Schröder–Catalan Matrix and Compactness of Matrix Operators on Its Associated Sequence Spaces

Abstract

In this article, the regular Schröder–Catalan matrix is constructed and acquired by benefiting Schröder and Catalan numbers. After that, two sequence spaces are introduced, described as the domain of Schröder–Catalan matrix. Additionally, some algebraic and topological properties of the spaces in question, such as completeness, inclusion relations, basis and duals, are examined. In the last two sections, the necessary and sufficient conditions of some matrix classes and compact operators related aforementioned spaces are presented.

1. Introduction

A large Schröder number $S_r$, which will be called as Schröder number in the remainder of study for short, represents the count of trails from the southwest corner $(0,0)$ of an $r\times r$ grid to the northeast corner $(r,r)$, utilizing just single steps north, northeast, or east, which do not rise above the southwest–northeast diagonal, where $r\in \mathbb{N}=\{0,1,2,3,\dots \}$. Some terms of the Schröder integer sequence are

$$1,2,6,22,90,394,1806,8558,\mathrm{41,586},\mathrm{206,098},\mathrm{1,037,718},\dots$$

and Schröder numbers satisfy the recurrence relation

$$S_{r+1}=S_r+\sum _{t=0}^r{S}_t{S}_{r-t}(r\ge 0,S_0=1).$$

Furthermore, the limit of the ratio of the consecutive terms of Schröder sequence is $3+\sqrt{2}$. To access Schröder numbers and related basic and more comprehensive information, readers can consult sources [1,2,3,4,5].

There are also fascinating numbers other than Schröder numbers, which have various applications in geometry, probability theory, combinatorics and number theory, such as the Catalan numbers $C_r$, whose first few terms are

$$1,1,2,5,14,42,132,429,1430,4862,\mathrm{16,796},\dots .$$

The Catalan number sequence was discovered by Leonhard Euler in 1751 while investigating the number of different ways to divide a polygon into triangles. There are several equally different ways of expressing Catalan numbers, but the basic relations for them are as follows:

$$C_r={\displaystyle \frac{1}{r+1}}\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{r}}\right)={\displaystyle \frac{\left(2r\right)!}{(r+1)!r!}},C_{r+1}=\sum _{t=0}^r{C}_t{C}_{r-t}\mathrm{and}{C}_{r+1}={\displaystyle \frac{4r+2}{r+2}}{C}_r,$$

where $r,t\in \mathbb{N}$, $C_0=1$, and the limit of the ratio of the consecutive terms of Catalan sequence is 4. Researchers wishing to obtain detailed information about Catalan numbers and the necessary basic concepts are advised to consult sources [6,7].

Moreover, the relationship between Schröder and Catalan numbers is given by the relation

$$S_r=\sum _{t=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{2r-t}{t}}\right)C_{r-t}.$$

Each subset of the space $\omega$ that contains all sequences whose elements are real numbers is called a sequence space. The classical sequence spaces mentioned in the study can be expressed as

$$\begin{array}{ccc}\hfill {\mathsf{\ell }}_p& :=& \left\{y=\left(y_t\right)\in \omega :\sum _{t=0}^{\infty }{\left|y_t\right|}^p<\infty \right\},{\mathsf{\ell }}_{\infty }:=\left\{y=\left(y_t\right)\in \omega :\underset{t\in \mathbb{N}}{sup}\left|y_t\right|<\infty \right\},\hfill \\ \hfill c& :=& \left\{y=\left(y_t\right)\in \omega :\underset{t\to \infty }{lim}{y}_t\mathrm{exists}\right\},c_0:=\left\{y=\left(y_t\right)\in \omega :\underset{t\to \infty }{lim}{y}_t=0\right\},\hfill \\ \hfill bs& :=& \left\{y=\left(y_t\right)\in \omega :\left(\sum _{t=0}^r{y}_t\right)\in {\mathsf{\ell }}_{\infty }\right\},cs:=\left\{y=\left(y_t\right)\in \omega :\left(\sum _{t=0}^r{y}_t\right)\in c\right\}\hfill \end{array}$$

for $0<p<\infty$. From now on, unless otherwise stated, ${\sum }_t$ will represent ${\sum }_{t=0}^{\infty }$ for brevity. A Banach space with continuous coordinates is named a BK-space. The spaces c, $c_0$ and ${\mathsf{\ell }}_{\infty }$ are BK-spaces with the norm ${\parallel u\parallel }_{{\mathsf{\ell }}_{\infty }}={sup}_{t\in \mathbb{N}}\left|u_t\right|$, and the space ${\mathsf{\ell }}_p$ is a BK-space with the norm ${\parallel u\parallel }_{{\mathsf{\ell }}_p}={\left({\sum }_t{\left|u_t\right|}^p\right)}^{{\displaystyle \frac{1}{p}}}$ for $1\le p<\infty$. Let $F_r$ denote the rth row of an infinite matrix $F=\left(f_{rt}\right)$ with real entries. In this case, the expression ${\left(Fy\right)}_r={\sum }_t{f}_{rt}{y}_t$ is called the F-transform of $y=\left(y_t\right)$ if the series converges for all $r\in \mathbb{N}$. The $\alpha$-, $\beta$- and $\gamma$-duals of a sequence space $\Delta$ are given the sets

$$\begin{array}{ccc}\hfill {\Delta }^{\alpha }& :=& \left\{\sigma =\left({\sigma }_t\right)\in \omega :\sigma y=\left({\sigma }_t{y}_t\right)\in {\mathsf{\ell }}_1\mathrm{for}\mathrm{all}y\in \Delta \right\},\hfill \\ \hfill {\Delta }^{\beta }& :=& \left\{\sigma =\left({\sigma }_t\right)\in \omega :\sigma y=\left({\sigma }_t{y}_t\right)\in cs\mathrm{for}\mathrm{all}y\in \Delta \right\},\hfill \\ \hfill {\Delta }^{\alpha }& :=& \left\{\sigma =\left({\sigma }_t\right)\in \omega :\sigma y=\left({\sigma }_t{y}_t\right)\in bs\mathrm{for}\mathrm{all}y\in \Delta \right\}.\hfill \end{array}$$

If the F-transform $Fy$ of every sequence y taken from $\Delta \subset \omega$ is in $\Pi \subset \omega$ for an infinite matrix F, then F is called a matrix transformation from $\Delta$ to $\Pi$, and the class of such matrices is represented by $(\Delta :\Pi )$. Furthermore, $F\in (\Delta :\Pi )$ if and only if $F_r\in {\Delta }^{\beta }$ and $Fy\in \Pi$ for all $y\in \Delta$. The matrix domain of an infinite matrix F on the sequence space $\Delta$ is represented by the set

$$\begin{array}{c}\hfill {\Delta }_F=\left\{y\in \omega :Fy\in \Delta \right\}.\end{array}$$

(1)

The normed spaces $\Delta$ and $\Pi$ are called linearly isomorphic spaces if there is a one-to-one, onto, and norm-preserving transformation between them, and they are denoted by $\Delta \cong \Pi$. Researchers who want to deal in detail with the basic topics of summability theory, sequence spaces, and matrix domains can refer to sources [8,9,10,11].

In recent years, the use of infinite matrices in obtaining new sequence spaces has become quite widespread. However, the idea of using special number sequences in obtaining infinite matrices is a newer approach. After the studies by Kara and Başarır [12,13] based on the Fibonacci integer sequence, generating new normed or paranormed sequence spaces by means of integer sequences appears to be an interesting idea. Thus, the integer sequences Lucas [14,15], Padovan [16], Leonardo [17], Catalan [18,19,20], Bell [21], Schröder [22,23], Mersenne [24] and Motzkin [25,26] were used to construct new infinite matrices and sequence spaces and at the same time some fundamental topics were examined in obtained sequence spaces.

In this study, firstly, we describe a novel matrix $S^{*}$ with the aid of both Schröder and Catalan numbers, and construct sequence spaces ${\mathsf{\ell }}_p\left(S^{*}\right)$ and ${\mathsf{\ell }}_{\infty }\left(S^{*}\right)$ as the domain of $S^{*}$. After that, some properties, such as regularity, completeness and inclusion relations, are examined. In the rest of the paper, the duals of the new spaces are calculated, and the characterization of some classes of matrices and compact operators is presented.

2. A New Regular Schröder–Catalan Matrix and Associated Sequence Spaces

In the current part, the regular Schröder–Catalan matrix $S^{*}$ is acquired by Schröder and Catalan numbers. Then, with the help of this matrix, BK-spaces ${\mathsf{\ell }}_p\left(S^{*}\right)$ and ${\mathsf{\ell }}_{\infty }\left(S^{*}\right)$ are obtained. After that, it is shown that ${\mathsf{\ell }}_p\left(S^{*}\right)\cong {\mathsf{\ell }}_p$ and ${\mathsf{\ell }}_{\infty }\left(S^{*}\right)\cong {\mathsf{\ell }}_{\infty }$ and, finally, the Schauder basis of ${\mathsf{\ell }}_p\left(S^{*}\right)$ and the inclusion relations of the spaces are presented.

The Schröder–Catalan matrix $S^{*}={\left(s_{rt}^{*}\right)}_{r,t\in \mathbb{N}}$ is constructed with the help of Schröder and Catalan numbers as follows:

$$\begin{array}{c}\hfill s_{rt}^{*}:=\left\{\begin{array}{ccc}{\displaystyle \left(\genfrac{}{}{0pt}{}{2r-t}{t}\right)\frac{C_{r-t}}{S_r}}& ,& \mathrm{if}0\le t\le r,\hfill \\ \\ {\displaystyle 0}& ,& \mathrm{if}t>r.\hfill \end{array}\right.\end{array}$$

(2)

$S^{*}={\left(s_{rt}^{*}\right)}_{r,t\in \mathbb{N}}$ is stated in matrix form as follows:

$$S^{*}:=\left[\begin{array}{cccccc}\left({\displaystyle \genfrac{}{}{0pt}{}{0}{0}}\right){\displaystyle \frac{C_0}{S_0}}& 0& 0& 0& 0& \cdots \\ \left({\displaystyle \genfrac{}{}{0pt}{}{2}{0}}\right){\displaystyle \frac{C_1}{S_1}}& \left({\displaystyle \genfrac{}{}{0pt}{}{1}{1}}\right){\displaystyle \frac{C_0}{S_1}}& 0& 0& 0& \cdots \\ \left({\displaystyle \genfrac{}{}{0pt}{}{4}{0}}\right){\displaystyle \frac{C_2}{S_2}}& \left({\displaystyle \genfrac{}{}{0pt}{}{3}{1}}\right){\displaystyle \frac{C_1}{S_2}}& \left({\displaystyle \genfrac{}{}{0pt}{}{2}{2}}\right){\displaystyle \frac{C_0}{S_2}}& 0& 0& \cdots \\ \left({\displaystyle \genfrac{}{}{0pt}{}{6}{0}}\right){\displaystyle \frac{C_3}{S_3}}& \left({\displaystyle \genfrac{}{}{0pt}{}{5}{1}}\right){\displaystyle \frac{C_2}{S_3}}& \left({\displaystyle \genfrac{}{}{0pt}{}{4}{2}}\right){\displaystyle \frac{C_1}{S_3}}& \left({\displaystyle \genfrac{}{}{0pt}{}{3}{3}}\right){\displaystyle \frac{C_0}{S_3}}& 0& \cdots \\ \left({\displaystyle \genfrac{}{}{0pt}{}{8}{0}}\right){\displaystyle \frac{C_4}{S_4}}& \left({\displaystyle \genfrac{}{}{0pt}{}{7}{1}}\right){\displaystyle \frac{C_3}{S_4}}& \left({\displaystyle \genfrac{}{}{0pt}{}{6}{2}}\right){\displaystyle \frac{C_2}{S_4}}& \left({\displaystyle \genfrac{}{}{0pt}{}{5}{3}}\right){\displaystyle \frac{C_1}{S_4}}& \left({\displaystyle \genfrac{}{}{0pt}{}{4}{4}}\right){\displaystyle \frac{C_0}{S_4}}& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right].$$

One point to note is that the Schröder–Catalan matrix $S^{*}$ is not symmetric, since its transpose is not equal to itself.

The $S^{*}$-transform of $y\in \omega$ is presented by

$$z_r:={\left(S^{*}y\right)}_r={\displaystyle \frac{1}{S_r}}\sum _{t=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{2r-t}{t}}\right)C_{r-t}{y}_t,(r\in \mathbb{N}).$$

(3)

Furthermore, from Theorem 2 of Deng and Yan [27], the inverse ${S^{*}}^{-1}=\left({s_{rt}^{*}}^{-1}\right)$ of the Schröder–Catalan matrix $S^{*}$ is computed as

$$\begin{array}{c}\hfill {s_{rt}^{*}}^{-1}:=\left\{\begin{array}{ccc}{\displaystyle {(-1)}^{r-t}\left(\genfrac{}{}{0pt}{}{2r}{r+t}\right)\frac{(2t+1)S_t}{(r+t+1)C_r}}& ,& \mathrm{if}0\le t\le r,\hfill \\ \\ {\displaystyle 0}& ,& \mathrm{if}t>r.\hfill \end{array}\right.\end{array}$$

(4)

After remembering that matrices that transform convergent sequences into convergent sequences by preserving the limit are termed as regular, let us give the relevant lemma and result.

Lemma 1.

$F=\left(f_{rt}\right)$ is regular iff

(i) 

${sup}_{r\in \mathbb{N}}{\sum }_t\left|f_{rt}\right|<\infty$;

(ii) 

${lim}_{r\to \infty }{\sum }_t{f}_{rt}=1$;

(iii) 

${lim}_{r\to \infty }{f}_{rt}=0$ for all $t\in \mathbb{N}$.

Theorem 1.

The Schröder–Catalan matrix $S^{*}$ is regular.

Proof. 

(i) and (ii) is clear by the relation

$$\sum _t\left|s_{rt}^{*}\right|=\sum _t{s}_{rt}^{*}=\sum _t\left({\displaystyle \genfrac{}{}{0pt}{}{2r-t}{t}}\right){\displaystyle \frac{C_{r-t}}{S_r}}=1.$$

Now, let us consider the series ${\sum }_r{s}_{rt}^{*}$, and examine the convergence of this. By the aid of the ratio test and the relation

$$\begin{array}{ccc}\hfill \underset{r\to \infty }{lim}\left|{\displaystyle \frac{s_{r+1,t}^{*}}{s_{rt}^{*}}}\right|& =& \underset{r\to \infty }{lim}{\displaystyle \frac{s_{r+1,t}^{*}}{s_{rt}^{*}}}\hfill \\ & =& \underset{r\to \infty }{lim}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2r+2-t}{t}\right)\frac{C_{r+1-t}}{S_{r+1}}}{\left(\genfrac{}{}{0pt}{}{2r-t}{t}\right)\frac{C_{r-t}}{S_r}}}\hfill \\ & =& \underset{r\to \infty }{lim}\left[{\displaystyle \frac{(2r-t+2)(2r-t+1)}{(2r-2t+2)(2r-2t+1)}}.{\displaystyle \frac{C_{r+1-t}}{C_{r-t}}}.{\displaystyle \frac{S_r}{S_{r+1}}}\right]\hfill \\ & =& {\displaystyle \frac{4}{3+2\sqrt{2}}}<1,\hfill \end{array}$$

it is reached that the series ${\sum }_r{s}_{rt}^{*}$ absolutely convergent, and

$$\underset{r\to \infty }{lim}{s}_{rt}^{*}=\underset{r\to \infty }{lim}\left({\displaystyle \genfrac{}{}{0pt}{}{2r-t}{t}}\right){\displaystyle \frac{C_{r-t}}{S_r}}=0.$$

Thus, Condition (iii) holds.

□

Now, we can introduce the sets ${\mathsf{\ell }}_p\left(S^{*}\right)$ and ${\mathsf{\ell }}_{\infty }\left(S^{*}\right)$ by

$$\begin{array}{ccc}\hfill {\mathsf{\ell }}_p\left(S^{*}\right)& =& \left\{y=\left(y_t\right)\in \omega :\sum _{r=0}^{\infty }{\left|{\displaystyle \frac{1}{S_r}}\sum _{t=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{2r-t}{t}}\right)C_{r-t}{y}_t\right|}^p<\infty \right\}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mathsf{\ell }}_{\infty }\left(S^{*}\right)& =& \left\{y=\left(y_t\right)\in \omega :\underset{r\in \mathbb{N}}{sup}\left|{\displaystyle \frac{1}{S_r}}\sum _{t=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{2r-t}{t}}\right)C_{r-t}{y}_t\right|<\infty \right\},\hfill \end{array}$$

for $1\le p<\infty$. Then, it can be said that ${\mathsf{\ell }}_p\left(S^{*}\right)$ and ${\mathsf{\ell }}_{\infty }\left(S^{*}\right)$ are the domains of $S^{*}$ on ${\mathsf{\ell }}_p$ and ${\mathsf{\ell }}_{\infty }$, respectively. It is stated in [28] that, for a triangle F and BK-space $\Delta$, ${\Delta }_F$ is a BK-space, too.

Theorem 2.

${\mathsf{\ell }}_p\left(S^{*}\right)$ and ${\mathsf{\ell }}_{\infty }\left(S^{*}\right)$ are BK-spaces with

$${\parallel y\parallel }_{{\mathsf{\ell }}_p\left(S^{*}\right)}={\left(\sum _{r=0}^{\infty }{\left|{\displaystyle \frac{1}{S_r}}\sum _{t=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{2r-t}{t}}\right)C_{r-t}{y}_t\right|}^p\right)}^{{\displaystyle \frac{1}{p}}}$$

and

$${\parallel y\parallel }_{{\mathsf{\ell }}_{\infty }\left(S^{*}\right)}=\underset{r\in \mathbb{N}}{sup}\left|{\displaystyle \frac{1}{S_r}}\sum _{t=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{2r-t}{t}}\right)C_{r-t}{y}_t\right|,$$

respectively.

Theorem 3.

${\mathsf{\ell }}_p\left(S^{*}\right)\cong {\mathsf{\ell }}_p$ and ${\mathsf{\ell }}_{\infty }\left(S^{*}\right)\cong {\mathsf{\ell }}_{\infty }$.

Proof. 

The mapping $\mathcal{G}:{\mathsf{\ell }}_p\left(S^{*}\right)\to {\mathsf{\ell }}_p$, $\mathcal{G}\left(y\right)=S^{*}y$ is linear. From $\mathcal{G}\left(y\right)=0\Rightarrow y=0$, $\mathcal{G}$ is incejtive.

Let us take $z=\left(z_t\right)\in {\mathsf{\ell }}_p$ and $y=\left(y_t\right)\in \omega$ with

$$y_t=\sum _{i=0}^t{(-1)}^{t-i}\left({\displaystyle \genfrac{}{}{0pt}{}{2t}{t+i}}\right){\displaystyle \frac{(2i+1)S_i}{(t+i+1)C_t}}{z}_i.(t\in \mathbb{N})$$

(5)

It is seen that $\mathcal{G}$ is surjective from the equation

$$\begin{array}{ccc}\hfill {\left(S^{*}y\right)}_r& =& {\displaystyle \frac{1}{S_r}}\sum _{t=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{2r-t}{t}}\right)C_{r-t}{y}_t\hfill \\ & =& {\displaystyle \frac{1}{S_r}}\sum _{t=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{2r-t}{t}}\right)C_{r-t}\sum _{i=0}^t{(-1)}^{t-i}\left({\displaystyle \genfrac{}{}{0pt}{}{2t}{t+i}}\right){\displaystyle \frac{(2i+1)S_i}{(t+i+1)C_t}}{z}_i\hfill \\ & =& z_r.\hfill \end{array}$$

Furthermore, $\mathcal{G}$ is norm keeper from the equality ${\parallel y\parallel }_{{\mathsf{\ell }}_p\left(S^{*}\right)}={\parallel S^{*}y\parallel }_{{\mathsf{\ell }}_p}$.

In other spaces, the result can be seen in a similar way. □

Definition 1.

Assume that $(\Delta ,\parallel .\parallel )$ is a normed sequence space, and $\left({\eta }_r\right)\in \Delta$. If, for any $y\in \Delta$, there exists a unique scalars’ sequence $\left({\tau }_r\right)$ as

$$∥y-\sum _{t=0}^r{\tau }_t{\eta }_t∥⟶0$$

for $r\to \infty$, then $\left({\eta }_r\right)$ is Schauder basis for Δ, and it is written as $y={\sum }_t{\tau }_t{\eta }_t$.

Let $e^r$ be a sequence whose other terms are zero, except for the rth term, which is 1. We conclude that the inverse image of the basis ${\left(e^{\left(r\right)}\right)}_{r\in \mathbb{N}}$ of ${\mathsf{\ell }}_p$ becomes the basis of ${\mathsf{\ell }}_p\left(S^{*}\right)$, since $\mathcal{G}:{\mathsf{\ell }}_p\left(S^{*}\right)\to {\mathsf{\ell }}_p$ is an isomorphism, given Theorem 3, so it will be given following result without proof.

Theorem 4.

The set ${\eta }^{\left(t\right)}=\left({\eta }_r^{\left(t\right)}\right)$ expressed by

$$\begin{array}{c}\hfill {\eta }_r^{\left(t\right)}:=\left\{\begin{array}{ccc}{\displaystyle {(-1)}^{r-t}\frac{(2t+1)}{(r+t+1)}\left(\genfrac{}{}{0pt}{}{2r}{r+t}\right)\frac{S_t}{C_r}}& ,& 0\le t\le r\hfill \\ \\ {\displaystyle 0}& ,& t>r\hfill \end{array}\right.\end{array}$$

is the Schauder basis for ${\mathsf{\ell }}_p\left(S^{*}\right)$. Furthermore, each $y\in {\mathsf{\ell }}_p\left(S^{*}\right)$ is represented uniquely by $y={\sum }_t{\tau }_t{\eta }^{\left(t\right)}$ for $1\le p<\infty$ and ${\tau }_t={\left(S^{*}y\right)}_t$.

Theorem 5.

The inclusion ${\mathsf{\ell }}_p\left(S^{*}\right)\subset {\mathsf{\ell }}_{p^{\prime }}\left(S^{*}\right)$ is strict for $1\le p<p^{\prime }<\infty$.

Proof. 

Let us take $y=\left(y_t\right)\in {\mathsf{\ell }}_p\left(S^{*}\right)$, such that $S^{*}y\in {\mathsf{\ell }}_p$. Furthermore, it is known that ${\mathsf{\ell }}_p\subset {\mathsf{\ell }}_{p^{\prime }}$ for $1\le p<p^{\prime }<\infty$. Then, it is seen that $S^{*}y\in {\mathsf{\ell }}_{p^{\prime }}$ and $y\in {\mathsf{\ell }}_{p^{\prime }}\left(S^{*}\right)$.

If $\tilde{z}=S^{*}\tilde{y}\in {\mathsf{\ell }}_{p^{\prime }}\setminus {\mathsf{\ell }}_p$ is chosen, it is reached that the inclusion is strict. □

Theorem 6.

${\mathsf{\ell }}_{\infty }\subset {\mathsf{\ell }}_{\infty }\left(S^{*}\right)$.

Proof. 

From the inequality

$$\begin{array}{ccc}\hfill {∥y∥}_{{\mathsf{\ell }}_{\infty }\left(S^{*}\right)}& =& \underset{r\in \mathbb{N}}{sup}\left|{\displaystyle \frac{1}{S_r}}\sum _{t=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{2r-t}{t}}\right)C_{r-t}{y}_t\right|\hfill \\ \hfill & \le & {∥y∥}_{\infty }\underset{r\in \mathbb{N}}{sup}\left|{\displaystyle \frac{1}{S_r}}\sum _{t=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{2r-t}{t}}\right)C_{r-t}\right|\hfill \\ \hfill & =& {∥y∥}_{\infty }<\infty \hfill \end{array}$$

for $y=\left(y_t\right)\in {\mathsf{\ell }}_{\infty }$, we see that $y\in {\mathsf{\ell }}_{\infty }\left(S^{*}\right)$. □

3. $\mathit{\alpha }$-, $\mathit{\beta }$- and $\mathit{\gamma }$-Duals

In this section, duals of new spaces will be found.

Now, let us list some conditions for the family of all finite subsets $\mathcal{D}$ of $\mathbb{N}$, $1<p<\infty$, and $q={\displaystyle \frac{p}{p-1}}$:

$$\begin{array}{ccc}& & \underset{t\in \mathbb{N}}{sup}\sum _r\left|f_{rt}\right|<\infty ,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& & \underset{t\in \mathbb{N}}{sup}\sum _r{\left|f_{rt}\right|}^p<\infty ,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}& & \underset{r,t\in \mathbb{N}}{sup}\left|f_{rt}\right|<\infty ,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& & \underset{r\to \infty }{lim}{f}_{rt}\mathrm{exists}\mathrm{for}\mathrm{all}t\in \mathbb{N},\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & \underset{r\to \infty }{lim}{f}_{rt}=0,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & \underset{D\in \mathcal{D}}{sup}\sum _t{\left|\sum _{r\in D}{f}_{rt}\right|}^q<\infty ,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & \underset{r\in \mathbb{N}}{sup}\sum _t{\left|f_{rt}\right|}^q<\infty ,\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& & \underset{D\in \mathcal{D}}{sup}\sum _r\left|\sum _{t\in D}{f}_{rt}\right|<\infty ,\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& & \underset{D\in \mathcal{D}}{sup}\sum _r{\left|\sum _{t\in D}{f}_{rt}\right|}^p<\infty ,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}& & \underset{r\in \mathbb{N}}{sup}\sum _t\left|f_{rt}\right|<\infty ,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & \underset{r\to \infty }{lim}\sum _t\left|f_{rt}\right|=\sum _t\left|\underset{r\to \infty }{lim}{f}_{rt}\right|,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & \underset{r\to \infty }{lim}\sum _t\left|f_{rt}\right|=0.\hfill \end{array}$$

(17)

It is now useful to present the Table 1 compiled from [29], which explains the necessary and sufficient conditions of the classical matrix classes:

Table 1. Conditions of $(\Delta :\Pi )$ for $\Delta ,\Pi \in \{{\mathsf{\ell }}_1,{\mathsf{\ell }}_p,{\mathsf{\ell }}_{\infty },c,c_0\}$ and $1<p<\infty$.

($\mathit{From}\downarrow :\mathit{to}\to$)	${\mathsf{\ell }}_1$	${\mathsf{\ell }}_{\mathit{p}}$	${\mathsf{\ell }}_{\mathit{\infty }}$	c	${\mathit{c}}_0$
${\mathsf{\ell }}_{\mathbf{1}}$	(6)	(7)	(8)	(8),(9)	(8),(10)
${\mathsf{\ell }}_{\mathit{p}}$	(11)	◃	(12)	(9),(12)	(10),(12)
${\mathsf{\ell }}_{\infty }$	(13)	(14)	(15)	(9),(16)	(17)
$\mathit{c}$	(13)	(14)	(15)	▹	▹
${\mathit{c}}_{\mathbf{0}}$	(13)	(14)	(15)	▹	▹

Undefined conditions are symbolized by “◃”, and out of scope conditions are symbolized by “▹”.

Theorem 7.

Let us consider the sets ${\mathcal{A}}_1-{\mathcal{A}}_3$ as

$$\begin{array}{ccc}\hfill {\mathcal{A}}_1& =& \left\{\sigma =\left({\sigma }_t\right)\in \omega :\underset{D\in \mathcal{D}}{sup}\sum _t{\left|\sum _{r\in D}{(-1)}^{r-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{r+t}}\right){\displaystyle \frac{(2t+1)S_t}{(r+t+1)C_r}}{\sigma }_t\right|}^q<\infty \right\},\hfill \\ \hfill {\mathcal{A}}_2& =& \left\{\sigma =\left({\sigma }_t\right)\in \omega :\underset{t\in \mathbb{N}}{sup}\sum _r\left|{(-1)}^{r-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{r+t}}\right){\displaystyle \frac{(2t+1)S_t}{(r+t+1)C_r}}{\sigma }_t\right|<\infty \right\},\hfill \\ \hfill {\mathcal{A}}_3& =& \left\{\sigma =\left({\sigma }_t\right)\in \omega :\underset{D\in \mathcal{D}}{sup}\sum _r\left|\sum _{t\in D}{(-1)}^{r-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{r+t}}\right){\displaystyle \frac{(2t+1)S_t}{(r+t+1)C_r}}{\sigma }_t\right|<\infty \right\}.\hfill \end{array}$$

In that case:

(i) 

${\left({\mathsf{\ell }}_p\left(S^{*}\right)\right)}^{\alpha }={\mathcal{A}}_1,(1<p<\infty )$;

(ii) 

${\left({\mathsf{\ell }}_1\left(S^{*}\right)\right)}^{\alpha }={\mathcal{A}}_2$;

(iii) 

${\left({\mathsf{\ell }}_{\infty }\left(S^{*}\right)\right)}^{\alpha }={\mathcal{A}}_3$.

Proof. 

(i) Let us define the infinite matrix $E=\left(e_{rt}\right)$ by

$$\begin{array}{c}\hfill e_{rt}:=\left\{\begin{array}{ccc}{\displaystyle {(-1)}^{r-t}\left(\genfrac{}{}{0pt}{}{2r}{r+t}\right)\frac{(2t+1)S_t}{(r+t+1)C_r}{\sigma }_r}& ,& \mathrm{if}0\le t\le r,\hfill \\ \\ {\displaystyle 0}& ,& \mathrm{if}t>r.\hfill \end{array}\right.\end{array}$$

For $y\in {\mathsf{\ell }}_p\left(S^{*}\right)$, and by the aid of (3), it is reached that

$$\begin{array}{ccc}\hfill {\sigma }_r{y}_r& =& {\sigma }_r\left(\sum _{t=0}^r{(-1)}^{r-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{r+t}}\right){\displaystyle \frac{(2t+1)S_t}{(r+t+1)C_r}}{z}_t\right)\hfill \\ & =& \sum _{t=0}^r\left({(-1)}^{r-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{r+t}}\right){\displaystyle \frac{(2t+1)S_t}{(r+t+1)C_r}}{\sigma }_r\right)z_t={\left(Ez\right)}_r.\hfill \end{array}$$

(18)

Then, by (18), it is concluded that $\sigma y=\left({\sigma }_r{y}_r\right)\in {\mathsf{\ell }}_1$, while $y\in {\mathsf{\ell }}_p\left(S^{*}\right)$ iff $Ez\in {\mathsf{\ell }}_1$ when $z\in {\mathsf{\ell }}_p$. Therefore, $\sigma \in {\left({\mathsf{\ell }}_p\left(S^{*}\right)\right)}^{\alpha }$ iff $E\in ({\mathsf{\ell }}_p:{\mathsf{\ell }}_1)$. So, according to Table 1, ${\left({\mathsf{\ell }}_p\left(S^{*}\right)\right)}^{\alpha }={\mathcal{A}}_1$ for $1<p<\infty$. The rest of the proof can be shown by similar reasoning. □

Theorem 8.

Let us consider the sets ${\mathcal{B}}_1-{\mathcal{B}}_4$ as

$$\begin{array}{ccc}\hfill {\mathcal{B}}_1& =& \left\{\sigma =\left({\sigma }_t\right)\in \omega :\underset{r\to \infty }{lim}\sum _{i=t}^r{(-1)}^{i-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2i}{i+t}}\right){\displaystyle \frac{(2t+1)S_t}{(i+t+1)C_i}}{\sigma }_i\mathit{exists} \mathit{for} \mathit{each}t\in \mathbb{N}\right\},\hfill \\ \hfill {\mathcal{B}}_2& =& \left\{\sigma =\left({\sigma }_t\right)\in \omega :\underset{r\in \mathbb{N}}{sup}\sum _t{\left|\sum _{i=t}^r{(-1)}^{i-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2i}{i+t}}\right){\displaystyle \frac{(2t+1)S_t}{(i+t+1)C_i}}{\sigma }_i\right|}^q<\infty \right\},\hfill \\ \hfill {\mathcal{B}}_3& =& \left\{\sigma =\left({\sigma }_t\right)\in \omega :\underset{r,t\in \mathbb{N}}{sup}\left|\sum _{i=t}^r{(-1)}^{i-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2i}{i+t}}\right){\displaystyle \frac{(2t+1)S_t}{(i+t+1)C_i}}{\sigma }_i\right|<\infty \right\},\hfill \\ \hfill {\mathcal{B}}_4& =& \left\{\sigma =\left({\sigma }_t\right)\in \omega :\underset{r\to \infty }{lim}\sum _t\left|\sum _{i=t}^r{(-1)}^{i-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2i}{i+t}}\right){\displaystyle \frac{(2t+1)S_t}{(i+t+1)C_i}}{\sigma }_i\right|\right.\hfill \\ & & \left.=\sum _t\left|\sum _{i=t}^{\infty }{(-1)}^{i-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2i}{i+t}}\right){\displaystyle \frac{(2t+1)S_t}{(i+t+1)C_i}}{\sigma }_i\right|\right\}.\hfill \end{array}$$

In that case:

(i) 

${\left({\mathsf{\ell }}_p\left(S^{*}\right)\right)}^{\beta }={\mathcal{B}}_1\cap {\mathcal{B}}_2$, $(1<p<\infty )$;

(ii) 

${\left({\mathsf{\ell }}_1\left(S^{*}\right)\right)}^{\beta }={\mathcal{B}}_1\cap {\mathcal{B}}_3$;

(iii) 

${\left({\mathsf{\ell }}_{\infty }\left(S^{*}\right)\right)}^{\beta }={\mathcal{B}}_1\cap {\mathcal{B}}_4$.

Proof. 

(i) Let us define the infinite matrix $O={\left(o_{rt}\right)}_{r,t\in \mathbb{N}}$ by

$$\begin{array}{c}\hfill o_{rt}:=\left\{\begin{array}{ccc}{\displaystyle \sum _{i=t}^r{(-1)}^{i-t}\left(\genfrac{}{}{0pt}{}{2i}{i+t}\right)\frac{(2t+1)S_t}{(i+t+1)C_i}{\sigma }_i}& ,& 0\le t\le r,\hfill \\ \\ {\displaystyle 0}& ,& t>r.\hfill \end{array}\right.\end{array}$$

(19)

For $y\in {\mathsf{\ell }}_p\left(S^{*}\right)$ and by the aid of (3), it is reached that

$$\begin{array}{ccc}\hfill {\psi }_r=\sum _{t=0}^r{\sigma }_t{y}_t& =& \sum _{t=0}^r{\sigma }_t\left(\sum _{i=0}^t{(-1)}^{t-i}\left({\displaystyle \genfrac{}{}{0pt}{}{2t}{t+i}}\right){\displaystyle \frac{(2i+1)S_i}{(t+i+1)C_t}}{z}_i\right)\hfill \\ & =& \sum _{t=0}^r\left(\sum _{i=t}^r{(-1)}^{i-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2i}{i+t}}\right){\displaystyle \frac{(2t+1)S_t}{(i+t+1)C_i}}{\sigma }_i\right)z_t\hfill \\ & =& {\left(Oz\right)}_r.\hfill \end{array}$$

(20)

Then, by (20), $\sigma y\in cs$ when $y\in {\mathsf{\ell }}_p\left(S^{*}\right)$ iff $\psi =\left({\psi }_r\right)\in c$ when $z\in {\mathsf{\ell }}_p$. Thus, $\sigma \in {\left\{{\mathsf{\ell }}_p\left(S^{*}\right)\right\}}^{\beta }$ iff $O\in \left({\mathsf{\ell }}_p:c\right)$. So, according to Table 1, the desired result is reached.

The rest of the proof can be shown by similar reasoning. □

The following result will be given without proof, since it can be proven in a similar way to the theorem just proved.

Theorem 9.

(i) 

${\left({\mathsf{\ell }}_p\left(S^{*}\right)\right)}^{\gamma }={\mathcal{B}}_2$ for $1<p<\infty$;

(ii) 

${\left({\mathsf{\ell }}_1\left(S^{*}\right)\right)}^{\gamma }={\mathcal{B}}_3$;

(iii) 

${\left({\mathsf{\ell }}_{\infty }\left(S^{*}\right)\right)}^{\gamma }={\mathcal{B}}_2$ for $q=1$.

4. Matrix Transformations

In this part, necessary and sufficient conditions for some classes of matrices containing new sequence spaces will be stated.

Theorem 10.

Assume that, the infinite matrices ${\mathcal{N}}^r=\left(n_{jt}^r\right)$ and $\mathcal{N}=\left(n_{rt}\right)$ are as follows:

$$\begin{array}{c}\hfill n_{jt}^r:=\left\{\begin{array}{ccc}{\displaystyle \sum _{i=t}^j{(-1)}^{i-t}\left(\genfrac{}{}{0pt}{}{2i}{i+t}\right)\frac{(2t+1)S_t}{(i+t+1)C_i}{f}_{ri}}& ,& 0\le t\le j,\hfill \\ \\ {\displaystyle 0}& ,& t>j\hfill \end{array}\right.\end{array}$$

(21)

and

$$\begin{array}{c}\hfill n_{rt}=\sum _{i=t}^{\infty }{(-1)}^{i-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2i}{i+t}}\right){\displaystyle \frac{(2t+1)S_t}{(i+t+1)C_i}}{f}_{ri}\end{array}$$

(22)

for all $r,t,j\in \mathbb{N}$. Then, for $\Delta ,\Pi \subset \omega$, $F=\left(f_{rt}\right)\in \left(\Delta \left(S^{*}\right):\Pi \right)$, if and only if ${\mathcal{N}}^r\in (\Delta :c)$ and $\mathcal{N}\in (\Delta :\Pi )$.

Proof. 

Assume that $F=\left(f_{rt}\right)\in \left(\Delta \left(S^{*}\right):\Pi \right)$ and $y\in \Delta \left(S^{*}\right)$. Then,

$$\begin{array}{ccc}\hfill \sum _{t=0}^j{f}_{rt}{y}_t& =& \sum _{t=0}^j{f}_{rt}\left(\sum _{i=0}^t{(-1)}^{t-i}\left({\displaystyle \genfrac{}{}{0pt}{}{2t}{t+i}}\right){\displaystyle \frac{(2i+1)S_i}{(t+i+1)C_t}}{z}_i\right)\hfill \\ & =& \sum _{t=0}^j\left(\sum _{i=t}^j{(-1)}^{i-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2i}{i+t}}\right){\displaystyle \frac{(2t+1)S_t}{(i+t+1)C_i}}{f}_{ri}\right)z_t=\sum _{t=0}^j{n}_{jt}^r{z}_t\hfill \end{array}$$

(23)

for all $j,r\in \mathbb{N}$. It must be ${\mathcal{N}}^r\in (\Delta :c)$, because $Fy$ exists. If we take limit for $j\to \infty$ on (23), it is reached that $Fy=\mathcal{N}z$. It is known that $Fy\in \Pi$; in that case, $\mathcal{N}z\in \Pi$. Thus, $\mathcal{N}\in (\Delta :\Pi )$.

Assume that ${\mathcal{N}}^r\in (\Delta :c)$ and $\mathcal{N}\in (\Delta :\Pi )$. In that case, it is seen that $n_{rt}\in {\Delta }^{\beta }$, which gives ${\left(f_{rt}\right)}_{t\in \mathbb{N}}\in {\left(\Delta \left(S^{*}\right)\right)}^{\beta }$. Consequently, $Fy$ exists for all $y\in \Delta \left(S^{*}\right)$. Thus, it is reached from (23) as $j\to \infty$ that $Fy=\mathcal{N}z$, and this implies that $F\in \left(\Delta \left(S^{*}\right):\Pi \right)$. □

Corollary 1.

The necessary and sufficient conditions for the classes $(\Delta \left(S^{*}\right):\Pi )$ can be seen in Table 2 for $\Delta \in \{{\mathsf{\ell }}_1,{\mathsf{\ell }}_p,{\mathsf{\ell }}_{\infty }\}$, $\Pi \in \{{\mathsf{\ell }}_1,{\mathsf{\ell }}_p,{\mathsf{\ell }}_{\infty },c,c_0\}$ and the matrices ${\mathcal{N}}^r$ and $\mathcal{N}$ expressed by (21) and (22).

1. (8),(9) hold with ${\mathcal{N}}^r$, (6) holds with $\mathcal{N}$.	2. (8),(9) hold with ${\mathcal{N}}^r$, (7) holds with $\mathcal{N}$.
3. (8),(9) hold with ${\mathcal{N}}^r$, (8) holds with $\mathcal{N}$.	4. (8),(9) hold with ${\mathcal{N}}^r$, (8),(9) hold with $\mathcal{N}$.
5. (8),(9) hold with ${\mathcal{N}}^r$, (8),(10) hold with $\mathcal{N}$.	6. (9),(12) hold with ${\mathcal{N}}^r$, (11) holds with $\mathcal{N}$.
7. (9),(12) hold with ${\mathcal{N}}^r$, (12) holds with $\mathcal{N}$.	8. (9),(12) hold with ${\mathcal{N}}^r$, (9),(12) hold with $\mathcal{N}$.
9. (9),(12) hold with ${\mathcal{N}}^r$, (10),(12) hold with $\mathcal{N}$.	10. (9),(16) hold with ${\mathcal{N}}^r$, (13) holds with $\mathcal{N}$.
11. (9),(16) hold with ${\mathcal{N}}^r$, (14) holds with $\mathcal{N}$.	12. (9),(16) hold with ${\mathcal{N}}^r$, (15) holds with $\mathcal{N}$.
13. (9),(16) hold with ${\mathcal{N}}^r$, (9),(16) hold with $\mathcal{N}$.	14. (9),(16) hold with ${\mathcal{N}}^r$, (17) holds with $\mathcal{N}$.

Table 2. Conditions of the classes $(\Delta \left(S^{*}\right):\Pi )$ for $\Delta \in \{{\mathsf{\ell }}_1,{\mathsf{\ell }}_p,{\mathsf{\ell }}_{\infty }\}$, $\Pi \in \{{\mathsf{\ell }}_1,{\mathsf{\ell }}_p,{\mathsf{\ell }}_{\infty },c,c_0\}$ and $1<p<\infty$.

($\mathit{From}\downarrow :\mathit{to}\to$)	${\mathsf{\ell }}_1$	${\mathsf{\ell }}_{\mathit{p}}$	${\mathsf{\ell }}_{\mathit{\infty }}$	c	${\mathit{c}}_0$
${\mathsf{\ell }}_{\mathbf{1}}\left({\mathit{S}}^{*}\right)$	1.	2.	3.	4.	5.
${\mathsf{\ell }}_{\mathit{p}}\left({\mathit{S}}^{*}\right)$	6.	◃	7.	8.	9.
${\mathsf{\ell }}_{\infty }\left({\mathit{S}}^{*}\right)$	10.	11.	12.	13.	14.

Table 2. Conditions of the classes $(\Delta \left(S^{*}\right):\Pi )$ for $\Delta \in \{{\mathsf{\ell }}_1,{\mathsf{\ell }}_p,{\mathsf{\ell }}_{\infty }\}$, $\Pi \in \{{\mathsf{\ell }}_1,{\mathsf{\ell }}_p,{\mathsf{\ell }}_{\infty },c,c_0\}$ and $1<p<\infty$.

($\mathit{From}\downarrow :\mathit{to}\to$)	${\mathsf{\ell }}_1$	${\mathsf{\ell }}_{\mathit{p}}$	${\mathsf{\ell }}_{\mathit{\infty }}$	c	${\mathit{c}}_0$
${\mathsf{\ell }}_{\mathbf{1}}\left({\mathit{S}}^{*}\right)$	1.	2.	3.	4.	5.
${\mathsf{\ell }}_{\mathit{p}}\left({\mathit{S}}^{*}\right)$	6.	◃	7.	8.	9.
${\mathsf{\ell }}_{\infty }\left({\mathit{S}}^{*}\right)$	10.	11.	12.	13.	14.

Theorem 11.

Let us assume that the infinite matrices $\tilde{F}=\left({\tilde{f}}_{rt}\right)$ and $F=\left(f_{rt}\right)$ are expressed by the equality

$$\begin{array}{ccc}\hfill {\tilde{f}}_{rt}& =& \sum _{j=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{2r-j}{j}}\right){\displaystyle \frac{C_{r-j}}{S_r}}{f}_{jt}.\hfill \end{array}$$

(24)

Then, $F\in \left(\Delta :\Pi \left(S^{*}\right)\right)$ if and only if $\tilde{F}\in \left(\Delta :\Pi \right)$, where $\Delta \in \left\{{\mathsf{\ell }}_1,{\mathsf{\ell }}_p,{\mathsf{\ell }}_{\infty },c,c_0\right\}$ and $\Pi \in \left\{{\mathsf{\ell }}_1,{\mathsf{\ell }}_p,{\mathsf{\ell }}_{\infty }\right\}$.

Proof. 

For a sequence $y=\left(y_t\right)\in \Delta$, it is obtained that

$$\sum _{t=0}^{\infty }{\tilde{f}}_{rt}{y}_t=\sum _{j=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{2r-j}{j}}\right){\displaystyle \frac{C_{r-j}}{S_r}}\sum _{t=0}^{\infty }{f}_{jt}{y}_t.$$

Thus, ${\tilde{F}}_r\left(y\right)=S_r^{*}\left(Fy\right)$ for all $r\in \mathbb{N}$, and $Fy\in \Pi \left(S^{*}\right)$ iff $\tilde{F}y\in \Pi$ for all $y\in \Delta$. Consequently, it is reached that $F\in \left(\Delta :\Pi \left(S^{*}\right)\right)$ if and only if $\tilde{F}\in \left(\Delta :\Pi \right)$. □

Corollary 2.

The necessary and sufficient conditions for the classes $(\Delta :\Pi \left(S^{*}\right))$ can be seen in Table 3 for the matrices $\tilde{F}=\left({\tilde{f}}_{rt}\right)$ and $F=\left(f_{rt}\right)$ expressed by (24).

Table 3. Conditions of the classes $(\Delta :\Pi \left(S^{*}\right))$ for $\Delta \in \left\{{\mathsf{\ell }}_1,{\mathsf{\ell }}_p,{\mathsf{\ell }}_{\infty },c,c_0\right\}$ and $\Pi \in \left\{{\mathsf{\ell }}_1,{\mathsf{\ell }}_p,{\mathsf{\ell }}_{\infty }\right\}$.

($\mathit{From}\downarrow :\mathit{to}\to$)	${\mathsf{\ell }}_1\left({\mathit{S}}^{*}\right)$	${\mathsf{\ell }}_{\mathit{p}}\left({\mathit{S}}^{*}\right)$	${\mathsf{\ell }}_{\mathit{\infty }}\left({\mathit{S}}^{*}\right)$
${\mathsf{\ell }}_{\mathbf{1}}$	(6)	(7)	(8)
${\mathsf{\ell }}_{\mathit{p}}$	(11)	◃	(12)
${\mathsf{\ell }}_{\infty }$	(13)	(14)	(15)
$\mathit{c}$	(13)	(14)	(15)
${\mathit{c}}_{\mathbf{0}}$	(13)	(14)	(15)

Hold(s) with $\tilde{F}=\left({\tilde{f}}_{rt}\right)$

5. Characterizations of Compact Operators

Assume that $\Delta$ is a normed space, and ${\mathcal{S}}_{\Delta }$ is the unit sphere in $\Delta$. The acronym ${\parallel y\parallel }_{\Delta }^{•}$ is expressed by

$${\parallel y\parallel }_{\Delta }^{•}=\underset{u\in {\mathcal{S}}_{\Delta }}{sup}\left|\sum _t{y}_t{u}_t\right|$$

for a BK-space $\Delta \supset \Theta$, and $y=\left(y_t\right)\in \omega$, provided that the series is finite, where $\Theta$ is space of all finite sequences. In that case, $y\in {\Gamma }^{\beta }$.

Lemma 2

([30]). The following hold:

(i) 

${\mathsf{\ell }}_{\infty }^{\beta }={\mathsf{\ell }}_1$ and ${\parallel y\parallel }_{{\mathsf{\ell }}_{\infty }}^{•}={\parallel y\parallel }_{{\mathsf{\ell }}_1}$, for all $y\in {\mathsf{\ell }}_1$.

(ii) 

${\mathsf{\ell }}_1^{\beta }={\mathsf{\ell }}_{\infty }$ and ${\parallel y\parallel }_{{\mathsf{\ell }}_1}^{•}={\parallel y\parallel }_{{\mathsf{\ell }}_{\infty }}$, for all $y\in {\mathsf{\ell }}_{\infty }$.

(iii) 

${\mathsf{\ell }}_p^{\beta }={\mathsf{\ell }}_q$ and ${\parallel y\parallel }_{{\mathsf{\ell }}_p}^{•}={\parallel y\parallel }_{{\mathsf{\ell }}_q}$, for all $y\in {\mathsf{\ell }}_q$ and $1<p<\infty$.

By $\mathcal{U}(\Delta :\Pi )$ is meant the family of all bounded linear operators from $\Delta$ to $\Pi$.

Lemma 3

([30]). Let us assume that Δ and Π are BK-spaces and $F\in (\Delta :\Pi )$. In that case, there is an operator ${\mathcal{V}}_F\in \mathcal{U}(\Delta :\Pi )$, such that ${\mathcal{V}}_F\left(y\right)=Fy$ for all $y\in \Delta$.

Lemma 4

([30]). Let us assume that $\Delta \supset \Theta$ is a BK-space. If $F\in (\Delta :\Pi )$, in that case, $\parallel {\mathcal{V}}_F{\parallel =\parallel F\parallel }_{(\Delta :\Pi )}={sup}_{r\in \mathbb{N}}{\parallel F_r\parallel }_{\Delta }^{•}<\infty$ for $\Pi \in \left\{c_0,c,{\mathsf{\ell }}_{\infty }\right\}$.

Let us assume that $\mathcal{B}$ is a bounded set in the metric space $\Delta$. Then, the Hausdorff measure of non-compactness of $\mathcal{B}$ is expressed as

$$\chi \left(\mathcal{B}\right)=inf\left\{\delta >0:\mathcal{B}\subset {\cup }_{k=0}^r\mathcal{Q}(y_k,n_k),y_k\in \Delta ,n_k<\delta ,r\in \mathbb{N}\right\},$$

where $\mathcal{Q}(y_k,n_k)$ is the open ball with center $y_k$ and radius $n_k$ for each $k=0,1,2,\dots ,r$. Comprehensive explanations of what has been discussed so far in this section can be obtained from source [30].

Theorem 12

([31]). Let us assume that $\mathcal{B}\subset {\mathsf{\ell }}_p$ is bounded, and an operator ${\mu }_k:{\mathsf{\ell }}_p⟶{\mathsf{\ell }}_p$ stated by ${\mu }_k\left(y\right)=(y_0,y_1,y_2,\dots ,y_k,0,0,\dots )$ for all $y=\left(y_k\right)\in {\mathsf{\ell }}_p$, $1\le p<\infty$ and $k\in \mathbb{N}$. In that case,

$$\chi \left(\mathcal{B}\right)=\underset{k\to \infty }{lim}\left(\underset{y\in \mathcal{B}}{sup}{\parallel (\mathcal{I}-{\mu }_k)\left(y\right)\parallel }_{{\mathsf{\ell }}_p}\right)$$

for the identity operator $\mathcal{I}$ on ${\mathsf{\ell }}_p$.

Let us assume that $\Delta$ and $\Pi$ are Banach spaces and linear operator $\mathcal{V}$ from $\Delta$ to $\Pi$. In that case, if $\left(\mathcal{V}\left(y\right)\right)$ has a convergent subsequence in $\Pi$ for all $y=\left(y_t\right)\in {\mathsf{\ell }}_{\infty }\cap \Delta$, it is said that $\mathcal{V}$ is compact. The acronym $\parallel \mathcal{V}\parallel \chi$ denotes the Hausdorff measure of non-compactness of operator $\mathcal{V}$. Furthermore, $\mathcal{V}$ is compact iff $\parallel \mathcal{V}\parallel \chi =0$.

Comprehensive information about compactness and Hausdorff measure of non-compactness can be obtained from [32,33,34].

Let us suppose the sequences $u=\left(u_t\right)$ and $\nu =\left({\nu }_t\right)$ as

$$\begin{array}{c}\hfill {\nu }_t=\sum _{i=t}^{\infty }{(-1)}^{i-t}\left({\displaystyle \genfrac{}{}{0pt}{}{2i}{i+t}}\right){\displaystyle \frac{(2t+1)S_t}{(i+t+1)C_i}}{u}_i\end{array}$$

(25)

for all $t\in \mathbb{N}$.

Lemma 5.

Assume that $u=\left(u_t\right)\in {\left({\mathsf{\ell }}_p\left(S^{*}\right)\right)}^{\beta }$ for $1\le p\le \infty$. In that case, $\nu =\left({\nu }_t\right)\in {\mathsf{\ell }}_q$ and

$$\begin{array}{ccc}& & \sum _t{u}_t{y}_t=\sum _t{\nu }_t{z}_t\hfill \end{array}$$

(26)

for all $y=\left(y_t\right)\in {\mathsf{\ell }}_p\left(S^{*}\right)$.

Lemma 6.

For $y=\left(y_t\right)$ with (25), the following are satisfied:

(i) 

${\parallel u\parallel }_{{\mathsf{\ell }}_{\infty }\left(S^{*}\right)}^{•}={\sum }_t\left|{\nu }_t\right|<\infty$ for all $u=\left(u_t\right)\in {\left({\mathsf{\ell }}_{\infty }\left(S^{*}\right)\right)}^{\beta }$.

(ii) 

${\parallel u\parallel }_{{\mathsf{\ell }}_1\left(S^{*}\right)}^{•}={sup}_t\left|{\nu }_t\right|<\infty$ for all $u=\left(u_t\right)\in {\left({\mathsf{\ell }}_1\left(S^{*}\right)\right)}^{\beta }$.

(iii) 

${\parallel u\parallel }_{{\mathsf{\ell }}_p\left(S^{*}\right)}^{•}={\left({\sum }_t{\left|{\nu }_t\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}<\infty$ for all $u=\left(u_t\right)\in {\left({\mathsf{\ell }}_p\left(S^{*}\right)\right)}^{\beta }$ and $1<p<\infty$.

Proof. 

It will be proved only for the first part.

(i) It is achieved from Lemma 5 that $\nu =\left({\nu }_t\right)\in {\mathsf{\ell }}_1$ and $\left(26\right)$ satisfies for $u=\left(u_t\right)\in {\left({\mathsf{\ell }}_{\infty }\left(S^{*}\right)\right)}^{\beta }$, and for all $y=\left(y_t\right)\in {\mathsf{\ell }}_{\infty }\left(S^{*}\right)$. It is seen that $y\in {\mathcal{S}}_{{\mathsf{\ell }}_{\infty }\left(S^{*}\right)}$ if and only if $z\in {\mathcal{S}}_{{\mathsf{\ell }}_{\infty }}$ from ${\parallel y\parallel }_{{\mathsf{\ell }}_{\infty }\left(S^{*}\right)}={\parallel z\parallel }_{{\mathsf{\ell }}_{\infty }}$ with (3). In that case, ${\parallel u\parallel }_{{\mathsf{\ell }}_{\infty }\left(S^{*}\right)}^{•}={sup}_{y\in {\mathcal{S}}_{{\mathsf{\ell }}_{\infty }\left(S^{*}\right)}}\left|{\sum }_t{u}_t{y}_t\right|={sup}_{z\in {\mathcal{S}}_{{\mathsf{\ell }}_{\infty }}}\left|{\sum }_t{\nu }_t{z}_t\right|={\parallel \nu \parallel }_{{\mathsf{\ell }}_{\infty }}^{•}.$ From Lemma 2, we see that ${\parallel u\parallel }_{{\mathsf{\ell }}_{\infty }\left(S^{*}\right)}^{•}={\parallel \nu \parallel }_{{\mathsf{\ell }}_{\infty }}^{•}={\parallel \nu \parallel }_{{\mathsf{\ell }}_1}={\sum }_t\left|{\nu }_t\right|<\infty$. □

Lemma 7

([32]). Assume that $\Delta \supset \Theta$ is a BK-space.

(i) 

If $F\in (\Delta :{\mathsf{\ell }}_{\infty })$, in this case, $0\le \parallel {\mathcal{V}}_F{\parallel }_{\chi }\le {lim\; sup}_r{\parallel F_r\parallel }_{\Delta }^{•}$ and ${\mathcal{V}}_F$ is compact if ${lim}_r{\parallel F_r\parallel }_{\Delta }^{•}=0$.

(ii) 

If $F\in (\Delta :c_0)$, in this case $\parallel {\mathcal{V}}_F{\parallel }_{\chi }={lim\; sup}_r{\parallel F_r\parallel }_{\Delta }^{•}$ and ${\mathcal{V}}_F$ is compact iff ${lim}_r{\parallel F_r\parallel }_{\Delta }^{•}=0$.

(iii) 

If $F\in (\Delta :{\mathsf{\ell }}_1)$, in this case,

$$\underset{k}{lim}\left(\underset{H\in {\mathcal{H}}_k}{sup}{∥\sum _{r\in H}{F}_r∥}_{\Delta }^{•}\right)\le {\parallel {\mathcal{V}}_F\parallel }_{\chi }\le 4.\underset{k}{lim}\left(\underset{H\in {\mathcal{H}}_k}{sup}{∥\sum _{r\in H}{F}_r∥}_{\Delta }^{•}\right)$$

and ${\mathcal{V}}_F$ is compact iff ${lim}_k\left({sup}_{H\in {\mathcal{H}}_k}{∥{\sum }_{r\in H}{F}_r∥}_{\Delta }^{•}\right)=0$. Here, $\mathcal{H}$ is the collection of all finite subsets of $\mathbb{N}$ and ${\mathcal{H}}_k$ is the subcollection of $\mathcal{H}$ including of subsets of $\mathbb{N}$ with entries which are bigger than k.

For the rest of the study, the existence of the relation $\left(22\right)$ between matrices $\mathcal{N}$ and F will be agreed by taking into account the convergence of the series.

Lemma 8.

Assume that the infinite matrix $F=\left(f_{rt}\right)$ and $\Pi \subset \omega$. If $F\in ({\mathsf{\ell }}_p\left(S^{*}\right):\Pi )$, in that case, $\mathcal{N}\in ({\mathsf{\ell }}_p:\Pi )$ and $Fy=\mathcal{N}z$ for all $y\in {\mathsf{\ell }}_p\left(S^{*}\right)$ and $1\le p\le \infty$.

Proof. 

We see that by the aid of Lemma 5. □

Theorem 13.

Assume that $1<p<\infty$. Then,

(i) 

If $F\in ({\mathsf{\ell }}_p\left(S^{*}\right):{\mathsf{\ell }}_{\infty })$, in this case,

$$0\le \parallel {\mathcal{V}}_F{\parallel }_{\chi }\le \underset{r}{lim\; sup}{\left(\sum _t{\left|n_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}$$

and ${\mathcal{V}}_F$ is compact if

$$\underset{r}{lim}{\left(\sum _t{\left|n_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}=0.$$

(ii) 

If $F\in ({\mathsf{\ell }}_p\left(S^{*}\right):c_0)$, then

$${\parallel {\mathcal{V}}_F\parallel }_{\chi }=\underset{r}{lim\; sup}{\left(\sum _t{\left|n_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}$$

and ${\mathcal{V}}_F$ is compact iff

$$\underset{r}{lim}{\left(\sum _t{\left|n_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}=0.$$

(iii) 

If $F\in ({\mathsf{\ell }}_p\left(S^{*}\right):{\mathsf{\ell }}_1)$, then

$$\underset{k}{lim}{\parallel F\parallel }_{({\mathsf{\ell }}_p\left(S^{*}\right):{\mathsf{\ell }}_1)}^{\left(k\right)}\le \parallel {\mathcal{V}}_F{\parallel }_{\chi }\le 4.\underset{k}{lim}{\parallel F\parallel }_{({\mathsf{\ell }}_p\left(S^{*}\right):{\mathsf{\ell }}_1)}^{\left(k\right)}$$

and ${\mathcal{V}}_F$ is compact iff

$$\underset{k}{lim}{\parallel F\parallel }_{({\mathsf{\ell }}_p\left(S^{*}\right):{\mathsf{\ell }}_1)}^{\left(k\right)}=0,$$

where ${\parallel F\parallel }_{({\mathsf{\ell }}_p\left(S^{*}\right):{\mathsf{\ell }}_1)}^{\left(k\right)}={sup}_{H\in {\mathcal{H}}_j}{\left({\sum }_t{\left|{\sum }_{r\in H}{n}_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}$ for all $k\in \mathbb{N}$.

Proof. 

(i) Assume that $F\in ({\mathsf{\ell }}_p\left(S^{*}\right):{\mathsf{\ell }}_{\infty })$ and $y=\left(y_t\right)\in {\mathsf{\ell }}_p\left(S^{*}\right)$. It is seen that $F_r\in {\left({\mathsf{\ell }}_p\left(S^{*}\right)\right)}^{\beta }$, because ${\sum }_t{f}_{rt}{y}_t$ converges for each $r\in \mathbb{N}$. By Lemma 6-(iii), ${\parallel F_r\parallel }_{{\mathsf{\ell }}_p\left(S^{*}\right)}^{•}={\left({\sum }_t{\left|n_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}$. Then, by Lemma 7-(i), the following expression is reached:

$$0\le \parallel {\mathcal{V}}_F{\parallel }_{\chi }\le \underset{r}{lim\; sup}{\left(\sum _t{\left|n_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}$$

and ${\mathcal{V}}_F$ is compact if

$$\underset{r}{lim}{\left(\sum _t{\left|n_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}=0.$$

(ii) Let $F\in ({\mathsf{\ell }}_p\left(S^{*}\right):c_0)$. It is known that ${\parallel F_r\parallel }_{{\mathsf{\ell }}_p\left(S^{*}\right)}^{•}={\left({\sum }_t{\left|n_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}$, and by Lemma 7-(ii), the following expression is reached:

$${\parallel {\mathcal{V}}_F\parallel }_{\chi }=\underset{r}{lim\; sup}{\left(\sum _t{\left|n_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}$$

and ${\mathcal{V}}_F$ is compact iff

$$\underset{r}{lim}{\left(\sum _t{\left|n_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}=0.$$

(iii) Let $F\in ({\mathsf{\ell }}_p\left(S^{*}\right):{\mathsf{\ell }}_1)$. By Lemma 6, $\parallel {\sum }_{r\in H}{F}_r{\parallel }_{{\mathsf{\ell }}_p\left(S^{*}\right)}^{•}={\parallel {\sum }_{r\in H}{\mathcal{N}}_r\parallel }_{{\mathsf{\ell }}_q}^{•}$. Thus, from Lemma 7-(iii), we see that

$$\underset{k}{lim}{\left(\underset{H\in {\mathcal{H}}_k}{sup}\sum _t{\left|\sum _{r\in H}{n}_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\le {\parallel {\mathcal{V}}_F\parallel }_{\chi }\le 4.\underset{k}{lim}{\left(\underset{H\in {\mathcal{H}}_k}{sup}\sum _t{\left|\sum _{r\in H}{n}_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}$$

and ${\mathcal{V}}_F$ is compact iff

$$\underset{k}{lim}{\left(\underset{H\in {\mathcal{H}}_k}{sup}\sum _t{\left|\sum _{r\in H}{n}_{rt}\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}=0.$$

□

Theorem 14.

(i) 

If $F\in ({\mathsf{\ell }}_{\infty }\left(S^{*}\right):{\mathsf{\ell }}_{\infty })$, in this case,

$$0\le \parallel {\mathcal{V}}_F{\parallel }_{\chi }\le \underset{r}{lim\; sup}\sum _t\left|n_{rt}\right|$$

and ${\mathcal{V}}_F$ is compact if

$$\underset{r}{lim}\sum _t\left|n_{rt}\right|=0.$$

(ii) 

If $F\in ({\mathsf{\ell }}_{\infty }\left(S^{*}\right):c_0)$, in this case,

$$\parallel {\mathcal{V}}_F{\parallel }_{\chi }=\underset{r}{lim\; sup}\sum _t\left|n_{rt}\right|$$

and ${\mathcal{V}}_F$ is compact iff

$$\underset{r}{lim}\sum _t\left|n_{rt}\right|=0.$$

(iii) 

If $F\in ({\mathsf{\ell }}_{\infty }\left(S^{*}\right):{\mathsf{\ell }}_1)$, in this case,

$$\underset{k}{lim}{\parallel F\parallel }_{({\mathsf{\ell }}_{\infty }\left(S^{*}\right):{\mathsf{\ell }}_1)}^{\left(k\right)}\le \parallel {\mathcal{V}}_F{\parallel }_{\chi }\le 4.\underset{k}{lim}{\parallel F\parallel }_{({\mathsf{\ell }}_{\infty }\left(S^{*}\right):{\mathsf{\ell }}_1)}^{\left(k\right)}$$

and ${\mathcal{V}}_F$ is compact iff

$$\underset{k}{lim}{\parallel F\parallel }_{({\mathsf{\ell }}_{\infty }\left(S^{*}\right):{\mathsf{\ell }}_1)}^{\left(k\right)}=0,$$

where ${\parallel F\parallel }_{({\mathsf{\ell }}_{\infty }\left(S^{*}\right):{\mathsf{\ell }}_1)}^{\left(k\right)}={sup}_{H\in {\mathcal{H}}_k}\left({\sum }_t\left|{\sum }_{r\in H}{n}_{rt}\right|\right)$.

Proof. 

It can be proved with similar approach to Theorem 13. □

Theorem 15.

(i) 

If $F\in ({\mathsf{\ell }}_1\left(S^{*}\right):{\mathsf{\ell }}_{\infty })$, in this case,

$$0\le \parallel {\mathcal{V}}_F{\parallel }_{\chi }\le \underset{r}{lim\; sup}\left(\underset{t}{sup}\left|n_{rt}\right|\right)$$

and ${\mathcal{V}}_F$ is compact if

$$\underset{r}{lim}\left(\underset{t}{sup}\left|n_{rt}\right|\right)=0.$$

(ii) 

If $F\in ({\mathsf{\ell }}_1\left(S^{*}\right):c_0)$, in this case,

$${\parallel {\mathcal{V}}_F\parallel }_{\chi }=\underset{r}{lim\; sup}\left(\underset{t}{sup}\left|n_{rt}\right|\right)$$

and ${\mathcal{V}}_F$ is compact iff

$$\underset{r}{lim}\left(\underset{t}{sup}\left|n_{rt}\right|\right)=0.$$

Proof. 

It can be proved with similar approach to Theorem 13. □

Lemma 9

([32]). Let $F\in (\Lambda :c)$. If Δ has AK property or $\Delta ={\mathsf{\ell }}_{\infty }$, in this case,

$${\displaystyle \frac{1}{2}}\underset{r}{lim\; sup}\parallel F_r{-f\parallel }_{\Delta }^{•}\le \parallel {\mathcal{V}}_F{\parallel }_{\chi }\le \underset{r}{lim\; sup}{\parallel F_r-f\parallel }_{\Delta }^{•}$$

and ${\mathcal{V}}_F$ is compact iff

$$\underset{r}{lim}{\parallel F_r-f\parallel }_{\Delta }^{•}=0,$$

where $f=\left(f_t\right)$ and $f_t={lim}_r{f}_{rt}$.

Theorem 16.

Assume that $F\in ({\mathsf{\ell }}_p\left(S^{*}\right):c)$ and $1<p<\infty$. In this case,

$${\displaystyle \frac{1}{2}}\underset{r}{lim\; sup}{\left(\sum _t{\left|n_{rt}-n_t\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\le {\parallel {\mathcal{V}}_F\parallel }_{\chi }\le \underset{r}{lim\; sup}{\left(\sum _t{\left|n_{rt}-n_t\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}$$

and ${\mathcal{V}}_F$ is compact iff

$$\underset{r}{lim}{\left(\sum _t{\left|n_{rt}-n_t\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}=0,$$

where $n=n_t={lim}_r{n}_{rt}$.

Proof. 

Let $F\in ({\mathsf{\ell }}_p\left(S^{*}\right):c)$. Then, $\mathcal{N}\in ({\mathsf{\ell }}_p:c)$ from Lemma 8. By Lemma 9,

$${\displaystyle \frac{1}{2}}\underset{r}{lim\; sup}\parallel {\mathcal{N}}_r{-n\parallel }_{{\mathsf{\ell }}_p}^{•}\le \parallel {\mathcal{V}}_F{\parallel }_{\chi }\le \underset{r}{lim\; sup}{\parallel {\mathcal{N}}_r-n\parallel }_{{\mathsf{\ell }}_p}^{•}.$$

Therefore, from Lemma 6-(iii),

$${\displaystyle \frac{1}{2}}\underset{r}{lim\; sup}{\left(\sum _t{\left|n_{rt}-n_t\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\le {\parallel {\mathcal{V}}_F\parallel }_{\chi }\le \underset{r}{lim\; sup}{\left(\sum _t{\left|n_{rt}-n_t\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}.$$

Consequently, by Lemma 9, ${\mathcal{V}}_F$ is compact iff

$$\underset{r}{lim}{\left(\sum _t{\left|n_{rt}-n_t\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}=0.$$

□

Theorem 17.

Let $F\in ({\mathsf{\ell }}_{\infty }\left(S^{*}\right):c)$, then

$${\displaystyle \frac{1}{2}}\underset{r}{lim\; sup}\left(\sum _t\left|n_{rt}-n_t\right|\right)\le {\parallel {\mathcal{V}}_F\parallel }_{\chi }\le \underset{r}{lim\; sup}\left(\sum _t\left|n_{rt}-n_t\right|\right)$$

and ${\mathcal{V}}_F$ is compact iff

$$\underset{r}{lim}\left(\sum _t\left|n_{rt}-n_t\right|\right)=0.$$

Proof. 

It can be proved with similar approach to Theorem 16. □

Theorem 18.

Let $F\in ({\mathsf{\ell }}_1\left(S^{*}\right):c)$, then

$${\displaystyle \frac{1}{2}}\underset{r}{lim\; sup}\left(\underset{t}{sup}\left|n_{rt}-n_t\right|\right)\le {\parallel {\mathcal{V}}_F\parallel }_{\chi }\le \underset{r}{lim\; sup}\left(\underset{t}{sup}\left|n_{rt}-n_t\right|\right)$$

and ${\mathcal{V}}_F$ is compact iff

$$\underset{r}{lim}\left(\underset{t}{sup}\left|n_{rt}-n_t\right|\right)=0.$$

Proof. 

It can be also proved with similar approach to Theorem 16. □

6. Conclusions

Taking advantage of the domains of infinite matrices while obtaining new normed or paranormed sequence spaces has become quite popular among researchers recently, and valuable studies have emerged with this idea. In this study, new sequence spaces were obtained as the domain of a new infinite Schröder–Catalan matrix obtained by using Schröder and Catalan numbers on the sequence spaces ${\mathsf{\ell }}_p$ and ${\mathsf{\ell }}_{\infty }$, and the topics of completeness, basis, inclusion relations, duals, matrix transformations and compact operators related to these spaces were examined in detail and new results were obtained. Examining the domains of the Schröder–Catalan matrix in other classical sequence spaces or obtaining new infinite matrices with the help of number sequences can be counted among our future plans.