Matrix Transformations of Double Convergent Sequences with Powers for the Pringsheim Convergence

Abstract

In 2004–2006, the corresponding double sequence spaces were defined for the Pringsheim and the bounded Pringsheim convergence by Gokhan and Colak. In 2009, Colak and Mursaleen characterized some classes of matrix transformations transforming the space of bounded Pringsheim convergent (to 0) double sequences with powers and the space of uniformly bounded double sequences with powers to the space of (bounded) Pringsheim convergent (to 0) double sequences. But many of their results appeared to be wrong. In 2024, we gave corresponding counterexamples and proved the correct results. Moreover, we gave the conditions for a wider class of matrices. As is well known, convergence of a double sequence in Pringsheim’s sense does not imply its boundedness. Assuming, in addition, boundedness for double sequences usually simplifies proofs. In this paper, we characterize matrix transformations transforming the space of Pringsheim convergent (to 0) double sequences with powers or the space of ultimately bounded double sequences with powers without assuming uniform boundedness.

1. Introduction

In [1], Maddox generalized the spaces $c_0,c,{\ell }_{\infty }$ by adding powers $p_k$ $(k\in \mathbb{N})$ in the definitions of the spaces to the terms of elements of sequences $\left(x_k\right)$. In [2,3,4], the corresponding double sequence spaces were defined for the Pringsheim and the bounded Pringsheim convergence. In [5], we additionally defined the corresponding double sequence space for the regular convergence. In [6], the authors characterized some classes of matrix transformations $(E,F)$ where E is the space of bounded Pringsheim convergent (to 0) double sequences with powers or the space of uniformly bounded double sequences with powers. But many of their results appeared to be wrong. In [5], we gave corresponding counterexamples and proved the correct results. In this paper, we characterize matrix transformations $(E,F)$ where E is the space of Pringsheim convergent (to 0) double sequences with powers or the space of ultimately bounded double sequences with powers.

Note that assuming uniform boundedness in double sequence spaces E greatly simplifies their structure, making them more analogous to single sequence spaces and easing the argumentation for four-dimensional matrices. In this paper, however, we drop the assumption of uniform boundedness, which results in more challenging argumentation and more interesting conditions for matrices.

First, let us introduce the notions and notation we need in this paper.

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

The following variable exponent spaces were defined by Maddox [1] and Nakano [7]:

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

where $\omega$ is the space of all complex (or real) sequences. When all the terms of $\left(p_k\right)$ are constant, we have ${\ell }_{\infty }\left(p\right)={\ell }_{\infty }$, $c\left(p\right)=c$ and $c_0\left(p\right)=c_0$, where ${\ell }_{\infty }$, c, $c_0$ are, respectively, the spaces of bounded, convergent, and null sequences.

These variable exponent spaces allow us to generalize classical convergence notions and accommodate cases with non-uniform growth conditions, making them particularly useful for studying summability transformations.

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

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

Pringsheim convergence is one of the fundamental notions of convergence for double sequences, with applications in signal processing, numerical methods, and random matrix theory. If, in addition, ${sup}_{k,l}\left|x_{kl}\right|<\infty$, or the limits ${lim}_k{x}_{kl}$ $(l\in \mathbb{N})$ and ${lim}_l{x}_{kl}$ $(k\in \mathbb{N})$ exist, then x is said to be boundedly convergent to a in Pringsheim’s sense (shortly, $bp$-convergent to a) and regularly convergent to a (shortly, r-convergent to a), respectively.

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 ${\left({\sum }_{k=1}^m{\sum }_{l=1}^n{x}_{kl}\right)}_{m,n}\in {\mathcal{C}}_{\nu }$. We also use the notations

$$\begin{array}{ccc}\hfill {\mathcal{M}}_u& :=& \left\{x\in \mathsf{\Omega }|\underset{k,l}{sup}\left|x_{kl}\right|<\infty \right\},\hfill \\ \hfill {\mathcal{M}}_p& :=& \left\{x\in \mathsf{\Omega }|\exists B\in \mathbb{N}:\underset{k,l>B}{sup}\left|x_{kl}\right|<\infty \right\},\hfill \\ \hfill {\mathcal{C}}_{\nu 0}& :=& \left\{x\in \mathsf{\Omega }|\nu -\underset{m,n}{lim}{x}_{mn}=0\right\},\hfill \\ \hfill {\mathcal{C}S}_{\nu }& :=& \left\{x\in \mathsf{\Omega }|{\left(\sum _{k,l=1}^{m,n}{x}_{kl}\right)}_{m,n}\in {\mathcal{C}}_{\nu }\right\}.\hfill \end{array}$$

Given a double sequence 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 [8].

Let $p=\left(p_{kl}\right)$ be a bounded double sequence of strictly positive numbers. Then, we set

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

$$\begin{array}{ccc}\hfill {\mathcal{C}}_{\nu }\left(p\right)& :=& \left\{x\in \mathsf{\Omega }|\exists a\in K:(x_{kl}-a)\in {\mathcal{C}}_{\nu 0}\left(p\right)\right\}(\nu \in \{p,bp,r\}).\hfill \end{array}$$

In [5], we verified that

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

Analogously, we will show that

$${\mathcal{M}}_p\left(p\right)=\bigcup _N\{x\in \mathsf{\Omega }:\left(|x_{kl}|N^{1/p_{kl}}\right)\in {\mathcal{M}}_p\}.$$

To see this, we use the same technique as in [9]. Indeed, if $x\in {\mathcal{M}}_p\left(p\right)$, then there are some $N,n_0\in \mathbb{N}$ with $|x_{kl}{|}^{p_{kl}}\le N$ ($k,l>n_0$); hence $|x_{kl}|N^{-1/p_{kl}}\le 1$ for $k,l>n_0$. Conversely, if $|x_{kl}|N^{-1/p_{kl}}\le M$ for some $M>0$, then for all $k,l>n_0$, we have $|x_{kl}{|}^{p_{kl}}\le M^{p_{kl}}N$, which is bounded since $\left(p_{kl}\right)$ is bounded. Hence, $x\in {\mathcal{M}}_p\left(p\right)$.

In [3], the authors verified that

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

Now, we will find the $\beta \left(\nu \right)$-dual of the space ${\mathcal{C}}_{p0}\left(p\right)$. Since the space ${\mathcal{C}}_{p0}\left(p\right)$ is solid, then ${\mathcal{C}}_{p0}{\left(p\right)}^{\alpha }={\mathcal{C}}_{p0}{\left(p\right)}^{\beta \left(\nu \right)}$ $(\nu \in \{p,bp,r\left\}\right)$ [10]. Since for every $l\in \mathbb{N}$ and $x=x{e}^l\in \mathsf{\Omega }$, we have $x\in {\mathcal{C}}_{p0}\left(p\right)$, so for every $y\in {\mathcal{C}}_{p0}{\left(p\right)}^{\alpha }$, we have ${\left(y_{kl}\right)}_k\in \phi$. Analogously, for every $k\in \mathbb{N}$ and $y\in {\mathcal{C}}_{p0}{\left(p\right)}^{\alpha }$, we have ${\left(y_{kl}\right)}_l\in \phi$, where $\phi$ is the space of all finite sequences.

For $z\in \phi$, let $l\left(z\right)$ be the index $k_0$ such that $z_{k_0}\ne 0$ and $z_k=0$ for $k>k_0$. For $z=0$, we set $l\left(z\right):=1$.

For $y\in {\mathcal{C}}_{p0}{\left(p\right)}^{\alpha }$, we define the map ${\phi }_y:\mathsf{\Omega }\to \omega$ in the following way: We set $k_1:=l_1:=l\left({\left(y_{1l}\right)}_l\right)$, ${\left({\phi }_y\left(x\right)\right)}_1:=x_{11},\dots ,{\left({\phi }_y\left(x\right)\right)}_{l_1}:=x_{1k_1}$. Then, $k_2:=l\left({\left(y_{k1}\right)}_k\right)$, $l_2:=l_1+k_2-1$, ${\left({\phi }_y\left(x\right)\right)}_{l_1+1}:=x_{21},\dots ,{\left({\phi }_y\left(x\right)\right)}_{l_2}:=x_{k_{2}1}$. We put $k_3:=l\left({\left(y_{2l}\right)}_l\right)$, $l_3:=l_2+k_3-1$, ${\left({\phi }_y\left(x\right)\right)}_{l_2+1}:=x_{22},\dots ,{\left({\phi }_y\left(x\right)\right)}_{l_3}:=x_{2k_3}$. Further, we set $k_4:=l\left({\left(y_{k2}\right)}_k\right)$, $l_4:=l_3+k_4-2$, ${\left({\phi }_y\left(x\right)\right)}_{l_3+1}:=x_{32},\dots ,{\left({\phi }_y\left(x\right)\right)}_{l_4}:=x_{k_{4}2}$. Continuing in the same way, we obtain the map ${\phi }_y:\mathsf{\Omega }\to \omega$ and let $\mathsf{\Phi }:\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ be the corresponding map of indexes. Note that ${\phi }_y\left({\mathcal{C}}_{p0}\left(p\right)\right)=c_0\left(\mathsf{\Phi }\left(p\right)\right)$. So, $y\in {\mathcal{C}}_{p0}{\left(p\right)}^{\alpha }$ iff

$${\phi }_y\left(y\right)\in c_0{\left(\mathsf{\Phi }\left(p\right)\right)}^{\alpha }=\bigcup _{N\in \mathbb{N}}\left\{\left(z_k\right):\sum _k\left|z_k\right|N^{-1/{\left(\mathsf{\Phi }\left(p\right)\right)}_k}<\infty \right\}.$$

Therefore,

$$\begin{array}{ccc}\hfill {\mathcal{C}}_{p0}{\left(p\right)}^{\alpha }& =& {\mathcal{C}}_{p0}{\left(p\right)}^{\beta \left(\nu \right)}=M_0^p\left(p\right)\hfill \\ & :=& M_0^2\left(p\right)\bigcap \left\{\left(y_{kl}\right):\forall l\in \mathbb{N}:{\left(y_{kl}\right)}_k\in \phi ,\forall k\in \mathbb{N}:{\left(y_{kl}\right)}_l\in \phi \right\},\hfill \end{array}$$

where

$$M_0^2\left(p\right):=\bigcup _{N\in \mathbb{N}}\left\{\left(y_{kl}\right):\sum _{k,l}\left|y_{kl}\right|N^{-1/p_{kl}}<\infty \right\}.$$

Since ${\mathcal{C}}_p\left(p\right)={\mathcal{C}}_{p0}\left(p\right)\oplus <e>$, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{C}}_p{\left(p\right)}^{\beta \left(\nu \right)}={({\mathcal{C}}_{p0}\left(p\right)\oplus <e>)}^{\beta \left(\nu \right)}=M^p\left(p\right)& :=& M_0^p\left(p\right)\bigcap {(<e>)}^{\beta \left(\nu \right)}\hfill \\ & =& M_0^p\left(p\right)\bigcap {\mathcal{C}\mathcal{S}}_{\nu }.\hfill \end{array}$$

Now, we will find the $\beta \left(\nu \right)$-dual of the space $M_p\left(p\right)$. The space $M_p\left(p\right)$ is solid so ${\mathcal{M}}_p{\left(p\right)}^{\alpha }={\mathcal{M}}_p{\left(p\right)}^{\beta \left(\nu \right)}$ $(\nu \in \{p,bp,r\left\}\right)$ [10]. We will verify that

$$\begin{array}{ccc}\hfill {\mathcal{M}}_p{\left(p\right)}^{\beta \left(\nu \right)}& =& M_{\infty }^p\left(p\right):=\hfill \\ & :=& M_{\infty }^2\left(p\right)\cap \left\{\left(y_{kl}\right):\forall l\in \mathbb{N}{\left(y_{kl}\right)}_k\in \phi \mathrm{and}\forall k\in \mathbb{N}{\left(y_{kl}\right)}_l\in \phi \right\}.\hfill \end{array}$$

First, in the same way as for ${\mathcal{C}}_{p0}\left(p\right)$, we see that for any $y\in {\mathcal{M}}_p{\left(p\right)}^{\alpha }$, its rows and columns are finite sequences. Next, since ${\mathcal{M}}_u\left(p\right)\subset {\mathcal{M}}_p\left(p\right)$, then ${\mathcal{M}}_p{\left(p\right)}^{\beta \left(\nu \right)}\subset {\mathcal{M}}_u{\left(p\right)}^{\beta \left(\nu \right)}=M_{\infty }^2\left(p\right)$.

Now, we will prove that if $y\in M_{\infty }^p\left(p\right)$ and $x\in {\mathcal{M}}_p\left(p\right)$, then ${\sum }_{k,l}\left|x_{kl}{y}_{kl}\right|<\infty$. Let $N\in \mathbb{N}$ be such that ${sup}_{k,l>N}{\left|x_{kl}\right|}^{p_{kl}}<\infty$. Let $\tilde{x}$ be defined by ${\tilde{x}}_{kl}:=x_{kl}$ for $k,l>N$ and ${\tilde{x}}_{kl}:=0$ otherwise. Then, $\tilde{x}\in {\mathcal{M}}_u\left(p\right)$. Hence,

$$\sum _{k,l}|{\tilde{x}}_{kl}{y}_{kl}|=\sum _{k,l=N+1}^{\infty }\left|x_{kl}{y}_{kl}\right|<\infty .$$

Since y has finite rows and columns, we have

$$\sum _k\sum _{l=1}^N|x_{kl}{y}_{kl}|<\infty ,\sum _l\sum _{k=1}^N\left|x_{kl}{y}_{kl}\right|<\infty .$$

Hence,

$$\begin{array}{c}\hfill \sum _{k,l}|x_{kl}{y}_{kl}|\le \sum _{k,l=N+1}^{\infty }|x_{kl}{y}_{kl}|+\sum _k\sum _{l=1}^N|x_{kl}{y}_{kl}|+\sum _{l=N+1}^{\infty }\sum _{k=1}^N\left|x_{kl}{y}_{kl}\right|<\infty .\end{array}$$

Let $A=\left(a_{mnkl}\right)$ be any four-dimensional scalar matrix and $\nu$ be some convergence notion of 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, if X is solid), we use the notation $A\in (X,Y)$. In addition, 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 $x\in X$. If $X,Y$ are sequence spaces and $B=\left(b_{nk}\right)$ is a two-dimensional matrix, we use the notation $B\in (X,Y)$ if and only if $Bx:={\left({\sum }_k{b}_{nk}{x}_k\right)}_n$ exists and $Bx\in Y$ whenever $x=\left(x_k\right)\in X$.

In this paper, we give the conditions for the classes of matrices ${(E,F)}_{\nu }$ where $E\in \{{\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_p\left(p\right),{\mathcal{M}}_p\left(p\right)\}$ and $F\in \left\{{\mathcal{M}}_p\left(q\right),\right.{\mathcal{M}}_u\left(q\right),{\mathcal{C}}_{p0}\left(q\right),{\mathcal{C}}_p\left(q\right),$ ${\mathcal{C}}_{bp0}\left(q\right),{\mathcal{C}}_{bp}\left(q\right),{\mathcal{C}}_{r0}\left(q\right),\left.{\mathcal{C}}_r\left(q\right)\right\}$.

2. Main Results

In what follows, we assume that $p=\left(p_{kl}\right)$, $q=\left(q_{kl}\right)$ are bounded double sequences of strictly positive numbers. To characterize matrix transformations of double convergent sequences with powers, we will use the following results from [5] supplementing Corollary 2.3 with characterization of matrices in $(X,{\mathcal{M}}_p\left(p\right))$ where X is a sequence space.

Lemma 1

[5]. Let X be a sequence space,

(a) $A\in (X,{\mathcal{C}}_{\nu 0}\left(q\right))$ iff $\forall L\in \mathbb{N}:\left(a_{mnkl}{L}^{1/q_{mn}}\right)\in (X,{\mathcal{C}}_{\nu 0})$.

(b) $A\in (X,{\mathcal{M}}_u\left(q\right))$ iff $\exists L\in \mathbb{N}:\left(a_{mnkl}{L}^{-1/q_{mn}}\right)\in (X,{\mathcal{M}}_u)$.

Proposition 1

[5]. Let $x=\left(x_{kl}\right)\in \mathsf{\Omega }$. Then,

(a) $x\in {\mathcal{C}}_p\left(p\right)$ iff $\exists p_x\in \mathbb{K}\forall$ index sequences $\left(k_i\right),\left(l_i\right)$:

$$\underset{i}{lim}{|x_{k_i{l}_i}-p_x|}^{p_{k_i{l}_i}}=0.$$

(b) $x\in {\mathcal{C}}_{bp}\left(p\right)$ iff $x\in {\mathcal{C}}_p\left(p\right)$ and $\forall \left(k_i\right),\left(l_i\right)$ in $\mathbb{N}$: $\left(x_{k_i{l}_i}\right)\in {\ell }_{\infty }\left(p_{k_i{l}_i}\right)$.

(c) $x\in {\mathcal{M}}_u\left(p\right)$ iff $\forall \left(k_i\right),\left(l_i\right)$ in $\mathbb{N}$: $\left(x_{k_i{l}_i}\right)\in {\ell }_{\infty }\left(p_{k_i{l}_i}\right)$.

(d) $x\in {\mathcal{M}}_p\left(p\right)$ iff ∀ index sequences $\left(k_i\right),\left(l_i\right)$: $\left(x_{k_i{l}_i}\right)\in {\ell }_{\infty }\left(p_{k_i{l}_i}\right)$.

Corollary 1

[5]. Let $B=\left(b_{mnk}\right)$ be a three-dimensional matrix and let X be a sequence space with $X\subset {\mathsf{\Omega }}_B$. Then, the following statements hold:

(a) $B\in (X,{\mathcal{C}}_p\left(p\right))$ iff ∀ index sequences $\left(m_i\right),\left(n_i\right):$ ${\left(b_{m_i{n}_{i}k}\right)}_{i,k}\in (X,c\left({\left(p_{m_i{n}_i}\right)}_i\right))$ and all these matrices are pairwise consistent on X.

(b) $B\in (X,{\mathcal{C}}_{bp}\left(p\right))$ iff $B\in (X,{\mathcal{C}}_p\left(p\right))$ and $\forall \left(m_i\right),\left(n_i\right)$ in $\mathbb{N}:$ ${\left(b_{m_i{n}_{i}k}\right)}_{i,k}\in (X,{\ell }_{\infty }\left({\left(p_{m_i{n}_i}\right)}_i\right))$.

(c) $B\in (X,{\mathcal{C}}_{p0}\left(p\right))$ iff ∀ index sequences $\left(m_i\right),\left(n_i\right):{\left(b_{m_i{n}_{i}k}\right)}_{i,k}\in (X,c_0\left({\left(p_{m_i{n}_i}\right)}_i\right))$.

(d) $B\in (X,{\mathcal{C}}_{bp0}\left(p\right))$ iff $B\in (X,{\mathcal{C}}_{p0}\left(p\right))$ and $\forall \left(m_i\right),\left(n_i\right)$ in $\mathbb{N}:$ ${\left(b_{m_i{n}_{i}k}\right)}_{i,k}\in (X,{\ell }_{\infty }\left({\left(p_{m_i{n}_i}\right)}_i\right))$.

(e) $B\in (X,{\mathcal{M}}_u\left(p\right))$ iff $\forall \left(m_i\right),\left(n_i\right)$ in $\mathbb{N}:$ ${\left(b_{m_i{n}_{i}k}\right)}_{i,k}\in (X,{\ell }_{\infty }\left({\left(p_{m_i{n}_i}\right)}_i\right))$.

(f) $B\in (X,{\mathcal{M}}_p\left(p\right))$ iff ∀ index sequences $\left(m_i\right),\left(n_i\right)$ in $\mathbb{N}:$ ${\left(b_{m_i{n}_{i}k}\right)}_{i,k}\in (X,{\ell }_{\infty }\left({\left(p_{m_i{n}_i}\right)}_i\right))$.

Now, we consider the main results of our paper. We used some conditions in more than one theorem, so we used the same numbering system instead of writing them repeatedly.

Theorem 1.

$A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{M}}_u\left(q\right))$ iff the following conditions hold:

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

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

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

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

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

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

Proof. 

Necessity. (i), (ii), and (v) follow since ${\left(a_{mnkl}\right)}_{k,l}\in {\mathcal{C}}_{p0}{\left(p\right)}^{\beta \left(\nu \right)}$ $(m,n\in \mathbb{N})$.

Since given $k\in \mathbb{N}$, the map ${\left(a_{mnkl}\right)}_{m,n,l}:\omega \to {\mathcal{M}}_u\left(q\right)$, then by Lemma 1 (b), $\forall L\in \mathbb{N}:{\left(a_{mnkl}{L}^{-1/q_{mn}}\right)}_{m,n,l}:\omega \to {\mathcal{M}}_u$. Let $L\in \mathbb{N}$ be fixed and consider the matrix $C:={\left(c_{mnl}\right)}_{m,n,l}={\left(a_{mnkl}{L}^{-1/q_{mn}}\right)}_{m,n,l}$. We can identify the double sequence space ${\mathcal{M}}_u$ with ${\ell }_{\infty }$ where $s=\left(s_i\right)=T\left(q\right)$ and the three-dimensional matrix C with the two-dimensional matrix $D=\left(d_{il}\right)$. Then, $D:w\to {\ell }_{\infty }$.

By Theorem 6.1 in [11] (Chap. 4), there exists $L\left(k\right)\in \mathbb{N}$ such that $d_{il}=0$ for $l>L\left(k\right)$ $(i\in \mathbb{N})$. Hence, $c_{mnl}=a_{mnkl}{L}^{-1/q_{mn}}=0$ for $l>L\left(k\right)$ $(m,n\in \mathbb{N})$. So (iii) follows. The statement (iv) follows in a similar way.

To prove (vi) in a similar way with the map ${\phi }_y$, we define the map ${\phi }_A:\mathsf{\Omega }\to \omega$ and let ${\mathsf{\Phi }}_A:\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ be the corresponding map of indexes, where we use $L\left(k\right)$ instead of $l\left({\left(y_{kl}\right)}_l\right)$ and $K\left(l\right)$ instead of $l\left({\left(y_{kl}\right)}_k\right)$. We note that ${\phi }_A\left({\mathcal{C}}_{p0}\left(p\right)\right)=c_0\left(r\right)$, where $r={\mathsf{\Phi }}_A\left(p\right)$. Let $B=\left(b_{mni}\right)$ be the three-dimensional matrix with ${\left(b_{mni}\right)}_i:={\phi }_A\left({\left(a_{mnkl}\right)}_{k,l}\right)$ $(m,n\in \mathbb{N})$. Then, $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{M}}_u\left(q\right))$ implies $B\in (c_0\left(r\right),{\mathcal{M}}_u\left(q\right))$. We can identify the double sequence space ${\mathcal{M}}_u\left(q\right)$ with ${\ell }_{\infty }\left(s\right)$ where $\left(s_i\right)=T\left(q\right)$ and the matrix B with the two-dimensional matrix $C=\left(c_{ik}\right)$. Then, $C\in (c_0\left(r\right),{\ell }_{\infty }\left(s\right))$. By Theorem 5.1, 13, in [12], this implies

$$\begin{array}{c}\hfill \exists M\in \mathbb{N}:\underset{i}{sup}{\left(\sum _k\left|c_{ik}\right|M^{-1/r_k}\right)}^{s_i}=\underset{m,n}{sup}{\left(\sum _{k,l}\left|a_{mnkl}\right|M^{-1/p_{kl}}\right)}^{q_{mn}}<\infty ,\end{array}$$

so (vi) follows.

Sufficiency. By (i), (ii), and (v), it follows that $Ax$ exists for every $x\in {\mathcal{C}}_{p0}\left(p\right)$. Suppose, on the contrary, that $Ax\notin {\mathcal{M}}_u\left(q\right)$ for some $x\in {\mathcal{C}}_{p0}\left(p\right)$. Then, $\left({\left[Ax\right]}_{m_i,n_i}\right)\notin {\ell }_{\infty }\left({\left(q_{m_i{n}_i}\right)}_i\right)$ for some sequences of integers $\left(m_i\right),\left(n_i\right)$ with $m_i+n_i\to \infty$. We consider the maps ${\phi }_A$ and ${\mathsf{\Phi }}_A$ defined in Necessity proof. Let $B=\left(b_{ij}\right)$ be the two-dimensional matrix with ${\left(b_{ij}\right)}_j:={\phi }_A\left({\left(a_{m_i{n}_{i}kl}\right)}_{k,l}\right)$ $(i\in \mathbb{N})$. Let $z:={\phi }_A\left(x\right)$ and $r={\mathsf{\Phi }}_A\left(p\right)$. Then, $z\in c_0\left(r\right)$. So $B\notin (c_0\left(r\right),{\ell }_{\infty }\left({\left(q_{m_i{n}_i}\right)}_i\right))$. Then, by Theorem 5.1, 13, in [12], the following condition does not hold:

$$\exists M\in \mathbb{N}:\underset{i}{sup}{\left(\sum _k|b_{ik}|M^{-1/r_k}\right)}^{q_{m_i{n}_i}}<\infty .$$

Hence, the condition (vi) does not hold. So the contradiction follows. □

Theorem 2.

(${\mathbf{a}}_{\mathbf{1}}$) $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_p\left(q\right))$ iff (i), (ii), (v), and the following conditions hold:

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

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

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

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

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

(${\mathbf{b}}_{\mathbf{1}}$) $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_{bp}\left(q\right))$ iff (i)-(v), (vii)-(xi) and the following condition hold:

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

(${\mathbf{c}}_{\mathbf{1}}$) $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_r\left(q\right))$ iff (i)-(v), (vii)-(xi) and the following conditions hold:

(xiii) $\exists \left({\beta }_{mkl}\right)$: ${lim}_n{|a_{mnkl}-{\beta }_{mkl}|}^{q_{mn}}=0$ $(m,k,l\in \mathbb{N});$

(xiv) $\exists \left({\gamma }_{nkl}\right)$: ${lim}_m{|a_{mnkl}-{\gamma }_{nkl}|}^{q_{mn}}=0$ $(n,k,l\in \mathbb{N});$

(xv) $\forall L,m\exists M:{sup}_n{L}^{1/q_{mn}}{\sum }_{k,l}|a_{mnkl}-{\beta }_{mkl}|M^{-1/p_{kl}}<\infty ;$

(xvi) $\forall L,n\exists M:{sup}_m{L}^{1/q_{mn}}{\sum }_{k,l}|a_{mnkl}-{\gamma }_{nkl}|M^{-1/p_{kl}}<\infty .$

Proof. 

(${\mathbf{a}}_{\mathbf{1}}$) Necessity. (vii) follows since $e^{kl}\in {\mathcal{C}}_{p0}\left(p\right)$ $(k,l\in \mathbb{N})$. (i), (ii), and (v) follow since ${\left(a_{mnkl}\right)}_{k,l}\in {\mathcal{C}}_{p0}{\left(p\right)}^{\beta \left(\nu \right)}=M_0^2\left(p\right)$ $(m,n\in \mathbb{N})$.

Suppose, on the contrary, that (viii) does not hold for some $k_0\in \mathbb{N}$. Then, we can find index sequences $\left(m_i\right),\left(n_i\right),\left(l_i\right)$ such that $a_{m_i{n}_i{k}_0{l}_i}\ne 0$ $(i\in \mathbb{N})$. Then, we define $x=\left(x_{kl}\right)$ in the following way: We set $x_{kl}:=0$ for $k\ne k_0$, $x_{k_{0}l}:=0$ for $l\notin \{l_i:i\in \mathbb{N}\}$, $x_{k_0{l}_1}:=1/a_{m_1{n}_1{k}_0{l}_1}$, and

$$x_{k_0{l}_i}:={\displaystyle \frac{1}{a_{m_i{n}_i{k}_0{l}_i}}}\left(i-\sum _{j=1}^{i-1}{a}_{m_i{n}_i{k}_0{l}_j}{x}_{k_0{l}_j}\right)(i\ge 2).$$

Then, ${\left(Ax\right)}_{m_i,n_i}=i$ $(i\in \mathbb{N})$. So $Ax\notin {\mathcal{M}}_p\left(q\right)\supset {\mathcal{C}}_p\left(q\right)$ and the contradiction follows. So (viii) follows. (ix) follows in a similar way.

To prove (x), suppose on the contrary that (x) does not hold; then,

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

Then, we can choose index sequences $\left(m_i\right),\left(n_i\right),\left(M_i\right)$ such that

$$\begin{array}{c}\hfill \underset{i}{sup}\sum _{k,l}\left|a_{m_i{n}_{i}kl}\right|M_i^{-1/p_{kl}}=\infty .\end{array}$$

(1)

Now, by (i), (ii), (viii), and (ix), for every $k\in \mathbb{N}$, there exists $L_1\left(k\right)\ge L\left(k\right)$ such that $a_{m_i{n}_{i}kl}=0$ for $l>L_1\left(k\right)$ $(i\in \mathbb{N})$, and for every $l\in \mathbb{N}$, there exists $K_1\left(l\right)\ge K\left(l\right)$ such that $a_{mnkl}=0$ for $k>K_1\left(l\right)$ $(i\in \mathbb{N})$.

In a similar way with the map ${\phi }_y$, we define the map ${\phi }_{m_i{n}_i}:\mathsf{\Omega }\to \omega$, where we use $L_1\left(k\right)$ instead of $l\left({\left(y_{kl}\right)}_l\right)$ and $K_1\left(k\right)$ instead of $l\left({\left(y_{kl}\right)}_k\right)$ and let ${\mathsf{\Phi }}_{m_i{n}_i}:\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ be the corresponding map of indexes. We note that ${\phi }_{m_i{n}_i}\left({\mathcal{C}}_{p0}\left(p\right)\right)=c_0\left(r\right)$, where $r={\mathsf{\Phi }}_{m_i{n}_i}\left(p\right)$. Let $B=\left(b_{qj}\right)$ be the two-dimensional matrix with ${\left(b_{qj}\right)}_j:={\phi }_{m_i{n}_i}\left({\left(a_{m_r{n}_{r}kl}\right)}_{k,l}\right)$ $(q\in \mathbb{N})$. Then, $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_p\left(q\right))$ implies $B\in (c_0\left(r\right),c\left({\left(q_{m_i{n}_i}\right)}_i\right))$. By Theorem 5.1, 9, in [12], this implies that

$$\exists M\in \mathbb{N}:\underset{i}{sup}\sum _k|b_{ik}|M^{-1/r_k}=\underset{i}{sup}\sum _{k,l}\left|a_{m_i{n}_{i}kl}\right|M^{-1/p_{kl}}<\infty ,$$

which contradicts (1).

To prove (xi), suppose on the contrary that (xi) does not hold; then,

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

Then, we can choose index sequences $\left(m_i\right),\left(n_i\right),\left(M_i\right)$ such that

$$\begin{array}{c}\hfill \underset{i}{sup}{L}_0^{1/q_{m_i{n}_i}}\sum _{k,l}|a_{m_i{n}_{i}kl}-a_{kl}|M_i^{-1/p_{kl}}=\infty .\end{array}$$

(2)

Defining the map ${\phi }_{m_i{n}_i}$, r and the two-dimensional matrix $B=\left(b_{ij}\right)$ as in the proof of (x), we have $B\in (c_0\left(r\right),c\left({\left(q_{m_i{n}_i}\right)}_i\right))$. Let $\left(b_k\right)={\phi }_{m_i{n}_i}\left(\left(a_{kl}\right)\right)$. By Theorem 5.1, 9, in [12], we have

$$\begin{array}{ccc}\hfill \forall L\in \mathbb{N}\exists M\in \mathbb{N}:& & \underset{i}{sup}{L}^{1/q_{m_i{n}_i}}\sum _k|b_{ik}-b_k|M^{-1/r_k}\hfill \\ & =& \underset{i}{sup}{L}^{1/q_{m_i{n}_i}}\sum _{k,l}|a_{m_i{n}_{i}kl}-a_{kl}|M^{-1/p_{kl}}<\infty ,\hfill \end{array}$$

which contradicts (2).

Sufficiency. By (i), (ii), and (v), it follows that $Ax$ exists for every $x\in {\mathcal{C}}_{p0}\left(p\right)$. Suppose, on the contrary, that $Ax\notin {\mathcal{C}}_p\left(q\right)$ for some $x\in {\mathcal{C}}_{p0}\left(p\right)$. Then, $\left({\left[Ax\right]}_{m_i,n_i}\right)\notin c\left({\left(q_{m_i{n}_i}\right)}_i\right)$ for some index sequences $\left(m_i\right),\left(n_i\right)$. In the same way as in the necessity proof of (x) and (xi), we define the map ${\phi }_{m_i{n}_i}$ and the matrix map $B=\left(b_{ij}\right)$. Then, by (i), we have

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

where $\left(b_k\right)={\phi }_{m_i{n}_i}\left(\left(a_{kl}\right)\right)$. Since $\left(a_{m_i{n}_{i}kl}\right)\notin ({\mathcal{C}}_{p0}\left(p\right),c\left({\left(q_{m_i{n}_i}\right)}_i\right))$, then $B\notin (c_0\left(r\right),c\left({\left(q_{m_i{n}_i}\right)}_i\right))$. So by Theorem 5.1, 9, in [12], at least one of the following conditions does not hold:

(a) $\exists M\in \mathbb{N}:{sup}_i{\sum }_k\left|b_{ik}\right|M^{-1/r_k}<\infty ;$

(b) $\forall L\in \mathbb{N}\exists M\in \mathbb{N}:{sup}_i{L}^{1/q_{m_i{n}_i}}{\sum }_k|b_{ik}-b_k|M^{-1/r_k}<\infty .$

Hence, at least one of conditions (x) and (xi) does not hold. So the contradiction follows.

(${\mathbf{b}}_{\mathbf{1}}$) The proof follows from ($a_1$) and Theorem 1.

(${\mathbf{c}}_{\mathbf{1}}$) Necessity. (i), (ii), (v), (vii), (viii), and (ix)–(xi) follow from ($a_1$), and (iii) and (xii) follow from ($b_1$), since ${\mathcal{C}}_r\left(q\right)\subset {\mathcal{C}}_{bp}\left(q\right)).$ Let $m\in \mathbb{N}$ be fixed. By (ii) and (viii), for every $k\in \mathbb{N}$, there exists $L_1\left(k\right)\ge L\left(k\right)$ such that $a_{mnkl}=0$ for $l>L_1\left(k\right)$ $(n\in \mathbb{N})$. In a similar way with the map ${\phi }_{m_i{n}_i}$ in the proof of ($a_1$), we define the maps ${\phi }^m:\mathsf{\Omega }\to \omega$, ${\mathsf{\Phi }}^m:\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ and the two-dimensional matrix $B=\left(b_{nj}\right)$ with ${\left(b_{nj}\right)}_j:={\phi }^m\left({\left(a_{mnkl}\right)}_{k,l}\right)$ $(n\in \mathbb{N})$. Then, ${\phi }^m\left({\mathcal{C}}_{p0}\left(p\right)\right)=c_0\left(r\right)$, where $r={\mathsf{\Phi }}^m\left(p\right)$. So $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_r\left(q\right))$ implies $B\in (c_0\left(r\right),c\left({\left(q_{mn}\right)}_n\right))$. By Theorem 5.1, 9, in [12], it follows that

(1) $\exists \left({\alpha }_k\right):{lim}_n{|b_{nk}-{\alpha }_k|}^{q_{mn}}=0$ $(k\in N);$

(2) $\forall L$ $\exists M$: ${sup}_n{L}^{1/q_{mn}}{\sum }_k|b_{nk}-{\alpha }_k|M^{-1/r_k}<\infty$.

Let ${\left({\beta }_{mkl}\right)}_{k,l}={\left({\phi }^m\right)}^{-1}\left({\alpha }_k\right).$ Then, (xiii) and (xv) follow. Analogously, we obtain (xiv) and (xvi).

Sufficiency. From (i), (ii), (v), (vii), (viii), and (ix)-(xi), it follows that ${\left(Ax\right)}_{mn}\in {\mathcal{C}}_p\left(q\right)$ for all $x\in {\mathcal{C}}_{p0}\left(p\right)$. Let $m\in \mathbb{N}$ be fixed. In the same way as in the proof of Necessity, we define the maps ${\phi }^m:\mathsf{\Omega }\to \omega$, ${\mathsf{\Phi }}^m:\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ and the 2 dimensional matrix $B=\left(b_{nj}\right)$. Then, ${\phi }^m\left({\mathcal{C}}_{p0}\left(p\right)\right)=c_0\left(r\right)$, where $r={\mathsf{\Phi }}^m\left(p\right)$.

Let $\left({\alpha }_k\right)={\phi }^m\left({\left({\beta }_{mkl}\right)}_{kl}\right).$ Then, by (xiii) and (xv) (cf. the proof of Necessity), the matrix B satisfies the conditions 1) and 2) in the Necessity proof. Hence, by Theorem 5.1, 9, in [12], $B\in (c_0\left(r\right),c\left({\left(q_{mn}\right)}_n\right))$, implying ${\left({\left(Ax\right)}_{mn}\right)}_n\in c\left({\left(q_{mn}\right)}_n\right)$ for each $x\in {\mathcal{C}}_{p0}\left(p\right)$. Analogously, we can prove ${\left({\left(Ax\right)}_{mn}\right)}_m\in c\left({\left(q_{mn}\right)}_m\right)$ for each $x\in {\mathcal{C}}_{p0}\left(p\right)$ $(n\in \mathbb{N})$. Hence, we obtain ${\left(Ax\right)}_{mn}\in {\mathcal{C}}_r\left(q\right)$ for each $x\in {\mathcal{C}}_{p0}\left(p\right)$. □

Theorem 3.

(${\mathbf{a}}_{\mathbf{2}}$) $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_{p0}\left(q\right))$ iff (i), (ii), (v), (viii), (ix), and the following conditions hold:

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

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

(${\mathbf{b}}_{\mathbf{2}}$) $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_{bp0}\left(q\right))$ iff (i), (ii), (v), (viii), (ix), (xii), and (xvii)–(xviii).

(${\mathbf{c}}_{\mathbf{2}}$) $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_{r0}\left(q\right))$ iff (i), (ii), (v), (viii), (ix), (xiii)–(xvi) and (xvii)–(xviii).

Proof. 

Since $e^{kl}\in {\mathcal{C}}_{p0}\left(p\right)$ $(k,l\in \mathbb{N})$, the proof follows from Theorem 2 by taking $a_{kl}=0$. □

Theorem 4.

$A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{M}}_p\left(q\right))$ iff (i), (ii), (v), (viii), (ix), and the following condition hold:

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

Proof. 

Necessity. (i), (ii), and (v) follow since ${\left(a_{mnkl}\right)}_{k,l}\in {\mathcal{C}}_{p0}{\left(p\right)}^{\beta \left(\nu \right)}=M_0^2\left(p\right)$ $(m,n\in \mathbb{N})$. Necessity of (viii) and (ix) follows in the same way as in Theorem 2.

To prove (xix) in the same way as in the proof of the necessity of (vi) in Theorem 1, we identify the space ${\mathcal{C}}_{p0}\left(p\right)$ with $c_0\left(r\right)$ and the matrix A with the three-dimensional matrix $B=\left(b_{mni}\right)$. Then, $A\in ({\mathcal{C}}_{p0}\left(p\right),{\mathcal{M}}_p\left(q\right))$ implies $B\in (c_0\left(r\right),{\mathcal{M}}_p\left(q\right))$. By Corollary 1 (f), this is equivalent to saying that for all index sequences $\left(m_i\right),\left(n_i\right)$, the two-dimensional matrix $C=\left(c_{ik}\right)=\left(b_{m_i{n}_{i}k}\right)$ is in $(c_0\left(r\right),{\ell }_{\infty }\left(q_{m_i{n}_i}\right))$. By Theorem 5.1, 13, in [12], this implies

$$\begin{array}{ccc}\hfill & & \forall \mathrm{index}\mathrm{sequences}\left(m_i\right),\left(n_i\right)\exists M\in \mathbb{N}:\hfill \\ & & \underset{i}{sup}{\left(\sum _k\left|c_{ik}\right|M^{-1/r_k}\right)}^{q_{m_i{n}_i}}=\underset{i}{sup}{\left(\sum _{k,l}\left|a_{m_i{n}_{i}kl}\right|M^{-1/p_{kl}}\right)}^{q_{m_i{n}_i}}<\infty .\hfill \end{array}$$

(3)

Suppose, on the contrary, that (xix) does not hold. Then, we can find index sequences $\left(m_i\right),\left(n_i\right)$ such that

$${\left(\sum _{k,l}\left|a_{m_i{n}_{i}kl}\right|i^{-1/p_{kl}}\right)}^{q_{m_i{n}_i}}>i(i\in \mathbb{N}).$$

Since for $j>i$, we obtain

$${\left(\sum _{k,l}\left|a_{m_j{n}_{j}kl}\right|i^{-1/p_{kl}}\right)}^{q_{m_j{n}_j}}\ge {\left(\sum _{k,l}\left|a_{m_j{n}_{j}kl}\right|j^{-1/p_{kl}}\right)}^{q_{m_j{n}_j}}>j,$$

so

$$\underset{j}{sup}{\left(\sum _{k,l}\left|a_{m_j{n}_{j}kl}\right|i^{-1/p_{kl}}\right)}^{q_{m_j{n}_j}}=\infty (i\in \mathbb{N}),$$

which contradicts (3). So (xix) follows.

Sufficiency. By (i), (ii), and (v), it follows that $Ax$ exists for every $x\in {\mathcal{C}}_{p0}\left(p\right)$. Assume that $Ax\notin {\mathcal{M}}_p\left(q\right)$ for some $x\in {\mathcal{C}}_{p0}\left(p\right)$. Then, $\left({\left[Ax\right]}_{m_i,n_i}\right)\notin {\ell }_{\infty }\left({\left(q_{m_i{n}_i}\right)}_i\right)$ for some index sequences $\left(m_i\right),\left(n_i\right)$. If we define the two-dimensional matrix $C=\left(c_{ij}\right)$ as in the proof of Necessity, we have $C\notin (c_0\left(r\right),{\ell }_{\infty }\left({\left(q_{m_i{n}_i}\right)}_i\right))$. Then, by Theorem 5.1, 13, in [12], $\forall M\in \mathbb{N}$, and we obtain

$$\underset{i}{sup}{\left(\sum _k\left|c_{ik}\right|M^{-1/r_k}\right)}^{q_{m_i{n}_i}}=\underset{i}{sup}{\left(\sum _{k,l}\left|a_{m_i{n}_{i}kl}\right|M^{-1/p_{kl}}\right)}^{q_{m_i{n}_i}}=\infty .$$

Then, the condition (xix) is not satisfied, so the contradiction follows. □

Theorem 5.

$A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{M}}_u\left(q\right))$ iff (i)-(iv) and the following condition hold:

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

Proof. 

Necessity. (i) and (ii) follow, since ${\mathcal{M}}_p{\left(p\right)}^{\beta \left(\nu \right)}=M_{\infty }^p\left(p\right)$. Since ${\mathcal{C}}_{p0}\left(p\right)\subset {\mathcal{M}}_p\left(p\right)$, we obtain (iii) and (iv) by Theorem 1. Since $M_u\left(p\right)\subset M_p\left(p\right)$, (xx) follows from Theorem 3.4 in [5].

Sufficiency. By (i), (ii), and (xx), it follows that $Ax$ exists for every $x\in {\mathcal{M}}_p\left(p\right)$. Assume that $Ax\notin {\mathcal{M}}_u\left(q\right)$ for some $x\in {\mathcal{M}}_p\left(p\right)$. Then, $\left({\left[Ax\right]}_{m_i,n_i}\right)\notin {\ell }_{\infty }\left({\left(q_{m_i{n}_i}\right)}_i\right)$ for some $\left(m_i\right),\left(n_i\right)\in \mathbb{N}$. If we define a two-dimensional matrix $B=\left(b_{ij}\right)$ and r as in the proof of sufficiency of Theorem 1, we have $B\notin ({\ell }_{\infty }\left(r\right),{\ell }_{\infty }\left({\left(q_{m_i{n}_i}\right)}_i\right))$. Then, by Theorem 5.1, 15, in [12],

$$\forall M\in \mathbb{N}:\underset{i}{sup}{\left(\sum _k\left|b_{ik}\right|M^{1/r_k}\right)}^{q_{m_i{n}_i}}=\infty .$$

So the condition (xx) does not hold and the contradiction follows. □

Theorem 6.

(${\mathbf{a}}_{\mathbf{3}}$) $A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{C}}_p\left(q\right))$ iff (i), (ii), (viii), (ix), (xx), and the following conditions hold:

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

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

(${\mathbf{b}}_{\mathbf{3}}$) $A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{C}}_{bp}\left(q\right))$ iff (i)–(iv), (viii), (ix), and (xx)–(xxii).

(${\mathbf{c}}_{\mathbf{3}}$) $A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{C}}_r\left(q\right))$ iff (i)–(iv), (viii), (ix), (xx), (xxii), (xxii), and the following conditions hold:

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

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

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

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

Proof. 

(${\mathbf{a}}_{\mathbf{3}}$) Necessity. (i), (ii), and (xx) follow, since ${\mathcal{M}}_p{\left(p\right)}^{\beta \left(\nu \right)}=M_{\infty }^p\left(p\right)$. Since ${\mathcal{C}}_{p0}\left(p\right)\subset {\mathcal{M}}_p\left(p\right)$, we obtain (viii) and (ix) by Theorem 2. Since $M_u\left(p\right)\subset M_p\left(p\right)$, (xxi) and (xxii) follow from Theorem 3.5 in [5].

Sufficiency. By (i), (ii), and (xx), it follows that $Ax$ exists for every $x\in {\mathcal{M}}_p\left(p\right)$. Now, let $x\in {\mathcal{M}}_p\left(p\right)$ be given. Let $N\in \mathbb{N}$ be such that ${sup}_{k,l>N}{\left|x_{kl}\right|}^{p_{kl}}<\infty$. We define $\tilde{x}$ by ${\tilde{x}}_{kl}:=x_{kl}$ for $k,l>N$ and ${\tilde{x}}_{kl}:=0$ otherwise. Then, $\tilde{x}\in M_u\left(p\right)$. Hence, $A\tilde{x}\in {\mathcal{C}}_p\left(q\right)$ by Theorem 3.5 in [5].

To prove that $A(x-\tilde{x})\in {\mathcal{C}}_p\left(q\right)$, we note that

$$x-\tilde{x}=\sum _{l=1}^{N}x{e}^l+\sum _{k=1}^{N}x{e}_k-\sum _{k=1}^N\sum _{l=1}^{N}x{e}^{kl}.$$

Since $x{e}^{kl}\in M_u\left(p\right)$, then $A\left(x{e}^{kl}\right)\in {\mathcal{C}}_p\left(q\right)$ by Theorem 3.5 in [5], $(k,l=1,\dots ,N)$.

For $l=1,\dots ,N$ from (v), we obtain

$${\left[A\left(x{e}^l\right)\right]}_{mn}={\left[\sum _{k=1}^{K\left(l\right)}A\left(x{e}^{kl}\right)\right]}_{mn}(m,n>K\left(l\right)),$$

so $A\left(x{e}^l\right)\in {\mathcal{C}}_p\left(q\right)$. Analogously, by (viii), $A\left(x{e}_k\right)\in {\mathcal{C}}_p\left(q\right)$ $(k=1,\dots ,N)$. Hence, $A(x-\tilde{x})\in {\mathcal{C}}_p\left(q\right)$. So, $Ax\in {\mathcal{C}}_p\left(q\right)$.

(${\mathbf{b}}_{\mathbf{3}}$) The proof follows from ($a_3$) and Theorem 5.

(${\mathbf{c}}_{\mathbf{3}}$) Necessity. Note that $A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{C}}_r\left(q\right))$ iff $A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{C}}_p\left(q\right))$, for all $n\in \mathbb{N}$, the three-dimensional matrix ${\left(a_{mnkl}\right)}_{m,k,l}$ maps ${\mathcal{M}}_p\left(p\right)$ to $c\left({\left(q_{mn}\right)}_m\right)$, and for all $m\in \mathbb{N}$, the three-dimensional matrix ${\left(a_{mnkl}\right)}_{n,k,l}$ maps ${\mathcal{M}}_p\left(p\right)$ to $c\left({\left(q_{mn}\right)}_n\right)$. By ($a_3$), the first statement is equivalent to (i), (ii), (viii), (ix) (xx), (xxi), and (xxii). Since $M_u\left(p\right)\subset M_p\left(p\right)$, (xxiii)–(xxvi) follow from Theorem 3.5(c) in [5].

Sufficiency. From (i), (ii), (viii), (ix) (xx), (xxi), and (xxii), it follows that ${\left(Ax\right)}_{mn}\in {\mathcal{C}}_p\left(q\right)$ for all $x\in {\mathcal{M}}_p\left(p\right)$. Let $m\in \mathbb{N}$ be fixed. In the same way as in the proof of Theorem 2, we define the maps ${\phi }^m:\mathsf{\Omega }\to \omega$, ${\mathsf{\Phi }}^m:\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ and the two-dimensional matrix $B=\left(b_{nj}\right)$. Then, ${\phi }^m\left({\mathcal{M}}_p\left(p\right)\right)={\ell }_{\infty }\left(r\right)$, where $r={\mathsf{\Phi }}^m\left(p\right)$.

Let $\left({\alpha }_k\right)={\phi }^m\left({\left({\beta }_{mkl}\right)}_{k,l}\right)$. By (xxiv) and (xxv), the matrix B satisfies the conditions

(1) $\forall M$: ${sup}_n{\sum }_k\left|b_{nk}\right|M^{1/r_k}<\infty$.

(2) $\exists \left({\alpha }_k\right)$ $\forall M$: ${lim}_n{\left({\sum }_k|b_{nk}-{\alpha }_k|M^{1/r_k}\right)}^{q_n}=0$.

Then, by Theorem 5.1, 11, in [12], $B\in ({\ell }_{\infty }\left(r\right),c\left({\left(q_{mn}\right)}_n\right))$, implying ${\left({\left(Ax\right)}_{mn}\right)}_n\in c\left({\left(q_{mn}\right)}_n\right)$ for each $x\in {\mathcal{M}}_p\left(p\right)$. Analogously, we can prove ${\left({\left(Ax\right)}_{mn}\right)}_m\in c\left({\left(q_{mn}\right)}_m\right)$ for each $x\in {\mathcal{M}}_p\left(p\right)$ $(n\in \mathbb{N})$. Hence, we obtain ${\left(Ax\right)}_{mn}\in {\mathcal{C}}_r\left(q\right)$ for each $x\in {\mathcal{M}}_p\left(p\right)$. □

Theorem 7.

(${\mathbf{a}}_{\mathbf{4}}$) $A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{C}}_{p0}\left(q\right))$ iff (i), (ii), (viii), (ix), (xx), and the following condition hold:

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

(${\mathbf{b}}_{\mathbf{4}}$) $A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{C}}_{bp0}\left(q\right))$ iff (i), (ii), (viii), (ix), (xx), and (xxvii).

(${\mathbf{c}}_{\mathbf{4}}$) $A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{C}}_{r0}\left(q\right))$ iff (i), (ii), (viii), (ix), (xx), and (xxiii)–(xxvii).

Proof. 

(${\mathbf{a}}_{\mathbf{4}}$) The proof follows in the same way as in Theorem 6 by applying Theorem 3.6 in [5].

(${\mathbf{b}}_{\mathbf{4}}$) The proof follows from ($a_4$) and Theorem 6 ($b_3$).

(${\mathbf{c}}_{\mathbf{4}}$) The proof follows from ($a_4$) and Theorem 6 ($c_3$). □

Theorem 8.

$A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{M}}_p\left(q\right))$ iff (i), (ii), (viii), (ix), (xx), and the following condition hold:

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

Proof. 

Necessity. The proof follows from Theorem 4 since ${\mathcal{C}}_{p0}\left(p\right)\subset {\mathcal{M}}_p\left(p\right)$ and ${\mathcal{M}}_p{\left(p\right)}^{\beta \left(\nu \right)}=M_{\infty }^p\left(p\right)$.

Sufficiency. By (i), (ii), (viii), (ix), and (xx), it follows that $Ax$ exists for every $x\in {\mathcal{M}}_p\left(p\right)$. Assume that $Ax\notin {\mathcal{M}}_p\left(q\right)$ for some $x\in {\mathcal{M}}_p\left(p\right)$. Then, $\left({\left[Ax\right]}_{m_i,n_i}\right)\notin {\ell }_{\infty }\left({\left(q_{m_i{n}_i}\right)}_i\right)$ for some index sequences $\left(m_i\right),\left(n_i\right)$. If we define a two-dimensional matrix $B=\left(b_{ij}\right)$ and r as in the proof of necessity of (x) and (xi) in Theorem 2, we have $B\notin ({\ell }_{\infty }\left(r\right),{\ell }_{\infty }\left({\left(q_{m_i{n}_i}\right)}_i\right))$. Then, by Theorem 5.1, 15, in [12], the condition (xxviii) does not hold. So the contradiction follows. □

Applying Theorems 1–3 and the fact that ${\mathcal{C}}_p\left(p\right)={\mathcal{C}}_{p0}\left(p\right)\oplus <e>$, we obtain the following theorems:

Theorem 9.

$A\in {({\mathcal{C}}_p\left(p\right),{\mathcal{M}}_u\left(q\right))}_{\nu }$ iff (i)–(vi) and the following condition hold:

(xxix) ${sup}_{m,n}{|\nu -{\sum }_{k,l}{a}_{mnkl}|}^{q_{mn}}<\infty .$

Theorem 10.

(${\mathbf{a}}_{\mathbf{5}}$) $A\in {({\mathcal{C}}_p\left(p\right),{\mathcal{C}}_p\left(q\right))}_{\nu }$ iff (i), (ii), (v), (vii)–(xi), and the following condition hold:

(xxx) $\exists \alpha \in \mathbb{R}:{lim}_{m,n}{|\nu -{\sum }_{k,l}{a}_{mnkl}-\alpha |}^{q_{mn}}=0$.

(${\mathbf{b}}_{\mathbf{5}}$) $A\in {({\mathcal{C}}_p\left(p\right),{\mathcal{C}}_{bp}\left(q\right))}_{\nu }$ iff (i)–(xii) and (xxix)–(xxx).

(${\mathbf{c}}_{\mathbf{5}}$) $A\in {({\mathcal{C}}_p\left(p\right),{\mathcal{C}}_r\left(q\right))}_{\nu }$ iff (i)–(v), (vii)–(xi), (xiii)–(xvi), and the following conditions hold:

(xxxi) $\exists \left({\alpha }_m\right)\in \mathbb{R}:{lim}_n{|{\sum }_{k,l}{a}_{mnkl}-{\alpha }_m|}^{q_{mn}}=0$ $(m\in \mathbb{N});$

(xxxii) $\exists \left({\delta }_n\right)\in \mathbb{R}:{lim}_m{|{\sum }_{k,l}{a}_{mnkl}-{\delta }_n|}^{q_{mn}}=0$ $(n\in \mathbb{N})$.

Theorem 11.

(${\mathbf{a}}_{\mathbf{6}}$) $A\in {({\mathcal{C}}_p\left(p\right),{\mathcal{C}}_{p0}\left(q\right))}_{\nu }$ iff (i), (ii), (v), (viii), (ix), (xvii), and the following conditions hold:

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

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

(${\mathbf{b}}_{\mathbf{6}}$) $A\in {({\mathcal{C}}_p\left(p\right),{\mathcal{C}}_{bp0}\left(q\right))}_{\nu }$ iff (i), (ii), (v), (vi), (viii), (ix), (xii), (xvii), (xxix), (xxxiii), and (xxxiv).

(${\mathbf{c}}_{\mathbf{6}}$) $A\in {({\mathcal{C}}_p\left(p\right),{\mathcal{C}}_{r0}\left(q\right))}_{\nu }$ iff (i), (ii), (v), (viii), (ix), (xiii)–(xvi), (xvii), (xxxiii), (xxxiv), and the following conditions hold:

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

(xxxvi) ${lim}_n{\left|{\sum }_{k,l}{a}_{mnkl}\right|}^{q_{mn}}=0$ $(m\in \mathbb{N})$.

Theorem 12.

$A\in {({\mathcal{C}}_p\left(p\right),{\mathcal{M}}_p\left(q\right))}_{\nu }$ iff (i), (ii), (v), (vi), (viii), (ix), and (xxix).

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

Example 1.

Let $p_{kl}=1/(k+l)$ and $q_{kl}=1$ $(k,l\in \mathbb{N})$. We consider the matrix $A=\left(a_{mnkl}\right)$ with $a_{mnkl}=0$ for $m,n,k>2^l$ or $m,n,l>2^k$ or $k>m$ or $l>n$ and $a_{mnkl}={(k!l!mn)}^{-1}$; otherwise, ($m,n,k,l\in \mathbb{N}$). Then, evidently, (i), (ii), (viii), and (ix) are satisfied. In addition,

$$\forall M:\underset{m,n>1}{sup}\sum _{k,l}\left|a_{mnkl}\right|M^{1/p_{kl}}\le {\displaystyle \frac{1}{2}}\sum _{k,l}{\displaystyle \frac{M^{k+l}}{k!l!}}<\infty ,$$

so (xx) and (xxi) are also satisfied in Theorem 6. For $\left(a_{kl}\right)=0$, we have

$$\forall M\in \mathbb{N}:\underset{m,n}{lim}{\left(\sum _{k,l}|a_{mnkl}-a_{kl}|M^{1/p_{kl}}\right)}^{q_{mn}}\le \underset{m,n}{lim}{\displaystyle \frac{1}{m+n}}\sum _{k,l}{\displaystyle \frac{M^{k+l}}{k!l!}}=0;$$

i.e., the condition (xxii) holds. So, by Theorem 6 ($a_3$), $A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{C}}_p\left(q\right))$.

On the other hand, the conditions (iii) and (iv) are not satisfied for this matrix (for example, $a_{1n1n}\ne 0(n\in \mathbb{N})$), so $A\notin (E,F)$ for $E\in \{{\mathcal{C}}_{p0}\left(p\right),{\mathcal{C}}_p\left(p\right),{\mathcal{M}}_p\left(p\right)\}$ and $F\in \{{\mathcal{M}}_u\left(q\right),{\mathcal{C}}_{bp}\left(q\right),{\mathcal{C}}_{bp0}\left(q\right),{\mathcal{C}}_r\left(q\right),{\mathcal{C}}_{r0}\left(q\right)\}$.

Example 2.

Let us change the previous example slightly. Again, let $p_{kl}=1/(k+l)$ and $q_{kl}=1$ $(k,l\in \mathbb{N})$. 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, ($m,n,k,l\in \mathbb{N}$). Now, in addition to (i), (ii), (viii), (ix), (xx), and (xxi), the conditions (iii) and (iv) are satisfied. So $A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{C}}_{bp}\left(q\right))$ by Theorem 6 ($b_3$).

Obviously, (xxiii) and (xxiv) hold. For $\left({\beta }_{mkl}\right)=\left(0\right)$ and any $M,m\in \mathbb{N}$, we obtain

$$\underset{n}{lim}{\left(\sum _{k,l}|a_{mnkl}-{\beta }_{mkl}|M^{1/p_{kl}}\right)}^{1/q_{mn}}\le \underset{n}{lim}{\displaystyle \frac{1}{m+n}}\sum _{k,l}{\displaystyle \frac{M^{k+l}}{k!l!}}=0,$$

so (xxv) is satisfied. Analogously, (xxvi) holds for $\left({\gamma }_{nkl}\right)=\left(0\right).$ So $A\in ({\mathcal{M}}_p\left(p\right),{\mathcal{C}}_r\left(q\right))$ by Theorem 6 ($c_3$).

For the reader’s convenience, we present our results in the diagram below (Table 1). The diagram indicates the theorem number where the characterization of $(E,F)$ can be found.

Table 1. Experimental results.

		F
E	${\mathcal{M}}_u\left(q\right)$	${\mathcal{C}}_p\left(q\right)$	${\mathcal{C}}_{p0}\left(q\right)$	${\mathcal{C}}_{bp}\left(q\right)$	${\mathcal{C}}_{bp0}\left(q\right)$	${\mathcal{C}}_r\left(q\right)$	${\mathcal{C}}_{r0}\left(q\right)$	${\mathcal{M}}_p\left(q\right)$
${\mathcal{C}}_{p0}\left(p\right)$	Theorem 1	Theorem 2	Theorem 3	Theorem 2	Theorem 3	Theorem 2	Theorem 3	Theorem 4
${\mathcal{C}}_p\left(p\right)$	Theorem 9	Theorem 10	Theorem 11	Theorem 10	Theorem 11	Theorem 10	Theorem 11	Theorem 12
${\mathcal{M}}_p\left(p\right)$	Theorem 5	Theorem 6	Theorem 7	Theorem 6	Theorem 7	Theorem 6	Theorem 7	Theorem 8