On some new matrix transformations

Abstract

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

1 Introduction

Let w denote the set of all real and complex sequences $x=(x_k)$. By $l_{\mathrm{\infty }}$ and c, we denote the Banach spaces of bounded and convergent sequences $x=(x_k)$ 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) $L(x)\ge 0$ if $n\ge 0$ (i.e., $x_n\ge 0$ for all n),

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

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

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 $\hat{c}$ denote the space of the almost convergent sequences. Lorentz [2] has shown that

$$\hat{c}=\{x\in l_{\mathrm{\infty }}:\underset{m}{lim}{d}_{m,n}(x)\text{ exists, uniformly in }n\},$$

where

$$d_{m,n}(x)=\frac{x_n+x_{n+1}+x_{n+2}+\cdots +x_{n+m}}{m+1}.$$

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 $(c,V_{\sigma })$, ${(c,V_{\sigma })}_{\mathrm{reg}}$ and $(l_{\mathrm{\infty }},V_{\sigma })$, 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 $(c(p),V_{\sigma })$, ${(c(p),V_{\sigma })}_{\mathrm{reg}}$ and $(l_{\mathrm{\infty }}(p),V_{\sigma })$ of matrices, which generalized the results due to Schaefer [4]. In [6], Mohiuddine and Aiyup defined the space $\omega (p,s)$ and obtained necessary and sufficient conditions to characterize the matrices of classes $(\omega (p,s),V_{\sigma })$, $({\omega }_p(s),V_{\sigma })$ and ${({\omega }_p(s),V_{\sigma })}_{\mathrm{reg}}$.

Matrix transformations between sequence spaces have also been discussed by Savaş and Mursaleen [7], Basarir and Savaş [8], Mursaleen [5, 9–16], Vatan and Simsek [17], Savaş [18–24], 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 $(\omega (p,s),V_{\sigma }^{\lambda })$, $({\omega }_p(s),V_{\sigma }^{\lambda })$ and ${({\omega }_p(s),V_{\sigma }^{\lambda })}_{\mathrm{reg}}$.

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) $\phi (x)\ge 0$ when the sequence $x=(x_k)$ has $x_k\ge 0$ for all k;

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

(iii) $\phi (x)=\phi (x_{\sigma (k)})$ 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 $x=(x_k)$ is σ-convergent if and only if $x\in V_{\sigma }$. For $\sigma (n)=n+1$, the set $V_{\sigma }$ is reduced to the set $\hat{c}$ of almost convergent sequences [2, 26].

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

$$V_{\sigma }=\{x\in l_{\mathrm{\infty }}:\underset{m}{lim}{t}_{mn}(x)=L,\text{ uniformly in }n;L=\sigma \text{-}limx\},$$

where

$$t_{mn}(x)=\frac{1}{m+1}\sum _{j=0}^m{T}^j{x}_n$$

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

If $p_k$ is real and positive, we define (see Maddox [27])

$$c_0(p)=\{x:\underset{k\to \mathrm{\infty }}{lim}|x_k{|}^{p_k}=0\}$$

and

$$c(p)=\{x:\underset{k\to \mathrm{\infty }}{lim}|x_k-l{|}^{p_k}=0\text{ for some }l\}.$$

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

$$\omega (p,s)=\{x:\frac{1}{n}\sum _{k=1}^n{k}^{-s}{|x_k-l|}^{p_k}\to 0,n\to \mathrm{\infty }\text{ for some }l,s\ge 0\},$$

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

The sequence space

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

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

We further define the following.

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

$${\lambda }_{m+1}\le {\lambda }_m+1,{\lambda }_1=1.$$

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

$$\begin{array}{c}{V}_{\sigma }^{\lambda }=\{x\in l_{\mathrm{\infty }}:\underset{m\to \mathrm{\infty }}{lim}{t}_{mn}(x)=L,\text{ uniformly in }n;L=(\sigma ,\lambda )\text{-}limx\},\hfill \\ t_{mn}(x)=\frac{1}{{\lambda }_m}\sum _{i\in I_m}{x}_{{\sigma }^i(n)},\hfill \end{array}$$

and $I_m=[m-{\lambda }_m+1,m]$. Note that $c\subset V_{\sigma }^{\lambda }\subset l_{\mathrm{\infty }}$. For $\sigma (n)=n+1,V_{\sigma }^{\lambda }$ reduces to the space ${\hat{V}}_{\lambda }$ of almost λ-convergent sequences [28]; and if we take $\sigma (n)=n+1$ and ${\lambda }_m=m$, then $V_{\sigma }^{\lambda }$ reduces to $\hat{c}$ (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.,

$$E^{+}=\{a:\sum _k{a}_k{x}_k\text{ converges for every }x\in E\}.$$

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

$$\mathbb{M}=\{a:\sum _{r=0}^{\mathrm{\infty }}\underset{r}{max}\{{(2^r\cdot N^{-1})}^{\frac{1}{p_k}}|a_k|\}<\mathrm{\infty }\text{ for some integer }N>1\},$$

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,

$${[\omega (p,s)]}^{\ast }=\{a:\sum _{r=0}^{\mathrm{\infty }}\underset{r}{max}\left\{{(2^r\cdot N^{-1})}^{\frac{1}{p_k}}\left|\frac{a_k}{s_k}\right|\right\}<\mathrm{\infty }\text{ for some integer }N>1\}.$$

3 Main results

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

We now characterize the matrices in the class $(c_0(p),V_{{\sigma }_0}^{\lambda }(p))$. We write

$$t_{m,n}(Ax)=\sum _{k}a(n,k,m)x_k,$$

where

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

Theorem 3.1 Let $0<p_k\le 1$, then $A\in (\omega (p,s),V_{\sigma }^{\lambda })$ if and only if

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

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

(ii) ${\alpha }_{(k)}={\{a_{nk}\}}_{n=1}^{\mathrm{\infty }}\in V_{\sigma }^{\lambda }$ for each k;

(iii) $\alpha ={\{{\sum }_k{a}_{nk}\}}_{n=1}^{\mathrm{\infty }}\in V_{\sigma }^{\lambda }$.

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

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

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

for each m, n (see [31]). Furthermore, if $f_{mn}(x)=t_{mn}(Ax)$, then ${\{f_{mn}\}}_m$ is a sequence of continuous linear functionals on $\omega (p,s)$ such that ${lim}_m{t}_{mn}(Ax)$ 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 $x\in \omega (p,s)$. We know that ${(a(n,k,m))}_k$ and $u_k$ are in ${\omega }^{+}(p,s)$ and that the series ${\sum }_{k}a(n,k,m)x_k$ and ${\sum }_k{u}_k{x}_k$ converge for each m, n. Write

$$c(n,k,m)=a(n,k,m)-u_k.$$

Then

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

by (ii) for some integer $k_0>0$, we have

$$\underset{m}{lim}\sum _{k\le k_0}c(n,k,m)(x_k-l)=0,\text{uniformly in }n,$$

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

$$\begin{array}{c}\underset{m,n}{sup}\sum _r\underset{r}{max}{(2^r\cdot B^{-1})}^{\frac{1}{p_k}}|c(n,k,m)|\le 2D_n,\hfill \\ \underset{m}{lim}\sum _{k\le k_0}\left|\frac{a(n,k,m)-u_k}{s_k}\right||s_k(x_k-l)|=0,\hfill \end{array}$$

uniformly in n, whence

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

 □

Theorem 3.2 Let $1\le p_k<\mathrm{\infty }$, then $A\in ({\omega }_p(s),V_{\sigma }^{\lambda })$ if and only if

(i) for every n

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

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

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

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

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

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

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

$$\sum _{r=0}^{\mathrm{\infty }}{2}^{\frac{r}{p}}{(\sum _r|s_k{u}_k|)}^{\frac{1}{q}}\le M(A)<\mathrm{\infty },$$

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 $A\in ({\omega }_p(s),V_{\sigma }^{\lambda })$.

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

$$t_{mn}(Ax)=\sum _{k}a(n,k,m)x_k$$

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

$$f_{nr}(x)=\sum _{r}a(n,k,m)x_k.$$

Then ${\{f_{nr}\}}_m$ is a sequence of continuous linear functionals on ${\omega }_p(s)$. Since

$$|f_{nr}(x)|\le {\left(\sum _r{\left|\frac{a(n,k,m)}{s_k}\right|}^q\right)}^{\frac{1}{q}}\cdot {\left(\sum _r{|s_k\cdot x_k|}^p\right)}^{\frac{1}{p}}\le 2^{\frac{r}{p}}{\left(\sum _r{\left|\frac{a(n,k,m)}{s_k}\right|}^q\right)}^{\frac{1}{q}}\cdot \parallel x\parallel ,$$

it follows that for each n,

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

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

$$\left|\frac{a(n,k,m)}{s_k}\right|\le K_{mn}\parallel x\parallel .$$

(3.1)

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

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

Hence, for each n,

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

(3.2)

Since $t_{mn}(Ax)$ is absolutely convergent, we get

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

so that

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

(3.3)

By virtue of (3.2) and (3.3),

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

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

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

which is (i). □

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