On the Domain of Four-Dimensional Forward Difference Matrix in Some Double Sequence Spaces

By $\Omega:=\{x=(x_{mn}):x_{mn}\in\mathbb{C},~{}~{}\forall m,n\in\mathbb{N}\}$, we denote the set of all complex valued double sequences; $\Omega$ is a vector space with coordinatewise addition and scalar multiplication and any vector subspace of $\Omega$ is called a double sequence space. A double sequence $x=(x_{mn})$ is called convergent in Pringsheim’s sense to a limit point $L$, if for every $\epsilon>0$ there exists a natural number $n_{0}=n_{0}(\epsilon)$ and $L\in\mathbb{C}$ such that $|x_{mn}-L|<\epsilon$ for all $m,n>n_{0}$, where $\mathbb{C}$ denotes the complex field; this is denoted by $L=p-\lim_{m,n\to\infty}x_{mn}$. The space of all double sequences that are convergent in the Pringsheim sense is denoted by $\mathcal{C}_{p}$ which is a linear space with coordinatewise addition and scalar multiplication. Mòricz [1] proved that the double sequence space $\mathcal{C}_{p}$ is a complete seminormed space with the seminorm

The space of all null double sequences in Pringsheim’s sense is denoted by $\mathcal{C}_{p0}$.

A double sequence $x=(x_{mn})$ of complex numbers is called bounded if $\|x\|_{\infty}=\sup_{m,n\in\mathbb{N}}|x_{mn}|<\infty$, where $\mathbb{N}=\{0,1,2,\cdots\}$, and the space of all bounded double sequences is denoted by $\mathcal{M}_{u}$, that is,

it is a Banach space with the norm $\|\cdot\|_{\infty}$.

Unlike as in the case of single sequences there are double sequences which are convergent in Pringsheim’s sense but unbounded. That is, the set $\mathcal{C}_{p}\setminus\mathcal{M}_{u}$ is not empty. Boos [2] defined the sequence $x=(x_{mn})$ by

which is obviously in $\mathcal{C}_{p}$, i.e., $p-\lim_{m,n\rightarrow\infty}x_{mn}=0$, but not in the set $\mathcal{M}_{u}$, i.e., $\|x\|_{\infty}=\sup_{m,n\in\mathbb{N}}|x_{mn}|=\infty$. Thus, $x\in\mathcal{C}_{p}\setminus\mathcal{M}_{u}$.

We also consider the set $\mathcal{C}_{bp}$ of double sequences which are both convergent in Pringsheim’s sense and bounded, that is,

The set $\mathcal{C}_{bp}$ is a Banach space with the norm

Hardy [3] called a sequence in the space $\mathcal{C}_{p}$ regularly convergent if it is a convergent single sequence with respect to each index. We denote the set of such double sequences by $\mathcal{C}_{r}$, that is,

where $c$ denotes the set of all convergent single sequences of complex numbers. Regular convergence requires the boundedness of double sequences; this is the main difference between regular convergence and the convergence in Pringsheim’s sense. We also use the notations $\mathcal{C}_{bp0}=\mathcal{M}_{u}\cap\mathcal{C}_{p0}$ and $\mathcal{C}_{r0}=\mathcal{C}_{r}\cap\mathcal{C}_{p0}$.

Throughout the text, unless otherwise stated we mean by the summation $\sum_{kl}x_{kl}$ without limits run from $0$ to $\infty$ is $\sum_{k,l=0}^{\infty}x_{kl}$.

The space $\mathcal{L}_{q}$ of all absolutely $q-$summable double sequences was introduced by Başar and Sever [4] as follows

which is a Banach space with the norm $\|\cdot\|_{q}$ defined by

Moreover, Zeltser [5] introduced the space $\mathcal{L}_{u}$ which is the special case of the space $\mathcal{L}_{q}$ for $q=1$.

The double sequence spaces $\mathcal{BS}$, $\mathcal{CS}_{\vartheta}$, where $\vartheta\in\{p,bp,r\}$, and $\mathcal{BV}$ were introduced by Altay and Başar [6]. The set $\mathcal{BS}$ of all double series whose sequences of partial sums are bounded is defined by

where the sequence $s_{mn}=\sum_{k,l=0}^{m,n}x_{kl}$ is the $(m,n)-th$ partial sum of the series. The series space $\mathcal{BS}$ is a Banach space with norm defined as

which is linearly isomorphic to the sequence space $\mathcal{M}_{u}$. The set $\mathcal{CS_{\vartheta}}$ of all series whose sequences of partial sums are $\vartheta-$convergent in Pringsheim’s sense is defined by

where $\vartheta\in\{p,bp,r\}$. The space $\mathcal{CS}_{p}$ is a complete seminormed space with the seminorm defined by

which is isomorphic to the sequence space $\mathcal{C}_{p}$. Moreover, the sets $\mathcal{CS}_{bp}$ and $\mathcal{CS}_{r}$ are also Banach spaces with the norm (1.2) and the inclusion $\mathcal{CS}_{r}\subset\mathcal{CS}_{bp}$ holds. The set $\mathcal{BV}$ of all double sequences of bounded variation is defined by Altay and Başar [6] as follows

The space $\mathcal{BV}$ is Banach space with the norm defined by

which is linearly isomorphic to the space $\mathcal{L}_{u}$ of absolutely convergent double series. Moreover, the inclusions $\mathcal{BV}\subset\mathcal{C_{\vartheta}}$ and $\mathcal{BV}\subset\mathcal{M}_{u}$ strictly hold.

Let $E$ be any double sequence space. Then,

Therefore, let $E_{1}$ and $E_{2}$ are arbitrary double sequences with $E_{2}\subset E_{1}$ then the inclusions $E_{1}^{\alpha}\subset E_{2}^{\alpha}$, $E_{1}^{\gamma}\subset E_{1}^{\alpha}$ and $E_{1}^{\beta(\vartheta)}\subset E_{1}^{\alpha}$ hold. But the inclusion $E_{1}^{\gamma}\subset E_{1}^{\beta(\vartheta)}$ does not hold, since $\mathcal{C}_{p}\setminus\mathcal{M}_{u}$ is not empty.

Let $A=(a_{mnkl})_{m,n,k,l\in\mathbb{N}}$ be an infinite four–dimensional matrix and $E_{1}$, $E_{2}\in\Omega$. We write

We say that $A$ defines a matrix transformation from $E_{1}$ to $E_{2}$ if

The $\vartheta-$summability domain $E_{A}^{(\vartheta)}$ of a four-dimensional infinite matrix $A$ in a double sequence space $E$ is defined by

which is a sequence space. The above notation (1.4) says that $A=(a_{mnkl})_{m,n,k,l\in\mathbb{N}}$ maps the space $E_{1}$ into the space $E_{2}$ if $E_{1}\subset(E_{2})_{A}^{(\vartheta)}$ and we denote the set of all four-dimensional matrices that map the space $E_{1}$ into the space $E_{2}$ by $(E_{1}:E_{2})$. Thus, $A\in(E_{1}:E_{2})$ if and only if the double series on the right side of (1.4) $\vartheta-$converges for each $m,n\in\mathbb{N}$, i.e, $A_{mn}\in(E_{1})^{\beta(\vartheta)}$ for all $m,n\in\mathbb{N}$ and we have $Ax\in E_{2}$ for all $x\in E_{1}$.

Adams [7] defined that the four-dimensional infinite matrix $A=(a_{mnkl})$ is a triangular matrix if $a_{mnkl}=0$ for $k>m$ or $l>n$ or both. We also say by [7] that a triangular matrix $A=(a_{mnkl})$ is called a triangle if $a_{mnmn}\neq 0$ for all $m,n\in\mathbb{N}$. One can be observed easily that if $A$ is triangle, then $E_{A}^{(\vartheta)}$ and $E$ are linearly isomorphic.

Wilansky [8, Theorem 4.4.2, p. 66] defined that if $E$ is a sequence space, then the continuous dual $E_{A}^{*}$ of the space $E_{A}$ is given by

Zeltser [9] stated the notations of the double sequences $e^{kl}=(e_{mn}^{kl}),e^{1},e_{k}$ and $e$ by

for all $k,l,m,n\in\mathbb{N}$.

The four-dimensional forward difference matrix $\Delta=(\delta_{mnkl})$ is defined by

for all $m,n,k,l\in\mathbb{N}$. The $\Delta-$transform of a double sequence $x=(x_{mn})$ is given by

for all $m,n\in\mathbb{N}$. We shall briefly discuss $\Delta^{-1}$ which is the inverse of four-dimensional forward difference matrix $\Delta$, where $(\Delta^{-1}\Delta)(x_{kl})=x_{kl}$. Let $\Delta^{-1}y_{kl}=x_{kl}$. Then we can show that $x_{kl}$ is a finite summation of the original double sequence $y_{kl}$.

If we write the equation (1.7) for $y_{00},y_{01},y_{10},...,y_{kl}$

Then we add the left hand sides up to $y_{00}+y_{01}+y_{10}+...+y_{kl}$

for all $k,l\in\mathbb{N}$. To be able to have $x_{kl}$ instead of having $x_{k+1,l+1}$ we must write it as

for all $k,l\in\mathbb{N}$. With this result we can introduce the role of inverse four-dimensional forward difference operator $\Delta^{-1}$ on the double sequence $y_{kl}$, where $x_{kl}=\Delta^{-1}y_{kl}$, as the $(k-1,l-1)^{th}-$partial sum of the double sequence $y_{kl}$ plus arbitrary constants on the first row and the first column of the double sequence $x=(x_{kl})$.

In this section, we introduce new double sequence spaces $\mathcal{M}_{u}(\Delta)$, $\mathcal{C}_{\vartheta}(\Delta)$, where $\vartheta\in\{bp,r\}$, as the matrix domains of the four-dimensional matrix of the forward differences in the sequence spaces $\mathcal{M}_{u}$ and $\mathcal{C}_{\vartheta}$ as follow;

where $y_{kl}=\Delta x_{kl}=(x_{kl}-x_{k+1,l}-x_{k,l+1}+x_{k+1,l+1})$ for all $k,l\in\mathbb{N}$.

Let $\vartheta=\{bp,bp0,r,r0\}$. We define the operator $P$ form $\lambda(\Delta)$ into itself, where $\lambda\in\{\mathcal{M}_{u},\mathcal{C}_{\vartheta}\}$ as

for all $x=(x_{kl})\in\lambda(\Delta)$. Clearly $P$ is a linear and bounded operator on $\lambda(\Delta)$.

Now we show that the four-dimensional forward difference operator $\Delta$ is a linear homeomorphism.

where the set $P(\lambda(\Delta))$ is defined by

and

Therefore, the spaces $P(\lambda(\Delta))$ and $\lambda$ are equivalent as topological spaces, and the $\Delta$ and $\Delta^{-1}$ are norm preserving and $\|\Delta\|=\|\Delta^{-1}\|=1$. We prove the following Lemma 2.2 for the case $\lambda=\mathcal{C}_{r0}$ by using the results in [1, Theorem 5., Remark 3., P.132]. Since the proofs of the other cases are similar to that of following Lemma 2.2, we left them as an exercise to the reader.