Algebraic duals of some sets of difference sequences defined by Orlicz functions

Abstract

In this paper, we first give a description of some sets of sequences generated by difference operators and defined by Orlicz functions. Then their algebraic duals such as the α-, β-, γ- and null-duals are computed.

MSC:46A45, 47N40, 65J99, 46A20.

1 Introduction

Let w denote the space of all scalar sequences, and any subspace of w is called a sequence space. Let ${\ell }_{\mathrm{\infty }}$, c and $c_0$ be the linear spaces of bounded, convergent and null sequences $x=(x_k)$ with complex terms, respectively, normed by ${\parallel x\parallel }_{\mathrm{\infty }}={sup}_k|x_k|$, where $k\in N=\{1,2,\dots \}$, the set of positive integers.

Lindenstrauss and Tzafriri [1] used the Orlicz function and introduced the sequence space ${\ell }_M$ as follows:

$${\ell }_M=\{(x_k)\in w:\sum _{k=1}^{\mathrm{\infty }}M\left(\frac{|x_k|}{\rho }\right)<\mathrm{\infty }\text{ for some }\rho >0\}.$$

They proved that ${\ell }_M$ is a Banach space normed by

$$\parallel (x_k)\parallel =inf\{\rho >0:\sum _{k=1}^{\mathrm{\infty }}M\left(\frac{|x_k|}{\rho }\right)\le 1\}.$$

Throughout this section X will denote one of the sequence spaces ${\ell }_{\mathrm{\infty }}$, c and $c_0$.

The notion of difference sequence spaces was introduced by Kizmaz [2]. For some other works on difference sequences, Orlicz functions and related literature, we refer to [3–8]. Let $v=(v_k)$ be any fixed sequence of non-zero complex numbers. Et and Esi [9] generalized the above sequence spaces to the following sequence spaces:

$$X\left({\mathrm{\Delta }}_v^m\right)=\{x=(x_k):\left({\mathrm{\Delta }}_v^m{x}_k\right)\in X\}$$

for $X={\ell }_{\mathrm{\infty }},c\text{ and }{c}_0$.

In this paper, for an Orlicz function M, we can have the following spaces in the line of the spaces studied by Mursaleen et al. [10]:

$$X(M,{\mathrm{\Delta }}_v^m)=\{x=(x_k):\left({\mathrm{\Delta }}_v^m{x}_k\right)\in X(M)\}.$$

In fact, we get the following spaces:

$$\begin{array}{c}{\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)=\{x=(x_k):\underset{k}{sup}M\left(\frac{|{\mathrm{\Delta }}_v^m{x}_k|}{\rho }\right)<\mathrm{\infty }\text{ for some }\rho >0\},\hfill \\ c(M,{\mathrm{\Delta }}_v^m)=\{x=(x_k):\underset{k\to \mathrm{\infty }}{lim}M\left(\frac{|{\mathrm{\Delta }}_v^m{x}_k-L|}{\rho }\right)=0\text{ for some }L\text{ and }\rho >0\},\hfill \\ c_0(M,{\mathrm{\Delta }}_v^m)=\{x=(x_k):\underset{k\to \mathrm{\infty }}{lim}M\left(\frac{|{\mathrm{\Delta }}_v^m{x}_k|}{\rho }\right)=0\text{ for some }\rho >0\},\hfill \end{array}$$

where ${\mathrm{\Delta }}_v^m{x}_k={\mathrm{\Delta }}_v^{m-1}{x}_k-{\mathrm{\Delta }}_v^{m-1}{x}_{k+1}$, ${\mathrm{\Delta }}_v^m{x}_k={\sum }_{i=0}^m{(-1)}^i\left(\genfrac{}{}{0ex}{}{m}{i}\right)v_{k+i}{x}_{k+i}$ for all $k\in N$.

Bektaş et al. [11] introduced the difference operator ${\mathrm{\Delta }}_v^{(m)}$ and defined it as follows:

$${\mathrm{\Delta }}_v^{(m)}{x}_k=\sum _{i=0}^m{(-1)}^i\left(\genfrac{}{}{0ex}{}{m}{i}\right)v_{k-i}{x}_{k-i}\text{for all }k\in N.$$

Using this difference operator, we can construct the following sequence space:

$$X(M,{\mathrm{\Delta }}_v^{(m)})=\{x=(x_k):\left({\mathrm{\Delta }}_v^{(m)}{x}_k\right)\in X(M)\}.$$

The operator

$${\mathrm{\Sigma }}^{(m)}:w\to w$$

is defined by

$${\mathrm{\Sigma }}^{(1)}{x}_k=\sum _{j=0}^k{x}_j(k=0,1,\dots ),{\mathrm{\Sigma }}^{(m)}={\mathrm{\Sigma }}^{(1)}o{\mathrm{\Sigma }}^{(m-1)}(m\ge 2)$$

and

$${\mathrm{\Sigma }}^{(m)}o{\mathrm{\Delta }}^{(m)}={\mathrm{\Delta }}^{(m)}o{\mathrm{\Sigma }}^{(m)}=\mathrm{id},\text{the identity on }w(\text{see [9, 12]}).$$

Now, for subsequent use, we slightly generalize the above definition as follows.

We define

$${\mathrm{\Sigma }}_v^{(m)}:w\to w$$

by

$${\mathrm{\Sigma }}_v^{(1)}{x}_k=\sum _{j=0}^k{v}_j{x}_j(k=0,1,\dots ),{\mathrm{\Sigma }}_v^{(m)}={\mathrm{\Sigma }}_v^{(m)}o{\mathrm{\Sigma }}_v^{(m-1)}(m\ge 2)$$

and

$$v^{-1}{\mathrm{\Sigma }}_v^{(m)}o{\mathrm{\Delta }}_v^{(m)}=v^{-1}{\mathrm{\Delta }}_v^{(m)}o{\mathrm{\Sigma }}_v^{(m)}=\mathrm{id},\text{the identity on }w\text{ and }{v}^{-1}=\left(v_k^{-1}\right).$$

Now, for $x\in X(M,{\mathrm{\Delta }}_v^m)$, let us define

$${\parallel x\parallel }_{\mathrm{\Delta }}=\sum _{i=1}^m|v_i{x}_i|+inf\{\rho >0:\underset{k}{sup}M\left(\frac{|{\mathrm{\Delta }}_v^m{x}_k|}{\rho }\right)\le 1\}.$$

It can be shown that $(X(M,{\mathrm{\Delta }}_v^m),{\parallel \cdot \parallel }_{\mathrm{\Delta }})$ is a BK-space.

Again for $x\in X(M,{\mathrm{\Delta }}_v^{(m)})$, let us define

$${\parallel x\parallel }_{{\mathrm{\Delta }}^{\prime }}=inf\{\rho >0:\underset{k}{sup}M\left(\frac{|{\mathrm{\Delta }}_v^{(m)}{x}_k|}{\rho }\right)\le 1\}.$$

It can be shown that $(X(M,{\mathrm{\Delta }}_v^{(m)}),{\parallel \cdot \parallel }_{{\mathrm{\Delta }}^{\prime }})$ is a BK-space.

It is trivial that $({\mathrm{\Delta }}_v^m{x}_k)\in X(M)$ if and only if $({\mathrm{\Delta }}_v^{(m)}{x}_k)\in X(M)$. Also the norms ${\parallel \cdot \parallel }_{\mathrm{\Delta }}$ and ${\parallel \cdot \parallel }_{{\mathrm{\Delta }}^{\prime }}$ are equivalent.

Let us define the operator

$$D:X(M,{\mathrm{\Delta }}_v^m)\to X(M,{\mathrm{\Delta }}_v^m)$$

by $Dx=(0,0,\dots ,x_{m+1},x_{m+2},\dots )$, where $x=(x_1,x_2,\dots ,x_m,\dots )$. It is trivial that D is a bounded linear operator on $X(M,{\mathrm{\Delta }}_v^m)$, $X={\ell }_{\mathrm{\infty }}$, c and $c_0$. Furthermore, the set

$$D\left[X(M,{\mathrm{\Delta }}_v^m)\right]=DX(M,{\mathrm{\Delta }}_v^m)=\{x=(x_k):x\in X(M,{\mathrm{\Delta }}_v^m),x_1=x_2=\cdots =x_m=0\}$$

is a subspace of $X(M,{\mathrm{\Delta }}_v^m)$ and normed by

$${\parallel x\parallel }_{\mathrm{\Delta }}=inf\{\rho >0:\underset{k}{sup}M\left(\frac{|{\mathrm{\Delta }}_v^m{x}_k|}{\rho }\right)\le 1\}\text{in }DX(M,{\mathrm{\Delta }}_v^m).$$

$DX(M,{\mathrm{\Delta }}_v^m)$ and $X(M)$ are equivalent as topological spaces since

$${\mathrm{\Delta }}_v^m:DX(M,{\mathrm{\Delta }}_v^m)\to X(M),$$

defined by

$${\mathrm{\Delta }}_v^{m}x=y=\left({\mathrm{\Delta }}_v^m{x}_k\right),$$

(1.1)

is a linear homeomorphism.

Moreover, obviously

$$\begin{array}{c}{\mathrm{\Delta }}_v^{(m)}:X(M,{\mathrm{\Delta }}_v^{(m)})\to X(M),{\mathrm{\Delta }}_v^{(m)}x=y=\left({\mathrm{\Delta }}_v^{(m)}{x}_k\right),\hfill \\ {\mathrm{\Sigma }}_v^{(m)}:X(M)\to X(M,{\mathrm{\Delta }}_v^{(m)}),{\mathrm{\Sigma }}_v^{(m)}x=y=\left({\mathrm{\Sigma }}_v^{(m)}{x}_k\right)\hfill \end{array}$$

are isometric isomorphisms for $X={\ell }_{\mathrm{\infty }},c\text{ and }{c}_0$.

Hence ${\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)$, $c(M,{\mathrm{\Delta }}_v^m)$ and $c_0(M,{\mathrm{\Delta }}_v^m)$ are isometrically isomorphic to ${\ell }_{\mathrm{\infty }}(M)$, $c(M)$ and $c_0(M)$, respectively.

Moreover, $X(M,{\mathrm{\Delta }}_v^i)\subset X(M,{\mathrm{\Delta }}_v^m)$ for $i=0,1,\dots ,m-1$, which follows from the following inequality and convexity of M:

$$M\left(\frac{|{\mathrm{\Delta }}_v^m{x}_k|}{2\rho }\right)\le \frac{1}{2}M\left(\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_k|}{\rho }\right)+\frac{1}{2}M\left(\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_{k+1}|}{\rho }\right).$$

Investigation of spaces is often combined with that of duals. The algebraic dual space is defined for all vector spaces. When defined for a topological vector space, there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space. For any finite-dimensional normed vector space or topological vector space, such as Euclidean n-space, the continuous dual and the algebraic dual coincide. This is, however, false for any infinite-dimensional normed space. For some related literature on duality relevant to this paper, we refer to [9, 11, 13, 14]. Our results of this paper will generalize few existing results as well as generate some new results in the literature of algebraic duality within the field of functional analysis.

2 Computation of algebraic duals

In this section we compute the α-, β-, γ- and N-duals of the spaces ${\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)$, $c(M,{\mathrm{\Delta }}_v^m)$ and $c_0(M,{\mathrm{\Delta }}_v^m)$.

Definition 2.1 [13]

Let X be a sequence space and define

$$\begin{array}{c}{X}^{\alpha }=\{a=(a_k):\sum _{k=1}^{\mathrm{\infty }}|a_k{x}_k|<\mathrm{\infty },\mathrm{\forall }x\in X\},\hfill \\ X^{\beta }=\{a=(a_k):\sum _{k=1}^{\mathrm{\infty }}{a}_k{x}_k\text{ is convergent},\mathrm{\forall }x\in X\},\hfill \\ X^{\gamma }=\{a=(a_k):\underset{n}{sup}|\sum _{k=1}^n{a}_k{x}_k|<\mathrm{\infty },\mathrm{\forall }x\in X\},\hfill \\ X^N=\{a=(a_k):\underset{k}{lim}{a}_k{x}_k=0,\mathrm{\forall }x\in X\},\hfill \end{array}$$

then $X^{\alpha }$, $X^{\beta }$, $X^{\gamma }$ and $X^N$ are called the α-, β-, γ- and N-(or null) duals of X, respectively. It is known that if $X\subset Y$, then $Y^{\eta }\subset X^{\eta }$ for $\eta =\alpha ,\beta ,\gamma \text{ and }N$.

Lemma 2.1 [2]

Let m be a positive integer. Then there exist positive constants $C_1$ and $C_2$ such that

$$C_1{k}^m\le \left(\genfrac{}{}{0ex}{}{m+k}{k}\right)\le C_2{k}^m,k=0,1,2,\dots .$$

Lemma 2.2 $x\in {\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)$ implies ${sup}_{k}M(\frac{|k^{-1}{\mathrm{\Delta }}_v^{m-1}{x}_k|}{\rho })<\mathrm{\infty }$ for some $\rho >0$.

Proof Let $x\in {\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)$, then

$$\underset{k}{sup}M\left(\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_k-{\mathrm{\Delta }}_v^{m-1}{x}_{k+1}|}{\rho }\right)<\mathrm{\infty }\text{for some }\rho >0.$$

Then there exists $U>0$ such that

$$M\left(\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_k-{\mathrm{\Delta }}_v^{m-1}{x}_{k+1}|}{\rho }\right)<U\text{for all }k\in N.$$

Taking $\eta =k\rho$, for an arbitrary fixed positive integer k, by the subadditivity of modulus, the monotonicity and convexity of M:

$$M\left(\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_1-{\mathrm{\Delta }}_v^{m-1}{x}_{k+1}|}{\eta }\right)\le \frac{1}{k}\sum _{l=1}^{k}M\left(\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_l-{\mathrm{\Delta }}_v^{m-1}{x}_{l+1}|}{\rho }\right)<U.$$

Then the above inequality, the inequality

$$\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_{k+1}|}{(k+1)\rho }\le \frac{1}{k+1}(\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_1|}{\rho }+k\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_1-{\mathrm{\Delta }}_v^{m-1}{x}_{k+1}|}{k\rho })$$

and the convexity of M imply

$$\begin{array}{rcl}M\left(\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_{k+1}|}{(k+1)\rho }\right)& \le & \frac{1}{k+1}(M\left(\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_1|}{\rho }\right)+kM\left(\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_1-{\mathrm{\Delta }}_v^{m-1}{x}_{k+1}|}{k\rho }\right))\\ \le & max\{M\left(\frac{|{\mathrm{\Delta }}_v^{m-1}{x}_1|}{\rho }\right),U\}<\mathrm{\infty }.\end{array}$$

Hence we have the desired result. □

Hence we have the following lemma.

Lemma 2.3

• (i) $x\in {\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)$ implies ${sup}_{k}M(\frac{|k^{-m}{v}_k{x}_k|}{\rho })<\mathrm{\infty }$ for some $\rho >0$,

• (ii) $x\in {\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)$ implies ${sup}_k{k}^{-m}|v_k{x}_k|<\mathrm{\infty }$.

Theorem 2.4 Let M be an Orlicz function. Then

• (i) ${[c_0(M,{\mathrm{\Delta }}_v^m)]}^{\alpha }={[c(M,{\mathrm{\Delta }}_v^m)]}^{\alpha }={[{\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)]}^{\alpha }=D_1$,

• (ii) $D_1^{\alpha }=D_2$,

where

$$\begin{array}{c}{D}_1=\{a=(a_k):\sum _{k=1}^{\mathrm{\infty }}{k}^m|v_k^{-1}{a}_k|<\mathrm{\infty }\},\hfill \\ D_2=\{b=(b_k):\underset{k}{sup}{k}^{-m}|v_k{b}_k|<\mathrm{\infty }\}.\hfill \end{array}$$

Proof (i) Let $a\in D_1$, then ${\sum }_{k=1}^{\mathrm{\infty }}{k}^m|v_k^{-1}{a}_k|<\mathrm{\infty }$. Now, for any $x\in {\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)$, we have ${sup}_k{k}^{-m}|v_k{x}_k|<\mathrm{\infty }$. Then we have

$$\sum _{k=1}^{\mathrm{\infty }}|a_k{x}_k|\le \underset{k}{sup}{k}^{-m}|v_k{x}_k|\sum _{k=1}^{\mathrm{\infty }}{k}^m\left|a_k{v}_k^{-1}\right|<\mathrm{\infty }.$$

Hence $a\in {[{\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)]}^{\alpha }$.

Conversely, suppose that $a\in {[X(M,{\mathrm{\Delta }}_v^m)]}^{\alpha }$ for $X=c\text{ and }{\ell }_{\mathrm{\infty }}$. Then ${\sum }_{k=1}^{\mathrm{\infty }}|a_k{x}_k|<\mathrm{\infty }$ for each $x\in X(M,{\mathrm{\Delta }}_v^m)$. So we can take

$$x_k=k^m{v}_k^{-1},k\ge 1.$$

Then

$$\sum _{k=1}^{\mathrm{\infty }}{k}^m\left|v_k^{-1}{a}_k\right|=\sum _{k=1}^{\mathrm{\infty }}|a_k{x}_k|<\mathrm{\infty }.$$

This implies that $a\in D_1$.

Again suppose that $a\in {[c_0(M,{\mathrm{\Delta }}_v^m)]}^{\alpha }$ and $a\notin D_1$. Then there exists a strictly increasing sequence $(n_i)$ of positive integers $n_i$ with $n_1<n_2<\cdots$ such that

$$\sum _{k=n_i+1}^{n_{i+1}}{k}^m\left|v_k^{-1}{a}_k\right|>i.$$

Define $x\in c_0(M,{\mathrm{\Delta }}_v^m)$ by

$$x_k=\{\begin{array}{ll}0,& 1\le k\le n_1,\\ k^m{v}_{k}sgn{a}_{k/i},& n_i<k\le n_{i+1}.\end{array}$$

Then we have

$$\begin{array}{rcl}\sum _{k=1}^{\mathrm{\infty }}|a_k{x}_k|& =& \sum _{k=n_1+1}^{n_2}|a_k{x}_k|+\cdots +\sum _{k=n_i+1}^{n_{i+1}}|a_k{x}_k|+\cdots \\ =& \sum _{k=n_1+1}^{n_2}{k}^m\left|v_k^{-1}{a}_k\right|+\cdots +\frac{1}{i}\sum _{k=n_i+1}^{n_{i+1}}{k}^m\left|v_k^{-1}{a}_k\right|+\cdots \\ >& 1+1+\cdots =\mathrm{\infty }.\end{array}$$

This contradicts $a\in {[c_0(M,{\mathrm{\Delta }}_v^m)]}^{\alpha }$. Hence $a\in D_1$. This completes the proof of (i).

• (ii) The proof is similar to that of part (i). □

If we take $v_k=1$ for all $k\in N$ in Theorem 2.4, then we obtain the following corollary.

Corollary 2.5 Let M be an Orlicz function. Then

• (i) ${[c_0(M,{\mathrm{\Delta }}^m)]}^{\alpha }={[c(M,{\mathrm{\Delta }}^m)]}^{\alpha }={[{\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}^m)]}^{\alpha }=E_1$,

• (ii) $E_1^{\alpha }=E_2$,

where

$$\begin{array}{c}{E}_1=\{a=(a_k):\sum _{k=1}^{\mathrm{\infty }}{k}^m|a_k|<\mathrm{\infty }\},\hfill \\ E_2=\{b=(b_k):\underset{k}{sup}{k}^{-m}|b_k|<\mathrm{\infty }\}.\hfill \end{array}$$

Theorem 2.6 Let M be an Orlicz function. Then ${[c(M,{\mathrm{\Delta }}_v^m)]}^N={[{\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)]}^N=F_1$, where

$$F_1=\{a=(a_k):\underset{k}{lim}{k}^m{v}_k^{-1}{a}_k=0\}.$$

Proof The proof is immediate using Lemma 2.3(ii). □

The following lemma will be used in the next theorem.

Lemma 2.7 [6]

Let $(p_n)$ be a sequence of positive numbers increasing monotonically to infinity.

• (i) If ${sup}_n|{\sum }_{v=1}^n{p}_v{a}_v|<\mathrm{\infty }$, then ${sup}_n|p_n{\sum }_{k=n+1}^{\mathrm{\infty }}{a}_k|<\mathrm{\infty }$.

• (ii) If ${\sum }_k{p}_k{a}_k$ is convergent, then ${lim}_n{p}_n{\sum }_{k=n+1}^{\mathrm{\infty }}{a}_k=0$.

Theorem 2.8 Let M be an Orlicz function and $c_0^{+}$ denote the set of all positive null sequences. Then

• (i) ${[D{\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)]}^{\beta }={[Dc(M,{\mathrm{\Delta }}_v^m)]}^{\beta }=G_1$,

• (ii) ${[D{c}_0(M,{\mathrm{\Delta }}_v^m)]}^{\beta }=G_2$,

• (iii) ${[D{\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)]}^{\gamma }={[Dc(M,{\mathrm{\Delta }}_v^m)]}^{\gamma }=H_1$,

• (iv) ${[D{c}_0(M,{\mathrm{\Delta }}_v^m)]}^{\gamma }=H_2$,

where

$$\begin{array}{c}{G}_1=\{a=(a_k):\sum _{k=1}^{\mathrm{\infty }}{a}_k{v}_k^{-1}\sum _{j=1}^{k-m}\left(\genfrac{}{}{0ex}{}{k-j-1}{m-1}\right)\mathit{\text{ is convergent}},\hfill \\ \phantom{G_1=}\sum _{k=1}^{\mathrm{\infty }}\left|\sum _{j=k+1}^{\mathrm{\infty }}{v}_j^{-1}{a}_j\right|\sum _{j=1}^{k-m+1}\left(\genfrac{}{}{0ex}{}{k-j-1}{m-2}\right)<\mathrm{\infty }\},\hfill \\ G_2=\{a=(a_k):\sum _{k=1}^{\mathrm{\infty }}{a}_k{v}_k^{-1}\sum _{j=1}^{k-m}\left(\genfrac{}{}{0ex}{}{k-j-1}{m-1}\right)u_j\mathit{\text{ converges and }}\hfill \\ \phantom{G_2=}\sum _{k=1}^{\mathrm{\infty }}\left|\sum _{j=k+1}^{\mathrm{\infty }}{v}_j^{-1}{a}_j\right|\sum _{j=1}^{k-m+1}\left(\genfrac{}{}{0ex}{}{k-j-1}{m-2}\right)u_j<\mathrm{\infty },\mathrm{\forall }u\in c_0^{+}\},\hfill \\ H_1=\{a=(a_k):\underset{n}{sup}|\sum _{k=1}^n{a}_k{v}_k^{-1}\sum _{j=1}^{k-m}\left(\genfrac{}{}{0ex}{}{k-j-1}{m-1}\right)|<\mathrm{\infty },\hfill \\ \phantom{H_1=}\sum _{k=1}^{\mathrm{\infty }}\left|\sum _{j=k+1}^{\mathrm{\infty }}{v}_j^{-1}{a}_j\right|\sum _{j=1}^{k-m+1}\left(\genfrac{}{}{0ex}{}{k-j-1}{m-2}\right)<\mathrm{\infty }\},\hfill \\ H_2=\{a=(a_k):\underset{n}{sup}|\sum _{k=1}^n{a}_k{v}_k^{-1}\sum _{j=1}^{k-m}\left(\genfrac{}{}{0ex}{}{k-j-1}{m-1}\right)u_j|<\mathrm{\infty },\hfill \\ \phantom{H_2=}\sum _{k=1}^{\mathrm{\infty }}\left|\sum _{j=k+1}^{\mathrm{\infty }}{v}_j^{-1}{a}_j\right|\sum _{j=1}^{k-m+1}\left(\genfrac{}{}{0ex}{}{k-j-1}{m-2}\right)u_j<\mathrm{\infty },\mathrm{\forall }u\in c_0^{+}\}.\hfill \end{array}$$

Proof We give the proof for part (i) for $D{\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)$, and the proof of other parts follows similarly using Lemma 2.7. For details, one may refer to [11].

For each $x\in D{\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)$, there exists one and only one $y=(y_k)\in {\ell }_{\mathrm{\infty }}(M)$ such that

$$x_k=v_k^{-1}\sum _{j=1}^{k-m}\left(\begin{array}{c}k-j-1\\ m-1\end{array}\right)y_j,y_{1-m}=y_{2-m}=\cdots =y_0=0$$

for sufficiently large k by (1.1). Let $a\in G_1$. Then, using the same technique as applied in [[11], p.429], we can show that $a\in {[D{\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)]}^{\beta }$.

Let $a\in {[D{\ell }_{\mathrm{\infty }}(M,{\mathrm{\Delta }}_v^m)]}^{\beta }$. Again, using the same technique as applied in [[11], pp.429-430], we can show that $a\in G_1$.

This completes the proof. □

3 Conclusion

Although we conclude this paper here, the following further suggestion remains open:

What is the N-dual of the space $c_0(M,{\mathrm{\Delta }}_v^m)$?