On the Double Sequence Space Hϑ as an Extension of Hahn Space h

Abstract

Double sequence spaces have become a significant area of research within functional analysis due to their applications in various branches of mathematics and mathematical physics. In this study, we investigate Hahn double sequence space denoted as ${\mathcal{H}}_{\vartheta }$, where $\vartheta \in p,bp,r$, as an extension of the Hahn sequence space $h$. Our investigation begins with an analysis of several topological properties of ${\mathcal{H}}_{\vartheta }$, apart from a comprehensive analysis of the relationship between Hahn double sequences and other classical double sequence spaces. The $\alpha -$ dual, algebraic dual and $\beta bp-$ dual, and $\gamma -$ dual of the space ${\mathcal{H}}_{\vartheta }$ are determined. Furthermore, we define the determining set of ${\mathcal{H}}_{\vartheta }$ and we state the conditions concerning the characterization of four-dimensional ($4\mathrm{D}$) matrix classes ${\mathcal{H}}_{\vartheta },\lambda$, where $\lambda ={\mathcal{H}}_{\vartheta },\mathcal{B}\mathcal{V},\mathcal{B}{\mathcal{V}}_{\vartheta 0},\mathcal{C}{\mathcal{S}}_{\vartheta },\mathcal{C}{\mathcal{S}}_{\vartheta 0},\mathcal{B}\mathcal{S}$ and $\mu ,{\mathcal{H}}_{\vartheta }$, where $\mu ={\mathcal{L}}_u,{\mathcal{C}}_{\vartheta 0},{\mathcal{C}}_{\vartheta },{\mathcal{M}}_u$. In conclusion, this research contributes to nonstandard investigation and various significant results in the space ${\mathcal{H}}_{\vartheta }$. The conducted results deepen the understanding of space ${\mathcal{H}}_{\vartheta }$ and open up new avenues for further research and applications in sequence space theory.

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1. Introduction, Notations, and Preliminaries

By the symbol $\mathrm{\Omega },$ we mean the set consisting of all complex-valued double sequences. That is,$$\begin{array}{}\text{(1)}& \mathrm{\Omega }=x={x_{jk}}_{j,k\in {\mathbb{N}}_0}:x_{jk}\mathbb{\in }\mathbb{C},\end{array}$$where the symbol $\mathbb{C}$ is the set of all complex numbers and $\mathbb{N}$ is the set of natural numbers. Any linear subspace of $\mathrm{\Omega }$ 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 $ϵ>0$, there exists a natural number $N=Nϵ$ and $L\mathbb{\in }\mathbb{C}$ such that $x_{mn}-L<ϵ$ for all $m,n>N$. Therefore, we write $L=p-{\lim }_{m,n⟶\infty }{x}_{mn}$. Moreover, a double sequence $x=x_{mn}$ is called regularly convergent if it is single convergent according to each index. Furthermore, a double sequence $x=x_{mn}$ is called bounded if $x_{\infty }={\sup }_{m,n\mathbb{\in }\mathbb{N}}{x}_{mn}<\infty$. Now, we can define the classical double sequence spaces, namely, ${\mathcal{M}}_u$, ${\mathcal{C}}_p$, ${\mathcal{C}}_{p0}$, ${\mathcal{C}}_{bp}$, ${\mathcal{C}}_{bp0}$, ${\mathcal{C}}_r$, ${\mathcal{C}}_{r0}$, and ${\mathcal{L}}_q$. The respective definitions are as follows: ${\mathcal{M}}_u$ represents the space of all bounded sequences, ${\mathcal{C}}_p$ contains sequences convergent in Pringsheim’s sense, ${\mathcal{C}}_{p0}$ denotes the null space of ${\mathcal{C}}_p$, ${\mathcal{C}}_{bp}$ consists of sequences in ${\mathcal{M}}_u\cap {\mathcal{C}}_p$, ${\mathcal{C}}_{bp0}={\mathcal{M}}_u\cap {\mathcal{C}}_{p0}$, ${\mathcal{C}}_r$ accommodates regular convergent sequences, ${\mathcal{C}}_{r0}$ denotes the null space of ${\mathcal{C}}_r$, and ${\mathcal{L}}_q$ encompasses $q-$ absolutely summable sequences, where $1\le q<\infty$. As a significant contribution by Móricz [1], it was shown that the spaces ${\mathcal{M}}_u$, ${\mathcal{C}}_{bp}$, ${\mathcal{C}}_{bp0}$, ${\mathcal{C}}_r$, and ${\mathcal{C}}_{r0}$ are Banach spaces when endowed with the supremum norm ${.}_{\infty }$.

A double sequence space $\mu$ is said to be a $DK-$ space if it is a local convex space, and all the seminorms defined on $\mu$ by $x=x_{kl}⟶x_{kl}$ are continuous. When such a $DK-$ space occupies a Frèchet topology, then it is called an $FDK-$ space. If we have a normed $FDK-$ space, then it is referred to as a $BDK-$ space. Three well-known examples of $BDK-$ spaces are ${\mathcal{M}}_u$, ${\mathcal{C}}_{bp}$, and ${\mathcal{C}}_r$, where the norm is given by$$\begin{array}{}\text{(2)}& x_{\infty }=\underset{k,l\mathbb{\in }\mathbb{N}}{\sup }{x}_{kl}<\infty .\end{array}$$

The double series is represented by the notation ${\sum }_{k,l=0}^{\infty }{x}_{kl}$ and is considered convergent with respect to $\vartheta$ if its $m{n}^{th}-$ partial sum $s_{mn}={\sum }_{k,l=0}^{m,n}{x}_{kl}$ converges as $m,n⟶\infty$, that is, there exists $s\mathbb{\in }\mathbb{R}$ in which $\vartheta -{\lim }_{m,n⟶\infty }{s}_{mn}={\sum }_{k,l=0}^{\infty }{x}_{kl}=s$.

The collection of $\vartheta -$ convergent series is represented as $\mathcal{C}{\mathcal{S}}_{\vartheta }$, where $\vartheta \in p,p0,bp,bp0,r,r0$. Moreover by $\mathcal{B}\mathcal{S}$, $\mathcal{B}\mathcal{V}$, and $\mathcal{B}{\mathcal{V}}_{\vartheta 0}=\mathcal{B}\mathcal{V}\cap {\mathcal{C}}_{\vartheta 0}$, we represent the set of all bounded series, a set comprising double sequences with bounded variation, and the null space of $\mathcal{B}\mathcal{V}$, respectively (see [2, 3]).

Sever [4] defined the residual component of a double series as follows:$$\begin{array}{}\text{(3)}& r_{mn}={\displaystyle \sum }_{k,l=m+1,0}^{\infty ,n}{x}_{kl}+{\displaystyle \sum }_{k,l=0,n+1}^{m,\infty }{x}_{kl}+{\displaystyle \sum }_{k,l=m+1,n+1}^{\infty ,\infty }{x}_{kl}.\end{array}$$

One can observe here that if a series is convergent by means of $\vartheta$, then the residual component of the series must be a null sequence.

Let $x^{m,n}$ denote the ${m,n}^{th}-$ section of a double sequence $x=x_{kl},$ expressed by$$\begin{array}{}\text{(4)}& x^{m,n}={\displaystyle \sum }_{k,l=1}^{m,n}{x}_{kl}{e}^{kl},\end{array}$$where $e^{kl}={e_{mn}^{kl}}_{m,n=1}^{\infty }$ for every $k,l\mathbb{\in }\mathbb{N}$ is defined with $e_{mn}^{mn}=1$ and $e_{mn}^{kl}=0$ for $k,l\ne m,n$, which was defined by Zeltser [5]. Let $\mu$ be an arbitrary double sequence space and suppose that it contains $\phi$, where $\phi =span{e}^{kl}$. Zeltser [5] defined the space $\mu$ to be an $AK\vartheta -$ space if each $x\in \mu$ has the $AK\vartheta -$ property in the set $\mu$, i.e., for an arbitrary $x=x_{kl}$, we have$$\begin{array}{}\text{(5)}& x=\vartheta -\underset{m,n⟶\infty }{\lim }{x}^{m,n}=\vartheta -\underset{m,n⟶\infty }{\lim }{\displaystyle \sum }_{k,l=1}^{m,n}{x}_{kl}{e}^{kl}={\displaystyle \sum }_{k,l=1}^{\infty }{x}_{kl}{e}^{kl}.\end{array}$$

Moreover, in the same paper [5], Zeltser introduced the double sequences $e^l,e_k$ and $e$ with the following definitions:

i. $e^l={\sum }_k{e}^{kl}$, where every element in the lth column is $1$, and the rest are $0.$

ii. $e_k={\sum }_l{e}^{kl}$, where all terms of the kth row are $1$, and the rest are $0$.

iii. $e={\sum }_{kl}{e}^{kl}$, where all terms are $1$.

For all $k,l,m,n\mathbb{\in }\mathbb{N}$, let $E$ be a double sequence space. Then, we define the differentiated and integrated spaces of $E$, respectively, as follows:$$\begin{array}{c}\text{(6)}& dE≔x=x_{kl}\in \mathrm{\Omega }:{\frac{1}{kl}{x}_{kl}}_{k,l\mathbb{\in }\mathbb{N}}\in E,\int E≔x=x_{kl}\in \mathrm{\Omega }:{kl{x}_{kl}}_{k,l\mathbb{\in }\mathbb{N}}\in E.\end{array}$$

Let $G=g_{mnkl}$ be a four-dimensional (4D) infinite matrix. Then, $G:\lambda ⟶\mu$ defines a matrix transformation., where $m,n,k,l\mathbb{\in }\mathbb{N}$. That is, $G$ maps $x\in \lambda$ into $Gx\in \mu$, where$$\begin{array}{}\text{(7)}& {Gx}_{mn}=\vartheta -{\displaystyle \sum }_{k,l}{g}_{mnkl}{x}_{kl},\end{array}$$for all $m,n\mathbb{\in }\mathbb{N}$.

Now, we define the sets ${\mathcal{P}}_{\infty }$, ${\mathcal{P}}_{\vartheta }$, and ${\mathcal{P}}_{\vartheta 0}$ in order to consider them in the main results of the paper.$$\begin{array}{c}\text{(8)}& {\mathcal{P}}_{\infty }≔x=x_{kl}\in \mathrm{\Omega }:\underset{m,n\mathbb{\in }\mathbb{N}}{\sup }\frac{1}{mn}{\displaystyle \sum }_{k,l=1}^{m,n}{x}_{kl}<\infty ,{\mathcal{P}}_{\vartheta }≔x=x_{kl}\in \mathrm{\Omega }:\vartheta -\underset{m,n⟶\infty }{\lim }\frac{1}{mn}{\displaystyle \sum }_{k,l=1}^{m,n}{x}_{kl}\text{exists},\\ {\mathcal{P}}_{\vartheta 0}≔x=x_{kl}\in \mathrm{\Omega }:\vartheta -\underset{m,n⟶\infty }{\lim }\frac{1}{mn}{\displaystyle \sum }_{k,l=1}^{m,n}{x}_{kl}=0.\end{array}$$

Then, the sets ${\mathcal{P}}_{\infty }$, ${\mathcal{P}}_{\vartheta }$, and ${\mathcal{P}}_{\vartheta 0}$ can be defined by the 4D matrix $T=t_{mnkl}$ domain in the sequence space $\mathcal{B}\mathcal{S}$, $\mathcal{C}{\mathcal{S}}_{\vartheta }$, and $\mathcal{C}{\mathcal{S}}_{\vartheta 0}$ where$$\begin{array}{}\text{(9)}& t_{mnkl}≔\begin{array}{cc}\frac{1}{mn},& 1\le k\le m,1\le l\le n,\\ 0,& \text{else}.\end{array}\end{array}$$

Consider $\vartheta =p,p0,bp,bp0,r,r0$. Then, the spaces ${\mathcal{P}}_{\infty }={\mathcal{B}\mathcal{S}}_T$, ${\mathcal{P}}_{\vartheta }={\mathcal{C}{\mathcal{S}}_{\vartheta }}_T$, and ${\mathcal{P}}_{\vartheta 0}={\mathcal{C}{\mathcal{S}}_{\vartheta 0}}_T$ are Banach spaces endowed with their natural norms and expressed as $$\begin{array}{}\text{(10)}& x_{{\mathcal{P}}_{\infty }}=\underset{m,n\mathbb{\in }\mathbb{N}}{\sup }\frac{1}{mn}{\displaystyle \sum }_{k,l=1}^{m,n}{x}_{kl},\end{array}$$respectively. Since ${\mathcal{C}}_p\backslash {\mathcal{M}}_u$ is not empty, we can easily see that the following inclusions:$$\begin{array}{c}\text{(11)}& {\mathcal{P}}_{\vartheta }\subset {\mathcal{P}}_{\infty },\mathcal{B}\mathcal{S}\subset {\mathcal{B}\mathcal{S}}_T,\\ \mathcal{C}{\mathcal{S}}_{\vartheta }\subset {\mathcal{C}{\mathcal{S}}_{\vartheta }}_T,\end{array}$$strictly hold, where $\vartheta =bp,bp0,r,r0$.

Now, we give a literature review focusing on the Hahn sequence space $h$ from a single sequence perspective, before starting the investigation of the Hahn double sequence space.

Hahn [6] defined the space $h$ as follows:$$\begin{array}{}\text{(12)}& h=x=x_k\in \omega :{\displaystyle \sum }_{k=1}^{\infty }k\mathrm{\Delta }{x}_k<\infty \cap c_0,\end{array}$$where $\mathrm{\Delta }$ represents the forward difference operator, i.e., $\mathrm{\Delta }{x}_k=x_k-x_{k+1}$ for all $k=1,2,3,\dots$. Here, $\omega$ denotes the set of sequences with complex values and $c_0$ refers to the set of sequences converging to $0$. Hahn [6] proved that $h$ is a $BK$ when equipped with the norm$$\begin{array}{}\text{(13)}& x={\displaystyle \sum }_{k=1}^{\infty }k\mathrm{\Delta }{x}_k+\underset{k\mathbb{\in }\mathbb{N}}{\sup }{x}_k,\end{array}$$for all $x={x_k}_{k=1}^{\infty }\in h$.

Rao [7] extended the study of $h$ and established that the Banach space $h$ has $AK$ property with the norm$$\begin{array}{}\text{(14)}& x={\displaystyle \sum }_{k=1}^{\infty }k{\mathrm{\Delta }x}_k<\infty ,\end{array}$$for all $x={x_k}_{k=1}^{\infty }\in h.$

Goes [8] investigated the generalized Hahn sequence space $h_d$ by considering an arbitrary negative or positive but not zero sequences $d={d_k}_{k=1}^{\infty }$ for all $k\mathbb{\in }\mathbb{N}$. The space $h_d$ is defined as follows:$$\begin{array}{}\text{(15)}& h_d=x=x_k\in \omega :{\displaystyle \sum }_{k=1}^{\infty }{d}_k\mathrm{\Delta }{x}_k<\infty \cap c_0.\end{array}$$

Most recently, Malkowsky, Rakočević, and Tuǧ [9] further extended the concept of the Hahn sequence space by studying $h_d$ with unbounded sequence $d={d_k}_{k=1}^{\infty }$, where $d_k<d_{k+1}$ and $d_k>0$ for all $k\mathbb{\in }\mathbb{N}$. Then, Tu$\breve{\mathrm{g}}$ et al. [10] studied the generalized $p-$ Hahn sequence space $h_d^p$, where $1<p<\infty$.

This research investigates the newly defined Hahn double sequence space ${\mathcal{H}}_{\vartheta }$ by focusing on its topological characteristics, inclusion relations, dual spaces, and matrix transformations. The study explores whether this space possesses completeness and Banach structure under specific norms and inclusion relations with classical double sequence spaces. Dual spaces will be characterized by deriving bounded functionals and analyzing their structural properties. Matrix transformations will be studied to determine necessary and sufficient conditions for $4\mathrm{D}$ matrices to map from the space ${\mathcal{H}}_{\vartheta }$ into some double sequence space or vice versa.

This work employs functional analysis, topology, and operator theory approaches, supported by some numerical and counterexamples to increase clarity and validate the proposed outcomes. This study aims to provide a comprehensive framework for understanding double sequence spaces and their applications by adding a new and very interesting double sequence space ${\mathcal{H}}_{\vartheta }$.

2. The Hahn Double Sequence Space ${\mathcal{H}}_{\vartheta }$

In the current section, we begin by introducing the Hahn double sequence space ${\mathcal{H}}_{\vartheta }$. Then, we study some topological structures exhibited by ${\mathcal{H}}_{\vartheta }$. In addition, we establish several strict inclusion relations among the known double sequence spaces within the space ${\mathcal{H}}_{\vartheta }$.

We define Hahn double sequence space, denoted as ${\mathcal{H}}_{\vartheta }$, in the following manner:$$\begin{array}{}\text{(16)}& {\mathcal{H}}_{\vartheta }≔x=x_{kl}\in \mathrm{\Omega }:{\displaystyle \sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}<\infty \cap {\mathcal{C}}_{\vartheta 0},\end{array}$$where $\mathrm{\Delta }$ represents the 4D forward difference operator, i.e., $\mathrm{\Delta }{x}_{jk}=x_{jk}-x_{j+1,k}-x_{j,k+1}+x_{j+1,k+1}$, and $\vartheta \in p,bp,r$.

Theorem 1.

i. The spaces ${\mathcal{H}}_{bp}$ and ${\mathcal{H}}_r$ are complete normed spaces due to the norm$$\begin{array}{}\text{(17)}& x_{{\mathcal{H}}_{\vartheta }}={\displaystyle \sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}<\infty .\end{array}$$

ii. The space ${\mathcal{H}}_p$ is a complete seminormed space, where the seminorm is given by$$\begin{array}{}\text{(18)}& x_{{\mathcal{H}}_p}={\displaystyle \sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}+\underset{N⟶\infty }{\lim }\underset{k,l\ge N}{\sup }{x}_{kl}.\end{array}$$

Proof.

i. Let $\vartheta =bp,r$. Showing that the norm defined in equation (17) for ${\mathcal{H}}_{\vartheta }$ satisfies the norm axioms is straightforward. Therefore, we only investigate the completeness of the space ${\mathcal{H}}_{\vartheta }$. We define a Cauchy sequence $x^i=x_{kl}^i\in {\mathcal{H}}_{\vartheta }$. Then, it gives$$\begin{array}{c}\text{(19)}& {\displaystyle \sum }_{k,l=0}^{\infty }kl\mathrm{\Delta }{x}_{kl}^i<\infty ,\vartheta -\underset{k,l⟶\infty }{\lim }{x}_{kl}^i=0,\text{\hspace{1em}for\hspace{0.17em}every\hspace{0.17em}fixed\hspace{0.17em}}i\mathbb{\in }\mathbb{N},\end{array}$$

  and $\forall ϵ>0$, $\exists Nϵ\mathbb{\in }\mathbb{N}$, in which$$\begin{array}{}\text{(20)}& {x^i-x^j}_{{\mathcal{H}}_{\vartheta }}<\frac{ϵ}{2},\end{array}$$

  for every $i,j\ge Nϵ$. For the fixed natural numbers $k,l$, and for all $q\ge k,p\ge l$ and all $i,j\ge Nϵ,$ one obtains$$\begin{array}{}\text{(21)}& {\displaystyle \sum }_{m,n=k,l}^{q,p}\mathrm{\Delta }{x}_{mn}^i-\mathrm{\Delta }{x}_{mn}^j\le {\displaystyle \sum }_{m,n=k,l}^{q,p}kl\mathrm{\Delta }{x}_{mn}^i-x_{mn}^j\le {x^i-x^j}_{{\mathcal{H}}_{\vartheta }}.\end{array}$$

  Thus, it can be deduced from (20) and (21) that$$\begin{array}{c}\text{(22)}& x_{kl}^i-x_{kl}^j={\displaystyle \sum }_{m,n=k,l}^{q,p}\mathrm{\Delta }{x}_{mn}^i-\mathrm{\Delta }{x}_{mn}^j+x_{q+1,p}^i-x_{q+1,p}^j+x_{q,p+1}^i-x_{q,p+1}^j-x_{q+1,p+1}^i-x_{q+1,p+1}^j<\frac{ϵ}{2}+x_{q+1,p}^i+x_{q+1,p}^j+x_{q,p+1}^i+x_{q,p+1}^j+x_{q+1,p+1}^i+x_{q+1,p+1}^j.\end{array}$$

  Now, letting limit as $q,p⟶\infty ,$ according to the $\vartheta -$ convergence, and since $x^i,x^j\in {\mathcal{C}}_{\vartheta 0}$ for all $i,j\mathbb{\in }\mathbb{N}$, one obtains$$\begin{array}{}\text{(23)}& x_{kl}^i-x_{kl}^j<ϵ,\end{array}$$

  for all $i,j\ge Nϵ$. Consequently, $x^i={x_{kl}^i}_{k,l=0}^{\infty }$ is a Cauchy sequence in $\mathbb{C}$ for every $k,l\mathbb{\in }\mathbb{N}$. As $\mathbb{C}$ is complete, $x^i=x_{kl}^i$ converges, i.e.,$$\begin{array}{}\text{(24)}& \vartheta -\underset{i⟶\infty }{\lim }{x}_{kl}^i=x_{kl},\end{array}$$

  where $x_{kl}\mathbb{\in }\mathbb{C}$ for all natural numbers $k,l$. Let $q,p\mathbb{\in }\mathbb{N}$ be given. Then, we obtain by (20) and (21) that$$\begin{array}{}\text{(25)}& {\displaystyle \sum }_{k,l=1}^{q,p}kl\mathrm{\Delta }{x}_{kl}^i-x_{kl}^j\le {x^i-x^j}_{{\mathcal{H}}_{\vartheta }}<\frac{ϵ}{2},\end{array}$$

  for all $i,j\mathbb{\in }\mathbb{N}$. Thus, the following inequality is obtained by letting the limit on (25) as $j⟶\infty$ and by (24):$$\begin{array}{}\text{(26)}& {\displaystyle \sum }_{k,l=1}^{q,p}kl\mathrm{\Delta }{x}_{kl}^i-x_{kl}\le \frac{ϵ}{2}.\end{array}$$

  Since $q,p\mathbb{\in }\mathbb{N}$ were arbitrary, we can write the following:$$\begin{array}{}\text{(27)}& {x^i-x}_{{\mathcal{H}}_{\vartheta }}<ϵ,\end{array}$$

  for all $i\ge Nϵ$.

  To show that $x\in {\mathcal{C}}_{\vartheta 0}$, we obtain the following inequality by using (24) and by letting $\vartheta -$ limit as $k,l⟶\infty$ for every $i\ge Nϵ$, that is:$$\begin{array}{}\text{(28)}& 0\le x_{kl}\le x_{kl}-x_{kl}^i+x_{kl}^i⟶0.\end{array}$$

  Therefore, $x\in {\mathcal{C}}_{\vartheta 0}$. By utilizing (27), it follows for every $n\ge Nϵ$ that$$\begin{array}{}\text{(29)}& x_{{\mathcal{H}}_{\vartheta }}\le {x-x^i}_{{\mathcal{H}}_{\vartheta }}+{x^i}_{{\mathcal{H}}_{\vartheta }}<\infty .\end{array}$$

  Thus, we conclude the proof by $x\in {\mathcal{H}}_{\vartheta }.$

ii. It closely resembles the proof of Theorem 1$\mathrm{i}$. Hence, we choose to omit it.

Theorem 2.

The space ${\mathcal{H}}_{\vartheta }$, where $\vartheta \in p,bp,r$, is an $AK\vartheta -$ space.

Proof.

Suppose that $\vartheta \in p,bp,r,$ and that $x\in {\mathcal{H}}_{\vartheta }$. Then, it follows that$$\begin{array}{c}\text{(30)}& {x-x^{m-1,n-1}}_{{\mathcal{H}}_{\vartheta }}={\displaystyle \sum }_{k,l=m,1}^{\infty ,n-1}kl\mathrm{\Delta }{x}_{kl}+{\displaystyle \sum }_{k,l=1,n}^{m-1,\infty }kl\mathrm{\Delta }{x}_{kl}+{\displaystyle \sum }_{k,l=m,n}^{\infty }kl\mathrm{\Delta }{x}_{kl}={\displaystyle \sum }_{k=m}^{\infty }{\displaystyle \sum }_{l=1}^{n-1}kl\mathrm{\Delta }{x}_{kl}+{\displaystyle \sum }_{l=n}^{\infty }{\displaystyle \sum }_{k=1}^{m-1}kl\mathrm{\Delta }{x}_{kl}+{\displaystyle \sum }_{k,l=m,n}^{\infty }kl\mathrm{\Delta }{x}_{kl},\end{array}$$for all $m,n\mathbb{\in }\mathbb{N}$. Then,$$\begin{array}{}\text{(31)}& \vartheta -\underset{m,n⟶\infty }{\lim }{\displaystyle \sum }_{k,l=m,n}^{\infty }kl\mathrm{\Delta }{x}_{kl}=0,\end{array}$$holds. Let us define the sequences $u=u_{ml}$ and $v=v_{kn}$ as$$\begin{array}{c}\text{(32)}& u_{ml}={\displaystyle \sum }_{k=1}^{m-1}kl\mathrm{\Delta }{x}_{kl},v_{kn}={\displaystyle \sum }_{l=1}^{n-1}kl\mathrm{\Delta }{x}_{kl},\end{array}$$for all $m,n,k,l\mathbb{\in }\mathbb{N}$, respectively. Then, one obtains$$\begin{array}{}\text{(33)}& {\displaystyle \sum }_{l=n}^{\infty }{\displaystyle \sum }_{k=1}^{m-1}kl\mathrm{\Delta }{x}_{kl}={\displaystyle \sum }_{l=n}^{\infty }{u}_{ml},\text{(34)}& {\displaystyle \sum }_{k=m}^{\infty }{\displaystyle \sum }_{l=1}^{n-1}kl\mathrm{\Delta }{x}_{kl}={\displaystyle \sum }_{k=m}^{\infty }{v}_{kn}.\end{array}$$

Since the series ${\sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}$ converges, then by (33) and (34), the series ${\sum }_{l=n}^{\infty }{u}_{ml}$ and ${\sum }_{k=m}^{\infty }{v}_{kn}$ also converge. Therefore, the general terms of the series approach to zero as $m,l⟶\infty$ and $k,n⟶\infty$, respectively. Hence, we conclude the proof by using (31),$$\begin{array}{}\text{(35)}& \vartheta -\underset{m,n⟶\infty }{\lim }{x-x^{m-1,n-1}}_{{\mathcal{H}}_{\vartheta }}=0,\end{array}$$as desired.

Theorem 3.

Each of the given assertions is strictly true.

i. ${\mathcal{H}}_{\vartheta }\subset \mathcal{B}\mathcal{V}$

ii. ${\mathcal{H}}_{\vartheta }\subset \mathcal{B}{\mathcal{V}}_{\vartheta 0}$

Proof.

i. Assume that $x=x_{kl}$ is in the space ${\mathcal{H}}_{\vartheta }$. Then, the sequence $x=x_{kl}$ will be in the space $\mathcal{B}\mathcal{V}$ since the following inequality:$$\begin{array}{}\text{(36)}& {\displaystyle \sum }_{k,l=1}^{\infty }\mathrm{\Delta }{x}_{kl}\le {\displaystyle \sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}<\infty ,\end{array}$$

  holds for every $k,l=1,2,3,\dots$. Therefore, the inclusion ${\mathcal{H}}_{\vartheta }\subset \mathcal{B}\mathcal{V}$ holds. Now, we must show that the set $\mathcal{B}\mathcal{V}\backslash {\mathcal{H}}_{\vartheta }$ is not empty. Consider $x=e$, then it is clear that $x\in \mathcal{B}\mathcal{V}\backslash {\mathcal{H}}_{\vartheta }$.

ii. First, we show that the inclusion ${\mathcal{H}}_{\vartheta }\subset \mathcal{B}{\mathcal{V}}_{\vartheta 0}$ holds. To do that, let us take a sequence $x=x_{kl}\in {\mathcal{H}}_{\vartheta }$, that is, $x\in {\mathcal{C}}_{\vartheta 0}$ and ${\sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}<\infty$. Since we can write inequality (36), then we can clearly see that $x\in \mathcal{B}\mathcal{V}$. Thus, $x\in \mathcal{B}{\mathcal{V}}_{\vartheta 0}$.

Now, to prove that the inclusion ${\mathcal{H}}_{\vartheta }\subset \mathcal{B}{\mathcal{V}}_{\vartheta 0}$ strictly holds, we must show that the set $\mathcal{B}{\mathcal{V}}_{\vartheta 0}\backslash {\mathcal{H}}_{\vartheta }$ is not empty. Consider $x=x_{kl}$ as given in the following expression:$$\begin{array}{}\text{(37)}& x_{kl}=\begin{array}{cc}\frac{1}{l},& k=1,\\ 0,& k\ge 2.\end{array}l=1,2,\dots .\end{array}$$

Clearly $x\in {\mathcal{C}}_{\vartheta 0}$. Therefore, the $\mathrm{\Delta }$ transform of the sequence $x$ is calculated as$$\begin{array}{}\text{(38)}& \mathrm{\Delta }{x}_{kl}=\begin{array}{cc}\frac{1}{ll+1},& k=1,\\ 0,& k\ge 2.\end{array}l=1,2,\dots .\end{array}$$

Hence, one obtains$$\begin{array}{}\text{(39)}& {\displaystyle \sum }_{k,l=1}^{\infty }\mathrm{\Delta }{x}_{kl}={\displaystyle \sum }_{l=1}^{\infty }\frac{1}{ll+1}=1<\infty .\end{array}$$

However, it is apparent that$$\begin{array}{}\text{(40)}& {\displaystyle \sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}={\displaystyle \sum }_{l=1}^{\infty }l\frac{1}{ll+1}={\displaystyle \sum }_{l=1}^{\infty }\frac{1}{l+1}=\infty .\end{array}$$

Therefore, $x\in \mathcal{B}\mathcal{V}$, but $x\notin {\mathcal{H}}_{\vartheta }$. It completes the proof.

Theorem 4.

Neither the space ${\mathcal{H}}_{\vartheta }$ nor the space ${\mathcal{L}}_u$ include each other.

Proof.

We provide two counterexamples to establish the theorem.

i. Choose a sequence $x=x_{kl},$ defined for all $k,l\mathbb{\in }\mathbb{N}$ as follows:$$\begin{array}{}\text{(41)}& x_{kl}=\begin{array}{cc}\frac{1}{kl},& k=l,\\ 0,& \text{otherwise}.\end{array}\end{array}$$

  It is clear that $x\in {\mathcal{C}}_{\vartheta 0}$. Moreover, we obtain that$$\begin{array}{}\text{(42)}& \mathrm{\Delta }{x}_{kl}=\begin{array}{cc}\frac{3kl+k+l+1}{klk+1l+1},& k=1,\forall l\mathbb{\in }\mathbb{N},\\ 2\frac{2kl+k+l+1}{klk+1l+1},& k\ge 2,\forall l\mathbb{\in }\mathbb{N},\end{array}\end{array}$$

  for all $k,l\mathbb{\in }\mathbb{N}$. Therefore,$$\begin{array}{}\text{(43)}& {\displaystyle \sum }_{k,l=1}^{\infty }{x}_{kl}={\displaystyle \sum }_{k=l=1}^{\infty }\frac{1}{kl}={\displaystyle \sum }_{r=1}^{\infty }\frac{1}{r^2}=\frac{{\pi }^2}{6}<\infty .\end{array}$$

  However,$$\begin{array}{}\text{(44)}& {\displaystyle \sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}={\displaystyle \sum }_{l=1}^{\infty }l\frac{2l+1}{ll+1}+{\displaystyle \sum }_{k=2}^{\infty }2k\frac{3k+2}{2kk+1}+{\displaystyle \sum }_{k=2}^{\infty }{\displaystyle \sum }_{l=2}^{\infty }2kl\frac{2kl+k+l+1}{klk+1l+1}=\infty .\end{array}$$

  Thus, the set ${\mathcal{L}}_u\backslash {\mathcal{H}}_{\vartheta }$ is nonempty.

ii. Take $x=x_{kl}$ defined by the following expression:$$\begin{array}{}\text{(45)}& x_{kl}=\begin{array}{cc}1,& k=1,\forall l\mathbb{\in }\mathbb{N},\\ 0,& \text{otherwise},\end{array}\end{array}$$

for all $k,l\mathbb{\in }\mathbb{N}$. Clearly, $x\in {\mathcal{C}}_{\vartheta 0}$. Moreover, we obtain that$$\begin{array}{}\text{(46)}& {\displaystyle \sum }_{k,l=1}^{\infty }{x}_{kl}={\displaystyle \sum }_{l=1}^{\infty }{x}_{1l}={\displaystyle \sum }_{k,l=1}^{\infty }1=\infty .\end{array}$$Thus, $x\notin {\mathcal{L}}_u$. However,$$\begin{array}{}\text{(47)}& {\displaystyle \sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}={\displaystyle \sum }_{l=1}^{\infty }l{x}_{1l}-x_{2l}-x_{1,l+1}+x_{2,l+1}={\displaystyle \sum }_{l=1}^{\infty }l{x}_{1l}-x_{1,l+1}=0<\infty ,\end{array}$$which shows $x\in {\mathcal{H}}_{\vartheta }$. Therefore, the set ${\mathcal{H}}_{\vartheta }\setminus {\mathcal{L}}_u$ is not empty, too.

The following corollary is immediate from the relations:$$\begin{array}{}\text{(48)}& {\displaystyle \sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}\le 4{\displaystyle \sum }_{k,l=1}^{\infty }kl{x}_{kl},\end{array}$$and$$\begin{array}{}\text{(49)}& x_{kl}=\frac{1}{{kl}^2}\in {\mathcal{H}}_{\vartheta }\backslash \int {\mathcal{L}}_u.\end{array}$$

Corollary 1.

The inclusion $\int {\mathcal{L}}_u\subset {\mathcal{H}}_{\vartheta }$ strictly holds.

3. The Dual Spaces of ${\mathcal{H}}_{\vartheta }$

Within this section, we undertake the computation of the algebraic dual, as well as $\alpha -$, $\beta bp-$, and $\gamma -$ dual of the space ${\mathcal{H}}_{\vartheta }$. Before starting the section’s main content, we perform some elementary calculations and establish the following lemmas, which are the keys in obtaining our main results.

Let us take $kl\mathrm{\Delta }{x}_{kl}=y_{kl}$, where $y\in {\mathcal{L}}_u$, $x\in {\mathcal{H}}_{\vartheta }$, and $k,l\mathbb{\in }\mathbb{N}$. Then, we write$$\begin{array}{c}\text{(50)}& \frac{1}{kl}{y}_{kl}=\mathrm{\Delta }{x}_{kl},\frac{1}{k+1l}{y}_{k+1,l}=\mathrm{\Delta }{x}_{k+1,l},\\ \frac{1}{kl+1}{y}_{k,l+1}=\mathrm{\Delta }{x}_{k,l+1},\\ \frac{1}{k+1l+1}{y}_{k+1,l+1}=\mathrm{\Delta }{x}_{k+1,l+1},\\ ⋮\\ \frac{1}{mn}{y}_{mn}=\mathrm{\Delta }{x}_{mn}.\end{array}$$

Adding both sides of the above equations ranging from $k,l$ to $m,n$, we obtain$$\begin{array}{}\text{(51)}& {\displaystyle \sum }_{i,j=k,l}^{m,n}\frac{y_{ij}}{ij}=x_{kl}-x_{m+1,l}-x_{k,n+1}+x_{m+1,n+1}.\end{array}$$Thus, by letting $\vartheta -$ limit over the equality in (51) as $m,n⟶\infty$, we get$$\begin{array}{}\text{(52)}& \vartheta -\underset{m,n⟶\infty }{\lim }{\displaystyle \sum }_{i,j=k,l}^{m,n}\frac{y_{ij}}{ij}={\displaystyle \sum }_{i,j=k,l}^{\infty }\frac{y_{ij}}{ij}=x_{kl},\end{array}$$since $\vartheta -{\lim }_{m,n⟶\infty }{x}_{mn}=0$.

Now, we write the following lemma with the specific case for $s=1$ from the results ([11], Theorem 4.1.(i), Theorem 4.2., and Theorem 4.3.(i)), respectively, as:

Lemma 1.

Each of the following assertions holds:

i. A 4D matrix $A\in {\mathcal{L}}_u,{\mathcal{M}}_u$ iff$$\begin{array}{}\text{(53)}& \underset{m,n,k,l}{\sup }{a}_{mnkl}<\infty .\end{array}$$

ii. A $4\mathrm{D}$ matrix $A\in {\mathcal{L}}_u,{\mathcal{L}}_u$ iff$$\begin{array}{}\text{(54)}& \underset{k,l\mathbb{\in }\mathbb{N}}{\sup }{\displaystyle \sum }_{m,n}{a}_{mnkl}<\infty .\end{array}$$

iii. A $4\mathrm{D}$ matrix $A\in {\mathcal{L}}_u,{\mathcal{C}}_{bp}$ iff the condition in (53) holds, and there exists $a_{kl}\mathbb{\in }\mathbb{C}$ satisfying$$\begin{array}{}\text{(55)}& bp-\underset{m,n⟶\infty }{\lim }{a}_{mnkl}=a_{kl}.\end{array}$$

Theorem 5.

Assume $\vartheta$ to be any one of the space $bp$, $p$, or $r$. Then, ${{\mathcal{H}}_{\vartheta }}^{\alpha }=h_1$, where$$\begin{array}{}\text{(56)}& h_1=a=a_{mn}\in \mathrm{\Omega }:\underset{k,l\mathbb{\in }\mathbb{N}}{\sup }\frac{1}{kl}{\displaystyle \sum }_{m,n=0}^{\infty }{a}_{mn}<\infty .\end{array}$$

Proof.

Let $a=a_{mn}\in \mathrm{\Omega }$ be given a double sequence and let us consider the following equality:$$\begin{array}{}\text{(57)}& a_{mn}{x}_{mn}={\displaystyle \sum }_{k,l=m,n}^{\infty }\frac{a_{mn}}{kl}{y}_{kl}={Dy}_{mn},\text{for\hspace{0.17em}every\hspace{0.17em}}m,n\mathbb{\in }\mathbb{N},\end{array}$$where $D=d_{mnkl}$ is defined by$$\begin{array}{}\text{(58)}& d_{mnkl}≔\begin{array}{cc}\frac{a_{mn}}{kl},& k\ge m,l\ge n,\\ 0,& \text{otherwise},\end{array}\end{array}$$for every natural number $m,n,k,l$. It follows from the equality in (57) with Lemma 1(ii) that $ax=a_{mn}{x}_{mn}\in {\mathcal{L}}_u$ whenever $x=x_{kl}\in {\mathcal{H}}_{\vartheta }$ if $Dy\in {\mathcal{L}}_u$ whenever $y=y_{kl}\in {\mathcal{L}}_u$. This shows that $a\in {{\mathcal{H}}_{\vartheta }}^{\alpha }$ whenever $x=x_{kl}\in {\mathcal{H}}_{\vartheta }$ if $D\in {\mathcal{L}}_u,{\mathcal{L}}_u$. That is to say that the condition stated in (54) is satisfied with $d_{mnkl}$ replacing $a_{mnkl}$. This proves that ${{\mathcal{H}}_{\vartheta }}^{\alpha }=h_1$.

Now, we give the following lemma for double sequences (cf. Wilansky [[12], Theorem 7.2.9] for single sequences).

Lemma 2.

Let $E$ be a $BDK-$ space with $AK\vartheta$, and $E^{\mathrm{\ast }}$ be the space of linear continuous functionals on $E$. Then, $E^{\mathrm{\ast }}$ can be identified with $E^{\beta \vartheta }$ by the isomorphism$$\begin{array}{c}\text{(59)}& T:E^{\mathrm{\ast }}⟶E^{\beta \vartheta },f⟶Tf={f{e}^{kl}}_{k,l=1}^{\infty }.\end{array}$$

Since $E$ has $AK\vartheta$ if every $f\in E^{\mathrm{\ast }}$ can be written in the form $fa={\sum }_{k,l=1}^{\infty }{a}_{kl}{b}_{kl}$, where $a=a_{kl}\in E$ and $b=b_{kl}\in E^{\beta \vartheta }$.

Proof.

First, we show that $T$ is a well-defined operator. Suppose $f\in E^{\mathrm{\ast }}$, we must show that $Tf\in E^{\beta \vartheta }$, that is, the sequence $Tf$ is $\beta \vartheta$-convergent. Since $E$ is a $BDK$ space with $AK\vartheta$, we can say that for each $f\in E^{\mathrm{\ast }}$, there exists a sequence $a\in E$ in which$$\begin{array}{}\text{(60)}& fx={\displaystyle \sum }_{k,l=1}^{\infty }{a}_{kl}{b}_{kl},\end{array}$$for some sequences $b\in E^{\beta \vartheta }$. Therefore,$$\begin{array}{}\text{(61)}& f{e}^{kl}={\displaystyle \sum }_{k,l=1}^{\infty }{a}_{kl}{b}_{kl}.\end{array}$$

Now, let us show that $Tf$ is $\vartheta$-convergent. This is seen from the fact that each component of $f{e}^{kl}$ is a convergent sequence in $E^{\beta \vartheta }$. Thus, $Tf\in E^{\beta \vartheta }$.

Moreover, the linearity of $T$ is clear. To show that $T$ is an isomorphism, we must show that it is bijective. Suppose $Tf=Tg$ for some $f,g\in E^{\mathrm{\ast }}$. Then, for any $x\in E$, $Tfx=Tgx$, which implies that $fx=gx$. Since $x$ is arbitrary, therefore, $T$ is injective. Now, let $h\in E^{\beta \vartheta }$. Since $E$ has $AK\vartheta$, for any $h\in E^{\beta \vartheta }$, there exists $f\in E^{\mathrm{\ast }}$ in which $Tf=h$. We have this because $E$ has $AK\vartheta$ if every $f\in E^{\mathrm{\ast }}$ can be written as $fa={\sum }_{k,l=1}^{\infty }{a}_{kl}{b}_{kl}$, where $a\in E$ and $b\in E^{\beta \vartheta }$. Therefore, $T$ is surjective, too. Hence, it is now clearly seen that $T:E^{\mathrm{\ast }}⟶E^{\beta \vartheta }$ is a well-defined linear isomorphism.

Theorem 6.

Let $\vartheta =bp,r$. The continuous dual ${\mathcal{H}}_{\vartheta }^{\mathrm{\ast }}$ and $\beta \vartheta -$ dual ${{\mathcal{H}}_{\vartheta }}^{\beta \vartheta }$ of the space ${\mathcal{H}}_{\vartheta }$ are norm isomorphic.

Proof.

Suppose that $\vartheta =bp,r$. It is established that ${\mathcal{H}}_{\vartheta }$ is a $BDK-$ space and has $AK\vartheta$ by Theorems 1 and 2, respectively. Moreover, the spaces ${\mathcal{H}}_{\vartheta }^{\mathrm{\ast }}$ and ${\mathcal{H}}_{\vartheta }^{\beta \vartheta }$ are isomorphic by Remark 3.3. Therefore, it is sufficient to prove the fact that$$\begin{array}{}\text{(62)}& f=a_{{\mathcal{P}}_{\infty }},\end{array}$$for all $f\in {\mathcal{H}}_{\vartheta }^{\mathrm{\ast }}$ represented by the sequence $a\in {\mathcal{P}}_{\infty }$ in which$$\begin{array}{}\text{(63)}& fx={\displaystyle \sum }_{k,l=1}^{\infty }{a}_{kl}{x}_{kl},\end{array}$$for all $x\in {\mathcal{H}}_{\vartheta }.$

Suppose that $f\in {\mathcal{H}}_{\vartheta }^{\mathrm{\ast }}$ and that $a=a_{kl}\in {\mathcal{P}}_{\infty }$ in which $fx={\sum }_{k,l=1}^{\infty }{a}_{kl}{x}_{kl}\text{\hspace{0.17em}for\hspace{0.17em}all\hspace{0.17em}}x\in {\mathcal{H}}_{\vartheta }$. Moreover, let $m,n\mathbb{\in }\mathbb{N}$ and $x\in {\mathcal{H}}_{\vartheta }.$ Therefore, by utilizing Abel’s double summability by parts, it follows that$$\begin{array}{c}\text{(64)}& f{x}^{m,n}={\displaystyle \sum }_{k,l=1}^{m,n}{a}_{kl}{x}_{kl}={\displaystyle \sum }_{k,l=1}^{m-1,n-1}{\mathrm{\Delta }}_{11}^{kl}{x}_{kl}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{ij}+{\displaystyle \sum }_{l=1}^{n-1}{\mathrm{\Delta }}_{01}^{kl}{x}_{ml}{\displaystyle \sum }_{i,j=1}^{m,l}{a}_{ij}+{\displaystyle \sum }_{k=1}^{m-1}{\mathrm{\Delta }}_{10}^{kl}{x}_{kn}{\displaystyle \sum }_{i,j=1}^{k,n}{a}_{ij}+x_{mn}{\displaystyle \sum }_{i,j=1}^{m,n}{a}_{ij},={\displaystyle \sum }_{k,l=1}^{m-1,n-1}kl{\mathrm{\Delta }}_{11}^{kl}{x}_{kl}\frac{1}{kl}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{ij}+{\displaystyle \sum }_{l=1}^{n-1}ml{\mathrm{\Delta }}_{01}^{kl}{x}_{ml}\frac{1}{ml}{\displaystyle \sum }_{i,j=1}^{m,l}{a}_{ij}+{\displaystyle \sum }_{k=1}^{m-1}kn{\mathrm{\Delta }}_{10}^{kl}{x}_{kn}\frac{1}{kn}{\displaystyle \sum }_{i,j=1}^{k,n}{a}_{ij}+mn{x}_{mn}\frac{1}{mn}{\displaystyle \sum }_{i,j=1}^{m,n}{a}_{ij},\end{array}$$where$$\begin{array}{c}\text{(65)}& {\mathrm{\Delta }}_{10}^{kl}{x}_{kl}=x_{kl}-x_{k+1,l},{\mathrm{\Delta }}_{01}^{kl}{x}_{kl}=x_{kl}-x_{k,l+1},\\ {\mathrm{\Delta }}_{11}^{kl}{x}_{kl}={\mathrm{\Delta }}_{10}^{kl}{\mathrm{\Delta }}_{01}^{kl}{x}_{kl}={\mathrm{\Delta }}_{01}^{kl}{\mathrm{\Delta }}_{10}^{kl}{x}_{kl}=\mathrm{\Delta }{x}_{kl}.\end{array}$$

It is known by Theorem 2 that the space ${\mathcal{H}}_{\vartheta }$ has $AK\vartheta$, and given $x\in {\mathcal{H}}_{\vartheta }$, we obtain that$$\begin{array}{c}\text{(66)}& 0\le mn{x}_{mn}={\displaystyle \sum }_{k,l=m,n}^{\infty }kl\mathrm{\Delta }{x}_{kl}\le {\displaystyle \sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}^{m,n}-x_{kl}+{\displaystyle \sum }_{k,l=m,n}^{\infty }kl\mathrm{\Delta }{x}_{kl}={x-x^{m,n}}_{{\mathcal{H}}_{\vartheta }}+{\displaystyle \sum }_{k,l=m,n}^{\infty }kl\mathrm{\Delta }{x}_{kl}⟶0asm,n⟶\infty .\end{array}$$

Hence, by using the continuity of $f$, we get that$$\begin{array}{}\text{(67)}& fx=\underset{m,n⟶\infty }{\lim }f{x}^{m,n}\le {\displaystyle \sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}{a}_{{\mathcal{P}}_{\infty }}=x_{{\mathcal{H}}_{\vartheta }}{a}_{{\mathcal{P}}_{\infty }},\end{array}$$and hence$$\begin{array}{}\text{(68)}& f\le a_{{\mathcal{P}}_{\infty }}.\end{array}$$Now, assume that $m,n\mathbb{\in }\mathbb{N}$, and that $x^{m,n}=e^{m,n}/mn$ which belongs to the space ${\mathcal{H}}_{\vartheta }$ and$$\begin{array}{}\text{(69)}& {x^{m,n}}_{{\mathcal{H}}_{\vartheta }}={\displaystyle \sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}^{m,n}=1.\end{array}$$

Therefore,$$\begin{array}{}\text{(70)}& {\displaystyle \sum }_{k,l=1}^{\infty }{a}_{kl}{x}_{kl}^{m,n}=\frac{1}{mn}{\displaystyle \sum }_{k,l=1}^{m,n}{a}_{kl}\le f.\end{array}$$

Since $m,n\mathbb{\in }\mathbb{N}$ were arbitrary and (69) holds, we obtain that$$\begin{array}{}\text{(71)}& a_{{\mathcal{P}}_{\infty }}\le f.\end{array}$$

Hence, we get the equality $a_{{\mathcal{P}}_{\infty }}=f$ by the results (68) and (71). The proof is thus completed.

Theorem 7.

${{\mathcal{H}}_{\vartheta }}^{\beta bp}={\mathcal{P}}_{\infty }$.

Proof.

Suppose that $a=a_{kl}\in \mathrm{\Omega }$ and $x=x_{kl}\in {\mathcal{H}}_{\vartheta }$. Then, we have$$\begin{array}{}\text{(72)}& {\displaystyle \sum }_{k,l=1}^{m,n}{a}_{kl}{x}_{kl}={\displaystyle \sum }_{k,l=1}^{m,n}{a}_{kl}{\displaystyle \sum }_{i,j=k,l}^{m,n}\frac{y_{ij}}{ij}={\displaystyle \sum }_{k,l=1}^{m,n}\frac{1}{kl}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{ij}{y}_{kl}={By}_{mn},\end{array}$$for all positive integers $m,n$, where $B=b_{mnkl}$ is defined by$$\begin{array}{}\text{(73)}& b_{mnkl}≔\begin{array}{cc}{\displaystyle \sum }_{i,j=1}^{k,l}\frac{a_{ij}}{kl},& k\ge m,l\ge n,\\ 0,& \text{otherwise},\end{array}\end{array}$$for all natural numbers $m,n,k,l$. Then, by using (72) and Lemma 1(iii), we obtain that $ax\in \mathcal{C}{\mathcal{S}}_{bp}$ whenever $x=x_{kl}\in {\mathcal{H}}_{\vartheta }$ if $By\in {\mathcal{C}}_{bp}$ whenever $y\in {\mathcal{L}}_u$. Therefore, $a\in {{\mathcal{H}}_{\vartheta }}^{\beta bp}$ if $B\in {\mathcal{L}}_u,{\mathcal{C}}_{bp}.$ As a result, the conditions in (53) and (55) hold with $b_{mnkl}$ instead of $a_{mnkl}$, that is,$$\begin{array}{}\text{(74)}& \underset{m,n,k,l\mathbb{\in }\mathbb{N}}{\sup }\frac{1}{kl}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{ij}<\infty ,\text{(75)}& bp-\underset{k,l⟶\infty }{\lim }\frac{1}{kl}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{ij}\text{exists}.\end{array}$$

This shows that $a=a_{kl}\in {\mathcal{P}}_{\infty }$ and $a=a_{kl}\in {\mathcal{P}}_{bp}$. Moreover, it is easy to see by (11) that ${\mathcal{P}}_{bp}\subset {\mathcal{P}}_{\infty }$. Thus, we conclude that ${{\mathcal{H}}_{\vartheta }}^{\beta bp}={\mathcal{P}}_{\infty }$.

Theorem 8.

${{\mathcal{H}}_{\vartheta }}^{\gamma }={\mathcal{P}}_{\infty }$.

Proof.

Utilizing a comparable approach to that used in Theorem 7, the proof can be summarized as follows.

We have from the equality in (72) and Lemma 1(i) that $ax\in \mathcal{B}\mathcal{S}$ whenever $x=x_{kl}\in {\mathcal{H}}_{\vartheta }$ if $By\in {\mathcal{M}}_u$ whenever $y\in {\mathcal{L}}_u$. Therefore, $a\in {{\mathcal{H}}_{\vartheta }}^{\gamma }$ if $B\in {\mathcal{L}}_u,{\mathcal{M}}_u$ in which the condition stated in (53) is satisfied with $b_{mnkl}$ replacing $a_{mnkl}$. Therefore, the condition in (74) holds which shows that $a=a_{kl}\in {\mathcal{P}}_{\infty }$. Thus, it gives the result that ${{\mathcal{H}}_{\vartheta }}^{\gamma }={\mathcal{P}}_{\infty }$.

4. Matrix Mappings From and Into the Space ${\mathcal{H}}_{\vartheta }$

Within this section, we represent the matrix transformations that operate both from and into the sequence space ${\mathcal{H}}_{\vartheta }$. First, we characterize the matrix classes ${\mathcal{H}}_{\vartheta },\lambda$, where $\lambda ={\mathcal{M}}_u,{\mathcal{L}}_u,{\mathcal{C}}_{bp}$. Then, we define determining a set of ${\mathcal{H}}_{\vartheta }$ and then we characterize the matrix class ${\mathcal{H}}_{\vartheta },{\mathcal{H}}_{\vartheta }$. Moreover, we determine the essential conditions of the matrix classes ${\mathcal{H}}_{\vartheta },\lambda$, where $\lambda =\mathcal{B}\mathcal{V},\mathcal{B}{\mathcal{V}}_{\vartheta 0},\mathcal{C}{\mathcal{S}}_{\vartheta },\mathcal{C}{\mathcal{S}}_{\vartheta 0},\mathcal{B}\mathcal{S}$ and $\mu ,{\mathcal{H}}_{\vartheta }$, where $\mu ={\mathcal{L}}_u,{\mathcal{C}}_{\vartheta 0},{\mathcal{C}}_{\vartheta },{\mathcal{M}}_u$.

In the entirety of this text, we presume that $A=a_{mnkl}$ is a $4\mathrm{D}$ matrix over $\mathbb{C}$.

Theorem 9.

$A=a_{mnkl}\in {\mathcal{H}}_{\vartheta },\lambda$ iff$$\begin{array}{}\text{(76)}& A_{mn}\in {\mathcal{H}}_{\vartheta }^{\beta \vartheta },\text{(77)}& E\in {\mathcal{L}}_u,\lambda ,\end{array}$$where the $4\mathrm{D}$ matrix $E=e_{mnkl}$ is given by$$\begin{array}{}\text{(78)}& e_{mnkl}≔\begin{array}{cc}\frac{1}{kl}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{mnij},& k\ge m,l\ge n,\\ 0,& \text{otherwise}.\end{array}\end{array}$$

Proof.

Assume that $A=a_{mnkl}\in {\mathcal{H}}_{\vartheta },\lambda$. Consequently, $A_{mn}x$ exists for every $x=x_{kl}\in {\mathcal{H}}_{\vartheta }$ and contains in $\lambda$ for every positive integer $m$ and $n$. The necessity of the condition stated in (76) is clear by considering the entries of the sequence $x=x_{kl}$ as $x_{kl}=e^{kl}$ for all $k,l\mathbb{\in }\mathbb{N}$.

Assume that $x=x_{kl}\in {\mathcal{H}}_{\vartheta }.$ Then, we deduce the subsequent equation as$$\begin{array}{}\text{(79)}& {\displaystyle \sum }_{k,l=1}^{m,n}{a}_{mnkl}{x}_{kl}={\displaystyle \sum }_{k,l=1}^{m,n}{a}_{mnkl}{\displaystyle \sum }_{i,j=k,l}^{m,n}\frac{y_{ij}}{ij}={\displaystyle \sum }_{k,l=1}^{m,n}{\displaystyle \sum }_{i,j=1}^{k,l}\frac{a_{mnij}}{kl}{y}_{kl}={Ey}_{mn},\end{array}$$where $E=e_{mnkl}$ is given as in (78). By letting the limit as $m,n⟶\infty$, we deduce the fact that $Ax=Ey$. In this case, we say that for every $x=x_{kl}\in {\mathcal{H}}_{\vartheta }$, $Ax\in \lambda$ exists for all natural numbers $m,n$ if for every $y=y_{kl}\in {\mathcal{L}}_u$, $Ey\in \lambda$ exists for all $m,n\mathbb{\in }\mathbb{N}.$ That is to say that $A=a_{mnkl}\in {\mathcal{H}}_{\vartheta },\lambda$ if $E\in {\mathcal{L}}_u,\lambda$, which indicates the requirement for (77).

Conversely, we assume that the conditions in (76) and (77) are satisfied and the relation (78) between the 4D infinite matrices $A=a_{mnkl}$ and $E=e_{mnkl}$ is given. Since the condition (76) holds, $Ax$ exists for every $x\in {\mathcal{H}}_{\vartheta }$. Moreover, the equation given in (79) and the condition in (77) show the existence of $Ax\in \lambda$. Therefore, $A=a_{mnkl}\in {\mathcal{H}}_{\vartheta },\lambda$.

Corollary 2.

Each of the following statements are valid:

i. $A=a_{mnkl}\in {\mathcal{H}}_{\vartheta },{\mathcal{M}}_u$ iff (76) holds, and the condition stated in (53) is satisfied with $e_{mnkl}$ replacing $a_{mnkl}$.

ii. $A=a_{mnkl}\in {\mathcal{H}}_{\vartheta },{\mathcal{L}}_u$ iff (76) holds, and the condition stated in (54) is satisfied with $e_{mnkl}$ replacing $a_{mnkl}$.

iii. $A=a_{mnkl}\in {\mathcal{H}}_{\vartheta },{\mathcal{C}}_{bp}$ iff (76) holds, and the conditions stated in (53) and (55) are satisfied with $e_{mnkl}$ replacing $a_{mnkl}$.

Definition 1.

See ([12], Definition 7.4.2) Let $X$ be a $BK$ space. A subset $E$ of the set $\varphi$ is called a determining set for $X$ if $DX={\overline{B}}_X\cap \varphi$ is the absolutely convex hull of $E$.

Theorem 10.

Consider the set $E$ consisting of all sequences in $\varphi$, each having nonzero elements given by$$\begin{array}{}\text{(80)}& \begin{array}{cccccc}0& 0& \dots & 0& \dots & 0\\ 0& 0& \dots & 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮& \ddots & ⋮\\ 0& 0& \dots & \frac{1}{mn}& \dots & 0\\ 0& 0& \dots & 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮& \ddots & ⋮\end{array},\end{array}$$where $1/mn$ occupies the ${m,n}^{th}-$ place and all other entries are zero. Then, it follows that $E$ is a determining set for ${\mathcal{H}}_{\vartheta }$.

Proof.

Let $D{\mathcal{H}}_{\vartheta }=x=x_{kl}\in \varphi :x_{{\mathcal{H}}_{\vartheta }}\le 1$, where $\varphi$ represents the space of all finite double sequences, and $KE$ is the absolutely convex hull of $E$. We must prove that $D{\mathcal{H}}_{\vartheta }=KE$ by showing that the inclusions$$\begin{array}{}\text{(81)}& D{\mathcal{H}}_{\vartheta }\subset KE,\text{(82)}& KE\subset D{\mathcal{H}}_{\vartheta },\end{array}$$hold. First, we show the inclusion (81). Suppose that $x\in D{\mathcal{H}}_{\vartheta }$. Since $x\in \varphi$, we obtain, for any positive inter $m$ and $n$ that$$\begin{array}{}\text{(83)}& x=x^{m,n}={\displaystyle \sum }_{k,l=1}^{m,n}{z}_{kl}{d}_{kl},\end{array}$$holds, where $d_{kl}=1/kl{e}^{kl}$ and $z_{kl}=kl\mathrm{\Delta }{x}_{kl}$ for any positive integer $k$ and $l$. This can inferred from the fact that $x_{m+1,l}=x_{k,n+1}=x_{m+1,n+1}=0$ for each $k,l\mathbb{\in }\mathbb{N}$ and$$\begin{array}{c}\text{(84)}& {\displaystyle \sum }_{k,l=1}^{m,n}{z}_{kl}{d}_{kl}={\displaystyle \sum }_{k,l=1}^{m,n}kl\mathrm{\Delta }{x}_{kl}{d}_{kl}={\displaystyle \sum }_{k,l=1}^{m,n}kl{x}_{kl}{d}_{kl}-{\displaystyle \sum }_{k,l=1}^{m-1,n}kl{x}_{k+1,l}{d}_{kl}-{\displaystyle \sum }_{k,l=1}^{m,n-1}kl{x}_{k,l+1}{d}_{kl}+{\displaystyle \sum }_{k,l=1}^{m-1,n-1}kl{x}_{k+1,l+1}{d}_{kl}\\ =x_{11}{d}_{11}+{\displaystyle \sum }_{l=2}^{n}l{x}_{1,l}{d}_{1,l}+{\displaystyle \sum }_{k=2}^{m}k{x}_{k,1}{d}_{k,1}+{\displaystyle \sum }_{k,l=2}^{m,n}kl{x}_{kl}{d}_{kl}-{\displaystyle \sum }_{k=2}^{m}k-1x_{k,1}{d}_{k-1,1}+{\displaystyle \sum }_{k,l=2}^{m,n}k-1l{x}_{kl}{d}_{k-1,l}\\ -{\displaystyle \sum }_{l=2}^{n}l-1x_{1,l}{d}_{1,l-1}+{\displaystyle \sum }_{k,l=2}^{m,n}kl-1x_{kl}{d}_{k,l-1}+{\displaystyle \sum }_{k,l=2}^{m,n}k-1l-1x_{kl}{d}_{k-1,l-1}\\ =x_{11}{e}^{11}+{\displaystyle \sum }_{k=2}^m{x}_{k,1}k{d}_{k,1}-k-1d_{k-1,1}+{\displaystyle \sum }_{l=2}^n{x}_{1,l}l{d}_{1,l}-l-1d_{1,l-1}\\ +{\displaystyle \sum }_{k,l=2}^{m,n}{x}_{kl}kl{d}_{kl}-k-1l{d}_{k-1,l}-kl-1d_{k,l-1}+k-1l-1d_{k-1,l-1}\\ =x_{11}{e}^{1,1}+{\displaystyle \sum }_{k=2}^m{x}_{k,1}{e}^{k,1}+{\displaystyle \sum }_{l=2}^n{x}_{1,l}{e}^{1,l}+{\displaystyle \sum }_{k,l=2}^{m,n}{x}_{kl}{e}^{kl}=x^{m,n}.\end{array}$$

Moreover, it is observed that$$\begin{array}{}\text{(85)}& {\displaystyle \sum }_{k,l=1}^{m,n}{z}_{kl}={\displaystyle \sum }_{k,l=1}^{m,n}kl\mathrm{\Delta }{x}_{kl}\le x_{{\mathcal{H}}_{\vartheta }}\le 1.\end{array}$$Therefore, $x\in KE$, which proves (81).

Now, suppose that $x\in KE$. Then, there are natural numbers $m,n$ such that$$\begin{array}{}\text{(86)}& x={\displaystyle \sum }_{k,l=1}^{m,n}{z}_{kl}{d}_{kl}\text{with}{\displaystyle \sum }_{k,l=1}^{m,n}{z}_{kl}\le 1.\end{array}$$

Since $x_{m+1,l}=x_{k,n+1}=x_{m+1,n+1}=0$ for each $k,l\mathbb{\in }\mathbb{N}$, that is, $x_{kl}=0$ for all $k\ge m+1$ or $l\ge n+1$ or both, by using the relation in (51), and for all $1\le k\le m$ and $1\le l\le n$, we obtain that$$\begin{array}{}\text{(87)}& {\displaystyle \sum }_{i,j=k,l}^{m,n}\frac{z_{ij}}{ij}={\displaystyle \sum }_{i,j=k,l}^{m,n}\mathrm{\Delta }{x}_{ij}=x_{kl}-x_{m+1,l}-x_{k,n+1}+x_{m+1,n+1}=x_{kl}.\end{array}$$Furthermore, since $x_{m+1,l}=x_{k,n+1}=x_{m+1,n+1}=0$ for each $k,l\mathbb{\in }\mathbb{N},$ we have that$$\begin{array}{}\text{(88)}& x_{{\mathcal{H}}_{\vartheta }}={\displaystyle \sum }_{k,l=1}^{\infty }kl\mathrm{\Delta }{x}_{kl}={\displaystyle \sum }_{k,l=1}^{m,n}{z}_{kl}\le 1.\end{array}$$

Hence, $x\in D{\mathcal{H}}_{\vartheta }$. Thus, the inclusion (82) is established. The result immediately follows from the inclusions (81) and (82).

Now, we give the following lemma for double sequence spaces concerning the notation in ([12], Theorem 8.3.4) as follows.

Lemma 3.

Let $X$ be a $BDK$ space with $AK\vartheta$, $E$ be a determining set for $X$, $Y$ be an $FDK$ space, and $A=a_{mnkl}$ be a 4D matrix. Suppose that either $X$ has $AK\vartheta$ or $A$ is row finite. Then, $A\in X,Y$ iff

i. The columns of $A$ belong to $Y$, that is, $A^{kl}={a_{mnkl}}_{m,n=1}^{\infty }\in Y$ for all $k,l\mathbb{\in }\mathbb{N}$.

ii. $LE$ is a bounded subset of $Y$, where $Lx=Ax$ for all $x\in E$.

Proof.

Assume that $A\in X,Y$. Then, $LE$ is bounded in $Y$, that is, ${Lx}_Y\le M{x}_X$ for all $x\in E$ and some positive constant $M$. For each fixed $k,l\mathbb{\in }\mathbb{N}$, let us consider the $k{l}^{th}$ column of $A$, which is denoted by $A^{kl}$. We need to show that $A^{kl}\in Y$. Now, for each $k,l,m,n\mathbb{\in }\mathbb{N}$, we have$$\begin{array}{}\text{(89)}& A^{kl}={\displaystyle \sum }_{m,n=1}^{\infty }{a}_{mnkl}{e}^{mn}.\end{array}$$Now, if we consider the $m{n}^{th}$ term in the series defined by $A^{kl}$, we will have $a_{mnkl}$. Then, by the definition of $\vartheta$-convergence,$$\begin{array}{}\text{(90)}& A^{kl}=\vartheta -\underset{m,n⟶\infty }{\lim }{A}^{kl}m,n,\end{array}$$where$$\begin{array}{}\text{(91)}& A^{kl}m,n={\displaystyle \sum }_{i,j=1}^{m,n}{a}_{ijkl}{e}^{ij}.\end{array}$$

Since $A\in X,Y$ and $A^{kl}m,n=L{x}^{m,n}$, where $x^{m,n}$ is a sequence in $X$. Therefore, $A^{kl}m,n\in Y$ for all natural numbers $m$ and $n$, and by definition of $FDK$ space, the sequence $A^{kl}$ is convergent with respect to $\vartheta$ and hence belongs to $Y$. Thus, condition (i) is satisfied. Moreover, $A^{kl}m,n=L{x}^{m,n}$, and $LE$ is bounded in $Y$, and it follows that $A^{kl}$ is bounded for all natural numbers $k$ and $l.$ Hence, condition (ii) suffices.

Conversely, assume that the conditions stated in (i) and (ii) hold true. Then, we have the following facts:

i. Since each column of $A$ belongs to $Y$, by definition of $X,Y$, we have $A^{kl}\in Y$ for all $k,l$.

ii. Since $LE\subset Y$ is bounded, $\exists M>0$ satisfies ${Lx}_Y\le M{x}_X$ for all $x\in E$.

Now, let $x\in X$ be arbitrary. By definition of $BDK$ space, $x=\vartheta -{\lim }_{m,n⟶\infty }{x}^{m,n}$, where $x^{m,n}\in E$ for all natural numbers $m$ and $n$. Then, we have $$\begin{array}{}\text{(92)}& Lx=\vartheta -\underset{m,n⟶\infty }{\lim }L{x}^{m,n}=\vartheta -\underset{m,n⟶\infty }{\lim }{A}^{m,n}.\end{array}$$

Since each $A^{kl}$ is bounded, it shows that $Lx$ is also bounded, that is, $A\in X,Y$. This ends the proof.

Theorem 11.

$A=a_{mnkl}\in {\mathcal{H}}_{\vartheta },{\mathcal{H}}_{\vartheta }$ iff$$\begin{array}{}\text{(93)}& \vartheta -\underset{m,n⟶\infty }{\lim }{a}_{mnkl}=0\forall k,l\mathbb{\in }\mathbb{N},\text{(94)}& A_{{\mathcal{H}}_{\vartheta },{\mathcal{H}}_{\vartheta }}=\underset{k,l\mathbb{\in }\mathbb{N}}{\sup }\frac{1}{kl}{\displaystyle \sum }_{m,n=1}^{\infty }{\displaystyle \sum }_{i,j=1}^{k,l}{\mathrm{\Delta }}_{mn}^{11}{a}_{mnij},\end{array}$$where ${\mathrm{\Delta }}_{mn}^{11}{a}_{mnij}=a_{mnij}-a_{m+1,nij}-a_{m,n+1,ij}+a_{m+1,n+1,ij}$.

Proof.

Since ${\mathcal{H}}_{\vartheta }$ is a $BDK-$ space with $AK\vartheta$ by Theorems 1 and 2, and $E=1/kl{e}^{kl}:k,l\mathbb{\in }\mathbb{N}$ is a determining set of ${\mathcal{H}}_{\vartheta }$ by Theorem 10, we apply Lemma 3 with $X=Y={\mathcal{H}}_{\vartheta }$ which is a $BDK-$ space with $AK\vartheta$ with its norm defined by (17). Let $k,l\mathbb{\in }\mathbb{N}$ and $d^{kl}=1/kl{e}^{kl}$. Then,$$\begin{array}{}\text{(95)}& A_{mn}{d}^{kl}={\displaystyle \sum }_{i,j=1}^{\infty }{a}_{mnij}{d}_{ij}^{kl}=\frac{1}{kl}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{mnij}.\end{array}$$

Therefore, we have two results as follows:$$\begin{array}{}\text{(96)}& {A_{mn}{d}^{kl}}_{{\mathcal{H}}_{\vartheta }}<\infty \text{\hspace{0.17em}for\hspace{0.17em}all}{d}^{kl}\in E,\text{(97)}& A_{mn}{d}^{kl}\in {\mathcal{C}}_{\vartheta 0}\text{\hspace{0.17em}for\hspace{0.17em}all}{d}^{kl}\in E.\end{array}$$

Then, we obtain by (96) that$$\begin{array}{c}\text{(98)}& {A_{mn}{d}^{kl}}_{{\mathcal{H}}_{\vartheta }}={\displaystyle \sum }_{m,n=1}^{\infty }mn{\mathrm{\Delta }}_{mn}^{11}{A}_{mn}{d}^{kl}={\displaystyle \sum }_{m,n=1}^{\infty }mn{A}_{mn}{d}^{kl}-A_{m+1,n}{d}^{kl}-A_{m,n+1}{d}^{kl}+A_{m+1,n+1}{d}^{kl}\\ ={\displaystyle \sum }_{m,n=1}^{\infty }mn\frac{1}{kl}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{mnij}-\frac{1}{kl}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{m+1,nij}-\frac{1}{kl}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{m,n+1,ij}+\frac{1}{kl}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{m+1,n+1,ij}\\ =\frac{1}{kl}{\displaystyle \sum }_{m,n=1}^{\infty }mn{\displaystyle \sum }_{i,j=1}^{k,l}{\mathrm{\Delta }}_{mn}^{11}{a}_{mnij},\end{array}$$for all $k,l\mathbb{\in }\mathbb{N}$. Thus, it is clearly seen that the conditions in (94) and (96) are equivalent.

Moreover, the condition (97) shows clearly that the condition in (93) also holds. This ends the proof.

The given result is obtained by considering the determining set $E$ of the double sequence space ${\mathcal{H}}_{\vartheta }$ and by utilizing Lemma 2. We employ the norm ${A_{mn}{d}^{kl}}_Y$ of the space $Y$, where $Y=\mathcal{B}\mathcal{V},\mathcal{B}{\mathcal{V}}_{\vartheta 0},\mathcal{C}{\mathcal{S}}_{\vartheta },\mathcal{C}{\mathcal{S}}_{\vartheta 0},\mathcal{B}\mathcal{S}$, respectively.

Corollary 3.

Let $\vartheta =p,bp,r$. Then, each of the following statements is valid for $4\mathrm{D}$ infinite matrix $A=a_{mnkl}$.

i. $A\in {\mathcal{H}}_{\vartheta },\mathcal{B}\mathcal{V}$ iff$$\begin{array}{}\text{(99)}& \underset{k,l\mathbb{\in }\mathbb{N}}{\sup }\frac{1}{kl}{\displaystyle \sum }_{m,n=0}^{\infty }{\displaystyle \sum }_{i,j=1}^{k,l}{\mathrm{\Delta }}_{mn}^{11}{a}_{mnij}<\infty .\end{array}$$

ii. $A\in {\mathcal{H}}_{\vartheta },\mathcal{B}{\mathcal{V}}_{\vartheta 0}$ iff (93) and (99) hold.

iii. $A\in {\mathcal{H}}_{\vartheta },\mathcal{C}{\mathcal{S}}_{\vartheta }$ iff$$\begin{array}{}\text{(100)}& \underset{s,t,k,l\mathbb{\in }\mathbb{N}}{\sup }\frac{1}{kl}{\displaystyle \sum }_{m,n=0}^{s,t}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{mnij}<\infty ,\text{(101)}& \vartheta -\underset{s,t⟶\infty }{\lim }\frac{1}{kl}{\displaystyle \sum }_{m,n=0}^{s,t}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{mnij}\text{exists\hspace{0.17em}for\hspace{0.17em}all\hspace{0.17em}}k,l\mathbb{\in }\mathbb{N}.\end{array}$$

iv. $A\in {\mathcal{H}}_{\vartheta },\mathcal{C}{\mathcal{S}}_{\vartheta 0}$ iff (100) holds and$$\begin{array}{}\text{(102)}& \vartheta -\underset{s,t⟶\infty }{\lim }\frac{1}{kl}{\displaystyle \sum }_{m,n=0}^{s,t}{\displaystyle \sum }_{i,j=1}^{k,l}{a}_{mnij}=0,\text{\hspace{1em}for\hspace{0.17em}all\hspace{0.17em}}k,l\mathbb{\in }\mathbb{N}.\end{array}$$

v. $A\in {\mathcal{H}}_{\vartheta },\mathcal{B}\mathcal{S}$ iff (100) holds.

Theorem 12.

Let us establish the relationship between the terms of the $4\mathrm{D}$ matrices $A=a_{mnkl}$ and $F=f_{mnkl}$ as follows:$$\begin{array}{}\text{(103)}& f_{mnkl}={\displaystyle \sum }_{i,j=m-1,n-1}^{m,n}{-1}^{m+n-i+j}mn{a}_{ijkl}=mn{\mathrm{\Delta }}_{11}^{mn}{a}_{mnkl}.\end{array}$$Then, the 4D matrix $A=a_{mnkl}\in \lambda ,{\mathcal{H}}_{\vartheta }$ if $F\in \lambda ,{\mathcal{L}}_u,$ where $\lambda$ is any given double sequence space.

Proof.

First, we assume that $A\in \lambda ,{\mathcal{H}}_{\vartheta }$. Therefore, for every $x\in \lambda$, $Ax$ exists, and contains in ${\mathcal{H}}_{\vartheta }$. Then, considering the relation stated in (103), we deduce that$$\begin{array}{}\text{(104)}& {\displaystyle \sum }_{k,l=0}^{s,t}{f}_{mnkl}{x}_{kl}={\displaystyle \sum }_{k,l=0}^{s,t}{\displaystyle \sum }_{i,j=m-1,n-1}^{m,n}{-1}^{m+n-i+j}mn{a}_{ijkl}{x}_{kl}={\displaystyle \sum }_{i,j=m-1,n-1}^{m,n}{-1}^{m+n-i+j}mn{\displaystyle \sum }_{k,l=0}^{s,t}{a}_{ijkl}{x}_{kl},\end{array}$$for all $m,n,s,t\mathbb{\in }\mathbb{N}$. By letting $s,t⟶\infty$, we deduce that$$\begin{array}{}\text{(105)}& {Fx}_{mn}={\displaystyle \sum }_{k,l}{f}_{mnkl}{x}_{kl}={\displaystyle \sum }_{i,j=m-1,n-1}^{m,n}{-1}^{m+n-i+j}mn{\displaystyle \sum }_{k,l}{a}_{ijkl}{x}_{kl}=mn{\displaystyle \sum }_{i,j=m-1,n-1}^{m,n}{-1}^{m+n-i+j}{Ax}_{ij}=mn{\mathrm{\Delta }}_{11}^{mn}{Ax}_{mn},\end{array}$$for all natural numbers $m$ and $n$. Consequently, it can be observed that $Ax\in {\mathcal{H}}_{\vartheta }$ whenever $x\in \lambda$ if $Fx\in {\mathcal{L}}_u$ whenever $x\in \lambda$. Therefore, we conclude that $A\in \lambda ,{\mathcal{H}}_{\vartheta }$ if $F\in \lambda ,{\mathcal{L}}_u$.

Let $\vartheta =bp,bp0,r,r0$. Since the proof can be performed by taking the determining set $E$ of the double sequence space ${\mathcal{H}}_{\vartheta }$, Lemma 3, and Theorem 12 into consideration, and by using the norm ${A_{mn}{d}^{kl}}_{{\mathcal{H}}_{\vartheta }}$ and the $BDK-$ space $X$ with $AK\vartheta$, where $X={\mathcal{L}}_u,{\mathcal{C}}_{\vartheta 0},{\mathcal{C}}_{\vartheta },{\mathcal{M}}_u$, respectively, we give the following theorem without its proof.

Theorem 13.

Assume $\vartheta =p,bp,r$. Then, each of the following assertions is valid for 4D infinite matrix $A=a_{mnkl}$:

i. $A\in {\mathcal{L}}_u,{\mathcal{H}}_{\vartheta }$ iff (93) holds and$$\begin{array}{}\text{(106)}& \underset{k,l\mathbb{\in }\mathbb{N}}{\sup }{\displaystyle \sum }_{m,n=1}^{\infty }mn{\mathrm{\Delta }}_{11}^{mn}{a}_{mnkl}<\infty .\end{array}$$

ii. $A\in {\mathcal{C}}_{\vartheta 0},{\mathcal{H}}_{\vartheta }$ iff (93) holds and$$\begin{array}{}\text{(107)}& \underset{K\mathbb{\subset }\mathbb{N}finite}{\sup }{\displaystyle \sum }_{m,n=1}^{\infty }mn{\displaystyle \sum }_{k,l\in K}{\mathrm{\Delta }}_{11}^{mn}{a}_{mnkl}<\infty .\end{array}$$

iii. $A\in {\mathcal{C}}_{\vartheta },{\mathcal{H}}_{\vartheta }$ iff (93) and (107) hold, and$$\begin{array}{}\text{(108)}& \underset{k,l\mathbb{\in }\mathbb{N}}{\sup }{\displaystyle \sum }_{m,n=1}^{\infty }mn{\displaystyle \sum }_{i,j=1}^{k,l}{\mathrm{\Delta }}_{11}^{mn}{a}_{mnij}<\infty .\end{array}$$

iv. $A\in {\mathcal{M}}_u,{\mathcal{H}}_{\vartheta }$ iff$$\begin{array}{}\text{(109)}& \underset{K\mathbb{\subset }\mathbb{N}finite}{\sup }{\displaystyle \sum }_{k,l=1}^{\infty }{\displaystyle \sum }_{m,n\mathbb{\in }\mathbb{N}}mn{\mathrm{\Delta }}_{11}^{mn}{a}_{mnkl}<\infty ,\text{(110)}& \vartheta -\underset{m,n⟶\infty }{\lim }{\displaystyle \sum }_{k,l=1}^{\infty }{a}_{mnkl}=0.\end{array}$$

5. Conclusion

Most recently, Demiriz and Duyar [13], Çapan and Başar [14], and Tu$\breve{\mathrm{g}}$, Rako$`\mathrm{c}$ evi$`\mathrm{c}$, and Malkowsky [2] introduced and studied the new spaces ${\mathcal{M}}_u\mathrm{\Delta }$ and $\mathcal{B}{\mathcal{V}}_q$ and $\mathcal{B}{\mathcal{V}}_{\vartheta 0}$, where $\vartheta =p,bp,r$, respectively. Within this work, we introduced $\text{Hahn}$ double sequence space ${\mathcal{H}}_{\vartheta }$, where $\vartheta =p,bp,r$ and we studied various topological and algebraic aspects, and we proved several containment relations. Moreover, we computed the algebraic dual and $\alpha -,\beta bp-,\gamma -$ duals of the Hahn double sequence space ${\mathcal{H}}_{\vartheta }$. Finally, we characterized the classes ${\mathcal{H}}_{\vartheta },\lambda$, where $\lambda ={\mathcal{H}}_{\vartheta },\mathcal{B}\mathcal{V},\mathcal{B}{\mathcal{V}}_{\vartheta 0},\mathcal{C}{\mathcal{S}}_{\vartheta },\mathcal{C}{\mathcal{S}}_{\vartheta 0},,\mathcal{B}\mathcal{S}$ and $\mu ,{\mathcal{H}}_{\vartheta }$, where $\mu ={\mathcal{L}}_u,{\mathcal{C}}_{\vartheta 0},{\mathcal{C}}_{\vartheta },{\mathcal{M}}_u$.

As an extension of this research, the computation of $4\mathrm{D}$ matrix domains within the Hahn double sequence space ${\mathcal{H}}_{\vartheta }$, and characterization of some novel matrix classes from and into ${\mathcal{H}}_{\vartheta }$ remain open for further exploration, and some can follow the studies [13, 15–17] recently published on double sequence theory.

Disclosure

The earlier version of the paper was submitted to ArXiv as a preprint (see [18]).

Funding

No funds were applied for this research.

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