# Generalization of the space $$l(p)$$ derived by absolute Euler summability and matrix operators

## Abstract

The sequence space $$l(p)$$ having an important role in summability theory was defined and studied by Maddox (Q. J. Math. 18:345–355, 1967). In the present paper, we generalize the space $$l(p)$$ to the space $$\vert E_{\phi }^{r} \vert (p)$$ derived by the absolute summability of Euler mean. Also, we show that it is a paranormed space and linearly isomorphic to $$l(p)$$. Further, we determine α-, β-, and γ-duals of this space and construct its Schauder basis. Also, we characterize certain matrix operators on the space.

## 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

$$A_{n} ( x ) =\sum_{v=0}^{\infty }a_{nv}x_{v}$$

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

$$X_{A}= \bigl\{ x= ( x_{n} ) \in \omega :A(x)\in X \bigr\} .$$
(1)

The α-, β-, and γ-duals of the space X are defined as follows:

\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}

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

$$\lim_{m\rightarrow \infty }x^{ [ m ] }=\lim_{m\rightarrow \infty }\sum _{v=0}^{m}x_{v}e^{(v)}=x$$

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

$$l(p)= \Biggl\{ x=(x_{n}):\sum_{n=1}^{\infty } \vert x_{n} \vert ^{p_{n}}< \infty \Biggr\}$$

is an FK space with AK with respect to its natural paranorm

$$g(x)= \Biggl( \sum_{n=0}^{\infty } \vert x_{n} \vert ^{p_{n}} \Biggr) ^{1/M},$$

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

$$\Vert x \Vert =\inf \Biggl\{ \delta >0:\sum_{n=0}^{\infty } \vert x_{n}/\delta \vert ^{p_{n}}\leq 1 \Biggr\}$$

([1921, 29]).

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 $$p_{n}=1$$.

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 )

$$\sum_{n=1}^{\infty } ( \phi_{n} ) ^{p_{n}-1} \bigl\vert A _{n} ( s ) -A_{n-1} ( s ) \bigr\vert ^{p_{n}}< \infty .$$

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 ) where Euler matrix $$E^{r}$$ is defined by

$$e_{nk}^{r}= \textstyle\begin{cases} {{{n}}\choose {{k}}}(1-r)^{n-k}r^{k}, &0\leq k\leq n , \\ 0,& k>n , \end{cases}$$

for $$0< r<1$$ and

$$e_{nk}^{1}= \textstyle\begin{cases} 0, &0\leq k< n , \\ 1, &k=n. \end{cases}$$

Also we refer the readers to the papers [7, 9, 30, 31, 35] for detailed terminology.

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 , Altay and Başar [2, 3], Yeşilkayagil and Başar  by defining as the domains of the band, Riesz, the factorable, and Nörlund matrices in the $$l(p)$$ (see also [1, 46, 1618, 2328]).

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.

## Needed lemmas

### Lemma 2.1

()

Let $$p= ( p_{v} )$$ and $$q= ( q_{v} )$$ be any two bounded sequences of strictly positive numbers.

1. (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)
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

$$\sup_{v}\sum_{n=0}^{\infty } \bigl\vert a_{nv}M^{-1/p_{v}} \bigr\vert ^{q_{n}}< \infty .$$
3. (iii)

If $$p_{v}\leq 1$$, then $$A\in ( l(p),c )$$ if and only if

$$(\mathrm{a})\quad \lim_{n}a_{nv}\textit{ exists for each }v,\quad\quad (\mathrm{b})\quad \sup_{n,v} \vert a_{nv} \vert ^{p_{v}}< \infty ,$$

and $$A\in ( l(p),l_{\infty } )$$ iff (b) holds.

4. (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

$$\sup_{n}\sum_{v=0}^{\infty } \bigl\vert a_{nv}M^{-1} \bigr\vert ^{p_{v}^{\ast }}< \infty ,$$

and $$A\in ( l(p),l_{\infty } )$$ iff (b) holds.

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

()

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

$$( 2C ) ^{-2}U_{p} [ A ] \leq L_{p} [ A ] \leq U_{p} [ A ] ,$$

where $$C=\max \{ 1,2^{H-1} \}$$, $$H=\sup _{v}p_{v}$$,

$$U_{p} [ A ] =\sum_{v=0}^{\infty } \Biggl( \sum_{n=0}^{\infty } \vert a_{nv} \vert \Biggr) ^{p_{v}}$$

and

$$L_{p} [ A ] =\sup \Biggl\{ \sum_{v=0}^{\infty } \biggl\vert \sum_{n\in K}a_{nv} \biggr\vert ^{p_{v}}:K\subset N\textit{ finite} \Biggr\} .$$

### Lemma 2.3

()

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

$$\hat{a}_{nv}=\sum_{j=v}^{\infty }a_{nj}s_{jv};\quad n,v=0,1, \ldots,$$

and

$$v_{mv}^{(n)}= \textstyle\begin{cases} \sum_{j=v}^{m}a_{nj}s_{jv},&0\leq v\leq m, \\ 0, &v>m. \end{cases}$$

## 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

$$\bigl\vert E_{\phi }^{r} \bigr\vert (p)= \Biggl\{ a\in \omega :\sum_{n=0}^{\infty }\phi_{n}^{p_{n}-1} \bigl\vert \bigtriangleup A _{n}^{r} ( s ) \bigr\vert ^{p_{n}}< \infty \Biggr\} ,$$

where

$$\bigtriangleup A_{n}^{r} ( s ) =A_{n}^{r}(s)-A_{n-1}^{r}(s)$$

and

$$A_{n}^{r}(s)=\sum_{k=0}^{n} {{{n}}\choose {{k}}}(1-r)^{n-k}r^{k}s _{k},\quad n \geq 0, \quad\quad A_{-1}^{r}(s)=0.$$

Also, a few calculations give

\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}

where

$$\sigma_{nm}= \textstyle\begin{cases} \sum_{k=m}^{n}(1-r)^{n-1-k}r^{k} [ {{{n-1}}\choose {{k-1}}}-r{{{n}}\choose {{k}}} ] , &1\leq m\leq n , \\ 0, & m>n. \end{cases}$$

Further, it follows by putting $$r=q(1+q)^{-1}$$

\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}

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

\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)

where

$$t_{nk}^{r}(\phi ,p)= \textstyle\begin{cases} \phi_{0}^{1/{p_{0}^{\ast }}}, &k=n=0 , \\ \phi_{n}^{1/{p_{n}^{\ast }}}{{{n-1}}\choose {{k-1}}}(1-r)^{n-k}r^{k}, & 1\leq k\leq n, \\ 0, &k>n. \end{cases}$$
(4)

Therefore, we can state the space $$\vert E_{\phi }^{r} \vert (p)$$ as follows:

$$\bigl\vert E_{\phi }^{r} \bigr\vert (p)= \Biggl\{ a=(a_{k}):\sum_{n=1}^{\infty } \Biggl\vert \phi_{n}^{1/{p_{n}^{\ast }}}\sum_{k=1}^{n} {{{n-1}}\choose {{k-1}}}(1-r)^{n-k}r^{k}a_{k} \Biggr\vert ^{p_{n}}< \infty \Biggr\} ,$$

or

$$\bigl\vert E_{\phi }^{r} \bigr\vert (p)= \bigl[ l(p) \bigr] _{T^{r}( \phi ,p)}$$

according to notation (1).

Further, since every triangle matrix has a unique inverse which is a triangle (see ), 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:

1. (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

$$\Vert x \Vert _{ \vert E_{\phi }^{r} \vert (p)}= \Biggl( \sum_{n=0}^{\infty } \bigl\vert T_{n}^{r}(\phi ,p) (x) \bigr\vert ^{p_{n}} \Biggr) ^{1/M},$$

where $$M=\max \{ 1,\sup p_{n} \}$$.

2. (b)

The space $$\vert E_{\phi }^{r} \vert (p)$$ is linearly isomorphic to the space $$l(p)$$, i.e., $$\vert E_{\phi }^{r} \vert (p) \cong l(p)$$.

3. (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 $$\vert E_{\phi }^{r} \vert (p)$$.

4. (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  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

$$\Vert x \Vert _{ \vert E_{\phi }^{r} \vert (p)}= \Biggl( \sum_{n=0}^{\infty } \bigl\vert T_{n}^{r}(\phi ,p) (x) \bigr\vert ^{p_{n}} \Biggr) ^{1/M}= \bigl\Vert T^{r}(\phi ,p) (x) \bigr\Vert _{l(p)}.$$

So, $$T^{r}(\phi ,p)$$ preserves the paranorm, which completes this part of the proof.

(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  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}
1. (i)

If $$p_{v}>1$$ for all v, then

$$\bigl\{ \bigl\vert E_{\phi }^{r} \bigr\vert (p) \bigr\} ^{\alpha }=D _{1}^{r},\quad\quad \bigl\{ \bigl\vert E_{\phi }^{r} \bigr\vert (p) \bigr\} ^{ \beta }=D_{4}^{r} \cap D_{3}^{r}, \quad\quad \bigl\{ \bigl\vert E_{\phi }^{r} \bigr\vert (p) \bigr\} ^{\gamma }=D_{4}^{r}.$$
2. (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

\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}

where $$D=(d_{nv})$$ is defined by

$$d_{nv} = \textstyle\begin{cases} \phi_{0}^{-1/p_{0}^{\ast }}a_{0},&n=v=0 , \\ \sum_{k=v}^{n}b_{k}^{(v)}a_{k},&1\leq v\leq n , \\ 0, &v>n. \end{cases}$$

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.

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

$$\hat{a}_{nv}=\sum_{j=v}^{\infty }a_{nj}b_{j}^{(v)}$$

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 $$n=0,1,\ldots$$ ,

\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)

and

\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

$$v_{mv}^{(n)}= \textstyle\begin{cases} \sum_{j=v}^{m}b_{j}^{(v)}a_{nj},&0\leq v\leq m, \\ 0,& v>m. \end{cases}$$

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

$$\sup_{v}\sum_{n=0}^{\infty } \bigl\vert M^{-1/p_{v}}f _{nv} \bigr\vert ^{q_{n}}< \infty ,$$

which completes the proof. □

### 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 $$n=0,1,\ldots$$ ,

\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)

and

\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 Â 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. □

## 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.

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## Acknowledgements

We thank the editor and referees for their careful reading, valuable suggestions and remarks.

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