v, then
\(A\in ( l(p),c ) \)
if and only if (a) (a) holds, and (b) there is a number
\(M>1 \)
such that
and
\(A\in ( l(p),l_{\infty } ) \)
iff (b) holds.
$$ \sup_{n}\sum_{v=0}^{\infty } \bigl\vert a_{nv}M^{-1} \bigr\vert ^{p_{v}^{\ast }}< \infty , $$
It may be noticed that condition (2) exposes a rather difficult condition in applications. The following lemma produces a condition to be equivalent to (2) and so the following lemma, which is more practical in many cases, will be used in the proofs of theorems.
Lemma 2.2
([33])
Let
\(A= ( a_{nv} ) \)
be an infinite matrix with complex numbers and
\(( p_{v} ) \)
be a bounded sequence of positive numbers. If
\(U_{p} [ A ] < \infty \)
or
\(L_{p} [ A ] <\infty \), then
where
\(C=\max \{ 1,2^{H-1} \} \), \(H=\sup _{v}p_{v}\),
and
$$ ( 2C ) ^{-2}U_{p} [ A ] \leq L_{p} [ A ] \leq U_{p} [ A ] , $$
$$ U_{p} [ A ] =\sum_{v=0}^{\infty } \Biggl( \sum_{n=0}^{\infty } \vert a_{nv} \vert \Biggr) ^{p_{v}} $$
$$ L_{p} [ A ] =\sup \Biggl\{ \sum_{v=0}^{\infty } \biggl\vert \sum_{n\in K}a_{nv} \biggr\vert ^{p_{v}}:K\subset N\textit{ finite} \Biggr\} . $$
Lemma 2.3
([22])
Let
X
be an
FK
space with
AK, T
be a triangle, S
be its inverse, and
Y
be an arbitrary subset of ω. Then we have
\(A\in ( X_{T},Y ) \)
if and only if
\(\widehat{A}\in ( X,Y ) \)
and
\(V^{(n)}\in ( X,c ) \)
for all
n, where
and
$$ \hat{a}_{nv}=\sum_{j=v}^{\infty }a_{nj}s_{jv};\quad n,v=0,1, \ldots, $$
$$ v_{mv}^{(n)}= \textstyle\begin{cases} \sum_{j=v}^{m}a_{nj}s_{jv},&0\leq v\leq m, \\ 0, &v>m. \end{cases} $$
3 Main theorems
In this section, we introduce the paranormed series space \(\vert E _{\phi }^{r} \vert (p)\) as the set of all series summable by the absolute summability method of Euler matrix and show that this space is linearly isomorphic to the space \(l(p)\). Also, we compute the Schauder base, α-, β-, and γ-duals of the space and characterize certain matrix transformations defined on that space.
First of all, we note that, by the definition of the summability \(\vert A,\phi_{n} \vert (p)\), we can write the space \(\vert E_{\phi }^{r} \vert (p)\) as
where
and
Also, a few calculations give
where
Further, it follows by putting \(r=q(1+q)^{-1}\)
Now, by considering \(T_{n}^{r}(\phi ,p)(a)=\phi_{n}^{1/{p_{n}^{\ast }}} \bigtriangleup A_{n}^{r} ( s ) \), we immediately get that \(T_{0}^{r}(\phi ,p)(a)=a_{0}\phi_{0}^{1/{p_{0}^{\ast }}}\) and
where
Therefore, we can state the space \(\vert E_{\phi }^{r} \vert (p)\) as follows:
or
according to notation (1).
$$ \bigl\vert E_{\phi }^{r} \bigr\vert (p)= \Biggl\{ a\in \omega :\sum_{n=0}^{\infty }\phi_{n}^{p_{n}-1} \bigl\vert \bigtriangleup A _{n}^{r} ( s ) \bigr\vert ^{p_{n}}< \infty \Biggr\} , $$
$$ \bigtriangleup A_{n}^{r} ( s ) =A_{n}^{r}(s)-A_{n-1}^{r}(s) $$
$$ A_{n}^{r}(s)=\sum_{k=0}^{n} {{{n}}\choose {{k}}}(1-r)^{n-k}r^{k}s _{k},\quad n \geq 0, \quad\quad A_{-1}^{r}(s)=0. $$
$$\begin{aligned} \bigtriangleup A_{n}^{r} ( s ) = & \sum _{m=0}^{n} \sum_{k=m}^{n} {{{n}}\choose {{k}}}(1-r)^{n-k}r^{k}a_{m}-\sum _{m=0}^{n-1}\sum_{k=m}^{n-1} {{{n-1}}\choose {{k}}}(1-r)^{n-1-k}r ^{k}a_{m} \\ = & \sum_{m=1}^{n}\sum _{k=m}^{n}(1-r)^{n-1-k} \biggl[ {{{n-1}}\choose {{k-1}}}-r{{{n}}\choose {{k}}} \biggr] r^{k}a_{m} \\ = & \sum_{m=1}^{n}\sigma_{nm}a_{m}, \end{aligned}$$
$$ \sigma_{nm}= \textstyle\begin{cases} \sum_{k=m}^{n}(1-r)^{n-1-k}r^{k} [ {{{n-1}}\choose {{k-1}}}-r{{{n}}\choose {{k}}} ] , &1\leq m\leq n , \\ 0, & m>n. \end{cases} $$
$$\begin{aligned} \sigma_{nm} = & (1+q)^{1-n}\sum _{k=m}^{n}q^{k} \biggl[ {{{n-1}}\choose {{k-1}}}-q(1+q)^{-1}{{{n}}\choose {{k}}} \biggr] \\ & \\ = & (1+q)^{-n}\sum_{k=m}^{n} \biggl[ q^{k} {{{n-1}}\choose {{k-1}}}-q^{k+1} {{{n-1}}\choose {{k}}} \biggr] \\ & \\ = & q^{m}(1+q)^{-n}{{{n-1}}\choose {{m-1}}}= {{{n-1}}\choose {{m-1}}}(1-r)^{n-m}r ^{m}. \end{aligned}$$
$$\begin{aligned} T_{n}^{r}(\phi ,p) (a) =&\phi_{n}^{1/{p_{n}^{\ast }}} \sum_{k=1}^{n}{{{n-1}}\choose {{k-1}}}(1-r)^{n-k}r^{k}a_{k} \\ =&\sum_{k=1}^{n}t_{nk}^{r}( \phi ,p)a_{k}, \end{aligned}$$
(3)
$$ t_{nk}^{r}(\phi ,p)= \textstyle\begin{cases} \phi_{0}^{1/{p_{0}^{\ast }}}, &k=n=0 , \\ \phi_{n}^{1/{p_{n}^{\ast }}}{{{n-1}}\choose {{k-1}}}(1-r)^{n-k}r^{k}, & 1\leq k\leq n, \\ 0, &k>n. \end{cases} $$
(4)
$$ \bigl\vert E_{\phi }^{r} \bigr\vert (p)= \Biggl\{ a=(a_{k}):\sum_{n=1}^{\infty } \Biggl\vert \phi_{n}^{1/{p_{n}^{\ast }}}\sum_{k=1}^{n} {{{n-1}}\choose {{k-1}}}(1-r)^{n-k}r^{k}a_{k} \Biggr\vert ^{p_{n}}< \infty \Biggr\} , $$
$$ \bigl\vert E_{\phi }^{r} \bigr\vert (p)= \bigl[ l(p) \bigr] _{T^{r}( \phi ,p)} $$
Further, since every triangle matrix has a unique inverse which is a triangle (see [36]), the matrix \(T^{r}(\phi ,p)\) has a unique inverse \(S^{r}(\phi ,p)=(s_{nk}^{r}(\phi ,p))\) given by
$$\begin{aligned} s_{nk}^{r}(\phi ,p)= \textstyle\begin{cases} \phi_{0}^{-1/{p_{0}^{\ast }}}, &k=n=0 , \\ \phi_{k}^{-1/{p_{k}^{\ast }}}{{{n-1}}\choose {{k-1}}}(r-1)^{n-k}r^{-n}, &1\leq k\leq n , \\ 0, &k>n. \end{cases}\displaystyle \end{aligned}$$
(5)
Before main theorems, note that if \(r=1\) and \(\phi_{n}=1\) for all \(n\geq 0\), the space \(\vert E_{\phi }^{r} \vert (p)\) is reduced to the space \(l(p)\).
Theorem 3.1
Let
\(0< r<1\)
and
\(p=(p_{n})\)
be a bounded sequence of non-negative numbers. Then:
(a)
The set
\(\vert E_{\phi }^{r} \vert (p)\)
becomes a linear space with the coordinate-wise addition and scalar multiplication, and also it is an
FK-space with respect to the paranorm
where
\(M=\max \{ 1,\sup p_{n} \} \).
$$ \Vert x \Vert _{ \vert E_{\phi }^{r} \vert (p)}= \Biggl( \sum_{n=0}^{\infty } \bigl\vert T_{n}^{r}(\phi ,p) (x) \bigr\vert ^{p_{n}} \Biggr) ^{1/M}, $$
(b)
The space
\(\vert E_{\phi }^{r} \vert (p)\)
is linearly isomorphic to the space
\(l(p)\), i.e., \(\vert E_{\phi }^{r} \vert (p) \cong l(p)\).
(c)
Define a sequence
\((b_{n}^{(v)})\)
by
\(S^{r} ( (e^{(v)}) ) = ( \sum_{v=0}^{n}s_{nv}^{r}(\phi ,p)e^{(v)} ) \). Then the sequence
\((b_{n}^{(v)})\)
is the Schauder base of the space
\(\vert E_{\phi }^{r} \vert (p)\).
(d)
The space
\(\vert E_{\phi }^{r} \vert (p)\)
is separable.
Proof
(a) The first part is a routine verification, so it is omitted. Since \(T^{r}(\phi ,p)\) is a triangle matrix and \(l(p)\) is an FK-space, it follows from Theorem 4.3.2 in [36] that \(\vert E_{\phi }^{r} \vert (p)= [ l(p) ] _{T^{r}( \phi ,p)}\) is an FK-space.
(b) We should show that there exists a linear bijection between the spaces \(\vert E_{\phi }^{r} \vert (p)\) and \(l(p)\). Now, consider \(T^{r}(\phi ,p): \vert E_{\phi }^{r} \vert (p) \rightarrow l(p)\) given by (3). Since the matrix corresponding this transformation is a triangle, it is obvious that \(T^{r}(\phi ,p)\) is a linear bijection. Furthermore, since \(T^{r}(\phi ,p)(x)\in l(p)\) for \(x\in \vert E_{\phi }^{r} \vert (p)\), we get
So, \(T^{r}(\phi ,p)\) preserves the paranorm, which completes this part of the proof.
$$ \Vert x \Vert _{ \vert E_{\phi }^{r} \vert (p)}= \Biggl( \sum_{n=0}^{\infty } \bigl\vert T_{n}^{r}(\phi ,p) (x) \bigr\vert ^{p_{n}} \Biggr) ^{1/M}= \bigl\Vert T^{r}(\phi ,p) (x) \bigr\Vert _{l(p)}. $$
(c) Since the sequence \((e^{(v)})\) is the Schauder base of the space \(l(p)\) and \(\vert E_{\phi }^{r} \vert (p)= [ l(p) ] _{T^{r}(\phi ,p)}\), it can be written from Theorem 2.3 in [15] that \(b^{(v)}=(S^{r}(\phi ,p)(e^{(v)}))\) is a Schauder base of the space \(\vert E_{\phi }^{r} \vert (p)\).
(d) Since the space \(\vert E_{\phi }^{r} \vert (p)\) is a linear metric space with a Schauder base, it is separable. □
Theorem 3.2
Let
\(0< r<1\). Define
$$\begin{aligned}& D_{1}^{r}= \Biggl\{ a\in \omega :\exists M>1, \sum _{v=0}^{\infty } \Biggl( \sum_{n=v}^{\infty } \bigl\vert M^{-1}b_{n}^{(v)}a_{n} \bigr\vert \Biggr) ^{p_{v}^{\ast }}< \infty \Biggr\} , \\& D_{2}^{r}= \Biggl\{ a\in \omega :\exists M>1,\sup _{v}M^{1/p_{v}}\sum_{n=v}^{\infty } \bigl\vert b_{n}^{(v)}a_{n} \bigr\vert < \infty \Biggr\} , \\& D_{3}^{r}= \Biggl\{ a\in \omega :\sum _{n=v}^{\infty }b_{n}^{(v)}a_{n} \textit{ converges for each } v \Biggr\} , \\& D_{4}^{r}= \Biggl\{ a\in \omega : \exists M>1, \sup _{n}\sum_{v=1} ^{n} \Biggl\vert \sum_{k=v}^{n}b_{k}^{(v)}a_{k}M^{-1} \Biggr\vert ^{p_{v}^{\ast }}< \infty \Biggr\} , \\& D_{5}^{r}= \Biggl\{ a\in \omega :\sup_{n,v} \Biggl\vert \sum_{k=v} ^{n}b_{k}^{(v)}a_{k} \Biggr\vert ^{p_{v}}< \infty \Biggr\} . \end{aligned}$$
(i)
If
\(p_{v}>1\)
for all
v, then
$$ \bigl\{ \bigl\vert E_{\phi }^{r} \bigr\vert (p) \bigr\} ^{\alpha }=D _{1}^{r},\quad\quad \bigl\{ \bigl\vert E_{\phi }^{r} \bigr\vert (p) \bigr\} ^{ \beta }=D_{4}^{r} \cap D_{3}^{r}, \quad\quad \bigl\{ \bigl\vert E_{\phi }^{r} \bigr\vert (p) \bigr\} ^{\gamma }=D_{4}^{r}. $$
(ii)
If
\(p_{v}\leq 1\)
for all
v, then
$$ \bigl\{ \bigl\vert E_{\phi }^{r} \bigr\vert (p) \bigr\} ^{\alpha }=D _{2}^{r}, \quad\quad \bigl\{ \bigl\vert E_{\phi }^{r} \bigr\vert (p) \bigr\} ^{ \beta }=D_{5}^{r} \cap D_{3}^{r},\quad\quad \bigl\{ \bigl\vert E_{\phi }^{r} \bigr\vert (p) \bigr\} ^{\gamma }=D_{5}^{r}. $$
Proof
To avoid the repetition of a similar statement, we only calculate β-duals of \(\vert E_{\phi }^{r} \vert (p)\).
(i) Let us recall that \(a\in \{ \vert E_{\phi }^{r} \vert (p) \} ^{\beta }\) if and only if \(ax\in cs\) whenever \(x\in \vert E_{\phi }^{r} \vert (p)\). Now, by using (5), it can be obtained that
where \(D=(d_{nv})\) is defined by
Since \(T^{r}(\phi ,p)(x)\in l(p)\) whenever \(x\in \vert E_{\phi } ^{r} \vert (p)\), \(a\in \{ \vert E_{\phi }^{r} \vert (p) \} ^{\beta }\) if and only if \(D\in (l(p),c)\). So, it follows from Lemma 2.1 that \(a\in D_{4} ^{r}\cap D_{3}^{r}\) if \(p_{v}>1\) for all v, and also \(a\in D_{5} ^{r}\cap D_{3}^{r}\) if \(p_{v}\leq 1\) for all v.
$$\begin{aligned} \sum_{k=0}^{n}a_{k}x_{k} = & T_{0}^{r}(\phi ,p) (x)\phi_{0} ^{-1/p_{0}^{\ast }}a_{0}+\sum_{k=1}^{n}a_{k} \sum_{v=1} ^{k}\phi_{v}^{-1/p_{v}^{\ast }} {{{k-1}}\choose {{v-1}}} ( r-1 ) ^{k-v}r^{-k}T_{v}^{r}( \phi ,p) (x) \\ = & T_{0}^{r}(\phi ,p) (x)\phi_{0}^{-1/p_{0}^{\ast }}a_{0}+ \sum_{v=1}^{n}\phi_{v}^{-1/p_{v}^{\ast }}T_{v}^{r}( \phi ,p) (x) \sum_{k=v}^{n}a_{k} {{{k-1}}\choose {{v-1}}} ( r-1 ) ^{k-v}r^{-k} \\ = & \sum_{v=0}^{n}d_{nv}T_{v}^{r}( \phi ,p) (x), \end{aligned}$$
$$ d_{nv} = \textstyle\begin{cases} \phi_{0}^{-1/p_{0}^{\ast }}a_{0},&n=v=0 , \\ \sum_{k=v}^{n}b_{k}^{(v)}a_{k},&1\leq v\leq n , \\ 0, &v>n. \end{cases} $$
The remaining part of the theorem can be similarly proved by Lemma 2.1. □
Theorem 3.3
Let
\(A= ( a_{nv} ) \)
be an infinite matrix of complex numbers, \(( \phi_{n} ) \)
and
\(( \psi _{n} ) \)
be sequences of positive numbers, \(p= ( p_{n} ) \)
and
\(q= ( q_{n} ) \)
be arbitrary bounded sequences of positive numbers with
\(p_{n}\leq 1\)
and
\(q_{n}\geq 1\)
for all n. Further, let the matrix
Â
be defined by
and
\(F=T^{r}(\psi ,q)\hat{A}\). Then
\(A\in ( \vert E_{\phi } ^{r} \vert (p), \vert E_{\psi }^{r} \vert (q) ) \)
if and only if there exists an integer
\(M>1\)
such that, for
\(n=0,1,\ldots\) ,
and
$$ \hat{a}_{nv}=\sum_{j=v}^{\infty }a_{nj}b_{j}^{(v)} $$
$$\begin{aligned}& \sum_{k=v}^{\infty }b_{k}^{(v)}a_{nk} \quad \textit{converges for each } v, \end{aligned}$$
(6)
$$\begin{aligned}& \sup_{m,v} \Biggl\vert \sum_{k=v}^{m}b_{k}^{(v)}a_{nk} \Biggr\vert ^{p_{v}}< \infty , \end{aligned}$$
(7)
$$\begin{aligned} \sup_{v}\sum_{n=0}^{\infty } \bigl\vert M^{-1/p_{v}}f_{nv} \bigr\vert ^{q_{n}}< \infty . \end{aligned}$$
(8)
Proof
Suppose that \(p_{v}\leq 1\), \(q_{v}\geq 1\) for all v. Note that \(\vert E_{\phi }^{r} \vert (p)= [ l(p) ] _{T^{r}(\phi ,p)}\) and \(\vert E_{\psi }^{r} \vert (q)= [ l(q) ] _{T^{r}(\psi ,q)}\). By Lemma 2.3, \(A\in ( \vert E _{\phi }^{r} \vert (p), \vert E_{\psi }^{r} \vert (q) ) \) if and only if \(\hat{A}\in ( l(p), \vert E_{\psi }^{r} \vert (q) ) \) and \(V^{(n)} \in ( l(p),c ) \), where the matrix \(V^{(n)}\) is defined by
One can see that since \(\hat{A}(x)\in \vert E_{\psi }^{r} \vert (q)= [ l(q) ] _{T^{r}(\psi ,q)}\) whenever \(x\in l(p)\), \(\hat{A}\in ( l(p), \vert E_{\psi }^{r} \vert (q) ) \) iff \(F=T^{r}(\psi ,q)\hat{A}\in ( l(p),l(q) ) \). Now, applying Lemma 2.1(ii) and (iii) to the matrices F and \(V^{(n)}\), it follows that \(V^{(n)}\in ( l(p),c ) \) iff, for \(n=0,1,\ldots\) , conditions (6) and (7) hold, and \(F\in ( l(p),l(q) ) \) iff there exists an integer M such that
which completes the proof. □
$$ v_{mv}^{(n)}= \textstyle\begin{cases} \sum_{j=v}^{m}b_{j}^{(v)}a_{nj},&0\leq v\leq m, \\ 0,& v>m. \end{cases} $$
$$ \sup_{v}\sum_{n=0}^{\infty } \bigl\vert M^{-1/p_{v}}f _{nv} \bigr\vert ^{q_{n}}< \infty , $$
Theorem 3.4
Assume that
\(A= ( a_{nv} ) \)
is an infinite matrix of complex numbers and
\(( \phi_{n} ) \), \(( \psi_{n} ) \)
are sequences of positive numbers. If
\(p= ( p _{n} ) \)
is an arbitrary bounded sequence of positive numbers such that
\(p_{n}>1\)
for all
n, and
\(H=T^{r}(\psi ,1)\hat{A}\), then
\(A\in ( \vert E_{\phi }^{r} \vert (p), \vert E_{ \psi }^{r} \vert (1) ) \)
if and only if there exists an integer
\(M>1\)
such that, for
\(n=0,1,\ldots\) ,
and
$$\begin{aligned}& \sum_{k=v}^{\infty }b_{k}^{(v)} a_{nk} \quad \textit{converges for each } v \end{aligned}$$
(9)
$$\begin{aligned}& \sup_{n}\sum_{v=0}^{\infty } \Biggl\vert \sum_{k=v}^{n}b _{k}^{(v)} a_{nk}M^{-1} \Biggr\vert ^{p_{v}^{\ast }}< \infty \end{aligned}$$
(10)
$$\begin{aligned} \sum_{v=0}^{\infty } \Biggl( \sum _{n=0}^{\infty } \bigl\vert M ^{-1}h_{nv} \bigr\vert \Biggr) ^{p_{v}^{\ast }}< \infty . \end{aligned}$$
(11)
Proof
Let \(p_{n}>1\) for all n. It is clear that \(\vert E _{\phi }^{r} \vert (p)= [ l(p) ] _{T^{r}(\phi ,p)}\) and \(\vert E_{\psi }^{r} \vert (1)=l_{T^{r}(\psi ,1)}\). So, by Lemma 2.3, we have \(A\in ( \vert E_{\phi }^{r} \vert (p), \vert E_{\psi }^{r} \vert (1) ) \) if and only if \(\hat{A}\in ( l(p), \vert E_{\psi }^{r} \vert (1) ) \) and \(V^{(n)}\in ( l(p),c ) \), where  and \(V^{(n)}\) are given in Theorem 3.3. If we take \(H=T^{r}(\psi ,1) \hat{A}\), then it is easily seen that \(\hat{A}\in ( l(p), \vert E _{\psi }^{r} \vert (1) ) \) iff \(H\in ( l(p),l_{1} ) \) because, if \(\hat{A}(x)\in \vert E_{\psi }^{r} \vert (1)\) for all \(x\in l_{1} ( p ) \), \(H(x)=T^{r}(\psi ,1)(\hat{A}(x)) \in l_{1}\). So, applying Lemma 2.1(iv) to the matrix \(V^{(n)}\), it is obtained that \(V^{(n)}\in ( l ( p ) ,c ) \) iff conditions (9) and (10) are satisfied. Again, if we apply Lemma 2.1(i) and Lemma 2.2 to the matrix H, then we have \(H\in ( l(p),l _{1} ) \) iff the last condition holds. □
4 Conclusion
The sequence spaces defined as domains of Riesz, factorable, Nörlund and S-matrices in the spaces \(l(p)\) and the space of series summable by the absolute Euler have been recently studied by several authors. In this paper, we have defined the new absolute Euler space \(\vert E _{\phi }^{r} \vert (p)\) and investigated some topological and algebraic properties such as isomorphism, duals, base, and also characterized certain matrix transformations on that space. So, we have extended some well-known results.
Acknowledgements
We thank the editor and referees for their careful reading, valuable suggestions and remarks.
Competing interests
The authors declare that they have no competing interests.
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