Matrix Transformations of Generalized Almost Convergent Double Sequences

Abstract

In this paper, we study matrix transformations on spaces of generalized almost convergent double sequences with powers. Extending classical results of Lorentz, Maddox, and Nanda, we characterize several classes of infinite matrices that map between Maddox’s double sequence spaces and spaces of almost convergent (to zero) double sequences with powers. Our results generalize earlier characterizations for single sequence spaces obtained by the authors in previous work, providing a structured framework for studying summability and convergence in higher dimensions.

1. Introduction

Matrix transformations of sequence spaces constitute a classical and well-developed topic in summability theory and functional analysis. Given an infinite matrix $A=\left(a_{nk}\right)$, such transformations describe linear operators acting on sequences via series of the form

$${\left(Ax\right)}_n=\sum _k{a}_{nk}{x}_k,$$

and their boundedness properties characterize how different modes of convergence and summability are preserved under linear mappings. Significant contributions in this direction include the works of Toeplitz, Lorentz, Maddox, and Nanda, among others [1,2,3,4,5].

In many applied and computational contexts, one naturally encounters two-dimensional (or, more generally, multidimensional) discrete data, such as images and signals sampled on a grid, finite-difference/finite-element discretizations of partial differential equations, and matrices of coefficients arising in bilinear transforms. In such settings the behavior of discrete operators is governed by their action on double sequences rather than on single sequences. This motivates a systematic study of the summability and stability properties of operators acting on double sequences, particularly matrix transformations between double sequence spaces with nonuniform (position-dependent) growth conditions.

In recent decades, this theory has been extended to generalized sequence spaces with variable exponents and to almost convergent sequences. However, the corresponding results for double sequences and, in particular, for matrix transformations acting between generalized almost convergent double sequence spaces, remain relatively limited. The present paper contributes to this line of research by providing complete characterizations of four-dimensional matrix transformations between such spaces.

The concept of almost convergence, introduced by Lorentz ([2]), has played an important role in summability theory and the study of sequence spaces. Unlike ordinary convergence, almost convergence captures limiting behavior through uniform Cesàro-type averages and allows meaningful limits for sequences that fail to converge in the classical sense. Over the years, this notion has been extended and refined in various directions, including generalized almost convergence and matrix transformations between sequence spaces.

Intuitively, the ‘almost limit’ of a double sequence is obtained by averaging over growing rectangular blocks in a way that is uniform with respect to shifts in the averaging window. Convergence of these two-dimensional Cesàro means expresses a robustness of the limit under local perturbations and translations of the indexing grid; thus, almost convergence detects a global averaged stability rather than pointwise behavior. In formulas, the means $t_{mnrs}\left(x\right)$ (defined below) serve exactly this averaging purpose: the requirement that $t_{mnrs}\left(x\right)$ converge uniformly in the shift indices $(r,s)$ formalizes the idea that the averaged value is insensitive to finite local translations.

Parallel to these developments, the study of double sequences has gained increasing attention due to their relevance in approximation theory, numerical analysis, summability methods, and multidimensional functional analysis. Double sequences naturally arise, for example, in discretizations of functions of several variables, in iterative schemes, and in the study of series and transforms indexed by multiple parameters. Consequently, extending convergence concepts from single sequences to double sequences is both natural and necessary.

Generalized almost convergence for double sequences provides a flexible framework that unifies several summability methods and allows for finer control over limiting processes. In particular, it enables the study of convergence behavior under multidimensional averaging procedures and matrix transformations that cannot be captured by classical convergence alone. Matrix transformations play a central role in this context, as they provide a powerful tool for describing how summability properties are preserved or altered under linear operators.

The aim of this paper is to study matrix transformations between spaces of generalized almost convergent double sequences and double sequence spaces with variable exponents. Extending earlier results for single sequences, we establish the necessary and sufficient conditions under which infinite matrices define bounded linear operators between these spaces. The obtained results generalize previous theorems by the authors to the two-dimensional setting.

To characterize matrix transformations involving almost convergent (to 0) sequences with powers in [6], the authors used matrix transformations involving convergent (to 0) sequences with powers mainly considered in [7]. To characterize matrix transformations involving almost convergent (to 0) double sequences with powers, we will use the mentioned results for single almost convergent (to 0) sequences with powers; matrix transformations involving convergent (to 0) double sequences with powers (see [8,9]), and the reduction method of matrix maps from a sequence space to a double sequence space to matrix maps between sequence spaces as described in [10].

First, we introduce the necessary notions and notation for the paper. We begin with notions and notation pertaining to sequence spaces.

Let $X,$ Y be two nonempty subsets of the space $\omega$ of all complex (or real) sequences, and let $A=\left(a_{nk}\right)$ be an infinite matrix of complex (or real) numbers. For every $x=\left(x_k\right)\in X$ and every $n\in \mathbb{N}$, we write

$${\left[Ax\right]}_n=\sum _k{a}_{nk}{x}_k.$$

We say that $A\in (X,Y)$ if and only if $Ax:={\left(\left[Ax\right]\right)}_n\in Y$ whenever $x\in X$. One can see that for a given $n\in \mathbb{N}$, for ${\left[Ax\right]}_n$ to exist for every $x\in X$, the sequence ${\left(a_{nk}\right)}_k$ must belong to

$$X^{\beta }=\left\{\left(y_k\right)\in \omega |\sum _k{x}_k{y}_k\mathrm{converges}\mathrm{for}\mathrm{every}x\in X\right\}.$$

Let $p=\left(p_k\right)$ be a sequence of strictly positive numbers. The following variable exponent spaces were defined by Maddox ([3]) and Nakano ([11]):

$$\begin{array}{cc}\hfill c_0\left(p\right)& =\{\left(x_k\right)\in \omega ||x_k{|}^{p_k}\to 0\},\hfill \\ \hfill c\left(p\right)& =\{\left(x_k\right)\in \omega ||x_k-l{|}^{p_k}\to 0\mathrm{for}\mathrm{some}l\},\hfill \\ \hfill \ell \left(p\right)& =\{\left(x_k\right)\in \omega |\sum _k|x_k{|}^{p_k}<\infty \},\hfill \\ \hfill {\ell }_{\infty }\left(p\right)& =\{\left(x_k\right)\in \omega |\underset{k}{sup}|x_k{|}^{p_k}<\infty \}.\hfill \end{array}$$

When all terms of $\left(p_k\right)$ are constant and equal to $p>0$, we have $\ell \left(p\right)={\ell }_p$, ${\ell }_{\infty }\left(p\right)={\ell }_{\infty }$, $c\left(p\right)=c$, and $c_0\left(p\right)=c_0$, where ${\ell }_p$, ${\ell }_{\infty }$, c, $c_0$ are the spaces of p-summable, bounded, convergent, and null sequences, respectively.

Note that (Theorem 6 in [12], Theorem 2 in [13])

$$\begin{array}{cc}\hfill c_0{\left(p\right)}^{\beta }& =M_0\left(p\right)=\bigcup _{N\in \mathbb{N}}\left\{\left(y_k\right):\sum _k\left|y_k\right|N^{-1/p_k}<\infty \right\},\hfill \\ \hfill M_0{\left(p\right)}^{\beta }& =\bigcap _{N\in \mathbb{N}}\left\{\left(y_k\right):\underset{k}{sup}\left|y_k\right|N^{1/p_k}<\infty \right\}\hfill \end{array}$$

and (Theorem 2 in [14])

$${\ell }_{\infty }{\left(p\right)}^{\beta }=M_{\infty }\left(p\right)=\bigcap _{N\in \mathbb{N}}\left\{\left(y_k\right):\sum _k\left|y_k\right|N^{1/p_k}<\infty \right\}.$$

Now we introduce the notions and notation for double sequence spaces.

Let e be the double sequence with all elements equal to 1, and let $e^{kl}$ be the double sequence with the $(k,l)$-th element equal to 1 and all others equal 0 $(k,l\in \mathbb{N})$. By an index sequence $\left(m_i\right)$, we mean an increasing sequence of integers.

A double sequence $x=\left(x_{kl}\right)$ of real or complex numbers is said to converge to the limit a in Pringsheim’s sense (shortly, p-converge to a) if

$$\forall \epsilon >0\exists N\in \mathbb{N}:k,l>N\Rightarrow \mid x_{kl}-a\mid <\epsilon .$$

Let $\mathsf{\Omega }$ denote the linear space of all double sequences. Linear subspaces of $\mathsf{\Omega }$ are called double sequence spaces.

For any notion of convergence $\nu$, the space of all $\nu$-convergent double sequences will be denoted by ${\mathcal{C}}_{\nu }$ and the limit of a $\nu$-convergent double sequence x by $\nu$ − ${lim}_{m,n}{x}_{mn}$. The sum of a double series ${\sum }_{k,l}{x}_{kl}$ is defined by $\nu -{\sum }_{k,l}{x}_{kl}:=\nu -{lim}_{m,n}{\sum }_{k=1}^m{\sum }_{l=1}^n{x}_{kl}$ whenever ${\left({\sum }_{k=1}^m{\sum }_{l=1}^n{x}_{kl}\right)}_{m,n}\in {\mathcal{C}}_{\nu }$. In the case of regular convergence, we skip the prefix $\nu$-. We consider the following double sequence spaces:

$$\begin{array}{ccc}\hfill {\mathcal{M}}_u& :=& \left\{x\in \mathsf{\Omega }|\underset{k,l}{sup}\left|x_{kl}\right|<\infty \right\},\hfill \\ \hfill {\mathcal{C}}_{\nu 0}& :=& \left\{x\in \mathsf{\Omega }|\nu -\underset{m,n}{lim}{x}_{mn}=0\right\}(\nu \in \{p,bp,r\}),\hfill \\ \hfill \mathcal{B}\mathcal{S}& :=& \left\{\left(x_{kl}\right)\in \mathsf{\Omega }:\underset{m,n}{sup}|\sum _{k,l=1}^{m,n}{x}_{kl}\right|<\infty \},\hfill \\ \hfill {\mathcal{C}S}_{\nu }& :=& \left\{x\in \mathsf{\Omega }|{\left(\sum _{k,l=1}^{m,n}{x}_{kl}\right)}_{m,n}\in {\mathcal{C}}_{\nu }\right\}(\nu \in \{p,bp,r\}),\hfill \\ \hfill \mathcal{B}V& :=& \{\left(x_{kl}\right)\in \mathsf{\Omega }:\sum _{k,l}|{\Delta }^{\left(11\right)}{x}_{kl}|<\infty ,\forall k\in \mathbb{N}:\sum _l\left|{\Delta }^{\left(01\right)}{x}_{kl}\right|<\infty ,\hfill \\ & & \forall l\in \mathbb{N}:\sum _k\left|{\Delta }^{\left(10\right)}{x}_{kl}\right|<\infty \}.\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\Delta }^{\left(10\right)}{x}_{kl}& :=x_{kl}-x_{k+1,l},{\Delta }^{\left(01\right)}{x}_{kl}:=x_{kl}-x_{k,l+1},\hfill \\ \hfill {\Delta }^{\left(11\right)}{x}_{kl}& :=x_{kl}-x_{k,l+1}-x_{k+1,l}+x_{k+1,l+1}.\hfill \end{array}$$

Given a double sequence space E, we define its $\alpha$- and $\beta \left(\nu \right)$-dual by

$$\begin{array}{ccc}& & E^{\alpha }:=\left\{u\in \mathsf{\Omega }:\sum _{k,l}\left|u_{kl}{x}_{kl}\right|<\infty ,\forall x\in E\right\},\hfill \\ & & E^{\beta \left(\nu \right)}:=\left\{u\in \mathsf{\Omega }:\nu -\sum _{k,l}{u}_{kl}{x}_{kl}\mathrm{exists}\mathrm{for}\mathrm{each}x\in E\right\},\hfill \end{array}$$

where $\nu \in \{p,bp,r\}$. For other notions and notations in the area of double sequences, we refer the reader to [15].

In [16], the authors showed that

$${\mathcal{B}V}^{\beta \left(\nu \right)}=D_1\cap D_2^{\nu },$$

where

$$\begin{array}{ccc}\hfill D_1& :=& \left\{u\in \mathsf{\Omega }|\underset{k,l,m,n}{sup}\left|\sum _{i,j=k,l}^{m,n}{u}_{ij}\right|<\infty \right\},\hfill \\ \hfill D_2^{\nu }& :=& \left\{u\in \mathsf{\Omega }|\forall k,l\in \mathbb{N}\exists \nu -\underset{m,n}{lim}\sum _{i,j=k,l}^{m,n}{u}_{ij}\right\}(\nu \in \{p,bp,r\}).\hfill \end{array}$$

It can be easily shown that $u\in D_1$ if and only if $u\in \mathcal{B}S$, and $u\in D_2^{\nu }$ if and only if $u\in {\mathcal{C}S}_r$$(\nu \in \{p,bp,r\left\}\right)$. Since ${\mathcal{C}S}_r\subset \mathcal{B}S$, it follows that

$${\mathcal{B}V}^{\beta \left(\nu \right)}={\mathcal{C}S}_r.$$

Let $p=\left(p_{kl}\right)$ be a double sequence of strictly positive numbers. We consider the following double sequence spaces with powers:

$$\begin{array}{ccc}\hfill {\mathcal{M}}_u\left(p\right)& :=& \left\{x\in \mathsf{\Omega }\left|\right(|x_{kl}{|}^{p_{kl}})\in {\mathcal{M}}_u\right\},\hfill \\ \hfill {\mathcal{C}}_{\nu 0}\left(p\right)& :=& \left\{x\in \mathsf{\Omega }\left|\right(|x_{kl}{|}^{p_{kl}})\in {\mathcal{C}}_{\nu 0}\right\}(\nu \in \{p,bp\}),\hfill \\ \hfill {\mathcal{C}}_{r0}\left(p\right)& :=& \{x\in {\mathcal{C}}_{p0}\left(p\right)|\forall l\in \mathbb{N}:{\left(x_{kl}\right)}_k\in c\left({\left(p_{kl}\right)}_k\right)\mathrm{and}\hfill \\ & & \forall k\in \mathbb{N}:{\left(x_{kl}\right)}_l\in c\left({\left(p_{kl}\right)}_l\right)\},\hfill \\ \hfill {\mathcal{C}}_{\nu }\left(p\right)& :=& \left\{x\in \mathsf{\Omega }|\exists a\in \mathbb{K}:(x_{kl}-a)\in {\mathcal{C}}_{\nu 0}\left(p\right)\right\}(\nu \in \{p,bp,r\}),\hfill \\ \hfill {\mathcal{L}}_u\left(p\right)& :=& \{\left(x_{kl}\right)\in \mathsf{\Omega }|\sum _{k,l}{\left|x_{kl}\right|}^{p_{kl}}<\infty \}.\hfill \end{array}$$

Most of these spaces were defined and studied by Gökhan and Çolak in [17,18,19]. In the proofs, we will use the representation ([8]):

$$\begin{array}{ccc}\hfill {\mathcal{C}}_{p0}\left(p\right)& =& \bigcap _N\left\{x\in \mathsf{\Omega }:\left(\right|x_{kl}\left|N^{1/p_{kl}}\right)\in {\mathcal{C}}_{p0}\right\}.\hfill \end{array}$$

(1)

In [8], the authors verified that

$$\begin{array}{ccc}\hfill {\mathcal{C}}_{p0}{\left(p\right)}^{\alpha }& =& {\mathcal{C}}_{p0}{\left(p\right)}^{\beta \left(\nu \right)}={\mathcal{M}}_0^p\left(p\right)\hfill \\ & :=& {\mathcal{M}}_0^2\left(p\right)\bigcap \left\{\left(y_{kl}\right):\forall l\in \mathbb{N}:{\left(y_{kl}\right)}_k\in \phi ,\forall k\in \mathbb{N}:{\left(y_{kl}\right)}_l\in \phi \right\},\hfill \end{array}$$

where

$${\mathcal{M}}_0^2\left(p\right):=\bigcup _{N\in \mathbb{N}}\{\left(y_{kl}\right):\sum _{k,l}\left|y_{kl}\right|N^{-1/p_{kl}}<\infty \},$$

and in [18] that

$${\mathcal{M}}_u{\left(p\right)}^{\alpha }={\mathcal{M}}_u{\left(p\right)}^{\beta \left(\nu \right)}={\mathcal{M}}_{\infty }^2\left(p\right):=\bigcap _{N\in \mathbb{N}}\left\{\left(y_{kl}\right):\sum _{k,l}\left|y_{kl}\right|N^{1/p_{kl}}<\infty \right\}.$$

Let $A=\left(a_{mnkl}\right)$ be any four-dimensional scalar matrix and let $\nu$ be some convergence notion for double sequences. We define

$${\mathsf{\Omega }}_A^{\left(\nu \right)}:=\left\{{x\in \mathsf{\Omega }|\forall m,n\in \mathbb{N}:\left[Ax\right]}_{mn}:=\nu -\sum _{k,l}{a}_{mnkl}{x}_{kl}\mathrm{exists}\right\}.$$

The map

$$A:{\mathsf{\Omega }}_A^{\left(\nu \right)}\to \mathsf{\Omega },x\mapsto Ax:={\left({\left[Ax\right]}_{mn}\right)}_{m,n}$$

is called a matrix map of type $\nu$. We use the notation $A\in {(X,Y)}_{\nu }$ if and only if A is a matrix map of type $\nu$ and $Ax\in Y$ whenever $x\in X$. If $X^{\beta \left(p\right)}=X^{\beta \left(bp\right)}=X^{\beta \left(r\right)}$ (for example, X is solid), we use the notation $A\in (X,Y)$.

In addition, if X is a double sequence space, Y is a single sequence space and $B=\left(b_{nkl}\right)$ is a three-dimensional matrix, we use the notation $B\in {(X,Y)}_{\nu }$ if and only if

$$x\in {\omega }_B^{\left(\nu \right)}:=\left\{y\in \mathsf{\Omega }|By:={\left(\nu -\sum _{k,l}{b}_{nkl}{y}_{kl}\right)}_n\mathrm{exists}\right\}$$

and $Bx\in Y$ whenever $x\in X$. As before if $X^{\beta \left(p\right)}=X^{\beta \left(bp\right)}=X^{\beta \left(r\right)}$ (for example if X is solid), we use the notation $B\in (X,Y)$.

If X is a sequence space, Y is a double sequence space and $B=\left(b_{mnk}\right)$ is a three-dimensional matrix, we use the notation $B\in (X,Y)$ if and only if

$$x\in {\mathsf{\Omega }}_B:=\left\{y\in \omega |By:={\left(\sum _k{b}_{mnk}{y}_k\right)}_{m,n}\mathrm{exists}\right\}$$

and $Bx\in Y$ whenever $x\in X$.

We set

$$t_{mnrs}\left(x\right)={\displaystyle \frac{1}{m+1}}{\displaystyle \frac{1}{n+1}}\sum _{i=0}^m\sum _{j=0}^n{x}_{r+i,s+j}(m,n,r,s\in \mathbb{N})$$

and note that

$$t_{mnrs}\left(Ax\right)=\nu -\sum _{k,l}a(m,n,r,s,k,l)x_{kl},$$

where

$$a(m,n,r,s,k,l)={\displaystyle \frac{1}{m+1}}{\displaystyle \frac{1}{n+1}}\sum _{i=0}^m\sum _{j=0}^n{a}_{r+i,s+j,k,l}(m,n,r,s,k,l\in \mathbb{N}).$$

We consider generalizations of the spaces of all almost convergent double sequences defined by Móricz and Rhoades [20]:

$${\mathcal{F}}^p=\{\left(x_{kl}\right)\in \mathsf{\Omega }:\underset{m,n}{lim}{t}_{mnrs}\left(x\right)\mathrm{exists}\mathrm{uniformly}\mathrm{in}r,s\}$$

and of all almost convergent to 0 double sequences

$${\mathcal{F}}_0^p=\{\left(x_{kl}\right)\in \mathsf{\Omega }:\underset{m,n}{lim}{t}_{mnrs}\left(x\right)=0\mathrm{uniformly}\mathrm{in}r,s\}$$

with indexes:

$${\mathcal{F}}^p\left(p\right)=\{\left(x_{kl}\right)\in \mathsf{\Omega }:\underset{m,n}{lim}|t_{mnrs}(x-le){|}^{p_{mn}}=0\mathrm{for}\mathrm{some}l\mathrm{uniformly}\mathrm{in}r,s\}$$

and

$${\mathcal{F}}_0^p\left(p\right)=\{\left(x_{kl}\right)\in \mathsf{\Omega }:\underset{m,n}{lim}|t_{mnrs}\left(x\right){|}^{p_{mn}}=0\mathrm{uniformly}\mathrm{in}r,s\}.$$

The following inclusions hold for these spaces.

Lemma 1. 

Let $p,q$ be bounded positive double sequences such that $0<p_{kl}\le q_{kl}$$(k,l\in \mathbb{N})$. Then, the following apply:

(a) 

${\mathcal{F}}_0^p\left(p\right)\subset {\mathcal{F}}_0^p\left(q\right)$.

(b) 

${\mathcal{F}}^p\left(p\right)\subset {\mathcal{F}}^p\left(q\right)$.

Proof. 

We prove part (b); part (a) follows by the same argument with $l=0$. Let $x\in {\mathcal{F}}^p\left(p\right)$ and let l be a number such that

$$\underset{m,n}{lim}{\left|t_{mnrs}(x-le)\right|}^{p_{mn}}=0\mathrm{uniformly}\mathrm{in}r,s.$$

Then $|t_{mnrs}{(x-le)|}^{p_{mn}}<1$ for sufficiently large $m,n$, so $|t_{mnrs}(x-le)|<1$ for sufficiently large $m,n$, and since $0<p_{mn}\le q_{mn}$, we have

$$|t_{mnrs}{(x-le)|}^{q_{mn}}\le {\left|t_{mnrs}(x-le)\right|}^{p_{mn}}\to 0\mathrm{uniformly}\mathrm{in}r,s.$$

□

As in the case of single sequences, almost every convergent double sequence is bounded ([20]). The inclusion ${\mathcal{F}}^p\left(p\right)\subset {\mathcal{M}}_u\left(p\right)$ does not hold in general. Set $p_{k1}:=k$, $p_{kl}:=1$$(l\ne 1)$, and $x_{kl}:={(-1)}^{k+l}2$$(k,l\in \mathbb{N})$. $x_{kl}:={(-1)}^{k+l}2$$(k,l\in \mathbb{N})$. Then $\left(x_{kl}\right)\in {\mathcal{F}}^p\left(p\right)\setminus {\mathcal{M}}_u\left(p\right)$. The following lemma demonstrates that ${\mathcal{F}}^p\left(p\right)\subset {\mathcal{M}}_u\left(p\right)$ holds when p is bounded.

Lemma 2. 

Let p be a bounded positive double sequence. Then ${\mathcal{F}}^p\left(p\right)\subset {\mathcal{M}}_u\subset {\mathcal{M}}_u\left(p\right)$.

Proof. 

Let $P:={sup}_{k,l}{p}_{kl}$ and $x\in {\mathcal{F}}^p\left(p\right)$. Then we also have $x\in {\mathcal{F}}^p\left(\tilde{p}\right)$, where ${\tilde{p}}_{kl}:=p_{kl}/P$$(k,l\in \mathbb{N})$. Hence, by Lemma 1, $x\in {\mathcal{F}}^p\left(e\right)={\mathcal{F}}^p\subset {\mathcal{M}}_u$. Therefore, by Theorem 1 (ii) in [18], $x\in {\mathcal{M}}_u\subset {\mathcal{M}}_u\left(p\right)$. □

In [6], we characterized the classes of two-dimensional matrices $(E,F)$ where $F\in \{f_0\left(q\right),f\left(q\right)\}$ and $E\in \{c_0\left(p\right),c\left(p\right),{\ell }_{\infty }\left(p\right),M_0\left(p\right),$$\ell \left(p\right),$$bv\}$.

In this paper, we will generalize these results to the corresponding double sequence spaces and four-dimensional matrices. More precisely, we will give the conditions for the classes of matrices ${(E,F)}_{\nu }$ where $F\in \{{\mathcal{F}}_0^p\left(q\right),{\mathcal{F}}^p\left(q\right)\}$ and

$$E\in \left\{{\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_p\left(p\right),{\mathcal{M}}_u\left(p\right),{\mathcal{M}}_0^2\left(p\right),{\mathcal{L}}_u\left(p\right),\mathcal{B}V\right\}.$$

Although the present work is mainly theoretical, the obtained results are closely related to several topics in matrix analysis and numerical linear algebra (see, for example, [21,22]). In particular, condition numbers quantify the sensitivity of solutions of linear systems and inverse problems with respect to data perturbations, which is directly connected to the boundedness and stability properties of the associated operators (see [23,24]). In this sense, infinite matrix transformations acting between generalized almost convergent double sequence spaces can be viewed as abstract models of discrete operators arising in numerical approximations, where stability plays a central role.

Moreover, generalized inverses, including the Moore–Penrose inverse and its weighted variants, as well as regularization techniques such as Tikhonov regularization, are fundamental tools in the analysis of ill-posed problems. The matrix transformation results established in this paper may, therefore, support further investigations of stability and robustness in regularized inverse problems.

It is worth noting that several results related to the summability and convergence of sequences can also be formulated within the general framework of ideal convergence (see [25] for the definition). While this approach provides a broad and unifying perspective, the present work focuses on generalized almost convergence, which is particularly well suited for obtaining explicit characterizations of infinite matrix transformations. This setting allows us to derive concrete operator conditions and matrix criteria that are less transparent in a purely ideal-theoretic formulation.

Furthermore, although the present study is connected to earlier works on almost convergence and matrix transformations, it is not merely a marginal modification of known results. The extension to generalized almost convergent double sequence spaces and to four-dimensional matrix transformations introduces substantial technical and conceptual differences that do not follow directly from the one-dimensional setting studied in [6]. In particular, the interaction between two-dimensional index structures and higher-dimensional matrices leads to new boundedness conditions and transformation criteria that do not follow directly from the one-dimensional setting.

2. Main Results

In what follows, we assume that $p=\left(p_{kl}\right)$ and $q=\left(q_{kl}\right)$ are bounded double sequences of strictly positive numbers. The boundedness of the exponent sequences $p=\left(p_{kl}\right)$, $q=\left(q_{kl}\right)$ is essential, since otherwise the set of almost convergent (to 0) double sequences with powers may fail to be linear. We also follow the order of the results in [6].

Example 1. 

Let $p_{kl}=k+l$ and consider the constant double sequence $x=\left(x_{kl}\right)$ defined by $x_{kl}=\frac{1}{2}$ for all $k,l\in \mathbb{N}$. Then for every $m,n,r,s\in \mathbb{N}$, we have

$$t_{mnrs}\left(x\right)={\displaystyle \frac{1}{m+1}}{\displaystyle \frac{1}{n+1}}\sum _{i=0}^m\sum _{j=0}^n{\displaystyle \frac{1}{2}}={\displaystyle \frac{1}{2}}.$$

Since $p_{mn}=m+n\to \infty$ as $m,n\to \infty$, it follows that

$$|t_{mnrs}\left(x\right){|}^{p_{mn}}={\left(\frac{1}{2}\right)}^{p_{mn}}\to 0,$$

uniformly in $r,s$. Hence, $x\in {\mathcal{F}}_0^p\left(p\right)$. On the other hand, for the sequence $2x$, we obtain $t_{mnrs}\left(2x\right)=1$, and therefore

$$|t_{mnrs}\left(2x\right){|}^{p_{mn}}=1^{p_{mn}}=1 /\to 0.$$

Thus, $2x\notin {\mathcal{F}}_0^p\left(p\right)$. Consequently, when the exponent sequence $p=\left(p_{kl}\right)$ is unbounded, the space ${\mathcal{F}}_0^p\left(p\right)$ is not closed under scalar multiplication and therefore need not be linear.

To characterize matrices in $({\mathcal{C}}_{p0}\left(p\right),{\mathcal{F}}_0^p\left(q\right))$, we use the characterizations of matrices in $({\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_{p0}\left(q\right))$ and $({\mathcal{C}}_{p0}\left(p\right),{\mathcal{M}}_u\left(q\right))$ (Theorems 2.6 and 2.4 in [9], respectively).

Proposition 1 

(Theorem 2.6 in [9]). $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_{p0}\left(q\right))$ if and only if the following conditions hold:

(i) 

${lim}_{m,n}{\left|a_{mnkl}\right|}^{q_{mn}}=0$$(k,l\in \mathbb{N});$

(ii) 

$\forall m,n,l\in \mathbb{N}:{\left(a_{mnkl}\right)}_k\in \phi ;$

(iii) 

$\forall m,n,k\in \mathbb{N}:{\left(a_{mnkl}\right)}_l\in \phi ;$

(iv) 

$\forall k\in \mathbb{N}$$\exists L\left(k\right)\in \mathbb{N}$ such that $a_{mnkl}=0$ for $m,n,l>L\left(k\right);$

(v) 

$\forall l\in \mathbb{N}$$\exists K\left(l\right)\in \mathbb{N}$ such that $a_{mnkl}=0$ for $m,n,k>K\left(l\right);$

(vi) 

$\forall m,n\in \mathbb{N}\exists M\in \mathbb{N}:{\sum }_{k,l}\left|a_{mnkl}\right|M^{-1/p_{kl}}<\infty ;$

(vii) 

$\forall L\in \mathbb{N}\exists M,D\in \mathbb{N}:{sup}_{m,n>D}{L}^{1/q_{mn}}{\sum }_{k,l}\left|a_{mnkl}\right|M^{-1/p_{kl}}<\infty .$

Proposition 2 

(Theorem 1 in [9]). $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{M}}_u\left(q\right))$ if and only if the following conditions hold:

(i) 

$\forall k,l\in \mathbb{N}:{sup}_{m,n}{\left|a_{mnkl}\right|}^{q_{mn}}<\infty ;$

(ii) 

$\forall m,n,l\in \mathbb{N}:{\left(a_{mnkl}\right)}_k\in \phi ;$

(iii) 

$\forall m,n,k\in \mathbb{N}:{\left(a_{mnkl}\right)}_l\in \phi ;$

(iv) 

$\forall k\in \mathbb{N}$$\exists L\left(k\right)\in \mathbb{N}$ such that $a_{mnkl}=0$ for $l>L\left(k\right)$$(m,n\in \mathbb{N});$

(v) 

$\forall l\in \mathbb{N}$$\exists K\left(l\right)\in \mathbb{N}$ such that $a_{mnkl}=0$ for $k>K\left(l\right)$$(m,n\in \mathbb{N});$

(vi) 

$\forall m,n\in \mathbb{N}\exists M\in \mathbb{N}:{\sum }_{k,l}\left|a_{mnkl}\right|M^{-1/p_{kl}}<\infty ;$

(vii) 

$\exists M\in \mathbb{N}:{sup}_{m,n}{\left({\sum }_{k,l}\left|a_{mnkl}\right|M^{-1/p_{kl}}\right)}^{q_{mn}}<\infty$.

Now, using these propositions, we find conditions for a matrix $A=\left(a_{mnkl}\right)$ to map ${\mathcal{C}}_{p0}\left(p\right)$ to ${\mathcal{F}}_0^p\left(q\right)$. For clarity, the conditions appearing in the characterizations are grouped into four types:

(A)

almost convergence-type conditions,

(C)

convergence-type conditions,

(B)

boundedness-type conditions (including restrictions on the support of the matrix entries),

(M)

modular control conditions.

Theorem 1. 

$A=\left(a_{mnkl}\right)$ is in $({\mathcal{C}}_{p0}\left(p\right),{\mathcal{F}}_0^p\left(q\right))$ if and only if the following conditions hold:

(A1) 

${lim}_{m,n}{sup}_{r,s}{\left|a(m,n,r,s,k,l)\right|}^{q_{mn}}=0$$(k,l\in \mathbb{N})$;

(B1) 

$\forall m,n,l\in \mathbb{N}:{\left(a_{mnkl}\right)}_k\in \phi ;$

(B2) 

$\forall m,n,k\in \mathbb{N}:{\left(a_{mnkl}\right)}_l\in \phi ;$

(B3) 

$\forall k\in \mathbb{N}$$\exists L\left(k\right)\in \mathbb{N}$ such that $a_{mnkl}=0$ for $l>L\left(k\right)$$(m,n\in \mathbb{N});$

(B4) 

$\forall l\in \mathbb{N}$$\exists K\left(l\right)\in \mathbb{N}$ such that $a_{mnkl}=0$ for $k>K\left(l\right)$$(m,n\in \mathbb{N});$

(M1) 

$\forall m,n\in \mathbb{N}\exists M\in \mathbb{N}:{\sum }_{k,l}\left|a_{mnkl}\right|M^{-1/p_{kl}}<\infty ;$

(M2) 

$\forall L\in \mathbb{N}\exists M,D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{r,s}{sup}{L}^{1/q_{mn}}\sum _{k,l}\left|a(m,n,r,s,k,l)\right|M^{-1/p_{kl}}<\infty .$$

Proof. 

Necessity. (A1) follows since $e^{kl}\in {\mathcal{C}}_{p0}\left(p\right)$$(k,l\in \mathbb{N})$. (B1), (B2) and (M1) follow since ${\left(a_{mnkl}\right)}_{k,l}\in {\mathcal{C}}_{p0}{\left(p\right)}^{\beta \left(\nu \right)}={\mathcal{M}}_0^2\left(p\right)$$(m,n\in \mathbb{N})$.

Since by Lemma 2 $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{F}}_0^p\left(q\right))$ implies $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{M}}_u\left(q\right))$, (B3) and (B4) follow from Proposition 2.

To prove (M2), suppose on the contrary that (M2) does not hold; that is,

$$\exists L_0\in \mathbb{N}\forall M,D\in \mathbb{N}:\underset{m,n>D}{sup}\underset{r,s}{sup}{L}_0^{1/q_{mn}}\sum _{k,l}\left|a(m,n,r,s,k,l)\right|M^{-1/p_{kl}}=\infty .$$

Then we can choose index sequences $\left(m_i\right),\left(n_i\right),\left(M_i\right)$ and sequences of integers $\left(r_i\right)$, $\left(s_i\right)$ such that

$$\begin{array}{c}\hfill \underset{i}{sup}{L}_0^{1/q_{m_i{n}_i}}\sum _{k,l}\left|a(m_i,n_i,r_i,s_i,k,l)\right|M_i^{-1/p_{kl}}>i(i\in \mathbb{N}).\end{array}$$

(2)

We consider the matrix $C=\left(c_{ijkl}\right)$ with $c_{iikl}=a(m_i,n_i,r_i,s_i,k,l)$ and $c_{ijkl}=0$ for $i\ne j$$(i,j,k,l\in \mathbb{N})$. Since

$$L_0^{1/q_{m_j{n}_j}}\sum _{k,l}|c_{jjkl}|M_i^{-1/p_{kl}}\ge L_0^{1/q_{m_j{n}_j}}\sum _{k,l}|c_{jjkl}|M_j^{-1/p_{kl}}>j(j\ge i),$$

we get

$$\forall M,D\in \mathbb{N}:\underset{i,j>D}{sup}{L}_0^{1/q_{m_i{n}_j}}\sum _{k,l}\left|c_{ijkl}\right|M^{-1/p_{kl}}=\infty .$$

Hence, by Proposition 1, the matrix C is not in $({\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_{p0}\left({\left(q_{m_i{n}_j}\right)}_{i,j}\right))$, so there exists $x\in {\mathcal{C}}_{p0}\left(p\right)$ such that $Cx\notin {\mathcal{C}}_{p0}\left({\left(q_{m_i{n}_j}\right)}_{i,j}\right)$.

Therefore

$$|\sum _{k,l}{c}_{iikl}{x}_{kl}{|}^{q_{m_i{n}_i}}=|\sum _{k,l}a(m_i,n_i,r_i,s_i,k,l)x_{kl}{|}^{q_{m_i{n}_i}} /\to 0.$$

Since

$$\underset{r,s}{sup}|\sum _{k,l}a(m_i,n_i,r,s,k,l)x_{kl}{|}^{q_{m_i{n}_i}}\ge |\sum _{k,l}a(m_i,n_i,r_i,s_i,k,l)x_{kl}{|}^{q_{m_i{n}_i}}(i\in \mathbb{N}),$$

then

$$\underset{r,s}{sup}|\sum _{k,l}a(m,n,r,s,k,l)x_{kl}{|}^{q_{mn}} /\to 0(m,n\to \infty ),$$

which implies $Ax\notin {\mathcal{F}}_0^p\left(q\right)$, and the contradiction follows.

Sufficiency. Suppose that the conditions (A1), (B1)–(B4) and (M1)–(M2) are satisfied; we shall prove that $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{F}}_0^p\left(q\right))$.

From (M2), it follows that $\exists M,D\in \mathbb{N}:$

$$\underset{r,s}{sup}\sum _{k,l}\left|a(m,n,r,s,k,l)\right|M^{-1/p_{kl}}<\infty (m,n>D).$$

So the matrix ${\left(a(m,n,r,s,k,l)\right)}_{r,s,k,l}$ satisfies the condition (vii) in Proposition 2 for $\left(q_{mn}\right)=\left(1\right)$$(m,n>D)$.

From (A1), (B1)–(B4) and (M1), it follows that the matrix ${\left(a(m,n,r,s,k,l)\right)}_{r,s,k,l}$ also satisfies the conditions (i)–(vi) in Proposition 2 for $\left(q_{mn}\right)=\left(1\right)$$(m,n>D)$.

Hence, the matrix ${\left(a(m,n,r,s,k,l)\right)}_{r,s,k,l}\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{M}}_u)$ for every $m,n>D$. Therefore, by Proposition 2 we have

$$\forall m,n>D:\underset{r,s}{sup}{\left|\sum _{k,l}a(m,n,r,s,k,l)x_{kl}\right|}^{q_{mn}}<\infty .$$

Suppose on the contrary that $Ax\notin {\mathcal{F}}_0^p\left(q\right)$ for some $x\in {\mathcal{C}}_{p0}\left(p\right)$. So

$$\underset{r,s}{sup}{\left|\sum _{k,l}a(m,n,r,s,k,l)x_{kl}\right|}^{q_{mn}} /\to 0.$$

Hence, we can find index sequences $\left(m_i\right),\left(n_i\right)$, sequences of integers $\left(r_i\right),\left(s_i\right)$, and $\epsilon >0$ such that

$${\left|\sum _{k,l}a(m_i,n_i,r_i,s_i,k,l)x_{kl}\right|}^{q_{m_i{n}_i}}>\epsilon (i\in \mathbb{N}).$$

We consider the matrix $C=\left(c_{ijkl}\right)$ with $c_{iikl}=a(m_i,n_i,r_i,s_i,k,l)$ and $c_{ijkl}=0$ for $i\ne j$$(i,j,k,l\in \mathbb{N})$.

Then the matrix C does not map ${\mathcal{C}}_{p0}\left(p\right)$ to ${\mathcal{C}}_{p0}\left({\left(q_{m_i{n}_j}\right)}_{i,j}\right)$.

Therefore, C does not satisfy at least one of the conditions (i)–(vii) in Proposition 1.

The condition (i) is fulfilled since

$$\begin{array}{cc}\hfill \underset{i,j}{lim}{\left|c_{ijkl}\right|}^{q_{m_i{n}_j}}& =\underset{i}{lim}{\left|a(m_i,n_i,r_i,s_i,k,l)\right|}^{q_{m_i{n}_i}}\hfill \\ \hfill & \le \underset{m,n}{lim}\underset{r,s}{sup}{\left|a(m,n,r,s,k,l)\right|}^{q_{mn}}=0(k,l\in \mathbb{N}).\hfill \end{array}$$

From (B1)–(B4) and (M1), it follows that the conditions (ii)–(vi) are satisfied in Proposition 1 for the matrix C.

Let us verify that the condition (vii) of Proposition 1 also holds.

By (M2), for a fixed $L\in \mathbb{N}$, we can find $M,D_0\in \mathbb{N}$ such that

$$\underset{m,n>D_0}{sup}\underset{r,s}{sup}{L}^{1/q_{mn}}\sum _{k,l}\left|a(m,n,r,s,k,l)\right|M^{-1/p_{kl}}<\infty .$$

Let D be such that $m_i,n_i>D_0$ for $i>D$; then

$$\begin{array}{cc}\hfill & \underset{i,j>D}{sup}{L}^{1/q_{m_i{n}_j}}\sum _{k,l}\left|c_{ijkl}\right|M^{-1/p_{kl}}\hfill \\ \hfill =& \underset{i>D}{sup}{L}^{1/q_{m_i{n}_i}}\sum _{k,l}\left|a(m_i,n_i,r_i,s_i,k,l)\right|M^{-1/p_{kl}}\hfill \\ \hfill \le & \underset{m,n>D_0}{sup}\underset{r,s}{sup}{L}^{1/q_{mn}}\sum _{k,l}\left|a(m,n,r,s,k,l)\right|M^{-1/p_{kl}}<\infty .\hfill \end{array}$$

Therefore, the condition (vii) in Proposition 1 also holds for the matrix C, and we reach a contradiction. □

Corollary 1. 

$A\in {({\mathcal{C}}_p\left(p\right),{\mathcal{F}}_0^p\left(q\right))}_{\nu }$ if and only if the conditions (A1), (B1)–(B4), and (M1)–(M2) in Theorem 1 are fulfilled as well as

(A2) 

${lim}_{m,n}{sup}_{r,s}{|\nu -{\sum }_{k,l}a(m,n,r,s,k,l)|}^{q_{mn}}=0.$

Proof. 

Necessity. The necessity of (A1), (B1)–(B4), and (M1)–(M2) follows from Theorem 1. We obtain the necessity of (A2) since $e\in {\mathcal{C}}_p\left(p\right)$.

Sufficiency. Let $x\in {\mathcal{C}}_p\left(p\right)$ be given. Then $y=x-le\in {\mathcal{C}}_{p0}\left(p\right)$ for some $l\in \mathbb{R}$. By Theorem 1, $Ay\in {\mathcal{F}}_0^p\left(q\right)$ and $e\in {\mathcal{F}}_0^p\left(q\right)$ by (A2). So $x=y+le\in {\mathcal{F}}_0^p\left(q\right)$ by the linearity of ${\mathcal{F}}_0^p\left(q\right)$. □

To find the conditions for a matrix A to be in $({\mathcal{C}}_{p0}\left(p\right),{\mathcal{F}}^p\left(q\right))$, we use the following result:

Proposition 3 

(Theorem 2.5 in [9]). $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_p\left(q\right))$ if and only if the following conditions hold:

(i) 

$\forall k,l\in \mathbb{N}$$\exists a_{kl}$: ${lim}_{m,n}{|a_{mnkl}-a_{kl}|}^{q_{mn}}=0;$

(ii) 

$\forall m,n,l\in \mathbb{N}:{\left(a_{mnkl}\right)}_k\in \phi ;$

(iii) 

$\forall m,n,k\in \mathbb{N}:{\left(a_{mnkl}\right)}_l\in \phi ;$

(iv) 

] $\forall k\in \mathbb{N}$$\exists L\left(k\right)\in \mathbb{N}$ such that $a_{mnkl}=0$ for $m,n,l>L\left(k\right);$

(v) 

$\forall l\in \mathbb{N}$$\exists K\left(l\right)\in \mathbb{N}$ such that $a_{mnkl}=0$ for $m,n,k>K\left(l\right);$

(vi) 

$\forall m,n\in \mathbb{N}\exists M\in \mathbb{N}:{\sum }_{k,l}\left|a_{mnkl}\right|M^{-1/p_{kl}}<\infty ;$

(vii) 

$\exists M,D\in \mathbb{N}$: ${sup}_{m,n>D}{\sum }_{k,l}\left|a_{mnkl}\right|M^{-1/p_{kl}}<\infty ;$

(viii) 

$\forall L$$\exists M,D\in \mathbb{N}$: ${sup}_{m,n>D}{L}^{1/q_{mn}}{\sum }_{k,l}|a_{mnkl}-a_{kl}|M^{-1/p_{kl}}<\infty$.

Applying this proposition, we obtain the conditions for a three-dimensional matrix to map ${\mathcal{C}}_{p0}\left(p\right)$ to $c\left(q\right)$.

Corollary 2. 

$A=\left(a_{nkl}\right)\in ({\mathcal{C}}_{p0}\left(p\right),c\left(q\right))$ if and only if the following conditions hold:

(i) 

$\forall k,l\in \mathbb{N}$$\exists a_{kl}$: ${lim}_n{|a_{nkl}-a_{kl}|}^{q_n}=0;$

(ii) 

$\forall n,l\in \mathbb{N}:{\left(a_{nkl}\right)}_k\in \phi ;$

(iii) 

$\forall n,k\in \mathbb{N}:{\left(a_{nkl}\right)}_l\in \phi ;$

(iv) 

$\forall k\in \mathbb{N}$$\exists L\left(k\right)\in \mathbb{N}$ such that $a_{nkl}=0$ for $n,l>L\left(k\right);$

(v) 

$\forall l\in \mathbb{N}$$\exists K\left(l\right)\in \mathbb{N}$ such that $a_{nkl}=0$ for $n,k>K\left(l\right);$

(vi) 

$\forall n\in \mathbb{N}\exists M\in \mathbb{N}:{\sum }_{k,l}\left|a_{nkl}\right|M^{-1/p_{kl}}<\infty ;$

(vii) 

$\exists M\in \mathbb{N}$: ${sup}_n{\sum }_{k,l}\left|a_{nkl}\right|M^{-1/p_{kl}}<\infty ;$

(viii) 

$\forall L$$\exists M\in \mathbb{N}$: ${sup}_n{L}^{1/q_n}{\sum }_{k,l}|a_{nkl}-a_{kl}|M^{-1/p_{kl}}<\infty$.

Now, using these proposition and corollary, we find conditions for a matrix $A=\left(a_{mnkl}\right)$ to map ${\mathcal{C}}_{p0}\left(p\right)$ to ${\mathcal{F}}^p\left(q\right)$.

We use the following notation: for $x\in {\mathcal{C}}_p\left(p\right)$ and $l\in \mathbb{C}$ such that $|x_{kl}{-l|}^{p_{kl}}\to 0$, we set ${\mathcal{C}}_p\left(p\right)-limx:=l$. Analogously, for $x\in {\mathcal{F}}^p\left(q\right)$ and $l\in \mathbb{C}$ such that ${lim}_{m,n}{\left|t_{mnrs}(x-le)\right|}^{p_{mn}}=0$ uniformly in $r,s$, we set

$${\mathcal{F}}^p\left(q\right)-limx:=l.$$

Theorem 2. 

$A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{F}}^p\left(q\right))$ if and only if

(A1) 

$\exists \left({\alpha }_{kl}\right):$${lim}_{m,n}{sup}_{r,s}{|a(m,n,r,s,k,l)-{\alpha }_{kl}|}^{q_{mn}}=0$$(k,l\in \mathbb{N})$;

(B1) 

$\forall m,n,l\in \mathbb{N}:{\left(a_{mnkl}\right)}_k\in \phi ;$

(B2) 

$\forall m,n,k\in \mathbb{N}:{\left(a_{mnkl}\right)}_l\in \phi ;$

(B3) 

$\forall k\in \mathbb{N}$$\exists L\left(k\right)\in \mathbb{N}$ such that $a_{mnkl}=0$ for $l>L\left(k\right)$$(m,n\in \mathbb{N});$

(B4) 

$\forall l\in \mathbb{N}$$\exists K\left(l\right)\in \mathbb{N}$ such that $a_{mnkl}=0$ for $k>K\left(l\right)$$(m,n\in \mathbb{N});$

(M1) 

$\forall m,n\in \mathbb{N}\exists M\in \mathbb{N}:{\sum }_{k,l}\left|a_{mnkl}\right|M^{-1/p_{kl}}<\infty ;$

(M2) 

$\exists M,D\in \mathbb{N}$: ${sup}_{m,n>D}{sup}_{r,s}{\sum }_{k,l}\left|a(m,n,r,s,k,l)\right|M^{-1/p_{kl}}<\infty ;$

(M3) 

$\forall L\in \mathbb{N}\exists M,D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{r,s}{sup}{L}^{1/q_{mn}}\sum _{k,l}|a(m,n,r,s,k,l)-{\alpha }_{kl}|M^{-1/p_{kl}}<\infty .$$

Moreover, in this case,

$$\begin{array}{c}\hfill {\mathcal{F}}^p\left(q\right)-limAx=\sum _{k,l}{\alpha }_{kl}{x}_{kl}(x\in {\mathcal{C}}_{p0}\left(p\right)).\end{array}$$

(3)

Proof. 

Necessity. The proof of (A1), (B1)–(B4), and (M1) follows in the same way as in Theorem 1.

To prove (M2), we suppose on the contrary that (M2) does not hold. Then

$$\forall M,D\in \mathbb{N}:\underset{m,n>D}{sup}\underset{r,s}{sup}\sum _{k,l}\left|a(m,n,r,s,k,l)\right|M^{-1/p_{kl}}=\infty .$$

Hence, we can choose index sequences $\left(m_i\right),\left(n_i\right),\left(M_i\right)$ and sequences of integers $\left(r_i\right),\left(s_i\right)$ such that

$$\begin{array}{c}\hfill \underset{i}{sup}\sum _{k,l}\left|a(m_i,n_i,r_i,s_i,k,l)\right|M_i^{-1/p_{kl}}>i(i\in \mathbb{N}).\end{array}$$

(4)

We consider the matrix $C=\left(c_{ikl}\right)$ with $c_{ikl}=a(m_i,n_i,r_i,s_i,k,l)$$(i,k,l\in \mathbb{N})$. Since

$$\sum _{k,l}|c_{jkl}|M_i^{-1/p_{kl}}\ge \sum _{k,l}\left|c_{jkl}\right|M_j^{-1/p_{kl}}>j(j\ge i),$$

we get

$$\forall M\in \mathbb{N}:\underset{i}{sup}\sum _{k,l}\left|c_{ikl}\right|M^{-1/p_{kl}}=\infty .$$

Hence, by Corollary 2, the matrix C is not in $({\mathcal{C}}_{p0}\left(p\right),c\left({\left(q_{m_i{n}_i}\right)}_i\right))$, so there exists $x\in {\mathcal{C}}_{p0}\left(p\right)$ such that $Cx\notin c\left({\left(q_{m_i{n}_i}\right)}_i\right)$.

Therefore,

$$|\sum _{k,l}{c}_{ikl}{x}_{kl}-l{|}^{q_{m_i{n}_i}}=|\sum _{k,l}a(m_i,n_i,r_i,s_i,k,l)x_{kl}-l{|}^{q_{m_i{n}_i}} /\to 0$$

for any $l\in \mathbb{R}$.

Since

$$\underset{r,s}{sup}|\sum _{k,l}a(m_i,n_i,r,s,k,l)x_{kl}-l{|}^{q_{m_i{n}_i}}\ge |\sum _{k,l}a(m_i,n_i,r_i,s_i,k,l)x_{kl}-l{|}^{q_{m_i{n}_i}}$$

for any $i\in \mathbb{N}$, then

$$\underset{r,s}{sup}|\sum _{k,l}a(m,n,r,s,k,l)x_{kl}-l{|}^{q_{mn}} /\to 0(m,n\to \infty ),$$

which implies $Ax\notin {\mathcal{F}}^p\left(q\right)$, a contradiction.

To prove the necessity of (M3), we first note that (A1) and (M2) imply that

$$\sum _{k,l}\left|{\alpha }_{kl}\right|M^{-1/p_{kl}}<\infty$$

for some $M\in \mathbb{N}$.

From (B3) and (B4) it follows that $\forall k\in \mathbb{N}$$\exists L\left(k\right)\in \mathbb{N}$ such that ${\alpha }_{kl}=0$ for $l>L\left(k\right)$ and $\forall l\in \mathbb{N}$$\exists K\left(l\right)\in \mathbb{N}$ such that ${\alpha }_{kl}=0$ for $k>K\left(l\right)$. Hence, $\left({\alpha }_{kl}\right)\in {\mathcal{C}}_{p0}{\left(p\right)}^{\beta \left(p\right)}$.

From the fact that $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{F}}^p\left(q\right))$, it follows that ${\left(a(m,n,r,s,k,l)\right)}_{m,n,k,l}:{\mathcal{C}}_{p0}\left(p\right)\to {\mathcal{C}}_p\left(q\right)$ for every $r,s\in \mathbb{N}$. Thus these matrices satisfy the conditions (i)–(viii) in Proposition 3.

From (M3) we get

$$\forall L,r,s\in \mathbb{N}\exists M,D\in \mathbb{N}:\underset{m,n>D}{sup}{L}^{1/q_{mn}}\sum _{k,l}|a(m,n,r,s,k,l)-{\alpha }_{kl}|M^{-1/p_{kl}}<\infty .$$

Altogether, by Proposition 1, we obtain ${(a(m,n,r,s,k,l)-{\alpha }_{kl})}_{m,n,k,l}:{\mathcal{C}}_{p0}\left(p\right)\to {\mathcal{C}}_{p0}\left(q\right)$ for every $r,s\in \mathbb{N}$.

Hence, for every $x\in {\mathcal{C}}_{p0}\left(p\right)$ we have

$$|\sum _{k,l}(a(m,n,r,s,k,l)-{\alpha }_{kl})x_{kl}{|}^{q_{mn}}\to 0(m,n\to \infty ;r,s\in \mathbb{N}).$$

On the other hand, since $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{F}}^p\left(q\right))$, for every $x\in {\mathcal{C}}_{p0}\left(p\right)$ we can find $l_x$ such that

$$\underset{r,s}{sup}|\sum _{k,l}a(m,n,r,s,k,l)x_{kl}-l_x{|}^{q_{mn}}\to 0.$$

So $l_x={\sum }_{k,l}{\alpha }_{kl}{x}_{kl}$ and

$$\underset{r,s}{sup}|\sum _{k,l}(a(m,n,r,s,k,l)-{\alpha }_{kl})x_{kl}{|}^{q_{mn}}\to 0.$$

Hence, the matrix $(a_{mnkl}-{\alpha }_{kl})$ is in $({\mathcal{C}}_{p0}\left(p\right),{\mathcal{F}}_0^p\left(q\right))$, thus the condition (M3) holds by Theorem 1.

Sufficiency. Suppose that the conditions (A1), (B1)–(B4), and (M1)–(M3) are satisfied; we shall prove that $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{F}}^p\left(q\right))$.

By Proposition 3, we have ${\left(a(m,n,r,s,k,l)\right)}_{m,n,k,l}:{\mathcal{C}}_{p0}\left(p\right)\to {\mathcal{C}}_p\left(q\right)$ for every $r,s\in \mathbb{N}$; so, in particular, ${\sum }_{k,l}a(m,n,r,s,k,l)x_{kl}$ converges for every $x\in {\mathcal{C}}_{p0}\left(p\right)$.

In the necessity proof, we verified that $\left({\alpha }_{kl}\right)\in {\mathcal{C}}_{p0}{\left(p\right)}^{\beta \left(p\right)}$.

By Theorem 1, the matrix $(a_{mnkl}-{\alpha }_{kl})$ is in $({\mathcal{C}}_{p0}\left(p\right),{\mathcal{F}}_0^p\left(q\right))$.

Let $x\in {\mathcal{C}}_{p0}\left(p\right)$ be given, then

$$\begin{array}{cc}\hfill & \underset{r,s}{sup}|\sum _{k,l}a(m,n,r,s,k,l)x_{kl}-\sum _{k,l}{\alpha }_{kl}{x}_{kl}{|}^{q_{mn}}\hfill \\ \hfill =& \underset{r,s}{sup}|\sum _{k,l}(a(m,n,r,s,k,l)-{\alpha }_{kl})x_{kl}{|}^{q_{mn}}\to 0.\hfill \end{array}$$

Hence, $Ax\in {\mathcal{F}}^p\left(q\right)$ and Formula (3) follows. □

Remark 1. 

The coefficients ${\alpha }_{kl}$ represent the asymptotic values of the Cesàro-averaged matrix entries

$$a(m,n,r,s,k,l)$$

as the averaging window $(m,n)$ grows. In this sense, they describe the limiting behavior of the matrix transformation in the almost convergence framework and provide the coefficients of the corresponding limiting operator.

Theorem 3. 

$A\in {({\mathcal{C}}_p\left(p\right),{\mathcal{F}}^p\left(q\right))}_{\nu }$ if and only if the conditions (A1), (B1)–(B4), and (M1)–(M3) in Theorem 2 hold and

(A2) 

$\exists \left(\alpha \right):$${lim}_{m,n}{sup}_{r,s}{|\nu -{\sum }_{k,l}a(m,n,r,s,k,l)-\alpha |}^{q_{mn}}=0$.

Moreover, in this case,

$$\begin{array}{c}\hfill {\mathcal{F}}^p\left(q\right)-limAx=\sum _{k,l}{\alpha }_{kl}\left(x_{kl}-{\mathcal{C}}_p\left(p\right)-limx\right)+\alpha ·{\mathcal{C}}_p\left(p\right)-limx\end{array}$$

(5)

for each $x\in {\mathcal{C}}_p\left(p\right)$.

Proof. 

Necessity and Sufficiency of (A1), (A2), (B1)–(B4), and (M1)–(M3) follow since

$${\mathcal{C}}_p\left(p\right)={\mathcal{C}}_{p0}\left(p\right)\oplus e.$$

To prove Formula (5), let $x\in {\mathcal{C}}_p\left(p\right)$ with $|x_{kl}{-l|}^{p_{kl}}\to 0$ be given. Then by Theorem 2

$$\underset{r,s}{sup}{\left|\sum _{k,l}(a(m,n,r,s,k,l)-{\alpha }_{kl})(x_{kl}-l)\right|}^{q_{mn}}\to 0.$$

Let $Q:={sup}_{k,l}{q}_{kl}$. In view of (A2) and

$$\begin{array}{cc}\hfill & {\left|\sum _{k,l}a(m,n,r,s,k,l)x_{kl}-\sum _{k,l}{\alpha }_{kl}(x_{kl}-l)-l\alpha \right|}^{q_{mn}/Q}\hfill \\ \hfill & \le {\left|\sum _{k,l}(a(m,n,r,s,k,l)-{\alpha }_{kl})(x_{kl}-l)\right|}^{q_{mn}/Q}\hfill \\ \hfill & +l{\left|\sum _{k,l}a(m,n,r,s,k,l)-\alpha \right|}^{q_{mn}/Q},\hfill \end{array}$$

(5) follows. □

To find the conditions for a matrix A to be in $({\mathcal{M}}_u\left(p\right),{\mathcal{F}}_0^p\left(q\right))$, we use the following results:

Proposition 4 

(Theorem 3.6 in [8]). $A\in ({\mathcal{M}}_u\left(p\right),{\mathcal{C}}_{p0}\left(q\right))$ if and only if the following conditions hold:

(i) 

$\forall M,m,n\in \mathbb{N}:{\sum }_{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}<\infty ;$

(ii) 

$\forall M\in \mathbb{N}:{lim}_{m,n}{\left({\sum }_{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}\right)}^{q_{mn}}=0.$

Proposition 5 

(Theorem 3.4 in [8]). $A\in ({\mathcal{M}}_u\left(p\right),{\mathcal{M}}_u\left(q\right))$ if and only if the following condition holds:

$$\forall M\in \mathbb{N}:\underset{m,n}{sup}{\left(\sum _{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}\right)}^{q_{mn}}<\infty .$$

(6)

Now, using these propositions we find conditions for a matrix $A=\left(a_{mnkl}\right)$ to map ${\mathcal{M}}_u\left(p\right)$ to ${\mathcal{F}}_0^p\left(q\right)$.

Theorem 4. 

$A\in ({\mathcal{M}}_u\left(p\right),{\mathcal{F}}_0^p\left(q\right))$ if and only if the following conditions hold:

(B1) 

$\forall M\in \mathbb{N}:{sup}_{m,n}{\left({\sum }_{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}\right)}^{q_{mn}}<\infty ;$

(A1) 

$\forall M\in \mathbb{N}:{lim}_{m,n}{sup}_{r,s}{\left({\sum }_{k,l}\left|a(m,n,r,s,k,l)\right|M^{1/p_{kl}}\right)}^{q_{mn}}=0.$

Proof. 

Necessity. (B1) follows from Lemma 2 and Proposition 5.

To prove the necessity of (A1), we first note that by Lemma 2 and Proposition 5 for $\left(q_{mn}\right)=e$, we have

$$\forall M\in \mathbb{N}:\underset{m,n}{sup}\sum _{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}<\infty .$$

Therefore,

$$\forall M\in \mathbb{N}:\underset{r,s}{sup}\sum _{k,l}\left|a(m,n,r,s,k,l)\right|M^{1/p_{kl}}<\infty (m,n\in \mathbb{N}).$$

(7)

Now, suppose on the contrary that

$$\underset{m,n}{lim}\underset{r,s}{sup}{\left(\sum _{k,l}\left|a(m,n,r,s,k,l)\right|M^{1/p_{kl}}\right)}^{q_{mn}}\ne 0$$

for some $M\in \mathbb{N}$.

Hence, we can find index sequences $\left(m_i\right),\left(n_i\right)$, sequences of integers $\left(r_i\right),\left(s_i\right)$, and $\epsilon >0$ such that

$${\left(\sum _{k,l}\left|a(m_i,n_i,r_i,s_i,k,l)\right|M^{1/p_{kl}}\right)}^{q_{m_i{n}_i}}>\epsilon (i\in \mathbb{N}).$$

We consider the matrix $C=\left(c_{ijkl}\right)$ with $c_{iikl}=a(m_i,n_i,r_i,s_i,k,l)$ and $c_{ijkl}=0$ for $i\ne j$$(i,j,k,l\in \mathbb{N})$.

Then, by Proposition 4, the matrix C does not map ${\mathcal{M}}_u\left(p\right)$ to ${\mathcal{C}}_{p0}\left({\left(q_{m_i{n}_j}\right)}_{i,j}\right)$.

Hence there exists $x\in {\mathcal{M}}_u\left(p\right)$ such that $Cx\notin {\mathcal{C}}_{p0}\left({\left(q_{m_i{n}_j}\right)}_{i,j}\right)$.

Then, in the same way as in the proof of Theorem 1, we obtain that $Ax\notin {\mathcal{F}}_0^p\left(q\right)$, which yields a contradiction.

Sufficiency. From (B1) it follows that $Ax$ exists for each $x\in {\mathcal{M}}_u\left(p\right)$.

From (A1) we obtain (7).

Hence by Proposition 5, the matrix ${\left(a(m,n,r,s,k,l)\right)}_{m,n,r,s}$ maps $({\mathcal{M}}_u\left(p\right),{\mathcal{M}}_u)$ for each $k,l\in \mathbb{N}$.

Therefore,

$$\underset{r,s}{sup}|\sum _{k,l}a(m,n,r,s,k,l)x_{kl}{|}^{q_{mn}}<\infty (x\in {\mathcal{M}}_u\left(p\right);m,n\in \mathbb{N}).$$

Now suppose on the contrary that $Ax\notin {\mathcal{F}}_0^p\left(q\right)$ for some $x\in {\mathcal{M}}_u\left(p\right)$.

In the same way as in Theorem 1, we obtain index sequences $\left(m_i\right),\left(n_i\right)$, sequences $\left(r_i\right),\left(s_i\right)$, and $\epsilon >0$ such that

$$|\sum _{k,l}a(m_i,n_i,r_i,s_i,k,l)x_{kl}{|}^{q_{m_i{n}_i}}>\epsilon .$$

We consider the matrix $C=\left(c_{ijkl}\right)$ as above.

Then C does not map ${\mathcal{M}}_u\left(p\right)$ to ${\mathcal{C}}_{p0}\left({\left(q_{m_i{n}_j}\right)}_{i,j}\right)$.

Hence C violates at least one of the conditions (i), (ii) in Proposition 4.

Condition (i) holds by (7), therefore

$$\begin{array}{cc}\hfill \underset{i}{lim}{\left(\sum _{k,l}\left|c_{iikl}\right|M^{-1/p_{kl}}\right)}^{q_{m_i{n}_i}}& =\underset{i}{lim}{\left(\sum _{k,l}\left|a(m_i,n_i,r_i,s_i,k,l)\right|M^{-1/p_{kl}}\right)}^{q_{m_i{n}_i}}\ne 0.\hfill \end{array}$$

Since

$$\begin{array}{cc}\hfill \underset{r,s}{sup}& {\left(\sum _{k,l}\left|a(m_i,n_i,r,s,k,l)\right|M^{-1/p_{kl}}\right)}^{q_{m_i{n}_i}}\hfill \\ \hfill & \ge {\left(\sum _{k,l}\left|a(m_i,n_i,r_i,s_i,k,l)\right|M^{-1/p_{kl}}\right)}^{q_{m_i{n}_i}}(i\in \mathbb{N}),\hfill \end{array}$$

we obtain

$$\underset{m,n}{lim}\underset{r,s}{sup}{\left(\sum _{k,l}\left|a(m,n,r,s,k,l)\right|M^{1/p_{kl}}\right)}^{q_{mn}}\ne 0.$$

This yields a contradiction. □

To find the conditions for a matrix A to be in $({\mathcal{M}}_u\left(p\right),{\mathcal{F}}^p\left(q\right))$, we use the following proposition:

Proposition 6 

(Theorem 3.5 in [8]). $A\in ({\mathcal{M}}_u\left(p\right),{\mathcal{C}}_p\left(q\right))$ if and only if the following conditions hold:

(i) 

$\forall M,m,n\in \mathbb{N}:{\sum }_{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}<\infty ;$

(ii) 

$\forall M\in \mathbb{N}\exists D\in \mathbb{N}:{sup}_{m,n>D}{\sum }_{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}<\infty ;$

(iii) 

$\exists \left(a_{kl}\right)\forall M\in \mathbb{N}:{lim}_{m,n}({\sum }_{k,l}|a_{mnkl}-a_{kl}{\left|M^{1/p_{kl}}\right)}^{q_{mn}}=0.$

Applying this theorem, we obtain the conditions for a three-dimensional matrix to map ${\mathcal{M}}_u\left(p\right)$ to $c\left(q\right)$.

Corollary 3. 

$A=\left(a_{nkl}\right)\in ({\mathcal{M}}_u\left(p\right),c\left(q\right))$ if and only if the following conditions hold:

(i) 

$\forall M,n\in \mathbb{N}:{\sum }_{k,l}\left|a_{nkl}\right|M^{1/p_{kl}}<\infty ;$

(ii) 

$\forall M\in \mathbb{N}:{sup}_n{\sum }_{k,l}\left|a_{nkl}\right|M^{1/p_{kl}}<\infty ;$

(iii) 

$\exists \left(a_{kl}\right)\forall M\in \mathbb{N}:{lim}_n({\sum }_{k,l}|a_{nkl}-a_{kl}{\left|M^{1/p_{kl}}\right)}^{q_n}=0.$

Now, using these proposition and corollary, we find conditions for a matrix $A=\left(a_{mnkl}\right)$ to map ${\mathcal{M}}_u\left(p\right)$ to ${\mathcal{F}}^p\left(q\right)$.

Theorem 5. 

$A\in ({\mathcal{M}}_u\left(p\right),{\mathcal{F}}^p\left(q\right))$ if and only if the following conditions hold:

(M1) 

$\forall M,m,n\in \mathbb{N}:{\sum }_{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}<\infty ;$

(B1) 

$\forall M\in \mathbb{N}\exists D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{r,s}{sup}\sum _{k,l}\left|a(m,n,r,s,k,l)\right|M^{1/p_{kl}}<\infty ;$$

(A1) 

$\exists \left({\alpha }_{kl}\right):$

$$\forall M\in \mathbb{N}:\underset{m,n}{lim}\underset{r,s}{sup}{\left(\sum _{k,l}|a(m,n,r,s,k,l)-{\alpha }_{kl}|M^{1/p_{kl}}\right)}^{q_{mn}}=0.$$

Moreover, in this case,

$$\begin{array}{c}\hfill {\mathcal{F}}^p\left(q\right)-limAx=\sum _{k,l}{\alpha }_{kl}{x}_{kl}(x\in \left({\mathcal{M}}_u\left(p\right)\right).\end{array}$$

(8)

Proof. 

Necessity. (M1) follows, since ${\mathcal{M}}_u{\left(p\right)}^{\beta \left(\nu \right)}={\mathcal{M}}_{\infty }^2\left(p\right)$.

In the same way as in Theorem 4, we obtain

$$\forall M\in \mathbb{N}:\underset{r,s}{sup}\sum _{k,l}\left|a(m,n,r,s,k,l)\right|M^{1/p_{kl}}<\infty (m,n\in \mathbb{N}).$$

(9)

To prove the necessity of (B1), suppose on the contrary that there exists $M\in \mathbb{N}$ such that

$$\forall D\in \mathbb{N}:\underset{m,n>D}{sup}\underset{r,s}{sup}\sum _{k,l}\left|a(m,n,r,s,k,l)\right|M^{1/p_{kl}}=\infty .$$

Hence, we can find index sequences $\left(m_i\right),\left(n_i\right)$ and integers $\left(r_i\right),\left(s_i\right)$ such that

$$\sum _{k,l}\left|a(m_i,n_i,r_i,s_i,k,l)\right|M^{1/p_{kl}}>i(i\in \mathbb{N}).$$

Define the matrix $C=\left(c_{ikl}\right)$ with $c_{ikl}=a(m_i,n_i,r_i,s_i,k,l)$. Then

$$\sum _{k,l}\left|c_{ikl}\right|M^{1/p_{kl}}>i(i\in \mathbb{N}).$$

Hence, by Corollary 3, the matrix C is not in $({\mathcal{M}}_u\left(p\right),c\left({\left(q_{m_i{n}_i}\right)}_i\right))$. Thus there exists $x\in {\mathcal{M}}_u\left(p\right)$ such that $Cx\notin c\left({\left(q_{m_i{n}_i}\right)}_i\right)$.

Now, in the same way as in the proof of (M2) of Theorem 2, we conclude that $Ax\notin {\mathcal{F}}^p\left(q\right)$, which yields a contradiction.

To prove the necessity of (A1), note first that $A\in ({\mathcal{M}}_u\left(p\right),{\mathcal{F}}^p\left(q\right))$ implies that the matrix ${\left(a(m,n,r,s,k,l)\right)}_{m,n,k,l}$ maps $({\mathcal{M}}_u\left(p\right),{\mathcal{C}}_p\left(q\right))$ for each $r,s\in \mathbb{N}$.

Hence, by Proposition 6,

$$\exists \left({\alpha }_{kl}^{rs}\right)\forall M\in \mathbb{N}:\underset{m,n}{lim}{\left(\sum _{k,l}|a(m,n,r,s,k,l)-{\alpha }_{kl}^{rs}|M^{1/p_{kl}}\right)}^{q_{mn}}=0.$$

Since $e^{kl}\in {\mathcal{M}}_u\left(p\right)$, there exists a sequence $\left({\alpha }_{kl}\right)$ such that

$$\underset{m,n}{lim}\underset{r,s}{sup}{|a(m,n,r,s,k,l)-{\alpha }_{kl}|}^{q_{mn}}=0(k,l\in \mathbb{N}).$$

From the inequality

$$|a(m,n,r,s,k,l)-{\alpha }_{kl}^{rs}{|}^{q_{mn}}\le {\left(\sum _{k,l}|a(m,n,r,s,k,l)-{\alpha }_{kl}^{rs}|1^{1/p_{kl}}\right)}^{q_{mn}},$$

we obtain

$$\underset{m,n}{lim}{|a(m,n,r,s,k,l)-{\alpha }_{kl}^{rs}|}^{q_{mn}}=0,$$

hence ${\alpha }_{kl}^{rs}={\alpha }_{kl}$.

Therefore, for $x\in {\mathcal{M}}_u\left(p\right)$ with $|x_{kl}{|}^{p_{kl}}\le M$, we have

$$\begin{array}{c}\hfill |\sum _{k,l}(a(m,n,r,s,k,l)-{\alpha }_{kl})x_{kl}{|}^{q_{mn}}\le {\left(\sum _{k,l}|a(m,n,r,s,k,l)-{\alpha }_{kl}|M^{1/p_{kl}}\right)}^{q_{mn}}\to 0.\end{array}$$

(10)

Hence, $A\in ({\mathcal{M}}_u\left(p\right),{\mathcal{F}}^p\left(q\right))$ implies ${\mathcal{F}}^p\left(q\right)$ − $limAx={\sum }_{k,l}{\alpha }_{kl}{x}_{kl}$.

Thus $(a_{mnkl}-{\alpha }_{kl})$ is in $({\mathcal{M}}_u\left(p\right),{\mathcal{F}}_0^p\left(q\right))$ and the necessity of (A1) follows from Theorem 4.

Sufficiency. From (A1) and (10) it follows that $(a_{mnkl}-{\alpha }_{kl})$ is in $({\mathcal{M}}_u\left(p\right),{\mathcal{F}}_0^p\left(q\right))$.

From (B1) and (A1) it follows that $\left({\alpha }_{kl}\right)\in {\mathcal{M}}_u{\left(p\right)}^{\beta \left(\nu \right)}$.

Hence the series ${\sum }_{k,l}{\alpha }_{kl}{x}_{kl}$ converges for every $x\in {\mathcal{M}}_u\left(p\right)$.

Therefore,

$$\begin{array}{cc}\hfill & \underset{m,n}{lim}\underset{r,s}{sup}|\sum _{k,l}a(m,n,r,s,k,l)x_{kl}-\sum _{k,l}{\alpha }_{kl}{x}_{kl}{|}^{q_{mn}}\hfill \\ \hfill =& \underset{m,n}{lim}\underset{r,s}{sup}|\sum _{k,l}(a(m,n,r,s,k,l)-{\alpha }_{kl})x_{kl}{|}^{q_{mn}}=0.\hfill \end{array}$$

Hence $Ax\in {\mathcal{F}}^p\left(q\right)$ and ${\mathcal{F}}^p\left(q\right)$ − $limAx={\sum }_{k,l}{\alpha }_{kl}{x}_{kl}$. □

To find the conditions for $A\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{F}}_0^p\left(q\right))$, we first find the conditions for a matrix to map ${\mathcal{M}}_0^2\left(p\right)$ into ${\mathcal{C}}_{p0}\left(q\right)$.

Theorem 6. 

$A\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{C}}_{p0}\left(q\right))$ if and only if

(C1) 

${lim}_{m,n}{\left|a_{mnkl}\right|}^{q_{mn}}=0(k,l\in \mathbb{N}),$

(M1) 

$\forall L,M\in \mathbb{N}\exists D\in \mathbb{N}:{sup}_{m,n>D}{sup}_{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}{L}^{1/q_{mn}}<\infty .$

Proof. 

We identify the double sequence space ${\mathcal{M}}_0^2\left(p\right)$ with the sequence space $M_0\left(r\right)$, where $r=T\left(p\right)$, via the isomorphism T between the spaces $\mathsf{\Omega }$ and $\omega$ introduced in [26], and the four-dimensional matrix A with the three-dimensional matrix $B=\left(b_{mnk}\right)$ (${\left(b_{mni}\right)}_i:=T\left({\left(a_{mnkl}\right)}_{k,l}\right)$$(m,n\in \mathbb{N})$).

By Corollary 3.3 (c) in [8], $B\in (M_0\left(r\right),{\mathcal{C}}_{p0}\left(q\right))$ if and only if for every index sequences $\left(m_i\right),\left(n_i\right)$, we have ${\left(b_{m_i{n}_{i}k}\right)}_{i,k}\in (M_0\left(r\right),c_0\left({\left(q_{m_i{n}_i}\right)}_i\right))$.

By Theorem 2.18 in [6], this is equivalent to the following:

(a) $\forall \mathrm{index}\mathrm{sequences}\left(m_i\right),\left(n_i\right):{lim}_i{\left|b_{m_i{n}_{i}k}\right|}^{q_{m_i{n}_i}}=0(k\in \mathbb{N}),$

(b) $\forall L,M\in \mathbb{N}\forall \mathrm{index}\mathrm{sequences}\left(m_i\right),\left(n_i\right):$

$$\underset{i,k}{sup}\left|b_{m_i{n}_{i}k}\right|M^{1/r_k}{L}^{q_{m_i{n}_i}}<\infty .$$

By Proposition 3 (e) in [10], the statement (a) is equivalent to

$$\underset{m,n}{lim}{\left|b_{mnk}\right|}^{q_{mn}}=0(k\in \mathbb{N}),$$

which is equivalent to (C1).

By Proposition 3 (a) in [10], the statement (b) is equivalent to

$$\begin{array}{c}\hfill \forall L,M\in \mathbb{N}\exists D\in \mathbb{N}:\underset{m,n>D}{sup}\underset{k}{sup}\left|b_{mnk}\right|M^{1/r_k}{L}^{1/q_{mn}}<\infty ,\end{array}$$

which is equivalent to (M1). □

Identifying the double sequence spaces ${\mathcal{M}}_0^2\left(p\right)$, ${\mathcal{M}}_u\left(q\right)$ with the sequence spaces $M_0\left(r\right)$ ($r=T\left(p\right)$), ${\ell }_{\infty }\left(s\right)$ ($s=T\left(q\right)$), and applying the conditions for $A\in (M_0\left(p\right),{\ell }_{\infty }\left(q\right))$ (Theorem 2.20 in [6]), we can find the conditions for $A\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{M}}_u\left(q\right))$:

Proposition 7 

(Theorem 2.20 in [6]). $A\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{M}}_u\left(q\right))$ if and only if

(i) 

$\exists L\in \mathbb{N}\forall M\in \mathbb{N}:{sup}_{m,n,k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}{L}^{-1/q_{mn}}<\infty$.

Now, using these theorems, we find conditions for a matrix $A=\left(a_{mnkl}\right)$ to map ${\mathcal{M}}_0^2\left(p\right)$ into ${\mathcal{F}}_0^p\left(q\right)$.

Theorem 7. 

$A\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{F}}_0^p\left(q\right))$ if and only if

(A1) 

${lim}_{m,n}{sup}_{r,s}{\left|a(m,n,r,s,k,l)\right|}^{q_{mn}}=0$$(k,l\in \mathbb{N})$;

(B1) 

$\forall L,M\in \mathbb{N}\exists D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{k,l,r,s}{sup}\left|a(m,n,r,s,k,l)\right|M^{1/p_{kl}}{L}^{1/q_{mn}}<\infty ;$$

(M1) 

$\forall M\in \mathbb{N}:{sup}_{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}<\infty$$(m,n\in \mathbb{N})$.

Proof. 

Necessity. The necessity of (A1) follows since $e^{kl}\in {\mathcal{M}}_0^2\left(p\right)$.

The necessity of (B1) follows in the same way as the necessity of (B1) in Theorem 5 by applying Theorem 6.

The necessity of (M1) follows since

$${\left(a_{mnkl}\right)}_{k,l}\in {\left({\mathcal{M}}_0^2\left(p\right)\right)}^{\beta \left(\nu \right)}=\bigcap _{M\in \mathbb{N}}\{\left(y_{kl}\right):\underset{k,l}{sup}|y_{kl}|M^{1/p_{kl}}\}(m,n\in \mathbb{N}).$$

Sufficiency. From (B1) it follows that

$$\forall M\in \mathbb{N}\exists D:\underset{k,l,r,s}{sup}\left|a(m,n,r,s,k,l)\right|M^{1/p_{kl}}<\infty (m,n>D).$$

Thus, by Proposition 7, the matrix ${\left(a(m,n,r,s,k,l)\right)}_{r,s,k,l}\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{M}}_u\left(q\right))$ for every $m,n>D$.

Therefore,

$$\underset{r,s}{sup}{\left|\sum _{k,l}a(m,n,r,s,k,l)x_{kl}\right|}^{q_{mn}}<\infty (x\in {\mathcal{M}}_0^2\left(p\right);m,n>D).$$

If we suppose, on the contrary, that $Ax\notin {\mathcal{F}}_0^p\left(q\right)$ for some $x\in {\mathcal{M}}_0^2\left(p\right)$, then in the same way as in Theorem 1, we obtain $\epsilon >0$, index sequences $\left(m_i\right),\left(n_i\right)$ and integers $\left(r_i\right),\left(s_i\right)$ such that

$${\left|\sum _{k,l}a(m_i,n_i,r_i,s_i,k,l)x_{kl}\right|}^{q_{m_i{n}_i}}>\epsilon .$$

We consider the matrix $C=\left(c_{ijkl}\right)$ with $c_{iikl}=a(m_i,n_i,r_i,s_i,k,l)$ and $c_{ijkl}=0$ for $i\ne j$.

Then the matrix C does not map ${\mathcal{M}}_0^2\left(p\right)$ to ${\mathcal{C}}_{p0}\left({\left(q_{m_i{n}_j}\right)}_{i,j}\right)$.

Therefore (cf. the proof of Theorem 1), the condition (B1) is not satisfied in Theorem 6 for the matrix C.

Thus there exist $L,M\in \mathbb{N}$ such that

$$\underset{i,k,l}{sup}|c_{iikl}|M^{1/p_{kl}}{L}^{1/q_{m_i{n}_i}}=\underset{i,k,l}{sup}|a(m_i,n_i,r_i,s_i,k,l)|M^{1/p_{kl}}{L}^{1/q_{m_i{n}_i}}=\infty .$$

By (M1),

$$\underset{k,l}{sup}\left|a(m_i,n_i,r_i,s_i,k,l)\right|M^{1/p_{kl}}{L}^{1/q_{m_i{n}_i}}<\infty (i\in \mathbb{N}).$$

Since

$$\begin{array}{cc}\hfill \underset{k,l,r,s}{sup}& |a(m_i,n_i,r,s,k,l)|M^{1/p_{kl}}{L}^{1/q_{m_i{n}_i}}\hfill \\ \hfill & \ge \underset{k,l}{sup}\left|a(m_i,n_i,r_i,s_i,k,l)\right|M^{1/p_{kl}}{L}^{1/q_{m_i{n}_i}}(i\in \mathbb{N}),\hfill \end{array}$$

it follows that

$$\underset{m,n>D}{sup}\underset{k,l,r,s}{sup}\left|a(m,n,r,s,k,l)\right|M^{1/p_{kl}}{L}^{1/q_{mn}}=\infty$$

for every $D\in \mathbb{N}$, which yields a contradiction. □

To obtain the conditions for $A=\left(a_{nk}\right)$ to be in $({\mathcal{M}}_0^2\left(p\right),{\mathcal{F}}^p\left(q\right))$, in the same way as for Theorem 7, we first derive the conditions for a matrix to map ${\mathcal{M}}_0^2\left(p\right)$ into ${\mathcal{C}}_p\left(q\right)$:

Theorem 8. 

$A\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{C}}_p\left(q\right))$ if and only if

(C1) 

$\exists \left(a_{kl}\right)$ such that

$$\underset{m,n}{lim}{|a_{mnkl}-a_{kl}|}^{q_{mn}}=0(k,l\in \mathbb{N});$$

(B1) 

$\forall L\in \mathbb{N}\exists D\in \mathbb{N}:{sup}_{m,n>D}{sup}_{k,l}\left|a_{mnkl}\right|L^{1/q_{mn}}<\infty ;$

(M1) 

$\forall L,M\in \mathbb{N}\exists D\in \mathbb{N}:{sup}_{m,n>D}{sup}_{k,l}|a_{mnkl}-a_{kl}|M^{1/p_{kl}}{L}^{1/q_{mn}}<\infty .$

Moreover, in this case:

$${\mathcal{C}}_p\left(q\right)-limAx=\sum _{k,l}{a}_{kl}{x}_{kl}(x\in {\mathcal{M}}_0^2\left(p\right)).$$

(11)

Proof. 

As in the proof of Theorem 6, we identify the double sequence space ${\mathcal{M}}_0^2\left(p\right)$ with the sequence space $M_0\left(r\right)$, where $r=T\left(p\right)$, and the four-dimensional matrix A with the three-dimensional matrix $B=\left(b_{mnk}\right)$.

By Corollary 3.3 (a) in [8], $B\in (M_0\left(r\right),{\mathcal{C}}_p\left(q\right))$ if and only if for every index sequences $\left(m_i\right),\left(n_i\right)$ the matrix ${\left(b_{m_i{n}_{i}k}\right)}_{i,k}\in (M_0\left(r\right),c\left({\left(q_{m_i{n}_i}\right)}_i\right))$ and all these matrices are pairwise consistent on $M_0\left(r\right)$.

By Theorem 2.22 in [6], this is equivalent to the following conditions:

(a) $\forall L\in \mathbb{N}$ ∀ index sequences $\left(m_i\right),\left(n_i\right)$

$$\underset{i,k}{sup}\left|b_{m_i{n}_{i}k}\right|L^{1/p_k}<\infty ;$$

(b) $\exists \left(b_k\right)$ such that

$$\underset{i}{lim}{|b_{m_i{n}_{i}k}-b_k|}^{q_{m_i{n}_i}}=0(k\in \mathbb{N});$$

(c) $\forall L,M\in \mathbb{N}$ ∀ index sequences $\left(m_i\right),\left(n_i\right)$

$$\underset{i,k}{sup}|b_{m_i{n}_{i}k}-b_k|M^{1/r_k}{L}^{1/q_{m_i{n}_i}}<\infty .$$

Moreover,

$$c\left(q\right)-lim{\left(Bx\right)}_{m_i{n}_i}=\sum _k{b}_k{x}_k(x\in M_0\left(r\right)).$$

(12)

By Proposition 3 (a) in [10], statement (a) is equivalent to

$$\forall L\in \mathbb{N}\exists D\in \mathbb{N}:\underset{m,n>D}{sup}\underset{k}{sup}\left|b_{mnk}\right|L^{1/q_{mn}}<\infty ,$$

which is equivalent to (B1).

By Proposition 3 (d) in [10], statement (b) is equivalent to

$$\underset{m,n}{lim}{|b_{mnk}-b_k|}^{q_{mn}}=0(k\in \mathbb{N}).$$

Denoting $\left(a_{kl}\right)=T^{-1}\left(\left(b_k\right)\right)$, we obtain (C1).

By Proposition 3 (a) in [10], statement (c) is equivalent to

$$\forall L,M\in \mathbb{N}\exists D\in \mathbb{N}:\underset{m,n>D}{sup}\underset{k}{sup}|b_{mnk}-b_k|M^{1/r_k}{L}^{1/q_{mn}}<\infty ,$$

which is equivalent to (M1).

Since for $s=\left(s_k\right)$, we have (cf. [7]):

$$\begin{array}{c}\hfill c_0\left(s\right)=\bigcap _N\{x\in \omega :(|x_k|N^{1/s_k})\in c_0\},\end{array}$$

(13)

(12) is equivalent to

$$\forall \left(m_i\right),\left(n_i\right)\forall L\in \mathbb{N}:$$

$$\underset{i}{lim}|{\left(Bx\right)}_{m_i{n}_i}-\sum _k{b}_k{x}_k|L^{1/q_{m_i{n}_i}}=0.$$

By Proposition 3 (d) in [10], this is equivalent to

$$\begin{array}{cc}\hfill & \underset{m,n}{lim}|{\left(Bx\right)}_{mn}-\sum _k{b}_k{x}_k|L^{1/q_{mn}}=0(x\in M_0\left(p\right))\hfill \\ \hfill \iff & \underset{m,n}{lim}|{\left(Ax\right)}_{mn}-\sum _{k,l}{a}_{kl}{x}_{kl}|L^{1/q_{mn}}=0(x\in {\mathcal{M}}_0^2\left(p\right)).\hfill \end{array}$$

So by (1), Formula (11) holds. □

Now, with the help of this theorem, we get the conditions for a matrix to be in $({\mathcal{M}}_0^2\left(p\right),{\mathcal{F}}^p\left(q\right))$.

Theorem 9. 

$A\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{F}}^p\left(q\right))$ if and only if

(A1) 

$\exists \left({\alpha }_{kl}\right)$ such that ${lim}_{m,n}{sup}_{r,s}{|a(m,n,r,s,k,l)-{\alpha }_{kl}|}^{q_{mn}}=0$$(k,l\in \mathbb{N})$;

(M1) 

$\forall L,M\in \mathbb{N}\exists D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{k,l,r,s}{sup}|a(m,n,r,s,k,l)-{\alpha }_{kl}|M^{1/p_{kl}}{L}^{1/q_{mn}}<\infty ;$$

(B1) 

$\forall M\in \mathbb{N}\exists D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{k,l,r,s}{sup}\left|a(m,n,r,s,k,l)\right|M^{1/p_{kl}}<\infty ;$$

(M2) 

$\forall M\in \mathbb{N}:{sup}_{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}<\infty$$(m,n\in \mathbb{N})$.

Moreover, in this case,

$${\mathcal{F}}^p\left(q\right)-limAx=\sum _{k,l}{\alpha }_{kl}{x}_{kl}(x\in {\mathcal{M}}_0^2\left(p\right)).$$

(14)

Proof. 

Necessity. The necessity of (A1) follows since $e^{kl}\in {\mathcal{M}}_0^2\left(p\right)$$(k,l\in \mathbb{N})$.

The necessity of (B1) follows in the same way as the necessity of (B1) in Theorem 5 by applying Theorem 8.

The necessity of (M2) follows since ${\left(a_{mnkl}\right)}_{k,l}\in {\left({\mathcal{M}}_0^2\left(p\right)\right)}^{\beta \left(\nu \right)}$ $(m,n\in \mathbb{N})$.

Since $A\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{F}}^p\left(q\right))$ implies ${\left(a(m,n,r,s,k,l)\right)}_{m,n,k,l}\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{C}}_p\left(q\right))$ for every $r,s\in \mathbb{N}$, by Theorem 8, for every $x\in {\mathcal{M}}_0^2\left(p\right)$ and $r,s\in \mathbb{N}$, we have

$$\begin{array}{cc}\hfill & {\left|\sum _{k,l}(a(m,n,r,s,k,l)-{\alpha }_{kl})x_{kl}\right|}^{q_{mn}}\hfill \\ \hfill =& {\left|\sum _{k,l}a(m,n,r,s,k,l)x_{kl}-\sum _{k,l}{\alpha }_{kl}{x}_{kl}\right|}^{q_{mn}}\to 0.\hfill \end{array}$$

Therefore,

$${\mathcal{F}}^p\left(q\right)-limAx=\sum _{k,l}{\alpha }_{kl}{x}_{kl}.$$

Hence ${(a_{mnkl}-{\alpha }_{kl})}_{m,n,k,l}\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{F}}_0^p\left(q\right))$, so (M1) holds by Theorem 7.

Sufficiency. Conditions (A1) and (B1) imply that ${sup}_{k,l}\left|{\alpha }_{kl}\right|M^{1/p_{kl}}<\infty$ for every $M\in \mathbb{N}$.

Hence, $\left({\alpha }_{kl}\right)\in {\left({\mathcal{M}}_0^2\left(p\right)\right)}^{\beta \left(\nu \right)}$.

By Theorem 7, it follows that ${(a_{mnkl}-{\alpha }_{kl})}_{m,n,k,l}\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{F}}_0^p\left(q\right))$.

So, for every $x\in {\mathcal{M}}_0^2\left(p\right)$,

$$\begin{array}{cc}\hfill & {\left|\sum _{k,l}a(m,n,r,s,k,l)x_{kl}-\sum _{k,l}{\alpha }_{kl}{x}_{kl}\right|}^{q_{mn}}\hfill \\ \hfill =& {\left|\sum _{k,l}(a(m,n,r,s,k,l)-{\alpha }_{kl})x_{kl}\right|}^{q_{mn}}\to 0.\hfill \end{array}$$

Hence, $A\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{F}}^p\left(q\right))$. □

Set $K_1:=\{(k,l)\in {\mathbb{N}}^2|p_{kl}\le 1\}$, $K_2:=\{(k,l)\in {\mathbb{N}}^2|p_{kl}>1\}$ and $p_{kl}^{\prime }:=p_{kl}/(p_{kl}-1)$$((k,l)\in K_2)$.

To obtain the conditions for a matrix to map ${\mathcal{L}}_u\left(p\right)$ into ${\mathcal{F}}^p\left(q\right)$, we first derive the conditions for $A\in ({\mathcal{L}}_u\left(p\right),{\mathcal{C}}_p\left(q\right))$.

Theorem 10. 

$A\in ({\mathcal{L}}_u\left(p\right),{\mathcal{C}}_p\left(q\right))$ if and only if

(B1) 

$\exists D\in \mathbb{N}:{sup}_{m,n>D}{sup}_{(k,l)\in K_1}{\left|a_{mnkl}\right|}^{p_{kl}}<\infty ;$

(B2) 

$\exists M,D\in \mathbb{N}:{sup}_{m,n>D}{\sum }_{(k,l)\in K_2}{\left|a_{mnkl}{M}^{-1}\right|}^{p_{kl}^{\prime }}<\infty ;$

(C1) 

$\exists \left(a_{kl}\right)$ such that ${lim}_{m,n}{|a_{mnkl}-a_{kl}|}^{q_{mn}}=0(k,l\in \mathbb{N});$

(M1) 

$\forall L\in \mathbb{N}\exists D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{(k,l)\in K_1}{sup}\left(\right|a_{mnkl}-a_{kl}{\left|L^{1/q_{mn}}\right)}^{p_{kl}}<\infty ;$$

(M2) 

$\forall L\in \mathbb{N}\exists M,D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\sum _{(k,l)\in K_2}\left(\right|a_{mnkl}-a_{kl}{\left|L^{1/q_{mn}}{M}^{-1}\right)}^{p_{kl}^{\prime }}<\infty .$$

To prove this theorem, we use the same reasoning as in Theorem 8 by applying Corollary 3.3 (a) in [8] and Theorem 5.1, part 8 in [7]. Now, with the help of this theorem and using a standard argument, we obtain the conditions for a matrix to be in $({\mathcal{L}}_u\left(p\right),{\mathcal{F}}^p\left(q\right))$.

Theorem 11. 

$A\in ({\mathcal{L}}_u\left(p\right),{\mathcal{F}}^p\left(q\right))$ if and only if

(B1) 

$\exists D\in \mathbb{N}:{sup}_{m,n>D}{sup}_{r,s}{sup}_{(k,l)\in K_1}{\left|a(m,n,r,s,k,l)\right|}^{p_{kl}}<\infty ;$

(B2) 

$\exists M,D\in \mathbb{N}:{sup}_{m,n>D}{sup}_{r,s}{\sum }_{(k,l)\in K_2}{\left|a(m,n,r,s,k,l)M^{-1}\right|}^{p_{kl}^{\prime }}<\infty ;$

(A1) 

$\exists \left({\alpha }_{kl}\right)$ such that ${lim}_{m,n}{sup}_{r,s}{|a(m,n,r,s,k,l)-{\alpha }_{kl}|}^{q_{mn}}=0(k,l\in \mathbb{N});$

(M1) 

$\forall L\in \mathbb{N}\exists D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{r,s}{sup}\underset{(k,l)\in K_1}{sup}\left(\right|a(m,n,r,s,k,l)-{\alpha }_{kl}{\left|L^{1/q_{mn}}\right)}^{p_{kl}}<\infty ;$$

(M2) 

$\forall L\in \mathbb{N}\exists M,D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{r,s}{sup}\sum _{(k,l)\in K_2}\left(\right|a(m,n,r,s,k,l)-{\alpha }_{kl}{\left|L^{1/q_{mn}}{M}^{-1}\right)}^{p_{kl}^{\prime }}<\infty .$$

To obtain the conditions for a matrix to map ${\mathcal{L}}_u\left(p\right)$ into ${\mathcal{F}}_0^p\left(q\right)$, we first derive the conditions for $A\in ({\mathcal{L}}_u\left(p\right),{\mathcal{C}}_{p0}\left(q\right))$.

Theorem 12. 

$A\in ({\mathcal{L}}_u\left(p\right),{\mathcal{C}}_{p0}\left(q\right))$ if and only if

(C1) 

${lim}_{m,n}{\left|a_{mnkl}\right|}^{q_{mn}}=0(k,l\in \mathbb{N});$

(M1) 

$\forall L\in \mathbb{N}\exists D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{(k,l)\in K_1}{sup}\left(\right|a_{mnkl}{\left|L^{1/q_{mn}}\right)}^{p_{kl}}<\infty ;$$

(M2) 

$\forall L\in \mathbb{N}\exists M,D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\sum _{(k,l)\in K_2}\left(\right|a_{mnkl}{\left|L^{1/q_{mn}}{M}^{-1}\right)}^{p_{kl}^{\prime }}<\infty .$$

To prove this theorem, we use the same reasoning as in Theorem 8 by applying Corollary 3.3 (c) in [8] and Theorem 5.1, part 4 in [7]. Now, with the help of this theorem and using a standard argument, we obtain the conditions for a matrix to be in $({\mathcal{L}}_u\left(p\right),{\mathcal{F}}_0^p\left(q\right))$.

Theorem 13. 

$A\in ({\mathcal{L}}_u\left(p\right),{\mathcal{F}}_0^p\left(q\right))$ if and only if

(A1) 

${lim}_{m,n}{sup}_{r,s}{\left|a(m,n,r,s,k,l)\right|}^{q_{mn}}=0(k,l\in \mathbb{N});$

(M1) 

$\forall L\in \mathbb{N}\exists D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{r,s}{sup}\underset{(k,l)\in K_1}{sup}\left(\right|a(m,n,r,s,k,l)|L^{1/q_{mn}}{)}^{p_{kl}}<\infty ;$$

(M2) 

$\forall L\in \mathbb{N}\exists M,D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{r,s}{sup}\sum _{(k,l)\in K_2}\left(\right|a(m,n,r,s,k,l)|L^{1/q_{mn}}{M}^{-1}{)}^{p_{kl}^{\prime }}<\infty .$$

To obtain the conditions for $A\in (\mathcal{B}V,{\mathcal{F}}^p\left(q\right))$, we use Theorem 9.

Theorem 14. 

$A\in (\mathcal{B}V,{\mathcal{F}}^p\left(q\right))$ if and only if

(B1) 

$\exists D\in \mathbb{N}:{sup}_{m,n>D}{sup}_{r,s,k,l}\left|{\sum }_{i=1}^k{\sum }_{j=1}^{l}a(m,n,r,s,i,j)\right|<\infty ;$

(A1) 

$\exists \left({\alpha }_{kl}\right)$ such that ${lim}_{m,n}{sup}_{r,s}{|a(m,n,r,s,k,l)-{\alpha }_{kl}|}^{q_{mn}}=0(k,l\in \mathbb{N});$

(M1) 

$\forall L\in \mathbb{N}\exists D\in \mathbb{N}:$

$$\underset{m,n>D}{sup}\underset{r,s,k,l}{sup}{\left(\left|\sum _{i=1}^k\sum _{j=1}^l(a(m,n,r,s,i,j)-{\alpha }_{ij})\right|L^{1/q_{mn}}\right)}^{p_{kl}}<\infty ;$$

(A2) 

$\exists \alpha$ such that ${lim}_{m,n}{sup}_{r,s}{\left|{\sum }_{k,l}a(m,n,r,s,k,l)-\alpha \right|}^{q_{mn}}=0.$

Moreover, in this case,

$$\begin{array}{c}\hfill {\mathcal{F}}^p\left(q\right)-limAx=\sum _{k,l}{\Delta }^{\left(11\right)}{x}_{kl}\sum _{i=1}^k\sum _{j=1}^l{\alpha }_{ij}+\alpha \underset{k,l}{lim}{x}_{kl}(x\in \mathcal{B}V).\end{array}$$

(15)

Proof. 

Necessity of (A1) and (A2) follows since $e,e^{kl}\in \mathcal{B}V$$(k,l\in \mathbb{N})$. To prove the necessity of (B1) and (M1), we note that by Abel’s summation formula for $x\in \mathcal{B}V$ and $i,j,m,n\in \mathbb{N}$, we have

$$\begin{array}{ccc}\hfill \sum _{k=1}^i\sum _{l=1}^j{a}_{mnkl}{x}_{kl}& =& \sum _{k=1}^{i-1}\sum _{l=1}^{j-1}{\Delta }^{\left(11\right)}{x}_{kl}{A}_{mnkl}\hfill \\ & +& \sum _{k=1}^{i-1}{\Delta }^{\left(10\right)}{x}_{kj}{A}_{mnkj}+\sum _{l=1}^{j-1}{\Delta }^{\left(01\right)}{x}_{il}{A}_{mnil}+x_{ij}{A}_{mnij},\hfill \end{array}$$

where

$$A_{mnkl}:=\sum _{s=1}^k\sum _{t=1}^l{a}_{mnst}(m,n,k,l\in \mathbb{N}).$$

Since $A\in (\mathcal{B}V,{\mathcal{F}}^p\left(q\right))$ implies ${\left(a_{mnkl}\right)}_{k,l}\in {\mathcal{B}V}^{\beta \left(\nu \right)}$$(m,n\in \mathbb{N})$, then ${sup}_{k,l}\left|A_{mnkl}\right|<\infty$ ($m,n\in \mathbb{N}$), so

$$\left|\sum _{k=1}^{i-1}{\Delta }^{\left(10\right)}{x}_{kj}{A}_{mnkj}\right|\le \underset{k,l}{sup}|A_{mnkl}|\sum _k\sum _{l=j}^{\infty }\left|{\Delta }^{\left(11\right)}{x}_{kl}\right|\to 0\mathrm{as}i,j\to \infty (m,n\in \mathbb{N}).$$

Analogously,

$$\sum _{l=1}^{j-1}{\Delta }^{\left(01\right)}{x}_{il}{A}_{mnil}\to 0\mathrm{as}i,j\to \infty (m,n\in \mathbb{N}).$$

Moreover, $A\in (\mathcal{B}V,{\mathcal{F}}^p\left(q\right))$ implies that the double series ${\sum }_{k,l}{a}_{mnkl}$ converges (regularly). By Proposition 8.11 in [27], we have $x\in {\mathcal{C}}_r$. So

$$\begin{array}{c}\hfill \sum _{k,l}{a}_{mnkl}{x}_{kl}=\sum _{k,l}{\Delta }^{\left(11\right)}{x}_{kl}{A}_{mnkl}+\underset{k,l}{lim}{x}_{kl}\sum _{k,l}{a}_{mnkl}.\end{array}$$

(16)

By considering $\mathcal{B}V\cap {\mathcal{C}}_{r0}$, we find that the matrix ${\left(A_{mnkl}\right)}_{m,n,k,l}$ is in $({\mathcal{L}}_u\left(p\right),{\mathcal{F}}^p\left(q\right))$. So the necessity of (B1) and (M1) follows from Theorem 11.

Sufficiency. Let $Q:={sup}_{k,l}{q}_{kl}$. Since

$${\left|\sum _{i=1}^k\sum _{j=1}^{l}a(m,n,r,s,i,j)-\sum _{i=1}^k\sum _{j=1}^l{\alpha }_{ij}\right|}^{{\displaystyle \frac{q_{mn}}{Q}}}\le \sum _{i=1}^k\sum _{j=1}^l{|a(m,n,r,s,i,j)-{\alpha }_{ij}|}^{{\displaystyle \frac{q_{mn}}{Q}}}$$

for any $k,l\in \mathbb{N}$, the matrix ${\left({\sum }_{i=1}^k{\sum }_{j=1}^l{a}_{mnij}\right)}_{m,n,k,l}$ is in $({\mathcal{L}}_u\left(p\right),{\mathcal{F}}^p\left(q\right))$ by Theorem 11.

Hence, the first summand in the right part of (16) is in ${\mathcal{F}}^p\left(q\right)$.

Moreover, from (A2) it follows that the second summand in the right part of (16) is in ${\mathcal{F}}^p\left(q\right)$.

Hence, $Ax\in {\mathcal{F}}^p\left(q\right)$ and (15) follows. □

From this theorem, we easily get conditions for a matrix to be in $(\mathcal{B}V,{\mathcal{F}}_0^p\left(q\right))$.

Corollary 4. 

A matrix $A=\left(a_{mnkl}\right)$ is in $A\in (\mathcal{B}V,{\mathcal{F}}_0^p\left(q\right))$ if and only if the conditions of Theorem 14 hold with ${\alpha }_{kl}=0$$(k,l\in \mathbb{N})$ and $\alpha =0$.

Proof. 

The result follows directly from Theorem 14 by applying (15) and using the identities ${\alpha }_{kl}={\mathcal{F}}^p\left(q\right)-limA{e}^{kl}$$(k,l\in \mathbb{N})$ and $\alpha ={\mathcal{F}}^p\left(q\right)-limAe$. □

We summarize our results in the

Table 1. Characterization of matrix transformations $(E,F)$.

	F
$\mathit{E}$	${\mathcal{F}}_{\mathbf{0}}^{\mathit{p}}\left(\mathit{q}\right)$	${\mathcal{F}}^{\mathit{p}}\left(\mathit{q}\right)$
${\mathcal{C}}_{p0}\left(p\right)$	Thm. 1	Thm. 2
${\mathcal{C}}_p\left(p\right)$	Cor. 1	Thm. 3
${\mathcal{M}}_u\left(p\right)$	Thm. 4	Thm. 5
${\mathcal{M}}_0^2\left(p\right)$	Thm. 7	Thm. 9
${\mathcal{L}}_u\left(p\right)$	Thm. 13	Thm. 16
$\mathcal{B}V$	Cor. 4	Thm. 14

. The number indicates the result where the characterization of $(E,F)$ can be found.

In the following examples, we illustrate how the previously discussed theorems can be applied to four-dimensional matrices.

Example 2. 

Let $p_{kl}=1/(k+l)$, $q_{kl}=1$$(k,l\in \mathbb{N})$. We consider the matrix $A=\left(a_{mnkl}\right)$ defined by $a_{mnkl}={\displaystyle \frac{1}{k!l!mn}}$ for $k\le m,l\le n$ and $a_{mnkl}=0$ otherwise ($m,n,k,l\in \mathbb{N}$). Then, $A\notin \left({\mathcal{C}}_{p0}\left(p\right)\right),{\mathcal{F}}_0^p\left(q\right))$, since conditions (B3), (B4) of Theorem 1 are not satisfied. Analogously, $A\notin \left({\mathcal{C}}_p\left(p\right)\right),{\mathcal{F}}_0^p\left(q\right))$, $A\notin \left({\mathcal{C}}_{p0}\left(p\right)\right),{\mathcal{F}}^p\left(q\right))$ and $A\notin \left({\mathcal{C}}_p\left(p\right)\right),{\mathcal{F}}^p\left(q\right))$.

Now we verify that both conditions of Theorem 4 are satisfied. Let $M\in \mathbb{N}$ be fixed, then

$$\begin{array}{cc}\hfill \underset{m,n}{sup}{\left(\sum _{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}\right)}^{q_{mn}}& =\underset{m,n}{sup}{\displaystyle \frac{1}{mn}}\left(\sum _{k=1}^m{\displaystyle \frac{M^k}{k!}}\right)\left(\sum _{l=1}^n{\displaystyle \frac{M^l}{l!}}\right)\hfill \\ \hfill & \le \underset{m,n}{sup}{\displaystyle \frac{{(e^M-1)}^2}{mn}}={(e^M-1)}^2.\hfill \end{array}$$

Hence, condition (B1) of Theorem 4 is satisfied. Moreover,

$${\left(\sum _{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}\right)}^{q_{mn}}\to 0\mathrm{as}m,n\to \infty .$$

Since the averaged element $a(m,n,r,s,k,l)$ is a Cesàro mean of terms that tend to 0, the limit of the supremum over $r,s$ is also 0. So, condition (A1) of Theorem 4 also follows, and we have $A\in ({\mathcal{M}}_u\left(p\right),{\mathcal{F}}_0^p\left(q\right))$. Consequently, $A\in ({\mathcal{M}}_u\left(p\right),{\mathcal{F}}^p\left(q\right))$.

Next we show that $A\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{F}}_0^p\left(q\right))$. Condition (A1) of Theorem 7 requires ${lim}_{m,n}{sup}_{r,s}\left|a(m,n,r,s,k,l)\right|=0$. For our matrix,

$$a_{mnkl}={\displaystyle \frac{1}{k!l!mn}}\to 0\mathrm{as}m,n\to \infty ,$$

and hence the Cesàro averages also tend to 0. Since ${\displaystyle \frac{M^{k+l}}{k!l!mn}}$ is bounded for any fixed M and $m,n\in \mathbb{N}$, condition (C1) of Theorem 7 follows. Condition (B1) of Theorem 7 requires boundedness of the weighted matrix elements. Since ${\displaystyle \frac{M^{k+l}}{k!l!mn}}$ is bounded for any fixed M, and the average preserves this, the condition holds. Therefore, by Theorem 7, $A\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{F}}_0^p\left(q\right))$, and consequently $A\in ({\mathcal{M}}_0^2\left(p\right),{\mathcal{F}}^p\left(q\right))$.

Now we verify that $A\in ({\mathcal{L}}_u\left(p\right),{\mathcal{F}}_0^p\left(q\right))$ (Theorem 13). Condition (A1) of Theorem 13 has already been verified. We check condition (M1) of Theorem 13 for $K_1={\mathbb{N}}^2$ (since $p_{kl}\le 1$):

$$\underset{m,n>D}{sup}\underset{r,s}{sup}\underset{k,l}{sup}{\left(\right|a(m,n,r,s,k,l)\left|L\right)}^{p_{kl}}<\infty .$$

First we estimate $\left|a\right(m,n,r,s,k,l\left)\right|$:

$$\begin{array}{cc}\hfill a(m,n,r,s,k,l)& \le {\displaystyle \frac{1}{k!l!}}·{\displaystyle \frac{1}{(m+1)(n+1)}}\sum _{i=0}^m\sum _{j=0}^n{\displaystyle \frac{1}{(r+i)(s+j)}}\hfill \\ \hfill & \le {\displaystyle \frac{1}{(m+1)(n+1)}}\left(\sum _{i=1}^{m+1}{\displaystyle \frac{1}{i}}\right)\left(\sum _{j=1}^{n+1}{\displaystyle \frac{1}{j}}\right)\hfill \\ & \sim {\displaystyle \frac{1}{(m+1)(n+1)}}(ln(m+1)+\gamma )(ln(n+1)+\gamma )<1.\hfill \end{array}$$

Thus, the term inside the parentheses is bounded by $L^{1/2}$, and condition (M1) of Theorem 13 follows. Condition (M2) of Theorem 13 holds trivially since $K_2=\varnothing$. Thus, by Theorem 13, $A\in ({\mathcal{L}}_u\left(p\right),{\mathcal{F}}_0^p\left(q\right))$, and consequently $A\in ({\mathcal{L}}_u\left(p\right),{\mathcal{F}}^p\left(q\right))$.

Finally, we will show that $A\in (\mathcal{B}V,{\mathcal{F}}_0^p\left(q\right))$. Condition (B1) of Theorem 14 involves the partial sums $A_{mnkl}={\sum }_{i=1}^k{\sum }_{j=1}^{l}a(m,n,r,s,i,j)$:

$$\begin{array}{cc}\hfill A_{mnkl}& \le {\displaystyle \frac{1}{(m+1)(n+1)}}\left(\sum _{i=1}^k{\displaystyle \frac{1}{i!}}\right)\left(\sum _{j=1}^l{\displaystyle \frac{1}{j!}}\right)\left(\sum _{\mu =0}^m{\displaystyle \frac{1}{r+\mu }}\right)\left(\sum _{\eta =0}^n{\displaystyle \frac{1}{s+\eta }}\right)\hfill \\ \hfill & <{(e-1)}^2·{\displaystyle \frac{1+ln(m+1)}{m+1}}·{\displaystyle \frac{1+ln(n+1)}{n+1}}\le {(e-1)}^2.\hfill \end{array}$$

So condition (B1) of Theorem 14 follows. We have seen already that condition (A1) of Theorem 14 is satisfied for ${\alpha }_{kl}=0$ ($k,l\in \mathbb{N}$). For condition (M1) of Theorem 14, we have:

$${\left(\left|\sum _{i=1}^k\sum _{j=1}^l(a(m,n,r,s,i,j)-{\alpha }_{kl})\right|L^{1/q_{mn}}\right)}^{p_{kl}}=\left(\right|A_{mnkl}{\left|L\right)}^{1/(k+l)},$$

and

$$\underset{m,n>D}{sup}\underset{r,s,k,l}{sup}\left(\right|A_{mnkl}{\left|L\right)}^{1/(k+l)}\le \underset{k,l}{sup}{\left({(e-1)}^{2}L\right)}^{1/(k+l)}={\left({(e-1)}^{2}L\right)}^{1/2}.$$

Condition (M2) of Theorem 14 requires

$$\underset{m,n}{lim}\underset{r,s}{sup}|\sum _{k,l}a(m,n,r,s,k,l)-\alpha |=0.$$

In a similar way to condition (B1), we get

$$\sum _{k,l}a(m,n,r,s,k,l)\le {(e-1)}^2·{\displaystyle \frac{1+ln(m+1)}{m+1}}·{\displaystyle \frac{1+ln(n+1)}{n+1}}.$$

So (A2) follows with $\alpha =0$. Hence, by Theorem 14, $A\in (\mathcal{B}V,{\mathcal{F}}_0^p\left(q\right))$ and consequently $A\in (\mathcal{B}V,{\mathcal{F}}^p\left(q\right))$.

We now modify the previous example slightly so that it satisfies Theorems 1–3, as well as Corollary 1.

Example 3. 

Let $p_{kl}=1/(k+l)$, $q_{kl}=1$$(k,l\in \mathbb{N})$. We consider the matrix $A=\left(a_{mnkl}\right)$ with $a_{mnkl}=0$ for $k>min\{m,2^l\}$ or $l>min\{2^k,n\}$ and $a_{mnkl}={(k!l!mn)}^{-1}$ otherwise ($m,n,k,l\in \mathbb{N}$) (cf. Example 2.17 in [9]).

Then, evidently, conditions (B1)–(B4) of Theorem 1 are satisfied. Condition (M1) holds since

$$\sum _{k,l}|a_{mnkl}|M^{-1/p_{kl}}=\sum _{k=1}^m\sum _{l=1}^n|a_{mnkl}|M^{-1/p_{kl}}<\infty .$$

Using the estimate

$$a(m,n,r,s,k,l)\lesssim {\displaystyle \frac{1}{(m+1)(n+1)}}(ln(m+1)+\gamma )(ln(n+1)+\gamma )$$

from the previous example, we obtain (A1). For condition (M2), we estimate

$$\begin{array}{cc}\hfill \mathsf{\Sigma }& =L^{1/q_{mn}}\sum _{k=1}^{\infty }\sum _{l=1}^{\infty }\left|a(m,n,r,s,k,l)\right|M^{-k-l}\hfill \\ \hfill & \le L{\displaystyle \frac{1}{(m+1)(n+1)}}\sum _{k=1}^{\infty }\sum _{l=1}^{\infty }{\displaystyle \frac{M^{-k}{M}^{-l}}{k!l!}}\left(\sum _{i=0}^m{\displaystyle \frac{1}{r+i}}\right)\left(\sum _{j=0}^n{\displaystyle \frac{1}{s+j}}\right)\hfill \\ \hfill & \le L{\displaystyle \frac{{(e^{-M}-1)}^2}{(m+1)(n+1)}}\left[1+ln\left(1+m\right)\right]\left[1+ln\left(1+n\right)\right]\le L{(e^{-M}-1)}^2.\hfill \end{array}$$

Hence, condition (M2) of Theorem 1 follows. Therefore, $A\in \left({\mathcal{C}}_{p0}\left(p\right)\right),{\mathcal{F}}_0^p\left(q\right))$, and consequently $A\in \left({\mathcal{C}}_p\left(p\right)\right),{\mathcal{F}}_0^p\left(q\right))$.

Condition (A2) of Corollary 1 follows in a similar way to condition (A2) of Theorem 14 in the previous example. Thus, by Corollary 1, $A\in {({\mathcal{C}}_p\left(p\right),{\mathcal{F}}_0^p\left(q\right))}_{\nu }$, and consequently $A\in \left({\mathcal{C}}_p\left(p\right)\right),{\mathcal{F}}^p{\left(q\right))}_{\nu }$.

3. Conclusions

In this paper, we have studied four-dimensional matrix transformations between spaces of generalized almost convergent double sequences with powers and Maddox-type double sequence spaces with variable exponents. The obtained characterizations extend classical one-dimensional results of Lorentz, Maddox, and Nanda to a genuinely two-dimensional framework.

A key feature of the analysis is the use of Cesàro-type averaging and the reduction method, which allows the treatment of matrix maps between double sequence spaces via suitable representations in terms of sequence spaces.

The presented framework unifies several known summability results and provides a systematic approach to multidimensional convergence under nonuniform growth conditions.

Future research may include further extensions to more general index sets, connections with ideal convergence, and potential applications to stability analysis of discrete operators arising in numerical analysis and approximation theory.