On some new vector valued sequence spaces $E(X, \lambda, p)$

1.   Introduction

Let $(X, \|\cdot\|)$ be a Banach space and $p = (p_k)$ be any bounded sequence of strictly positive real numbers. Let $w(X)$ be the set of all sequences $x = x_k$ with $x_k\in X$. Bounded, convergent and null vector-valued sequence spaces in $X$ are denoted by $\ell_\infty(X)$, $c(X)$, and $c_0(X)$, respectively. These sequence spaces are linear subspace of $w(X)$. When $X = \mathbb{R}$ or $\mathbb{C}$ (the real or complex numbers) it is used the familiar notations $\ell_\infty$, $c$ and $c_0$. Here, $\ell_\infty(X)$, $c(X)$, and $c_0(X)$ are Banach spaces with the norm

where $x_k\in X$ for $k = 1, 2, ...$. The set

is the unit ball. $B(X, Y)$ denotes the set of all bounded operators from a Banach space $X$ into the Banach space $Y$. If $T\in B(X, Y)$, the operator norm of $T$ is

In this paper, to indicate the continuous dual of $X$ we use the notation $X^{*}$ instead of $B(X, \mathbb{C})$. Also, $e$ and $e(x)$ denote the sequences $(1, 1, 1, ...)$ and $(x, x, x, ...)$, respectively.

In the classical theory of matrix transformations, one of the basic problems is the characterization of matrices that map a sequence space $E$ into a sequence space $F$. The first step of this characterization is the determination of the Köthe-Toeplitz duals of $E$. Also, the Köthe-Toeplitz duals of $E$ are called $\beta$-dual and $\alpha$-dual of $E$.

Let $X$ and $Y$ be Banach spaces and $(A_k)$ be the sequence of operators $A_k$ defined from $X$ to $Y$, which are linear but not necessarily bounded. Then, the $\beta$-dual and $\alpha$-dual of $E$ are defined as follows:

Generalized Köthe-Toeplitz duals of $X$-valued sequence spaces $\ell_\infty(X)$, $c(X)$ and $c_0(X)$ were defined by Maddox ^[6].

Let $A {\rm{{x}}} = (A_{nk})$ be an infinite matrix of linear, but not necessarily bounded, operators $A_{nk}$ from $X$ to $Y$. Suppose $E$ is a nonempty subset of $w(X)$ and suppose $F$ is a nonempty subset of $w(Y)$. We define the matrix classes $(E, F)$ by saying that $A\in (E, F)$ if and only if, for every ${\rm{{x}}} = (x_k)\in E$,

converges for each $n$ in the norm topology of $Y$ and the sequence

$X$-valued paranormed sequence spaces $c_0(X, p)$, $c(X, p)$, $\ell_\infty(X, p)$ and $\ell(X, p)$ that are the generalization of Maddox spaces $c_0(p)$, $c(p)$, $\ell_\infty(p)$ and $\ell(p)$ were first defined by Rath ^[1]. Rath also determined the generalized Köthe-Toeplitz duals of $c_0(X, p)$, $c(X, p)$, $\ell_\infty(X, p)$ and $\ell(X, p)$. Suantai ^[8] and Suantai and Sudsukh ^[7] gave the matrix characterizations $(\xi(X, p), c(q))$ where

Srivastava and Srivastava ^[2] introduced the set $c_0(X, \lambda, p)$ and determined the generalized Köthe-Toeplitz duals of these spaces where $\lambda = (\lambda_k)$ is any sequence of non-zero complex numbers.

Now, we define the new $X$-valued sequence spaces as follows:

The spaces $c(X, \lambda, p)$ and $\ell_\infty(X, \lambda, p)$ are $BK$ spaces with the norm

where

and

$c(X, \lambda, p)$ and $\ell_\infty(X, \lambda, p)$ are generalization of several sequence spaces. Some of these generalizations are as follows:

If $\lambda_k = 1$ for all $k$ and $X = \mathbb{C}$, then we obtain spaces $c(p)$ and $\ell_\infty(p)$ introduced by Maddox in ^[5].

If $\lambda_k = 1$ for all $k$, then we obtain spaces $c(X, p)$ and $\ell_\infty(X, p)$ introduced by Rath in ^[1] and also introduced by Suantai and Sudsukh ^[7].

If $\lambda_k = 1$ and $p_k = 1$ for all $k$, then we obtain spaces $c(X)$ and $\ell_\infty(X)$ introduced by Maddox in ^[6].

2.   Köthe-Toeplitz duals

This section is concerned with the generalized Köthe-Toeplitz duals of the spaces $c(X, \lambda, p)$, $\ell_\infty(X, \lambda, p)$.

Now we define the group norm and consider an inequality to be used to calculate the $\beta$- duals of our spaces. If $(F_k)$ is a sequence in $B(X, Y)$, the group norm of $(T_k)$ is

where the supremum is taken over all $n\in \mathbb{N}$ and all choices of $x_k\in S$. Inequality

holds for any $x_k$ and all $n\geq1$, and all non-negative integers $p$, where

is called $n^{th}$ tail of $(F_n)$, see [5].

Some results related to Köthe-Toeplitz duals of the space $c_0(X, \lambda, p)$ were obtained by Srivastava and Srivastava ^[2] as follows:

Lemma 1. ${{A}} = (A_k)\in c_0^\alpha(X, \lambda, p)$ if and only if

(i) There exist an integer $m\geq1$ such that $A_k\in B(X, Y)$ for each $k\geq m$.

(ii) There exist an integer $N > 1$ such that $\sum_{k = m}^{\infty}|\lambda_k|^{-1}\|A_k\|N^{-r_k} < \infty$.

Lemma 2. Let $(T_k)$ is a sequence in $B(X, Y)$. Then, exactly one of the following is true:

(i) $\|R_k\| = \infty$ for all $k\geq1$.

(ii) $\|R_k\| < \infty$ for all $k\geq1$.

Lemma 3. ${{A}} = (A_k)\in c_0^\beta(X, \lambda, p)$ if and only if

(i) There exist an integer $m\geq1$ such that $A_k\in B(X, Y)$ for each $k\geq m$.

(ii) $\|R_m(\lambda, N)\| < \infty$ for some $N > 1$, where

Theorem 1. $\; \; {{A}} = (A_k)\in c^\beta(X, \lambda, p)$ if and only if

(i) ${{A}} = (A_k)\in c_0^\beta(X, \lambda, p)$.

(ii) $\sum_{k = 1}^\infty \lambda_{k}^{-1}A_kx$ converges for each $x\in X$.

Proof. Suppose ${\rm{{x}}}\in c(X, \lambda, p)$ and suppose (i) and (ii) hold. We can write

where

See Theorem 3.9 ^[9]. Then, we can write

for all $k\in\mathbb{N}$ where the sequence $z$ in $X$ and the sequence $(y_k)$ in $c_0(X, \lambda, p)$.

Therefore, $\sum_{k = 1}^\infty \lambda_k^{-1}A_kz$ is convergent by condition (ii) and $\sum_{k = 1}^\infty A_ky_k$ is convergent by Lemma 3. Thus we have that $\sum_{k = 1}^\infty A_kx_k$ converges.

Since $c^\beta(X, \lambda, p)\subset c_0^\beta(X, \lambda, p)$ condition (i) follows from Lemma 3. If we take any $x\in c(X, \lambda, p)$ and $y\in c_0(X, \lambda, p)$, then $\sum_{k = 1}^\infty A_kx_k$ and $\sum_{k = 1}^\infty A_ky_k$ are converge. Morever, we consider

for all $k\in\mathbb{N}$ where the sequence $z$ in $X$ and the sequence $(y_k)$ in $c_0(X, \lambda, p)$. Hence, $\sum_{k = 1}^\infty \lambda_{k}^{-1}A_kz$ converges for each $z\in X$.

If we take $\lambda_k = 1$ and $p_k = 1$ for all $k$ in the Theorem 1, then we have Maddox ^[6] Proposition 3.2 as follows:

Corollary 1. $(A_k)\in c^\beta(X)$ if and only if

(i) There exists $m\in \mathbb{N}$ such that $A_k\in B(X, Y)$ for all $k\geq m$.

(ii) $\sum_{k = 1}^\infty A_k$ converges in the strong operator topology.

Theorem 2. $\; \; {\rm{{A}}} = (A_k)\in \ell_\infty^\beta(X, \lambda, p)$ if and only if

(i) There exist an integer $m\geq1$ such that $A_k\in B(X, Y)$ for each $k\geq m$.

(ii) $\|R_m(\lambda, \mathcal{N})\|\rightarrow 0$ for all $N > 1$, where

Proof. Let (i) and (ii) hold and $x\in\ell_\infty(X, \lambda, p)$. Then, there exists an integer $N_0 > 1$ such that $\|x_k\| < |\lambda_k|^{-1}N_0^{r_k}$ for all $k\in\mathbb{N}$. If we use inequality (2.2), then we obtain

Hence, $\sum_{k = 1}^\infty A_kx_k$ is convergent since $Y$ is a Banach space.

Since $\ell_\infty^\beta(X, \lambda, p)\subset c_0^\beta(X, \lambda, p)$ Lemma 3.2 yields condition (i). Now, suppose that (ii) fails, say

Then, there exist integer $0 < m\leq m_{1}\leq n_{1}$ and $z_{m_{1}}, z_{m_{1}+1}, ..., z_{n_{1}}$ in $S$ such that

Choose $n_{1} < m_{2}$ such that $\|R_{m_{2}}(\lambda)\| > 2\kappa$. Then, there exist $m_{2}\leq n_{2}$ and $z_{m_{2}}, z_{m_{2}+1}, ..., z_{n_{2}}$ in $S$ such that

Now, by proceeding in this way, we can define the sequence $x = (x_k)$ as follows

for all natural numbers $i$. Then, we have that ${\rm{{x}}}\in \ell_\infty(X, \lambda, p)$, since

On the other hand $\sum_{k = 0}^{\infty}A_kx_k$ is divergent, that is a contradiction. Hence the proof is completed.

The Köthe-Toeplitz duals of $c_0(X, \lambda, p)$, $c(X, \lambda, p)$ and $\ell_\infty(X, \lambda, p)$ can be rewritten in the case of $A_k\in B(X, Y)$ for all integers $k$. For these concept let define the following sets, which are the subspace of $w(B(X, Y))$:

Srivastava and Srivastava ^[2] showed that in the Theorem 3.1

Moreover, they showed that if $Y = \mathbb{C}$ and $f_k\in X^*$ for $k\geq1$, then

in the Theorem 3.5.

The following result can now be proved in the same way as Theorem 3.1 in ^[2].

Theorem 3. $\ell_\infty^\alpha(X, \lambda, p) = M_\infty(B(X, Y), \lambda, p)$.

Theorem 4. Let $Y$ be complex field $\mathbb{C}$ and $f = (f_k)$ be a sequence of continuous linear functionals on $X$. Then,

Proof.

from Theorem 3. Now, we show that

Following Srivastava and Srivastava ^[2] as in Theorem 3.5; suppose

then we obtain

for some $N > 1$. Also, there exists $z_k\in S$ such that $\|f_k\| < 2|f_k(z_k)|$ for each $k\geq1$. If define the sequence ${x} = (x_k)$ as

for $k\in S(N)$, then the sequence ${x} = (x_k)$ in $\ell_\infty(X, \lambda, p)$ but

These show that $f = (f_k)\notin \ell_\infty^\beta(X, \lambda, p)$, so $\ell_\infty^\beta(X, \lambda, p)\subset M_\infty(X^*, \lambda, p)$.

On the other hand $\ell_\infty^\alpha(X, \lambda, p)\subset\ell_\infty^\beta(X, \lambda, p)$ since $\mathbb{C}$ is complete.

Taking ${\rm{f}} = (f_k)\in B(X, \mathbb{C})$ instead of ${\rm{{A}}} = (A_k)\in B(X, Y)$ in Theorem 1, the following result can be obtained.

Theorem 5. Let ${f} = (f_k)$ be a sequence such that $f_k\in X$ for all $k\in\mathbb{N}$. Then, ${{f}} = (f_k)\in c^\beta(X, \lambda, p)$ if and only if

(i) $\sum_{k = 1}^{\infty}|\lambda_k|^{-1}\|f_k\|N^{-1/p_{k}} < \infty$ for some $N\in\mathbb{N}$.

(ii) $\sum_{k = 1}^\infty \lambda_{k}^{-1}f_k(x)$ converges for each $x\in X$.

3.   Matrix transformations

In this section, some matrix classes will be characterized using similar methods developed by Wu and Liu ^[11] and Suantai and Sudsukh ^[7]. Throughout these section, $F = (f_k^n)$ be an infinite matrix and $f_k^n\in X^*$ for all $k, n\in\mathbb{N}$. Moreover, let $u = (u_k)$ and $v = (v_k)$ be scalar sequences and $u_k\neq 0$, $v_k\neq 0$ and we can define two sets as follow:

and

An $X$-valued sequence space $E$ is solid when $x = (x_k)\in E$ and $\|y_k\|\leq\|x_k\|$ for all $k\in\mathbb{N}$ together imply that $y = (y_k)\in E$.

Now we give two lemmas, their proof can be found in ^[7].

Lemma 4. Let $E$ be a solid $X$-valued sequence space which is an $F$K-space and contain $\Phi(X)$. Then

Lemma 5. Let $E$ and $E_n$ be $X$-valued sequence spaces, and $F$ and $F_n$ be scalar-valued sequence spaces for all $n\in\mathbb{N}$. Then,

(i) $(\bigcup_{n = 1}^{\infty}E_n, F) = \bigcap_{n = 1}^{\infty}(E_n, F).$ (ii) $(E, \bigcap_{n = 1}^{\infty}F_n) = \bigcap_{n = 1}^{\infty}(E, F_n).$ (iii) $(E_1+E_2, F) = (E_1, F)\bigcap(E_2, F).$ (iv) $(E_u, F_v) = _{v}(E, F)_{u^{^-1}}$.

Theorem 6. ${{F}} = (f_k^n)\in (c_0(X, \lambda, p), c(q))$ if and only if there is a sequence $(f_k)\subset X^*$ such that

(i) $\sum_{k = 1}^{\infty}|\lambda_k|^{-1}\|f_k\|N^{-1/p_k} < \infty$ for some $N\in\mathbb{N}$.

(ii) $m^{1/q_n}(f_k^n-f_k)\rightarrow0$ as $n\rightarrow \infty$ for every $k, m\in\mathbb{N}$.

(iii) $m^{1/q_n}\sum_{k = 1}^{\infty}\|f_j^n\|r^{1/p_k}\rightarrow0$ as $n, r\rightarrow \infty$ for each $m\in\mathbb{N}$.

Proof. For the sufficiency assume that

Since

from ^[10] (page-301), then we obtain

Now, we can choose

and

such that $F = D+E$ by Lemma 4. In addition, $(h_k^n(x))_{n = 1}^{\infty}\in \langle e\rangle$ for all $x\in X$ and $k\in\mathbb{N}$ since $\Phi(X)\subset c_0(X, \lambda, p)$. These yields that $h^{n}_{k} = h^{n+1}_{k}$ for all $k, n\in\mathbb{N}$. Let choose $f_k = h_{k}^{1}$, so we obtain

We know that

from Theorem 0 in ^[10], which yields that

for all $m\in\mathbb{N}$ by Lemma 5(ii) and (iv). If we use Theorem 2.4 in ^[11], then conditions (ii) and (iii) hold. Further $\sum_{k = 1}^{\infty}f_k(x_k)$ converges for all

since

So $(i)$ holds by Theorem 3.5 in ^[2].

For the necessity suppose that $(f_k)$ be a sequence such that $f_k\in X^*$ for all $k\in\mathbb{N}$ and conditions (i)–(iii) hold. Also, let $F = D+E$ where

and $E = (f_k)_{n, k}$. Hence

by conditions (i), (ii), Proposition 3.1 (ⅱ) and Theorem 2.4 in ^[11]. Moreover, $\sum_{k = 1}^{\infty}f_k(x_k)$ converges for all

by proposition 3.3 and the condition (i). Hence $E\in (c_0(X, \lambda, p), \langle { \rm{e}}\rangle)$. So Lemma 4 yields

As an easy consequence of Theorem 4.3 in ^[7], we have

Lemma 6. ${{F}} = (f_k^n)\in (\ell_\infty(X, \lambda), c(q))$ if and only if there is a sequence $(f_k)\subset X^*$ such that

(i) $\sum_{k = 1}^{\infty}|\lambda_k|^{-1}\|f_k\| < \infty$.

(ii) $m^{1/q_n}(f_k^n-f_k)\rightarrow0$ as $n\rightarrow \infty$ for every $k, m\in\mathbb{N}$.

(iii) $\sum_{j = k+1}^{\infty}m^{1/q_n}\|f_j^n-f_j\|r^{1/p_k}\rightarrow0$ as $k\rightarrow \infty$ uniformly on $n\in\mathbb{N}$, for each $m, r\in\mathbb{N}$.

Lemma 7. $\ell_\infty(X, \lambda, p) = \bigcup_{m = 1}^{\infty}\ell_\infty(X, \lambda)_{(m^{-1/p_k})}.$

Proof. Let ${\rm{{x}}}\in\ell_\infty(X, \lambda, p)$, then there is some $m\in\mathbb{N}$ such that $\sup_k\|\lambda_kx_k\|^{p_k} < m$. Therefore,

for all $k\in\mathbb{N}$. Hence,

Conversely, suppose that

Then, there are some $m\in\mathbb{N}$ and $K > 1$ such that

for all $k\in\mathbb{N}$. Hence, this yields

for all $k\in\mathbb{N}$, where $t = \sup_kp_k$, so that ${\rm{{x}}}\in\ell_\infty(X, \lambda, p)$.

Theorem 7. $\; \; {{F}} = (f_k^n)\in (\ell_\infty(X, \lambda, p), c(q))$ if and only if there is a sequence $(f_k)\subset X^*$ such that

(i) $\sum_{k = 1}^{\infty}|\lambda_k|^{-1}\|f_k\|N^{1/p_k} < \infty$ for all $N\in\mathbb{N}$.

(ii) $r^{1/q_n}(m^{1/p_k}f_k^n-f_k)\rightarrow0$ as $n\rightarrow \infty$ for every $k, m\in\mathbb{N}$.

(iii) $r^{1/q_n}\sum_{j = k+1}^{\infty}\|m^{1/p_j}f_j^n-f_j\|s^{1/p_j}\rightarrow0$ as $k\rightarrow \infty$ uniformly on $n\in\mathbb{N}$, for each $m, r, s\in\mathbb{N}$.

Proof. Let consider the Lemmas 5–7. Then, we derive from two-sided implication $Fx$ is in $c(q)$ whenever $x\in\ell_\infty(X, \lambda, p)$ if and only if

So we have for the necessary and sufficient (i)–(iii) hold.

Theorem 8. ${{F}} = (f_k^n)\in (c(X, \lambda, p), c(q))$ if and only if there is a sequence $(f_k)\subset X^*$ such that

(i) $\sum_{k = 1}^{\infty}|\lambda_k|^{-1}\|f_k\|N^{-1/p_k} < \infty$ for some $N\in\mathbb{N}$.

(ii) $m^{1/q_n}(f_k^n-f_k)\rightarrow ^{$\textrm{w}$^{*}}0$ as $n\rightarrow \infty$ for every $k, m\in\mathbb{N}$.

(iii) $m^{1/q_n}\sum_{k = 1}^{\infty}\|f_j^n\|r^{1/p_k}\rightarrow0$ as $n, r\rightarrow \infty$ for each $m\in\mathbb{N}$.

(iv) $(\sum\limits_{k = 1}^{\infty}f^{n}_{k}(x))_{n}\in c(q)$ for all $x\in X$.

Proof. As in the proof of Theorem 1

Hence, Lemma 5 show that

if and only if

and

Then,

if and only if the conditions (i)–(iii) hold, follows easily from Theorem 6. Also, one can show that

if and only if (iv) holds.

4.   Conclusions

The computation of Köthe-Toeplitz duals of sequence spaces is an important field of study in order to determine the matrix transformation classes between two spaces. Many researchers try to derive new sequence spaces using well-known sequence spaces. They also investigate for matrix transformation classes between the proposed new sequence spaces and classical sequence spaces by identifying the duals of the spaces they have obtained.

Many different methods have been used to obtain new sequence spaces. One of these is paranormed sequence spaces defined by Maddox. Maddox has constructed new sequence spaces from classical sequence spaces $\ell_\infty$, $c_0$ and $c$. These sequence spaces are $\ell_\infty(p)$, $c_0(p)$, $c_0(p)$ where

and

With these sequence spaces defined by Maddox a wide working area has been opened. An other method to obtaining new sequence spaces is "matrix domain" method. For an arbitrary sequence space $X$, the matrix domain of an infinite matrix $A$ in $X$ is defined by

as in ^[3,4]. The other method is to generate a new sequence space from classical sequence spaces using any scalar sequence $\lambda = (\lambda_k)$. For example, Srivastava and Srivastava^[2] introduced the sequence space

The researchers can obtain new sequence spaces by using above methods.

In this study, we introduced generalized vector valued sequence spaces $c(X, \lambda, p)$ and $\ell_\infty(X, \lambda, p)$. We determined the Köthe-Toeplitz duals of these spaces and characterized some matrix classes $(E(X, \lambda, p), c(q))$ where $E\in \{\ell_\infty, c\}$. These spaces are the generalization of some sequence spaces given in the references.

Acknowledgments

I would like to thank the editor and anonymous reviewers for their insightful and constructive comments and suggestions that have been helpful in providing a better version of the present work.

Conflict of interest

Author declares no conflict of interest in this paper.