# On some new matrix transformations

## Abstract

In this paper, we characterize some matrix classes $\left(\omega \left(p,s\right),{V}_{\sigma }^{\lambda }\right)$, $\left({\omega }_{p}\left(s\right),{V}_{\sigma }^{\lambda }\right)$ and ${\left({\omega }_{p}\left(s\right),{V}_{\sigma }^{\lambda }\right)}_{\mathrm{reg}}$ under appropriate conditions.

## 1 Introduction

Let w denote the set of all real and complex sequences $x=\left({x}_{k}\right)$. By ${l}_{\mathrm{\infty }}$ and c, we denote the Banach spaces of bounded and convergent sequences $x=\left({x}_{k}\right)$ normed by $\parallel x\parallel ={sup}_{k}|{x}_{k}|$, respectively. A linear functional L on ${l}_{\mathrm{\infty }}$ is said to be a Banach limit [1] if it has the following properties:

(1) $L\left(x\right)\ge 0$ if $n\ge 0$ (i.e., ${x}_{n}\ge 0$ for all n),

(2) $L\left(e\right)=1$, where $e=\left(1,1,\dots \right)$,

(3) $L\left(Dx\right)=L\left(x\right)$, where the shift operator D is defined by $D\left({x}_{n}\right)=\left\{{x}_{n+1}\right\}$.

Let B be the set of all Banach limits on ${l}_{\mathrm{\infty }}$. A sequence $x\in {\ell }_{\mathrm{\infty }}$ is said to be almost convergent if all Banach limits of x coincide. Let $\stackrel{ˆ}{c}$ denote the space of the almost convergent sequences. Lorentz [2] has shown that

where

${d}_{m,n}\left(x\right)=\frac{{x}_{n}+{x}_{n+1}+{x}_{n+2}+\cdots +{x}_{n+m}}{m+1}.$

The study of regular, conservative, coercive and multiplicative matrices is important in the theory of summability. In [3], King used the concept of the almost convergence of a sequence introduced by Lorentz to define more general classes of matrices than those of regular and conservative ones.

In [4], Schaefer defined the concepts of σ-conservative, σ-regular and σ-coercive matrices and characterized the matrix classes $\left(c,{V}_{\sigma }\right)$, ${\left(c,{V}_{\sigma }\right)}_{\mathrm{reg}}$ and $\left({l}_{\mathrm{\infty }},{V}_{\sigma }\right)$, where ${V}_{\sigma }$ denotes the set of all bound sequences, all of whose invariant means (or σ-means) are equal. In [5], Mursaleen characterized the classes $\left(c\left(p\right),{V}_{\sigma }\right)$, ${\left(c\left(p\right),{V}_{\sigma }\right)}_{\mathrm{reg}}$ and $\left({l}_{\mathrm{\infty }}\left(p\right),{V}_{\sigma }\right)$ of matrices, which generalized the results due to Schaefer [4]. In [6], Mohiuddine and Aiyup defined the space $\omega \left(p,s\right)$ and obtained necessary and sufficient conditions to characterize the matrices of classes $\left(\omega \left(p,s\right),{V}_{\sigma }\right)$, $\left({\omega }_{p}\left(s\right),{V}_{\sigma }\right)$ and ${\left({\omega }_{p}\left(s\right),{V}_{\sigma }\right)}_{\mathrm{reg}}$.

Matrix transformations between sequence spaces have also been discussed by Savaş and Mursaleen [7], Basarir and Savaş [8], Mursaleen [5, 916], Vatan and Simsek [17], Savaş [1824], Vatan [25] and many others.

In this paper we characterize the matrix classes from this space to the space ${V}_{\sigma }^{\lambda }$, i.e., we obtain necessary and sufficient conditions to characterize the matrices of classes $\left(\omega \left(p,s\right),{V}_{\sigma }^{\lambda }\right)$, $\left({\omega }_{p}\left(s\right),{V}_{\sigma }^{\lambda }\right)$ and ${\left({\omega }_{p}\left(s\right),{V}_{\sigma }^{\lambda }\right)}_{\mathrm{reg}}$.

## 2 Preliminaries

Let σ be a one-to-one mapping from the set N of natural numbers into itself. A continuous linear functional φ on ${l}_{\mathrm{\infty }}$ is said to be an invariant mean or σ-mean if and only if

(i) $\phi \left(x\right)\ge 0$ when the sequence $x=\left({x}_{k}\right)$ has ${x}_{k}\ge 0$ for all k;

(ii) $\phi \left(e\right)=1$;

(iii) $\phi \left(x\right)=\phi \left({x}_{\sigma \left(k\right)}\right)$ for all $x\in {l}_{\mathrm{\infty }}$.

Let ${V}_{\sigma }$ denote the set of bounded sequences all of whose σ-means are equal. We say that a sequence $x=\left({x}_{k}\right)$ is σ-convergent if and only if $x\in {V}_{\sigma }$. For $\sigma \left(n\right)=n+1$, the set ${V}_{\sigma }$ is reduced to the set $\stackrel{ˆ}{c}$ of almost convergent sequences [2, 26].

If $x=\left({x}_{n}\right)$, write $Tx=\left({x}_{\sigma \left(n\right)}\right)$. It is easy to show that

where

${t}_{mn}\left(x\right)=\frac{1}{m+1}\sum _{j=0}^{m}{T}^{j}{x}_{n}$

and ${\sigma }^{m}\left(n\right)$ denotes the m th iterate of σ at n.

If ${p}_{k}$ is real and positive, we define (see Maddox [27])

${c}_{0}\left(p\right)=\left\{x:\underset{k\to \mathrm{\infty }}{lim}|{x}_{k}{|}^{{p}_{k}}=0\right\}$

and

The classes $\left(\omega \left(p,s\right),{V}_{\sigma }\right)$, $\left({\omega }_{p}\left(s\right),{V}_{\sigma }\right)$ and ${\left({\omega }_{p}\left(s\right),{V}_{\sigma }\right)}_{\mathrm{reg}}$ have been defined by Mohiuddine and Aiyup [6] and, for $p=\left({p}_{k}\right)$ with ${p}_{k}>0$, the space $\omega \left(p,s\right)$ is defined for $s\ge 0$ by

where $s=\left({s}_{k}\right)$ is an arbitrary sequence with ${s}_{k}\ne 0$ ($k=1,2,\dots$). If ${p}_{k}=p$, for each k, we have $\omega \left(p,s\right)={\omega }_{p}\left(s\right)$.

The sequence space

$\omega \left(p\right)=\left\{x:\frac{1}{n}\sum _{k=1}^{n}{|{x}_{k}-l|}^{{p}_{k}}\to 0,n\to \mathrm{\infty }\right\}$

for some l, which has been investigated by Maddox is the special case of $\omega \left(p,s\right)$ which corresponds to $s=0$. Obviously $\omega \left(p\right)\subset \omega \left(p,s\right)$.

We further define the following.

Let $\lambda =\left({\lambda }_{m}\right)$ be a non-decreasing sequence of positive numbers tending to ∞ such that

${\lambda }_{m+1}\le {\lambda }_{m}+1,\phantom{\rule{1em}{0ex}}{\lambda }_{1}=1.$

A sequence $x=\left({x}_{k}\right)$ of real numbers is said to be $\left(\sigma ,\lambda \right)$-convergent to a number L if and only if $x\in {V}_{\sigma }^{\lambda }$, where

and ${I}_{m}=\left[m-{\lambda }_{m}+1,m\right]$. Note that $c\subset {V}_{\sigma }^{\lambda }\subset {l}_{\mathrm{\infty }}$. For $\sigma \left(n\right)=n+1,{V}_{\sigma }^{\lambda }$ reduces to the space ${\stackrel{ˆ}{V}}_{\lambda }$ of almost λ-convergent sequences [28]; and if we take $\sigma \left(n\right)=n+1$ and ${\lambda }_{m}=m$, then ${V}_{\sigma }^{\lambda }$ reduces to $\stackrel{ˆ}{c}$ (see [29]). Further, if we take ${\lambda }_{m}=m$, then ${V}_{\sigma }^{\lambda }$ reduces to ${V}_{\sigma }$.

If E is a subset of ω, then we write ${E}^{+}$ for a generalized Köthe-Toeplitz dual of E; i.e.,

If $0<{p}_{k}\le 1$, then ${\omega }^{+}\left(p\right)=\mathbb{M}$, where

and max is the maximum taken over ${2}^{r}\le k<{2}^{r+1}$ (see Theorem 4, [30]).

If X is a topological linear space, we denote by ${X}^{\ast }$ the continuous dual of X; i.e., the set of all continuous linear functionals on X. Obviously,

## 3 Main results

Let X and Y be two nonempty subsets of the space w of complex sequences. Let $A=\left({a}_{nk}\right)$ ($n,k=1,2,\dots$) be an infinite matrix of complex numbers. We write $Ax=\left({A}_{n}\left(x\right)\right)$ if ${A}_{n}\left(x\right):={\sum }_{k}{a}_{nk}{x}_{k}$ converges for each n. (Throughout, ${\sum }_{k}$ will denote summation over k from $k=1$ to $k=\mathrm{\infty }$.) If $x=\left({x}_{k}\right)\in X$ implies that $Ax=\left({A}_{n}\left(x\right)\right)\in Y$, we say that A defines a (matrix) transformation from X to Y and we denote it by $A:X\to Y$. By $\left(X,Y\right)$ we mean the class of matrices A such that $A:X\to Y$.

We now characterize the matrices in the class $\left({c}_{0}\left(p\right),{V}_{{\sigma }_{0}}^{\lambda }\left(p\right)\right)$. We write

${t}_{m,n}\left(Ax\right)=\sum _{k}a\left(n,k,m\right){x}_{k},$

where

$a\left(n,k,m\right)=\frac{1}{{\lambda }_{m}}\sum _{i\in {I}_{m}}{a}_{{\sigma }^{i}\left(n\right),k}.$

Theorem 3.1 Let $0<{p}_{k}\le 1$, then $A\in \left(\omega \left(p,s\right),{V}_{\sigma }^{\lambda }\right)$ if and only if

(i) there exists an integer $B>1$ such that for every n

${D}_{n}=\underset{m}{sup}\sum _{r=0}^{\mathrm{\infty }}\underset{r}{max}{\left({2}^{r}\cdot {B}^{-1}\right)}^{\frac{1}{{p}_{k}}}|\frac{a\left(n,k,m\right)}{{s}_{k}}|<\mathrm{\infty };$

(ii) ${\alpha }_{\left(k\right)}={\left\{{a}_{nk}\right\}}_{n=1}^{\mathrm{\infty }}\in {V}_{\sigma }^{\lambda }$ for each k;

(iii) $\alpha ={\left\{{\sum }_{k}{a}_{nk}\right\}}_{n=1}^{\mathrm{\infty }}\in {V}_{\sigma }^{\lambda }$.

In this case the $\sigma \text{-}lim$ of Ax is $\left(limx\right)\left[u-{\sum }_{k}{u}_{k}\right]+{\sum }_{k}{u}_{k}{x}_{k}$ for every $x\in w\left(p,s\right)$, where $u=\sigma \text{-}lima$ and ${u}_{k}=\sigma \text{-}lim{a}_{\left(k\right)}$, $k=1,2,\dots$ .

Proof Suppose that $A\in \left(\omega \left(p,s\right),{V}_{\sigma }^{\lambda }\right)$. Define ${e}^{k}=\left(0,0,\dots ,1,0,\dots \right)$ having 1 in the k th coordinate sequence. Since e and ${e}^{k}$ are in $\omega \left(p,s\right)$, necessity of (ii) and (iii) is clear. We know that ${\sum }_{k}a\left(n,k,m\right){x}_{k}$ converges for each m, n and $x\in \omega \left(p,s\right)$. Therefore ${\left(a\left(n,k,m\right)\right)}_{k}\in {\omega }^{+}\left(p,s\right)$ and

$\sum _{r=0}^{\mathrm{\infty }}\underset{r}{max}{\left({2}^{r}\cdot {B}^{-1}\right)}^{\frac{1}{{p}_{k}}}|\frac{a\left(n,k,m\right)}{{s}_{k}}|<\mathrm{\infty }$

for each m, n (see [31]). Furthermore, if ${f}_{mn}\left(x\right)={t}_{mn}\left(Ax\right)$, then ${\left\{{f}_{mn}\right\}}_{m}$ is a sequence of continuous linear functionals on $\omega \left(p,s\right)$ such that ${lim}_{m}{t}_{mn}\left(Ax\right)$ exists. Therefore, by using the Banach-Steinhaus theorem, the necessity of (i) follows immediately.

Conversely, suppose that the conditions (i), (ii) and (iii) hold and $x\in \omega \left(p,s\right)$. We know that ${\left(a\left(n,k,m\right)\right)}_{k}$ and ${u}_{k}$ are in ${\omega }^{+}\left(p,s\right)$ and that the series ${\sum }_{k}a\left(n,k,m\right){x}_{k}$ and ${\sum }_{k}{u}_{k}{x}_{k}$ converge for each m, n. Write

$c\left(n,k,m\right)=a\left(n,k,m\right)-{u}_{k}.$

Then

$\sum _{k}a\left(n,k,m\right){x}_{k}=\sum _{k}{u}_{k}{x}_{k}+l\sum _{k}c\left(n,k,m\right)+\sum _{k}c\left(n,k,m\right)\left({x}_{k}-l\right)$

by (ii) for some integer ${k}_{0}>0$, we have

where l is the limit of x for $x\in \omega \left(p,s\right)$. Since

$\begin{array}{c}\underset{m,n}{sup}\sum _{r}\underset{r}{max}{\left({2}^{r}\cdot {B}^{-1}\right)}^{\frac{1}{{p}_{k}}}|c\left(n,k,m\right)|\le 2{D}_{n},\hfill \\ \underset{m}{lim}\sum _{k\le {k}_{0}}|\frac{a\left(n,k,m\right)-{u}_{k}}{{s}_{k}}||{s}_{k}\left({x}_{k}-l\right)|=0,\hfill \end{array}$

uniformly in n, whence

$\underset{n}{lim}\sum _{k}a\left(n,k,m\right){x}_{k}-l\cdot u+\sum _{k}{u}_{k}\left({x}_{k}-l\right).$

□

Theorem 3.2 Let $1\le {p}_{k}<\mathrm{\infty }$, then $A\in \left({\omega }_{p}\left(s\right),{V}_{\sigma }^{\lambda }\right)$ if and only if

(i) for every n

$M\left(A\right)=\underset{m}{sup}\sum _{r}{2}^{\frac{r}{p}}{\left(\sum _{r}{|\frac{a\left(n,k,m\right)}{{s}_{k}}|}^{q}\right)}^{\frac{1}{q}}<\mathrm{\infty },$

where ${p}^{-1}+{q}^{-1}=1$;

(ii) ${\alpha }_{\left(k\right)}\in {V}_{\sigma }^{\lambda }$ for each k;

(iii) $\alpha \in {V}_{\sigma }^{\lambda }$.

Proof Assume that the conditions are satisfied and let $x\in {\omega }_{p}\left(s\right)$. Then

$|{t}_{mn}\left(Ax\right)|\le \sum _{r=0}^{\mathrm{\infty }}\sum _{r}|\frac{a\left(n,k,m\right){s}_{k}{x}_{k}}{{s}_{k}}|\le \sum _{r=0}^{\mathrm{\infty }}{\left(\sum _{r}{|\frac{a\left(n,k,m\right)}{{s}_{k}}|}^{q}\right)}^{\frac{1}{q}}\cdot {\left(\sum _{r}{|{x}_{k}|}^{p}\right)}^{\frac{1}{p}},$

and hence ${t}_{mn}\left(Ax\right)$ is absolutely and uniformly convergent for each m, n. Note that (i) and (ii) imply that

$\sum _{r=0}^{\mathrm{\infty }}{2}^{\frac{r}{p}}{\left(\sum _{r}|{s}_{k}{u}_{k}|\right)}^{\frac{1}{q}}\le M\left(A\right)<\mathrm{\infty },$

so that by Hölder’s inequality, ${\sum }_{k}{u}_{k}{x}_{k}<\mathrm{\infty }$. Now as in the converse part of Theorem 3.1, it follows that $A\in \left({\omega }_{p}\left(s\right),{V}_{\sigma }^{\lambda }\right)$.

Conversely, suppose that $A\in \left({\omega }_{p}\left(s\right),{V}_{\sigma }^{\lambda }\right)$. Since ${e}^{k}$ and e are in ${\omega }_{p}\left(s\right)$, the necessity of (ii) and (iii) is clear. For the necessity of (i), suppose that

${t}_{mn}\left(Ax\right)=\sum _{k}a\left(n,k,m\right){x}_{k}$

exits for each n whenever $x\in {\omega }_{p}\left(s\right)$. Then, for each n and $r\ge 0$, write

${f}_{nr}\left(x\right)=\sum _{r}a\left(n,k,m\right){x}_{k}.$

Then ${\left\{{f}_{nr}\right\}}_{m}$ is a sequence of continuous linear functionals on ${\omega }_{p}\left(s\right)$. Since

$|{f}_{nr}\left(x\right)|\le {\left(\sum _{r}{|\frac{a\left(n,k,m\right)}{{s}_{k}}|}^{q}\right)}^{\frac{1}{q}}\cdot {\left(\sum _{r}{|{s}_{k}\cdot {x}_{k}|}^{p}\right)}^{\frac{1}{p}}\le {2}^{\frac{r}{p}}{\left(\sum _{r}{|\frac{a\left(n,k,m\right)}{{s}_{k}}|}^{q}\right)}^{\frac{1}{q}}\cdot \parallel x\parallel ,$

it follows that for each n,

$\underset{j}{lim}\sum _{r=0}^{j}{f}_{mr}\left(x\right)={t}_{mn}\left(Ax\right)$

is in the dual space ${\omega }_{p}^{\ast }$. Hence there exists a ${K}_{mn}$ such that

$|\frac{a\left(n,k,m\right)}{{s}_{k}}|\le {K}_{mn}\parallel x\parallel .$
(3.1)

For each n and any integer $j>0$, define $x\in {\omega }_{p}\left(s\right)$ as in [30] (Theorem 7, p.173), we get

$\sum _{r=0}^{j}{2}^{\frac{r}{p}}{\left(\sum _{r}{|\frac{a\left(n,k,m\right)}{{s}_{k}}|}^{q}\right)}^{\frac{1}{q}}\le {K}_{mn}.$

Hence, for each n,

$\sum _{r=0}^{\mathrm{\infty }}{2}^{\frac{r}{p}}{\left(\sum _{r}{|\frac{a\left(n,k,m\right)}{{s}_{k}}|}^{q}\right)}^{\frac{1}{q}}\le {K}_{mn}<\mathrm{\infty }.$
(3.2)

Since ${t}_{mn}\left(Ax\right)$ is absolutely convergent, we get

$|{t}_{mn}\left(Ax\right)|\le \sum _{r=0}^{\mathrm{\infty }}{2}^{\frac{r}{p}}{\left(\sum _{r}{|\frac{a\left(n,k,m\right)}{{s}_{k}}|}^{q}\right)}^{\frac{1}{q}}\parallel x\parallel ,$

so that

${K}_{mn}\le \sum _{r=0}^{\mathrm{\infty }}{2}^{\frac{r}{p}}{\left(\sum _{r}{|\frac{a\left(n,k,m\right)}{{s}_{k}}|}^{q}\right)}^{\frac{1}{q}}.$
(3.3)

By virtue of (3.2) and (3.3),

${K}_{mn}=\sum _{r=0}^{\mathrm{\infty }}{2}^{\frac{r}{p}}{\left(\sum _{r}{|\frac{a\left(n,k,m\right)}{{s}_{k}}|}^{q}\right)}^{\frac{1}{q}}.$

Finally, by (Theorem 11, [30], p.114) for every n, the existence of ${lim}_{m}{t}_{mn}\left(Ax\right)$ on ${\omega }_{p}\left(s\right)$ implies that

$\underset{m}{sup}{K}_{mn}=\underset{m}{sup}\sum _{r=0}^{\mathrm{\infty }}{2}^{\frac{r}{p}}{\left(\sum _{r}{|\frac{a\left(n,k,m\right)}{{s}_{k}}|}^{q}\right)}^{\frac{1}{q}}<\mathrm{\infty },$

which is (i). □

Theorem 3.3 Let $0<{p}_{k}<\mathrm{\infty }$, then $A\in {\left({\omega }_{p}\left(s\right),{V}_{\sigma }\right)}_{\mathrm{reg}}$ if and only if conditions (i), (ii) with $\sigma \text{-}lim=0$ and (iii) with $\sigma \text{-}lim=+1$ of Theorem  3.2 hold.

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## Acknowledgements

Dedicated to Professor Hari M Srivastava.

We would like to thank the referees for their valuable comments.

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Correspondence to Rahmet Savaş.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

Both authors completed the paper together. Both authors read and approved the final manuscript.

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Savaş, R., Savaş, E. On some new matrix transformations. J Inequal Appl 2013, 254 (2013). https://doi.org/10.1186/1029-242X-2013-254

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• DOI: https://doi.org/10.1186/1029-242X-2013-254

### Keywords

• Banach Space
• Dual Space
• Matrix Transformation
• Sequence Space
• Nonempty Subset