Mappings Ideal of Type the Domain of Generalized p,q-Euler Matrix in c0 and c

Abstract

We develop new Banach sequence spaces $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ derived by the domain of generalized $p,q$-Euler matrix $E^{a,b}p,q$ in the spaces of null and convergent sequences, respectively. We investigate some topological properties and inclusion natures related to these spaces. We construct bases and obtain $\alpha$, $\beta$, and $\gamma$-duals of the spaces $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$. Certain classes of matrix transformations are characterized from $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ to $\mathsf{Z}\in {\ell }_{\infty },c,c_0,{\ell }_1,{\ell }_k$. We obtain essential conditions of compactness of operators from $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ to $\mathsf{Z}\in {\ell }_{\infty },c,c_0,{\ell }_1,bs,cs,c{s}_0$. Finally, under a definite functional $\mathrm{\varrho }$ and a weighted sequence of positive reals $\delta$, we define a new sequence space ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$. Certain geometric and topological properties of this space along with the eigenvalue distribution of mapping ideals due to this space and $s$-numbers are investigated.

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1. Introduction and Preliminaries

Throughout this paper, the set of all real-valued sequences is symbolised by $s$. A sequence space is a linear subspace of $s$. Few examples of classical sequence spaces which are used frequently in this study are$$\begin{array}{c}\text{(1)}& c=z=z_r\in s:\underset{r⟶\infty }{\lim }{z}_r\text{exists},c_0=z=z_r\in s:\underset{r⟶\infty }{\lim }{z}_r=0,\\ {\ell }_{\infty }=z=z_r\in s:\underset{r\in {\mathbb{N}}_0}{\sup }{z}_r<\infty ,{\ell }_k=z=z_r\in s:{\displaystyle \sum _{r=0}^{\infty }{z_r}^k}<\infty ,1\le k<\infty ,\\ cs=z=z_r\in s:\underset{r⟶\infty }{\lim }{\displaystyle \sum _{v=0}^r{z}_v}\text{\hspace{0.17em}exists},bs=z=z_r\in s:\underset{r\in {\mathbb{N}}_0}{\sup }{\displaystyle \sum _{v=0}^r{z}_v}<\infty \text{and\hspace{0.17em}}\\ c{s}_0=z=z_r\in s:\underset{r⟶\infty }{\lim }{\displaystyle \sum _{v=0}^r{z}_v}=0.\end{array}$$

Here and in what follows, ${\mathrm{\mathbb{N}}}_0=\mathrm{0,1,2},\dots$. A Banach space is a complete normed space. Furthermore, a Banach space having continuous coordinates is called a $BK$-space. The sequence spaces $c_0$ and $c$ are $BK$ spaces endowed with supremum norm defined by $z_{c_0}=z_c={\text{sup}}_{r\in {\mathrm{\mathbb{N}}}_0}{z}_r$.

Let $\mathrm{\Lambda }={\lambda }_{rv}$ be an infinite matrix consisting of real entries and $\mathsf{Z}$ and $\mathsf{W}$ be any two sequence spaces. We denote the sequence in the $r^{th}$ row of the matrix $\mathrm{\Lambda }$ by ${\mathrm{\Lambda }}_r$. The notation $\mathrm{\Lambda }z={\mathrm{\Lambda }z}_r={\displaystyle {\sum }_{v=0}^{\infty }{\lambda }_{rv}{z}_v}$ is called $\mathrm{\Lambda }$-transform of $z=z_r$, given that ${\displaystyle {\sum }_{v=0}^{\infty }{\lambda }_{rv}{z}_v}$ converges $r\in {\mathrm{\mathbb{N}}}_0$. Furthermore, if for each $z\in \mathsf{Z}$, $\mathrm{\Lambda }z\in \mathsf{W}$, then $\mathrm{\Lambda }$ is called a matrix transformation from $\mathsf{Z}$ to $\mathsf{W}$. We use the symbol $\mathsf{Z},\mathsf{W}$ to mean the family of matrix transformation from $\mathsf{Z}$ to $\mathsf{W}$. Additionally, $\mathrm{\Lambda }={\lambda }_{rv}$ is said to be a triangle if ${\lambda }_{rv}=0$ for $v>r$ and ${\lambda }_{rr}\ne 0$.

The domain of $\mathrm{\Lambda }$ in $\mathsf{Z}$, denoted by ${\mathsf{Z}}_{\mathrm{\Lambda }}$, is given by$$\begin{array}{}\text{(2)}& {\mathsf{Z}}_{\Lambda }=z\in \mathsf{Z}:\Lambda z\in \mathsf{Z}.\end{array}$$

The theory of matrix domain is very crucial in constructing sequence spaces. It is known that the matrix domain ${\mathsf{Z}}_{\mathrm{\Lambda }}$ is itself a sequence space. Moreover, if $\mathsf{Z}$ is a $BK$ space and $\mathrm{\Lambda }$ is a triangle, then we enjoy the celebrated property that ${\mathsf{Z}}_{\mathrm{\Lambda }}$ is also a $BK$-space endowed with the norm $z_{{\mathsf{Z}}_{\mathrm{\Lambda }}}={\mathrm{\Lambda }z}_{\mathsf{Z}}$. The literature contains several studies concerning formation of sequence spaces (or Banach sequence spaces) by using domain of special matrices. For good sources of studies, we refer the readers to [1–6] and textbook [7].

1.1. $p,q$-Calculus

The $p,q$-calculus is a recently developed theory that generalizes several concepts in the field of mathematics, engineering, and physics. In particular, in the field of mathematics, $p,q$-calculus is widely utilized in operator theory, approximation theory, algebra, special functions, combinatorics, etc. $p,q$-calculus or $p,q$-analog is an advancement of the classical notions or concepts or mathematical expression by using two independent parameters $p$ and $q$. When $p=1$, the $p,q$-analog of a mathematical expression reduces to $q$-analog (see [8]) which further reduces to the original mathematical expression when $q⟶1^{-}$. For more details, we refer the readers to [9, 10] for studies on $p,q$-calculus and [8] for those on $q$-calculus.

1.1.1. Relevant Notions in $p,q$-Calculus

Definition 1 (see [10]).

The $p,q$-number $0<q<p\le 1$ is given by$$\begin{array}{}\text{(3)}& v_{p,q}=\begin{array}{cc}\frac{p^v-q^v}{p-q},\hfill & v=\mathrm{1,2,3},\dots ,\hfill \\ 0,\hfill & v=0.\hfill \end{array}\end{array}$$

It is well known that when $p=1$, $v_{p,q}$ reduces to its $q$ version $v_q$ (cf. [8]).

Definition 2 (see [9]).

The $p,q$-binomial coefficient is given by$$\begin{array}{}\text{(4)}& {\begin{array}{c}r\\ v\end{array}}_{p,q}=\begin{array}{cc}\frac{r_{p,q}!}{{r-v}_{p,q}!v_{p,q}!},\hfill & r\ge v,\hfill \\ 0,\hfill & v>r.\hfill \end{array}\end{array}$$

Here the symbol $v_{p,q}!$ is the $p,q$-factorial of $v$ which is expressed by$$\begin{array}{}\text{(5)}& v_{p,q}!=v_{p,q}{v-1}_{p,q}\dots 2_{p,q}{1}_{p,q}.\end{array}$$

Lemma 1 (see [11]).

The $p,q$-binomial formula is expressed as$$\begin{array}{c}\text{(6)}& {a+b}_{p,q}^r=\begin{array}{cc}a+bqa+pb{q}^{2}a+p^{2}b\dots q^{r-1}a+p^{r-1}b,\hfill & r=\mathrm{1,2,3},\dots ,\hfill \\ 1,\hfill & r=0,\hfill \end{array}={\displaystyle \sum _{v=0}^r{\begin{array}{c}r\\ v\end{array}}_{p,q}}{p}^{\begin{array}{c}r-v\\ 2\end{array}}{q}^{\begin{array}{c}v\\ 2\end{array}}{a}^v{b}^{r-v}.\end{array}$$

Clearly when $p=1$, $p,q$-binomial formula reduces to $q$-binomial formula (cf. [8]). The readers will be interested in these papers [8–12] to have greater idea about $p,q$ (or $q$)-calculus.

1.2. Euler Sequence Space

Let $0<a<1$; then, the Euler mean $E^a=e_{rv}^a$ is defined by$$\begin{array}{}\text{(7)}& e_{rv}^a=\begin{array}{cc}\begin{array}{c}r\\ v\end{array}{1-a}^{r-v}{b}^v,\hfill & 0\le v\le r,\hfill \\ 0,\hfill & v>r.\hfill \end{array}\end{array}$$

Let $0<a<1$ and $a+b\ne 1$; then, the binomial matrix $E^{a,b}=e_{rv}^{a,b}$ is given by$$\begin{array}{}\text{(8)}& e_{rv}^{a,b}=\begin{array}{cc}\frac{1}{{a+b}^r}\begin{array}{c}r\\ v\end{array}{a}^r{b}^{r-v},\hfill & 0\le v\le r,\hfill \\ 0,\hfill & v>r.\hfill \end{array}\end{array}$$

It is clear that when $a+b=1$, then $E^{a,b}$ reduces to $E^a$. A good number of publications can be found in the literature concerning construction of sequence spaces using Euler mean $E^a$ and binomial matrix $E^{a,b}$. We mention in Table 1 some celebrated papers dealing with Euler spaces. However, before proceeding to the table, we first define backward difference operators ${\mathrm{\Delta }}^B={\delta }_{rv}^B$, ${\mathrm{\Delta }}^{Bm}={\delta }_{rv}^{Bm}$, ${\mathrm{\Delta }}^{Bq}={\delta }_{rv}^{Bq}$, $B^m=b_{rv}^m$, and $B_n^m=b_{rv}^{m,n}$ that are used in the table:$$\begin{array}{c}\text{(9)}& {\delta }_{rv}^B=\begin{array}{cc}{-1}^{r-v},\hfill & r-1\le v\le r,\hfill \\ 0,\hfill & v>r.\hfill \end{array}{\delta }_{rv}^{Bm}=\begin{array}{cc}{-1}^{r-v}\begin{array}{c}m\\ r-v\end{array},\hfill & \text{max}0,r-m\le v\le r,\hfill \\ 0,\hfill & 0\le v<\text{max}0,r-m\text{or}v>r.\hfill \end{array}\\ b_{rv}^m=\begin{array}{cc}\begin{array}{c}m\\ r-v\end{array}{a}^{m-r+v}{b}^{r-v},\hfill & \text{max}0,r-m\le v\le r,\hfill \\ 0,\hfill & 0\le v\le \text{max}0,r-m\text{or}v>r,\hfill \end{array}\\ b_{rv}^{m,n}=\begin{array}{cc}\begin{array}{c}m\\ r-v\end{array}{a}^{m-r+v}{b}^{r-v}{n}_v,\hfill & \text{max}0,r-m\le v\le r,\hfill \\ 0,\hfill & 0\le v\le \text{max}0,r-m\text{or}v>r,\hfill \end{array}\\ {\delta }_{rv}^{Bq}=\begin{array}{cc}{-1}^{r-v}\frac{\Gamma q+1}{r-v!\Gamma q-r+v+1},\hfill & \text{max}0\le v\le r,\hfill \\ 0,\hfill & v>r,\hfill \end{array}\end{array}$$where $m\in {\mathrm{\mathbb{N}}}_0$, $q\in \mathrm{\mathbb{N}}$, and $n=n_r$ is any fixed sequence of real numbers.

Table 1

Euler spaces.

Euler spaces	References
$e_0^a={c_0}_{E^a}$, $e_c^a=c_{E^a}$	[1]
$e_k^a={{\ell }_k}_{E^a}$, $e_{\infty }^a={{\ell }_{\infty }}_{E^a}$	[2, 4]
$e_0^a{\mathrm{\Delta }}^B={c_0}_{E^a{\mathrm{\Delta }}^B}$, $e_c^a{\mathrm{\Delta }}^B=c_{E^a{\mathrm{\Delta }}^B}$	[13]
$e_0^a{\mathrm{\Delta }}^{Bm}={c_0}_{E^a{\mathrm{\Delta }}^{Bm}}$, $e_c^a{\mathrm{\Delta }}^{Bm}=c_{E^a{\mathrm{\Delta }}^{Bm}}$, $e_{\infty }^a{\mathrm{\Delta }}^{Bm}={{\ell }_{\infty }}_{E^a{\mathrm{\Delta }}^{Bm}}$	[14]
$e_k^a{\mathrm{\Delta }}^{Bq}={{\ell }_k}_{E^a{\mathrm{\Delta }}^{Bq}}$, $e_0^a{\mathrm{\Delta }}^{Bq}={c_0}_{E^a{\mathrm{\Delta }}^{Bq}}$, $e_c^a{\mathrm{\Delta }}^{Bq}=c_{E^a{\mathrm{\Delta }}^{Bq}}$, $e_{\infty }^r{\mathrm{\Delta }}^{Bq}={{\ell }_{\infty }}_{E^a{\mathrm{\Delta }}^{Bq}}$	[15]
$e^{a}k={\ell k}_{E^a}$	[16]
$e_0^a{\mathrm{\Delta }}^B,k={c_0}_{E^a{\mathrm{\Delta }}^B}$, $e_c^a{\mathrm{\Delta }}^B,k=c_{E^a{\mathrm{\Delta }}^B}$ and $e_{\infty }^a{\mathrm{\Delta }}^B,k={{\ell }_{\infty }}_{E^a{\mathrm{\Delta }}^B}$	[17]
$e_0^a{\mathrm{\Delta }}^{Bm},k={c_{0}k}_{E^a{\mathrm{\Delta }}^{Bm}}$, $e_c^r{\mathrm{\Delta }}^{Bm},k={ck}_{E^a{\mathrm{\Delta }}^{Bm}}$, $e_{\infty }^a{\mathrm{\Delta }}^{Bm},k={{\ell }_{\infty }k}_{E^a{\mathrm{\Delta }}^{Bm}}$	[18]
$f_0{E}^a={f_0}_{E^a}$, $f{E}^a=f_{E^a}$	[19]
$e_0^a{B}^m={c_0}_{E^a{B}^m}$, $e_c^a{B}^m=c_{E^a{B}^m}$, $e_{\infty }^a{B}^m={{\ell }_{\infty }}_{E^a{B}^m}$	[20]
$e_0^a{B}_n^m={c_0}_{E^a{B}_n^m}$, $e_c^a{B}_n^m=c_{E^a{B}_n^m}$	[21]
$e_k^{a,b}={{\ell }_k}_{E^{a,b}}$, $e_0^{a,b}={c_0}_{E^{a,b}}$, $e_c^{a,b}=c_{E^{a,b}}$, $e_{\infty }^{a,b}={{\ell }_{\infty }}_{E^{a,b}}$	[22, 23]
$e_0^{a,b}{B}^m={c_0}_{E^{a,b}{B}^m}$, $e_c^{a,b}{B}^m=c_{E^{a,b}{B}^m}$, $e_{\infty }^{a,b}{B}^m={{\ell }_{\infty }}_{E^{a,b}{B}^m}$	[24]
$e_0^{a,b}{\mathrm{\Delta }}^{Bq}={c_0}_{E^{a,b}{\mathrm{\Delta }}^{Bq}}$, $e_c^{a,b}{\mathrm{\Delta }}^{Bq}=c_{E^{a,b}{\mathrm{\Delta }}^{Bq}}$, $e_{\infty }^{a,b}{\mathrm{\Delta }}^{Bq}={{\ell }_{\infty }}_{E^{a,b}{\mathrm{\Delta }}^{Bq}}$	[25]
$e_k^{a,b}{\mathrm{\Delta }}^{Bq}={{\ell }_k}_{E^{a,b}{\mathrm{\Delta }}^{Bq}}$	[5]

1.3. $q$ or $p,q$-Sequence Spaces and Motivation

For $0<q<1$, $q$-Cesàro matrix $C^q=c_{rv}^q$ [12] is defined by$$\begin{array}{}\text{(10)}& c_{rv}^q=\begin{array}{cc}\frac{q^v}{{r+1}_q},\hfill & 0\le v\le r,\hfill \\ 0,\hfill & v>r.\hfill \end{array}\end{array}$$

The development of sequence spaces using post-quantum (or $p,q$-) analog of well-known matrices is a new study in the field of sequence spaces. We have seen some studies concerning construction of sequence spaces using the quantum analog of special matrices. For instance, $q$-Cesàro sequence spaces $X_0^q$ and $X_c^q$ are studied by Demiriz and Şahin [26] generated by $q$-Cesàro matrix in $c_0$ and $c$, respectively. As a natural continuation, Yaying et al. [6] developed its domains $X_k^q$ and $X_{\infty }^q$ in ${\ell }_k$ and ${\ell }_{\infty }$, respectively, and studied associated operator ideals. More recently, Yaying et al. [27] constructed $p,q$-Euler sequence spaces $e_k^{a,b}p,q={{\ell }_k}_{E^{a,b}p,q}$ and $e_{\infty }^{a,b}p,q={{\ell }_{\infty }}_{E^{a,b}p,q}$ and studied certain geometric properties of these spaces. Yaying et al. [28] introduced $q$-Euler sequence spaces $e_0^{a,b}q={c_0}_{E^{a,b}q}$ and $e_c^{a,b}q=c_{E^{a,b}q}$.

We are motivated by the studies made in [27, 28] to advance this theory of developing sequence spaces by applying $p,q$-calculus in $c$ and $c_0$. The main objective of this paper is to study generalized Euler spaces $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ due to $p,q$-Euler matrix $E^{a,b}p,q$ in $c_0$ and $c$, respectively. We obtain some topological properties and inclusion nature and determine bases for the spaces $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$. In Section 3, we determine $\alpha$, $\beta$, and $\gamma$-duals of $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$. In Section 4, certain matrix transformations are characterized from $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ to $\mu \in {\ell }_{\infty },c,c_0,{\ell }_1,cs,c{s}_0,bs$. Section 5 is devoted to the computation of essential conditions of compactness for operators on $e_0^{a,b}p,q$. Finally, in Sections 6 and 7, under a definite functional $\mathrm{\varrho }$ and a weighted sequence of positive reals $\delta$, we define new sequence space ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$. We exhibit some geometric and topological natures of this space and eigenvalue distribution of mapping ideals constructed by this space and $s$-numbers.

2. Banach Spaces $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$

The generalized $p,q$-Euler matrix $E^{a,b}p,q=e_{rv}^{a,b}$ of order $a,b$ is defined by (cf. [27])$$\begin{array}{}\text{(11)}& e_{rv}^{a,b}=\begin{array}{cc}\frac{1}{{a+b}_{p,q}^r}{\begin{array}{c}r\\ v\end{array}}_{p,q}{p}^{\begin{array}{c}r-v\\ 2\end{array}}{q}^{\begin{array}{c}v\\ 2\end{array}}{a}^v{b}^{r-v},\hfill & 0\le v\le r,\hfill \\ 0,\hfill & v>r,\hfill \end{array}\end{array}$$where $0<q<p\le 1$ and $a$ and $b$ are nonnegative real numbers. Equivalently,$$\begin{array}{}\text{(12)}& E^{a,b}p,q=\begin{array}{ccccc}1& 0& 0& 0& \dots \\ \frac{b}{{a+b}_{p,q}}& \frac{a}{{a+b}_{p,q}}& 0& 0& \dots \\ \frac{p{b}^2}{{a+b}_{p,q}^2}& \frac{p+qab}{{a+b}_{p,q}^2}& \frac{q{a}^2}{{a+b}_{p,q}^2}& 0& \dots \\ \frac{p^3{b}^3}{{a+b}_{p,q}^3}& \frac{p{p}^2+pq+q^{2}a{b}^2}{{a+b}_{p,q}^3}& \frac{q{p}^2+pq+q^2{a}^{2}b}{{a+b}_{p,q}^3}& \frac{q^3{a}^3}{{a+b}_{p,q}^3}& \dots \\ ⋮& ⋮& ⋮& ⋮& \ddots \end{array}.\end{array}$$

We notice that $\underset{r⟶\infty }{\lim }{e}_{rv}^{a,b}=0$ and ${\displaystyle {\sum }_{v=0}^r{e}_{rv}^{a,b}}=1r\in {\mathrm{\mathbb{N}}}_0$. Thus, $E^{a,b}p,q$ is regular. Furthermore, we may term $E^{a,b}p,q$ as the $p,q$ analog of the binomial matrix $E^{a,b}$. The definition so provided is justifiable as one may see that the matrix $E^{a,b}p,q$ reduces to binomial matrix $E^{a,b}$ when $p=1$ and $q$ approaches 1. It is worth mentioning that $E^{a,b}p,q$ reduces to the triangle $E^{a,b}q$ when $p=1$ with entries ${\begin{array}{c}r\\ v\end{array}}_q{q}^{\begin{array}{c}v\\ 2\end{array}}{a}^v{b}^{r-v}/{a+b}_q^r$. We call $E^{a,b}q$ as the $q$-analog of the binomial matrix $E^{a,b}$. Moreover, when $b=1-a$, then the matrix $E^{a,b}p,q$ reduces to triangle $E^{a}p,q$ with entries $1/{a+1-a}_{p,q}^r{\begin{array}{c}r\\ v\end{array}}_{p,q}{p}^{\begin{array}{c}r-v\\ 2\end{array}}{q}^{\begin{array}{c}v\\ 2\end{array}}{a}^v{1-a}^{r-v}$.

The generalized $p,q$-Euler spaces $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ are defined by$$\begin{array}{c}\text{(13)}& e_0^{a,b}p,q=z=z_v\in s:\underset{r⟶\infty }{\lim }\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\begin{array}{c}r\\ v\end{array}}_{p,q}}{p}^{\begin{array}{c}r-v\\ 2\end{array}}{q}^{\begin{array}{c}v\\ 2\end{array}}{a}^v{b}^{r-v}{z}_v=0,e_c^{a,b}p,q=z=z_v\in s:\underset{r⟶\infty }{\lim }\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\begin{array}{c}r\\ v\end{array}}_{p,q}}{p}^{\begin{array}{c}r-v\\ 2\end{array}}{q}^{\begin{array}{c}v\\ 2\end{array}}{a}^v{b}^{r-v}{z}_v\text{\hspace{0.17em}exists}.\end{array}$$

In view of (2), we noticed that $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ can also be defined in the form$$\begin{array}{}\text{(14)}& e_0^{a,b}p,q={c_0}_{E^{a,b}p,q},e_c^{a,b}p,q=c_{E^{a,b}p,q}.\end{array}$$

We emphasize here that $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ reduce to the below given spaces depending on the choice of $p,q$ and $a,b$:

(1) If $p=1$, $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ reduce to $q$-binomial spaces $e_0^{a,b}q={c_0}_{E^{a,b}q}$ and $e_c^{a,b}q=c_{E^{a,b}q}$, respectively, due to Yaying et al. [28].

(2) If $p=1$ and $q⟶1^{-}$, $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ reduce to binomial spaces $e_0^{a,b}$ and $e_c^{a,b}$, respectively, due to Bişgin [22].

(3) If $p=1$ and $b=1-a$, $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ reduce to $q$-Euler space $e_0^{a}q={c_0}_{E^{a}q}$ and $e_c^{a}q=c_{E^{a}q}$, respectively.

(4) If $p=1$ and $q⟶1^{-}$, $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ reduce to Euler spaces $e_0^a$ and $e_c^a$, respectively, due to Altay and Başar [1].

(5) If $b=1-a$, $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ reduce to $p,q$-Euler spaces $e_0^{a}p,q={c_0}_{E^{a}p,q}$ and $e_c^{a}p,q=c_{E^{a}p,q}$.

Consider the sequence $w=w_v$ defined in terms of the sequence $z=z_v$ by$$\begin{array}{}\text{(15)}& w_r={E^{a,b}p,qz}_r=\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\begin{array}{c}r\\ v\end{array}}_{p,q}}{p}^{\begin{array}{c}r-v\\ 2\end{array}}{q}^{\begin{array}{c}v\\ 2\end{array}}{a}^v{b}^{r-v}{z}_v,r\in {\mathrm{\mathbb{N}}}_0.\end{array}$$

Then, $w=w_r$ is called $E^{a,b}p,q$-transform of $z=z_v$. Due to (15), we get that$$\begin{array}{}\text{(16)}& z_r={\displaystyle \sum _{v=0}^r{-1}^{r-v}}\frac{{\begin{array}{c}r\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}r-v\\ 2\end{array}}{b}^{r-v}{a+b}_{p,q}^v}{a^r{q}^{\begin{array}{c}r\\ 2\end{array}}}{w}_{v}r\in {\mathrm{\mathbb{N}}}_0.\end{array}$$

Theorem 1.

The spaces$e_0^{a,b}p,q$and$e_c^{a,b}p,q$are$BK$spaces because$$\begin{array}{}\text{(17)}& z_{e_0^{a,b}p,q}=z_{e_c^{a,b}p,q}={\text{sup}}_{r\in {\mathrm{\mathbb{N}}}_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\begin{array}{c}r\\ v\end{array}}_{p,q}}{p}^{\begin{array}{c}r-v\\ 2\end{array}}{q}^{\begin{array}{c}v\\ 2\end{array}}{a}^v{b}^{r-v}{z}_v.\end{array}$$

Proof.

This is a routine work.

Remark 1.

(i) In the case $p=1$, $e_0^{a,b}q$ and $e_c^{a,b}q$ are $BK$spaces because$$\begin{array}{}\text{(18)}& z_{e_0^{a,b}q}=z_{e_c^{a,b}q}={\text{sup}}_{r\in {\mathrm{\mathbb{N}}}_0}\frac{1}{{a+b}_q^r}{\displaystyle \sum _{v=0}^r{\begin{array}{c}r\\ v\end{array}}_q}{q}^{\begin{array}{c}v\\ 2\end{array}}{a}^v{b}^{r-v}{z}_v,\end{array}$$

  as obtained by Yaying et al. [28].

(ii) In the case $p=1$ and $q⟶1$, $e_0^{a,b}$ and $e_c^{a,b}$ are $BK$ because$$\begin{array}{}\text{(19)}& z_{e_0^{a,b}}=z_{e_c^{a,b}}={\text{sup}}_{r\in {\mathrm{\mathbb{N}}}_0}\frac{1}{{a+b}^r}{\displaystyle \sum _{v=0}^r\begin{array}{c}r\\ v\end{array}}{a}^v{b}^{r-v}{z}_v,\end{array}$$

  as deduced by Bişgin [22].

Theorem 2.

The spaces$e_0^{a,b}p,q\cong c_0$and$e_c^{a,b}p,q\cong c$.

Proof.

Since the proof of both spaces is similar, we provide the proof of the first case only. Note that the mapping $\pi :e_0^{a,b}p,q⟶c_0$ defined by $\pi z=E^{a,b}p,qz$ for all $z\in e_0^{a,b}p,q$ is linear and $1-1$. Let $w=w_r$ be any sequence in $c_0$ and $z=z_r$ be as defined in (16). Then, one obtains$$\begin{array}{c}\text{(20)}& \underset{r⟶\infty }{\lim }\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\begin{array}{c}r\\ v\end{array}}_{p,q}}{p}^{\begin{array}{c}r-v\\ 2\end{array}}{q}^{\begin{array}{c}v\\ 2\end{array}}{a}^v{b}^{r-v}{z}_v=\underset{r⟶\infty }{\lim }\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\begin{array}{c}r\\ v\end{array}}_{p,q}}{p}^{\begin{array}{c}r-v\\ 2\end{array}}{q}^{\begin{array}{c}v\\ 2\end{array}}{a}^v{b}^{r-v}{\displaystyle \sum _{l=0}^v{-1}^{v-l}}\frac{{\begin{array}{c}v\\ l\end{array}}_{p,q}{q}^{\begin{array}{c}v-l\\ 2\end{array}}{t}^{v-l}{a+b}_{p,q}^l}{a^v{q}^{\begin{array}{c}v\\ 2\end{array}}}{w}_l\\ =\underset{r⟶\infty }{\lim }{w}_r=0.\end{array}$$

Thus, $z\in e_0^{a,b}p,q$ and $\pi :e_0^{a,b}p,q⟶c_0$ is onto. This proves that the space $e_0^{a,b}p,q\cong c_0$.

Remark 2.

(i) In the case$p=1$, the spaces$e_0^{a,b}q\cong c_0$and$e_c^{a,b}q\cong c$as obtained by Yaying et al. [28].

(ii) In the case$p=1$and$q⟶1^{-}$, the spaces$e_0^{a,b}\cong c_0$and$e_c^{a,b}\cong c$as obtained by Bişgin [22].

Theorem 3.

The following inclusions strictly hold:

(a) $e_0^{a,b}p,q\subset e_c^{a,b}p,q$.

(b) $c_0\subset e_0^{a,b}p,q$and$c\subset e_c^{a,b}p,q$.

Proof.

The inclusion $a$ is obvious. Hence, we present the proof for inclusion $b$. We know that the matrix $E^{a,b}p,q$ is regular; therefore, the inclusion $c_0\subset e_0^{a,b}p,q$ holds. Now, let us consider the sequence $z=1,-\mathrm{1,1},-1,\dots$; then, $E^{a,b}p,q-$ transform of $z$, that is, $E^{a,b}p,q=-a+b^r/{a+b}_{p,q}^{r}r\in {\mathrm{\mathbb{N}}}_0$, is a sequence in the space $c_0$, although $z$ is not a sequence in $c_0$. This proves that the inclusion $c_0\subset e_0^{a,b}p,q$ strictly holds. The proof for the later case is similar to the above discussion and hence the details are excluded.

We recall that the matrix domain ${\mathsf{Z}}_{\mathrm{\Lambda }}$ has a basis iff $\mathsf{Z}$ has basis, where $\mathrm{\Lambda }$ is a triangle (see [29]). In view of this fact, one obtains the following result.

Theorem 4.

Let${ϵ}^{v}p,q={ϵ}_r^{v}p,q\in e_0^{a,b}p,qv\in {\mathrm{\mathbb{N}}}_0$defined by$$\begin{array}{}\text{(21)}& {\mathrm{ϵ}}_r^{v}p,q=\begin{array}{cc}{-1}^{r-v}\frac{{\begin{array}{c}r\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}r-v\\ 2\end{array}}{b}^{r-v}{a+b}_{p,q}^v}{a^r{q}^{\begin{array}{c}r\\ 2\end{array}}},\hfill & v\le r,\hfill \\ 0,\hfill & v>r.\hfill \end{array}\end{array}$$

Then, the set

(a) ${ϵ}^{0}p,q,{ϵ}^{1}p,q,{ϵ}^{2}p,q,\dots$ forms a basis of $e_0^{a,b}p,q$ and every $z\in e_0^{p,q}p,q$ has a unique representation $z={\displaystyle {\sum }_{v=0}^{\infty }{w}_v{z}^v}p,q$.

(b) $e,{ϵ}^{0}p,q,{ϵ}^{1}p,q,{ϵ}^{2}p,q,\dots$ forms a basis of $e_c^{a,b}p,q$, where $e=\mathrm{1,1,1},\dots$, and every $z\in e_c^{a,b}p,q$ has a unique representation $z=\xi e+{\displaystyle {\sum }_{v=0}^{\infty }{w}_v-\xi {ϵ}^v}p,q$, where $\xi ={\lim }_{v⟶\infty }{w}_v={\lim }_{v⟶\infty }{E^{a,b}p,qz}_v$.

3. $\alpha$, $\beta$, and $\gamma$-Duals

In the present section, we compute $\alpha$, $\beta$, and $\gamma$-duals of $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$. Because the techniques of the proof for both spaces $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ are the same, we present the proof for the space $e_0^{a,b}p,q$ only.

Definition 3.

The $\alpha$, $\beta$, and $\gamma -$duals of $\mathsf{Z}\subset s$ are given by$$\begin{array}{c}\text{(22)}& {\mathsf{Z}}^{\alpha }=\varsigma ={\varsigma }_r\in s:\varsigma z={\varsigma }_r{z}_r\in {\ell }_1\forall z\in \mathsf{Z},{\mathsf{Z}}^{\beta }=\varsigma ={\varsigma }_r\in s:\varsigma z={\varsigma }_r{z}_r\in cs\forall z\in \mathsf{Z},\\ {\mathsf{Z}}^{\gamma }=\varsigma ={\varsigma }_r\in s:\varsigma z={\varsigma }_r{z}_r\in bs\forall z\in \mathsf{Z},\end{array}$$respectively.

Now we present well-known lemmas due to Stielglitz and Tietz [30] which are used for proving our theorems. In the rest of the paper, $\mathrm{ℛ}$ represents the family of all finite subsets of ${\mathrm{\mathbb{N}}}_0$.

Lemma 2.

$\mathrm{\Lambda }={\lambda }_{rv}\in c_0,{\ell }_1$iff$$\begin{array}{}\text{(23)}& {\text{sup}}_{R\in \mathrm{ℛ}}{\displaystyle \sum _{v=0}^{\infty }{\displaystyle \sum _{r\in R}{\lambda }_{rv}}}<\infty .\end{array}$$

Lemma 3.

$\mathrm{\Lambda }={\lambda }_{rv}\in c_0,c$iff$$\begin{array}{}\text{(24)}& {\text{sup}}_{r\in {\mathrm{\mathbb{N}}}_0}{\displaystyle \sum _{v=0}^r{\lambda }_{rv}}<\infty ,\text{(25)}& \underset{r⟶\infty }{\lim }{\lambda }_{rv}\text{\hspace{0.17em}exists\hspace{0.17em}for\hspace{0.17em}each\hspace{0.17em}v}\in {\mathrm{\mathbb{N}}}_{\text{0}}.\end{array}$$

Lemma 4.

$\mathrm{\Lambda }={\lambda }_{rv}\in c_0,{\ell }_{\infty }$iff (24) holds.

Theorem 5.

Define the set$d_{1}p,q$by$$\begin{array}{}\text{(26)}& d_{1}p,q=\varsigma ={\varsigma }_r\in s:{\text{sup}}_{R\in \mathrm{ℛ}}{\displaystyle \sum _{v=0}^{\infty }{\displaystyle \sum _{r\in R}{-1}^{r-v}}\frac{{\begin{array}{c}r\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}r-v\\ 2\end{array}}{b}^{r-v}{a+b}_{p,q}^v}{a^r{q}^{\begin{array}{c}r\\ 2\end{array}}}{\varsigma }_r}<\infty .\end{array}$$

Then, ${e_0^{a,b}p,q}^{\alpha }={e_c^{a,b}p,q}^{\alpha }=d_{1}p,q$.

Proof.

Let ${\varsigma }_r\in s$. Then, by using equality (16), we have$$\begin{array}{c}\text{(27)}& {\varsigma }_r{z}_r={\displaystyle \sum _{v=0}^r{-1}^{r-v}}\frac{{\begin{array}{c}r\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}r-v\\ 2\end{array}}{b}^{r-v}{a+b}_{p,q}^v}{a^r{q}^{\begin{array}{c}r\\ 2\end{array}}}{\varsigma }_r{w}_v={{\Phi }^{a,b}p,qw}_r,r\in {\mathrm{\mathbb{N}}}_0,\end{array}$$where ${\mathrm{\Phi }}^{a,b}p,q={\varphi }_{rv}^{a,b}p,q$ is a triangle defined by$$\begin{array}{}\text{(28)}& {\varphi }_{rv}^{a,b}p,q=\begin{array}{cc}{-1}^{r-v}\frac{{\begin{array}{c}r\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}r-v\\ 2\end{array}}{b}^{r-v}{a+b}_{p,q}^v}{a^r{q}^{\begin{array}{c}r\\ 2\end{array}}}{\varsigma }_r,\hfill & 0\le v\le r,\hfill \\ 0,\hfill & v>r.\hfill \end{array}\end{array}$$

Thus, we observe on using (27) that $\varsigma z={\varsigma }_r{z}_r\in {\ell }_1$ whenever $z\in e_0^{a,b}p.q$ iff ${\mathrm{\Phi }}^{a,b}p,qw\in {\ell }_1$ whenever $w\in c_0$. Thus, we realize that $\varsigma ={\varsigma }_r\in {e_0^{p,q}p,q}^{\alpha }$ iff $E^{a,b}p,q\in c_0,{\ell }_1$. Thus, using Lemma 2, we conclude that ${e_0^{a,b}p,q}^{\alpha }=d_{1}p,q$.

Theorem 6.

Define the sets$d_{2}p,q$,$d_{3}p,q$, and$d_{4}p,q$by$$\begin{array}{c}\text{(29)}& d_{2}p,q=\varsigma ={\varsigma }_r\in s:{\displaystyle \sum _{l=v}^{\infty }{-1}^{l-v}}\frac{{\begin{array}{c}l\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}l-v\\ 2\end{array}}{b}^{l-v}{a+b}_{p,q}^v}{a^l{q}^{\begin{array}{c}l\\ 2\end{array}}}{\varsigma }_l\text{exists\hspace{0.17em}}\forall \text{v}\in {\mathrm{\mathbb{N}}}_{\text{0}},d_{3}p,q=\varsigma ={\varsigma }_r\in s:{\text{sup}}_{r\in {\mathbb{N}}_0}{\displaystyle \sum _{v=0}^r{\displaystyle \sum _{l=v}^r{-1}^{l-v}\frac{{\begin{array}{c}l\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}l-v\\ 2\end{array}}{b}^{l-v}{a+b}_{p,q}^v}{a^l{q}^{\begin{array}{c}l\\ 2\end{array}}}{\varsigma }_l}}<\infty ,\\ d_{4}p,q=\varsigma ={\varsigma }_r\in s:\underset{r⟶\infty }{\lim }{\displaystyle \sum _{v=0}^r{\displaystyle \sum _{l=v}^r{-1}^{l-v}}}\frac{{\begin{array}{c}l\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}l-v\\ 2\end{array}}{b}^{l-v}{a+b}_{p,q}^v}{a^l{q}^{\begin{array}{c}l\\ 2\end{array}}}{\varsigma }_l\text{exists}.\end{array}$$

Then, ${e_0^{a,b}p,q}^{\beta }=d_{2}p,q\cap d_{3}p,q$ and ${e_c^{a,b}p,q}^{\beta }=d_{2}p,q\cap d_{3}p,q\cap d_{4}p,q$.

Proof.

Let ${\varsigma }_r\in s$ and $w=w_r$ be the $E^{a,b}p,q$-transform of the sequence $z=z_r$. Then, using equality (16), we get$$\begin{array}{c}\text{(30)}& {\displaystyle \sum _{v=0}^r{\varsigma }_v{z}_v}={\displaystyle \sum _{v=0}^r{\displaystyle \sum _{l=0}^v{-1}^{v-l}}\frac{{\begin{array}{c}v\\ l\end{array}}_{p,q}{q}^{\begin{array}{c}v-l\\ 2\end{array}}{b}^{v-l}{a+b}_{p,q}^l}{a^v{q}^{\begin{array}{c}v\\ 2\end{array}}}{w}_l}{\varsigma }_v={\displaystyle \sum _{v=0}^r{\displaystyle \sum _{l=v}^r{-1}^{l-v}}\frac{{\begin{array}{c}l\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}l-v\\ 2\end{array}}{b}^{l-v}{a+b}_{p,q}^v}{a^l{q}^{\begin{array}{c}l\\ 2\end{array}}}{\varsigma }_l}{w}_v\\ \text{(31)}& ={{\Psi }^{a,b}p,qw}_r,r\in {\mathrm{\mathbb{N}}}_0,\end{array}$$where ${\mathrm{\Psi }}^{a,b}p,q={\psi }_{rv}^{a,b}p,q$ is given by$$\begin{array}{}\text{(32)}& {\psi }_{rv}^{a,b}p,q=\begin{array}{cc}{\displaystyle \sum _{l=v}^r{-1}^{l-v}}\frac{{\begin{array}{c}l\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}l-v\\ 2\end{array}}{b}^{l-v}{a+b}_{p,q}^v}{a^l{q}^{\begin{array}{c}l\\ 2\end{array}}}{\varsigma }_l,\hfill & 0\le v\le r,\hfill \\ 0,\hfill & v>r.\hfill \end{array}\end{array}$$

In the light of (31), one obtains that $\varsigma z={\varsigma }_r{z}_r\in cs$ whenever $z=z_r\in e_0^{a,b}p,q$ iff ${\mathrm{\Psi }}^{a,b}p,qw\in c$ whenever $w=w_v\in c_0$. Thus, $\varsigma ={\varsigma }_r\in {e_0^{a,b}p,q}^{\beta }$ iff ${\mathrm{\Psi }}^{a,b}p,q\in c_0,c$. Thus, one obtains on using Lemma 3 that$$\begin{array}{}\text{(33)}& {\text{sup}}_{r\in {\mathrm{\mathbb{N}}}_0}{\displaystyle \sum _{v=0}^r{\psi }_{rv}^{a,b}p,q}<\infty \text{\hspace{0.17em}and\hspace{0.17em}}\underset{r⟶\infty }{\lim }{\psi }_{rv}^{a,b}p,q\text{exists\hspace{0.17em}}\forall v\in {\mathrm{\mathbb{N}}}_0.\end{array}$$

Thus, $e_0^{a,b}p,q=d_{2}p,q\cap d_{3}p,q$.

Theorem 7.

${e_0^{a,b}p,q}^{\gamma }={e_c^{a,b}p,q}^{\gamma }=d_{3}p,q$.

Proof.

The proof is similar to the previous theorem except that Lemma 4 is used in place of Lemma 3.

4. Matrix Transformations

Now we characterize certain class of matrix transformations from $e_0^{a,b}p,q$ and $e_c^{a,b}p,q$ to $\mathsf{W}\in {\ell }_{\infty },c,c_0,{\ell }_1,{\ell }_k$.

Theorem 8.

Let$\mathsf{Z}\subset s$be arbitrary. Then, $\mathrm{\Lambda }={\lambda }_{rv}\in e_0^{a,b}p,q,\mathsf{Z}$(or, respectively, $e_c^{a,b}p,q,\mathsf{Z}$) if and only if${\mathrm{\Omega }}^r={\omega }_{mv}^r\in c_0,c$(or, respectively, $c,c$) for each$r\in {\mathrm{\mathbb{N}}}_0$, and$\mathrm{\Omega }={\omega }_{rv}\in c_0,\mathsf{Z}$(or, respectively, $c,\mathsf{Z}$), where$$\begin{array}{c}\text{(34)}& {\omega }_{mv}^r=\begin{array}{cc}0,\hfill & v>m,\hfill \\ {\displaystyle \sum _{l=v}^m{-1}^{l-v}}\frac{{\begin{array}{c}l\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}l-v\\ 2\end{array}}{b}^{l-v}{a+b}_{p,q}^v}{a^l{q}^{\begin{array}{c}l\\ 2\end{array}}}{\lambda }_{rl},\hfill & 0\le v\le m,\hfill \end{array}{\omega }_{rv}={\displaystyle \sum _{l=v}^{\infty }{-1}^{l-v}}\frac{{\begin{array}{c}l\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}l-v\\ 2\end{array}}{b}^{l-v}{a+b}_{p,q}^v}{a^l{q}^{\begin{array}{c}l\\ 2\end{array}}}{\lambda }_{rl}.\end{array}$$for all $r,v\in {\mathrm{\mathbb{N}}}_0$.

Proof.

This is straightforward from ([3], Theorem 8).

Now, in the light of Theorem 8 and recalling the celebrated results of Stielglitz and Tietz [30], we get the following results.

Corollary 1.

The following statements hold:

(1) $\mathrm{\Lambda }\in e_0^{a,b}p,q,{\ell }_{\infty }$iff$$\begin{array}{}\text{(35)}& {\text{sup}}_{m\in {\mathrm{\mathbb{N}}}_0}{\displaystyle \sum _{v=0}^{\infty }{\omega }_{mv}^r}<\infty ,\text{(36)}& \underset{m⟶\infty }{\lim }{\omega }_{mv}^r\text{exists\hspace{0.17em}for\hspace{0.17em}allv}\in {\mathrm{\mathbb{N}}}_{\text{0}},\text{(37)}& {\text{sup}}_{r\in {\mathrm{\mathbb{N}}}_0}{\displaystyle \sum _{v=0}^{\infty }{\omega }_{rv}}<\infty .\end{array}$$

(2) $\mathrm{\Lambda }\in e_0^{a,b}p,q,c$iff (35) and (36) hold, and$$\begin{array}{}\text{(38)}& {\text{sup}}_{r\in {\mathrm{\mathbb{N}}}_0}{\displaystyle \sum _{v=0}^{\infty }{\omega }_{rv}}<\infty ,\text{(39)}& \underset{r⟶\infty }{\lim }{\omega }_{rv}\text{exists\hspace{0.17em}for\hspace{0.17em}all\hspace{0.17em}v}\in {\mathrm{\mathbb{N}}}_{\text{0}}.\end{array}$$

(3) $\mathrm{\Lambda }\in e_0^{a,b}p,q,c_0$iff (35)–(37) hold, and$$\begin{array}{}\text{(40)}& \underset{r⟶\infty }{\lim }{\omega }_{rv}=0\text{for\hspace{0.17em}all\hspace{0.17em}v}\in {\mathrm{\mathbb{N}}}_{\text{0}}.\end{array}$$

(4) $\mathrm{\Lambda }\in e_0^{a,b}p,q,{\ell }_1$iff (35) and (36) hold, and$$\begin{array}{}\text{(41)}& {\text{sup}}_{R\in \mathrm{ℛ}}{\displaystyle \sum _{v=0}^{\infty }{\displaystyle \sum _{r\in R}{\omega }_{rv}}}<\infty .\end{array}$$

(5) $\mathrm{\Lambda }\in e_0^{a,b}p,q,{\ell }_k$iff (35) and (36) hold, and$$\begin{array}{}\text{(42)}& {\text{sup}}_{V\in \mathrm{ℛ}}{\displaystyle \sum _{r=0}^{\infty }{{\displaystyle \sum _{v\in V}{\omega }_{rv}}}^k}<\infty k>1.\end{array}$$

Corollary 2.

The following statements hold:

(1) $\mathrm{\Lambda }\in e_c^{a,b}p,q,{\ell }_{\infty }$iff (35), (36) and (38) hold, and$$\begin{array}{}\text{(43)}& \underset{m⟶\infty }{\lim }{\displaystyle \sum _{v=0}^{\infty }{\omega }_{mv}^r}.\end{array}$$

(2) $\mathrm{\Lambda }\in e_c^{a,b}p,q,c$iff (35)–(37), (39), and (43) hold, and$$\begin{array}{}\text{(44)}& \underset{r⟶\infty }{\lim }{\displaystyle \sum _{v=0}^r{\omega }_{rv}}\text{exists}.\end{array}$$

(3) $\mathrm{\Lambda }\in e_c^{a,b}p,q,c_0$iff (35)–(37), (39), and (43) hold.$$\begin{array}{}\text{(45)}& \underset{r⟶\infty }{\lim }{\displaystyle \sum _{v=0}^r{\omega }_{rv}}=0.\end{array}$$

(4) $\mathrm{\Lambda }\in e_c^{a,b}p,q,{\ell }_1$iff (35), (36), (41), and (43) hold.

(5) $\mathrm{\Lambda }\in e_c^{a,b}p,q,{\ell }_k$iff (35), (36), (42), and (43) hold.

Başar and Altay [31] developed an important result that assists in characterizing matrix mappings between two sequence spaces.

Lemma 5 (see [31]).

Let $\mathrm{\Lambda }$ be an infinite matrix, $\mathrm{\Omega }$ be a triangle, and $\mathsf{Z}$ and $\mathsf{W}$ be any two sequence spaces. Then, $\mathrm{\Lambda }\in \mathsf{Z},{\mathsf{W}}_{\mathrm{\Omega }}$if and only if$\mathrm{\Omega }\mathrm{\Lambda }\in \mathsf{Z},\mathsf{W}$.

Let $\mathrm{\Lambda }={\lambda }_{rv}$ be an infinite matrix. In the light of Lemma 5 together with Corollaries 1 and 2, we obtain following classes of matrix mappings.

Corollary 3.

Define the matrix$C^{\alpha }=c_{rv}^{\alpha }$by$$\begin{array}{}\text{(46)}& c_{rv}^{\alpha }={\displaystyle \sum _{m=0}^r\frac{\begin{array}{c}\alpha +r-m-1\\ r-m\end{array}}{\begin{array}{c}\alpha +r\\ r\end{array}}{\lambda }_{mv}}\alpha >1,\end{array}$$for all $r,v\in {\mathrm{\mathbb{N}}}_0$. Then, the necessary and sufficient condition that $\mathrm{\Lambda }$ is in any one of the families $e_0^{a,b}p,q,C_k^{\alpha }$, $e_0^{a,b}p,q,C_{\infty }^{\alpha }$, $e_c^{a,b}p,q,C_k^{\alpha }$, and $e_c^{a,b}p,q,C_{\infty }^{\alpha }$ is determined from the respective ones in Corollaries 1 and 2, by replacing the elements of $\mathrm{\Lambda }$ with those of $C^{\alpha }$, where $C_k^{\alpha }$ and $C_{\infty }^{\alpha }$ are generalized Cesàro sequence spaces of order $\alpha$ given by Roopaei et al. [32].

Corollary 4.

Let$\mathrm{\Lambda }={\lambda }_{rv}$be an infinite matrix and define the matrix$\tilde{C}=C_{rv}$by$$\begin{array}{}\text{(47)}& {\tilde{C}}_{rv}={\displaystyle \sum _{m=0}^r\frac{C_m{C}_{r-m}}{C_{r+1}}}{\lambda }_{mv}m,v\in {\mathrm{\mathbb{N}}}_0,\end{array}$$where$C_r$represent sequence of Catalan numbers. Then, the necessary and sufficient condition that$\mathrm{\Lambda }$belongs to one of the classes$e_0^{a,b}p,q,c_0\tilde{C}$,$e_0^{a,b}p,q,c\tilde{C}$,$e_c^{a,b}p,q,c_0\tilde{C}$, and$e_c^{a,b}p,q,c\tilde{C}$is determined from the respective ones inCorollaries 1 and 2by replacing the elements of$\mathrm{\Lambda }$with those of$\tilde{C}$, where$c\tilde{C}$and$c_0\tilde{C}$are Catalan sequence spaces given by İlkhan [33].

Corollary 5.

Define$\tilde{\lambda }={\tilde{\lambda }}_{rv}$by$$\begin{array}{}\text{(48)}& {\tilde{\lambda }}_{rv}={\displaystyle \sum _{m=0}^r{\lambda }_{mv}},r,v\in {\mathrm{\mathbb{N}}}_0.\end{array}$$

Then, the necessary and sufficient condition that $\mathrm{\Lambda }$ belongs to one of the families $e_0^{a,b}p,q,bs$, $e_0^{a,b}p,q,cs$, $e_0^{a,b}p,q,c{s}_0$, $e_c^{a,b}p,q,bs$, $e_c^{a,b}p,q,cs$, and $e_c^{a,b}p,q,c{s}_0$ is determined by replacing the elements of $\mathrm{\Lambda }$ with those of $\tilde{\lambda }$ from the respective ones in Corollaries 1 and 2.

5. Compactness by Hmnc

Let $\mathsf{Z}$ and $\mathsf{W}$ be any two Banach spaces. We represent the set of all bounded linear operators $T:\mathsf{Z}⟶\mathsf{W}$ by $B\mathsf{Z},\mathsf{W}$, which itself is a Banach space endowed with the norm $T={\text{sup}}_{z\in B\mathsf{Z}}Tz$, where the notation $B_{\mathsf{Z}}$ denotes open ball in $\mathsf{Z}$. Define ${\varsigma }_{\mathsf{Z}}^{†}={\text{sup}}_{z\in B_{\mathsf{Z}}}{\displaystyle {\sum }_{v=0}^{\infty }{\varsigma }_v{z}_v}$. In this case, we realize that $\varsigma ={\varsigma }_v\in {\mathsf{Z}}^{\beta }$, given that the supremum exists.

Now we recall the definitions of compact operator and Hausdorff measure of noncompactness (or Hmnc in short) of a bounded set.

Definition 4.

An operator $T:\mathsf{Z}⟶\mathsf{W}$ is compact if the domain of $\mathsf{Z}$ is all of $\mathsf{Z}$ and the sequence $T{z}_r$ has a convergent subsequence in $\mathsf{W}$ for every bounded sequence $z_r$ in $\mathsf{Z}$.

Definition 5.

Let $J$ be a bounded subset of a metric space $\mathsf{Z}$. Then, Hmnc of $J$ is defined by$$\begin{array}{}\text{(49)}& \chi J=\inf \epsilon >0:J\subset {\cup }_{v=0}^{r}B{z}_v,a_v,z_v\in \mathsf{Z},a_v<\epsilon v=\mathrm{0,1,2},\dots ,r,r\in {\mathrm{\mathbb{N}}}_0,\end{array}$$where $B{z}_v,a_v$ is an open sphere with centre at $z_v$ and radius $a_v$.

The Hmnc plays a very crucial role in determining the compactness of operators between the $BK$ spaces. An operator $T:\mathsf{Z}⟶\mathsf{W}$ is compact iff $T_{\chi }=0$, where $T_{\chi }$ represents Hmnc of $T$ given by $T_{\chi }=\chi T{B}_{\mathsf{Z}}$. We refer the readers to [25, 34, 35].

Let $\mathsf{Z}$ and $\mathsf{W}$ be any two $BK$-spaces. Then, $\mathrm{\Lambda }\in \mathsf{Z},\mathsf{W}$ corresponds a linear operator $T_{\mathrm{\Lambda }}\in B\mathsf{Z},\mathsf{W}$, where $T_{\mathrm{\Lambda }}z=\mathrm{\Lambda }z\forall z\in \mathsf{Z}$ (see ([7], Theorem 3.2.4(a))). Additionally, if $\mathsf{Z}\supset \sigma$, then $T_{\mathrm{\Lambda }}={\mathrm{\Lambda }}_{\mathsf{Z},\mathsf{W}}={\text{sup}}_{r\in {\mathrm{\mathbb{N}}}_0}{|{\mathrm{\Lambda }}_r}_{\mathsf{Z}}^{†}<\infty$ (see ([34], Theorem 1.23)), where $\sigma$ is the set of sequences that end in zeroes.

Lemma 6.

${\ell }_{\infty }^{\beta }=c^{\beta }=c_0^{\beta }={\ell }_1$. Furthermore, if$\mathsf{Z}\in {\ell }_{\infty },c,c_0$, then$z_{\mathsf{Z}}^{†}=z_{{\ell }_1}$.

Lemma 7 (see ([34], Theorem 2.15)).

Let$J\subset c_0$be bounded and define the operator${\pi }_r:c_0⟶c_0$by${\pi }_r{z}_0,z_1,z_2\dots =z_0,z_1,z_2\dots ,z_r,\mathrm{0,0},\dots \forall z=z_r\in c_0$; then,$$\begin{array}{}\text{(50)}& \chi J=\underset{r⟶\infty }{\lim }{\text{sup}}_{r\in J}I-{\pi }_{r}z,\end{array}$$where $I$ is the identity operator on $c_0$.

Lemma 8 (see ([35], Theorem 7)).

Let$\mathsf{Z}\supset \sigma$be a$BK-$space. Then, one has the following statements:

(a) If$\mathrm{\Lambda }\in \mathsf{Z},c_0$, then${T_{\mathrm{\Lambda }}}_{\chi }=\underset{r⟶\infty }{\lim \sup }{{\mathrm{\Lambda }}_r}_{\mathsf{Z}}^{†}$and$T_{\mathrm{\Lambda }}$is compact iff$\underset{r⟶\infty }{\lim }{{\mathrm{\Lambda }}_r}_{\mathsf{Z}}^{†}=0$.

(b) If$\mathsf{Z}$has$AK$and$\mathrm{\Lambda }\in \mathsf{Z},c$, then$$\begin{array}{}\text{(51)}& \frac{1}{2}{\lim \sup }_{r⟶\infty }{{\Lambda }_r-\lambda }_{\mathsf{Z}}^{†}\le {T_{\Lambda }}_{\chi }\le {\lim \sup }_{r⟶\infty }{{\Lambda }_r-\lambda }_{\mathsf{Z}}^{†},\end{array}$$

  and$T_{\mathrm{\Lambda }}$is compact iff${\lim }_{r⟶\infty }{{\mathrm{\Lambda }}_r-\lambda }_{\mathsf{Z}}^{†}=0,$where$\lambda ={\lambda }_r$with${\lambda }_r={\lim }_{r⟶\infty }{\lambda }_{rv}\forall v\in {\mathrm{\mathbb{N}}}_0$.

(c) If$\mathrm{\Lambda }\in \mathsf{Z},{\ell }_{\infty }$, then$0\le {T_{\mathrm{\Lambda }}}_{\chi }\le {\lim \sup }_{r⟶\infty }{{\mathrm{\Lambda }}_r}_{\mathsf{Z}}^{†}$and$T_{\mathrm{\Lambda }}$is compact if${\lim }_{r⟶\infty }{{\mathrm{\Lambda }}_r}_{\mathsf{Z}}^{†}=0.$

We use the symbol ${\mathrm{ℛ}}_m$ to indicate the subfamily of $\mathrm{ℛ}$ with elements that are greater than $m$.

Lemma 9 (see ([35], Theorem 3.11)).

Let$\mathsf{Z}\supset \sigma$be a$BK-$space. If$\mathrm{\Lambda }\in \mathsf{Z},{\ell }_1$, then$$\begin{array}{}\text{(52)}& \underset{m⟶\infty }{\lim }{\text{sup}}_{R\in {\mathrm{ℛ}}_m}{{\displaystyle \sum _{r\in R}{\Lambda }_r}}_{\mathsf{Z}}^{†}\le {T_{\Lambda }}_{\chi }\le 4\cdot \underset{m⟶\infty }{\lim }{\text{sup}}_{R\in {\mathrm{ℛ}}_m}{{\displaystyle \sum _{r\in R}{\Lambda }_r}}_{\mathsf{Z}}^{†}.\end{array}$$

$T_{\mathrm{\Lambda }}$ is compact iff ${\lim }_{m⟶\infty }\underset{R\in {\mathrm{ℛ}}_m}{\text{sup}}{{\displaystyle {\sum }_{r\in R}{\mathrm{\Lambda }}_r}}_{\mathsf{Z}}^{†}=0$.

Lemma 10 (see ([35], Theorem 4.4, Corollary 3)).

Let$\mathsf{Z}\supset \sigma$be a$BK-$space and let$$\begin{array}{}\text{(53)}& {\Lambda }_{bs}^r={{\displaystyle \sum _{v=0}^r{\Lambda }_v}}_{\mathsf{Z}}^{†}.\end{array}$$

Then, the following statements hold:

(a) If $\mathrm{\Lambda }\in \mathsf{Z},c{s}_0$, then ${T_{\mathrm{\Lambda }}}_{\chi }={\lim \sup }_{r⟶\infty }{\mathrm{\Lambda }}_{\mathsf{Z},bs}^r$ and $T_{\mathrm{\Lambda }}$ is compact iff ${\lim }_{r⟶\infty }{\mathrm{\Lambda }}_{\mathsf{Z},bs}^r=0.$

(b) If $\mathsf{Z}$ has $AK$ and $\mathrm{\Lambda }\in \mathsf{Z},cs$, then$$\begin{array}{}\text{(54)}& \frac{1}{2}\underset{r⟶\infty }{\lim \sup }{{\displaystyle \sum _{v=0}^r{\Lambda }_v}-\tilde{\lambda }}_{\mathsf{Z}}^{†}\le {T_{\Lambda }}_{\chi }\le \underset{r⟶\infty }{\lim \sup }{{\displaystyle \sum _{v=0}^r{\Lambda }_v}-\tilde{\lambda }}_{\mathsf{Z}}^{†}.\end{array}$$

  $T_{\mathrm{\Lambda }}$ is compact iff ${\lim \sup }_{r⟶\infty }{{\displaystyle {\sum }_{v=0}^r{\mathrm{\Lambda }}_v-\tilde{\lambda }}}_{\mathsf{Z}}^{†}=0,$ where $\tilde{\lambda }={\tilde{\lambda }}_v$ with ${\tilde{\lambda }}_v={\lim }_{r⟶\infty }{\displaystyle {\sum }_{m=0}^r{\lambda }_{mv}}$ for all $v\in {\mathrm{\mathbb{N}}}_0$.

(c) If $\mathrm{\Lambda }\in \mathsf{Z},bs$, then $0\le {T_{\mathrm{\Lambda }}}_{\chi }\le {\lim \sup }_{r⟶\infty }{\mathrm{\Lambda }}_{\mathsf{Z},bs}^r$ and $T_{\mathrm{\Lambda }}$ is compact if ${\lim }_{r⟶\infty }{\mathrm{\Lambda }}_{\mathsf{Z},bs}^r=0.$

Lemma 11.

If$\mathrm{\Lambda }\in e_0^{a,b}p,q,\mathsf{Z}$, then$\mathrm{\Omega }\in c_0,\mathsf{Z}$and$\mathrm{\Lambda }z=\mathrm{\Omega }w$for all$z\in e_0^{a,b}p,q$, where the matrix$\mathrm{\Omega }={\omega }_{rv}$is defined in (34).

Proof.

Let $\mathrm{\Lambda }\in e_0^{a,b}p,q,\mathsf{Z}$ and $z\in e_0^{a,b}p,q$. Then, ${\mathrm{\Lambda }}_r={{\lambda }_{rv}}_{v\in {\mathbb{N}}_0}\in {e_0^{a,b}p,q}^{\beta }\forall r\in {\mathrm{\mathbb{N}}}_0$. One obtains$$\begin{array}{c}\text{(55)}& {\Omega w}_v={\displaystyle \sum _{v=0}^{\infty }{\omega }_{rv}{w}_v}={\displaystyle \sum _{v=0}^{\infty }{\displaystyle \sum _{m=v}^{\infty }{-1}^{m-v}}\frac{{\begin{array}{c}m\\ v\end{array}}_{p,q}{q}^{\begin{array}{c}m-v\\ 2\end{array}}{b}^{m-v}{a+b}_{p,q}^v}{a^m{q}^{m/2}}{\lambda }_{rm}}\\ \cdot \frac{1}{{a+b}_{p,q}^v}{\displaystyle \sum _{l=0}^v{\begin{array}{c}v\\ l\end{array}}_{p,q}}{p}^{\begin{array}{c}v-l\\ 2\end{array}}{q}^{\begin{array}{c}l\\ 2\end{array}}{a}^l{b}^{v-l}{z}_l\\ ={\displaystyle \sum _{v=0}^{\infty }{\lambda }_{rv}}{z}_v={\Lambda z}_{v}v\in {\mathrm{\mathbb{N}}}_0.\end{array}$$

Thus, ${\mathrm{\Omega }}_r\in {\ell }_1$ for each $r\in {\mathbb{N}}_0$ and $\mathrm{\Omega }w\in \mathsf{Z}$. Consequently, $\mathrm{\Omega }\in c_0,\mathsf{Z}$.

Theorem 9.

The following statements hold:

(a) If$\mathrm{\Lambda }\in e_0^{a,b}p,q,c_0$, then${T_{\mathrm{\Lambda }}}_{\chi }=\underset{r⟶\infty }{\lim \sup }{\displaystyle {\sum }_{v=0}^{\infty }{\omega }_{rv}}$.

(b) If$\mathrm{\Lambda }\in e_0^{a,b}p,q,c$, then$$\begin{array}{}\text{(56)}& \frac{1}{2}{\lim \sup }_{r⟶\infty }{\displaystyle \sum _{v=0}^{\infty }{\omega }_{rv}-\omega }\le {T_{\Lambda }}_{\chi }\le {\lim \sup }_{r⟶\infty }{\displaystyle \sum _{v=0}^{\infty }{\omega }_{rv}-\omega },\end{array}$$

  where $\omega ={\omega }_v$ and ${\omega }_v={\lim }_{r⟶\infty }{\omega }_{rv}\forall v\in {\mathrm{\mathbb{N}}}_0$.

(c) If$\mathrm{\Lambda }\in e_0^{a,b}p,q,{\ell }_{\infty }$, then$0\le {T_{\mathrm{\Lambda }}}_{\chi }\le {\lim \sup }_{r⟶\infty }{\displaystyle {\sum }_{v=0}^{\infty }{\omega }_{rv}}$.

(d) If$\mathrm{\Lambda }\in e_0^{a,b}p,q,{\ell }_1$, then$$\begin{array}{}\text{(57)}& {\lim }_{m⟶\infty }{\Lambda }_{e_0^{a,b}p,q,{\ell }_1}^m\le {T_{\Lambda }}_{\chi }\le 4{\lim }_{m⟶\infty }{\Lambda }_{e_0^{a,b}p,q,{\ell }_1}^m,\end{array}$$

  where ${\mathrm{\Lambda }}_{e_0^{a,b}p,q,{\ell }_1}^m={\text{sup}}_{R\in {\mathrm{ℛ}}_m}{\displaystyle {\sum }_{v=0}^{\infty }{\displaystyle {\sum }_{r\in R}{\omega }_{rv}}}.$

(e) If$\mathrm{\Lambda }\in e_0^{a,b}p,q,c{s}_0$, then${T_{\mathrm{\Lambda }}}_{\chi }={\lim \sup }_{r⟶\infty }{\displaystyle {\sum }_{v=0}^{\infty }{\displaystyle {\sum }_{l=0}^r{\omega }_{lv}}}$.

(f) If$\mathrm{\Omega }\in e_0^{a,b}p,q,cs$, then$$\begin{array}{}\text{(58)}& \frac{1}{2}\underset{r⟶\infty }{\lim \sup }{\displaystyle \sum _{v=0}^{\infty }{\displaystyle \sum _{l=0}^r{\omega }_{lv}}-\tilde{\omega }}\le {T_{\Lambda }}_{\chi }\le \underset{r⟶\infty }{\lim \sup }{\displaystyle \sum _{v=0}^{\infty }{\displaystyle \sum _{l=0}^r{\omega }_{lv}}-\tilde{\omega }},\end{array}$$

  where $\tilde{\omega }={\tilde{\omega }}_v$ with ${\tilde{\omega }}_v={\lim }_{r⟶\infty }{\displaystyle {\sum }_{l=0}^r{\omega }_{lv}}\forall v\in {\mathbb{N}}_0$.

(g) If$\mathrm{\Lambda }\in e_0^{a,b}p,q,bs$, then$0\le {T_{\mathrm{\Lambda }}}_{\chi }\le {\lim \sup }_{r⟶\infty }{\displaystyle {\sum }_{v=0}^{\infty }{\displaystyle {\sum }_{l=0}^r{\omega }_{lv}}}$.

Proof.

(a) Let $\mathrm{\Lambda }\in e_0^{a,b}p,q,c_0$. We observe that$$\begin{array}{}\text{(59)}& {{\Lambda }_r}_{e_0^{a,b}p,q}^{†}={{\Omega }_r}_{c_0}^{†}={{\Omega }_r}_{{\ell }_1}={\displaystyle \sum _{v=0}^{\infty }{\omega }_{rv}},r\in {\mathbb{N}}_0.\end{array}$$

  One obtains on engaging Lemma 8 (a) that$$\begin{array}{}\text{(60)}& {T_{\Lambda }}_{\chi }=\underset{r⟶\infty }{\lim \sup }{\displaystyle \sum _{v=0}^{\infty }{\omega }_{rv}}.\end{array}$$

(b) We observe that$$\begin{array}{}\text{(61)}& {{\Omega }_r-\omega }_{c_0}^{†}={{\Omega }_r-\omega }_{{\ell }_1}={\displaystyle \sum _{v=0}^r{\omega }_{rv}-{\omega }_v}r\in {\mathbb{N}}_0.\end{array}$$

  Now, let $\mathrm{\Lambda }\in e_0^{a,b}p,q,c$; then, by using Lemma 11, one obtains that $\mathrm{\Omega }\in c_0,c$. By engaging Lemma 8 (b), we deduce that$$\begin{array}{}\text{(62)}& \frac{1}{2}\underset{r⟶\infty }{\lim \sup }{{\Omega }_r-\omega }_{c_0}^{†}\le {T_{\Lambda }}_{\chi }\le \underset{r⟶\infty }{\lim \sup }{{\Omega }_r-\omega }_{c_0}^{†},\end{array}$$

  which in the light of (62) yields$$\begin{array}{}\text{(63)}& \frac{1}{2}\underset{r⟶\infty }{\lim \sup }{\displaystyle \sum _{v=0}^{\infty }{\omega }_{rv}-{\omega }_v}\le {T_{\Lambda }}_{\chi }\le \underset{r⟶\infty }{\lim \sup }{\displaystyle \sum _{v=0}^{\infty }{\omega }_{rv}-{\omega }_v},\end{array}$$

  which is the required result.

(c) The proof is similar to the proof of (a). Hence, the details are excluded.

(d) We have$$\begin{array}{}\text{(64)}& {{\displaystyle \sum _{r\in R}{\Omega }_r}}_{c_0}^{†}={{\displaystyle \sum _{r\in R}{\Omega }_r}}_{{\ell }_1}={\displaystyle \sum _{v=0}^{\infty }{\displaystyle \sum _{r\in R}{\omega }_{rv}}}.\end{array}$$

  Let $\mathrm{\Lambda }\in e_0^{a,b}p,q,{\ell }_1$. Then, Lemma 11 implies that $\mathrm{\Omega }\in c_0,{\ell }_1$. Hence, by using Lemma 9, we get$$\begin{array}{}\text{(65)}& \underset{m⟶\infty }{\lim }{\text{sup}}_{R\in {\mathrm{ℛ}}_m}{{\displaystyle \sum _{r\in R}{\Omega }_r}}_{c_0}^{†}\le {T_{\Lambda }}_{\chi }\le 4\cdot \underset{m⟶\infty }{\lim }{\text{sup}}_{R\in {\mathrm{ℛ}}_m}{{\displaystyle \sum _{r\in R}{\Omega }_r}}_{c_0}^{†},\end{array}$$

  which further reduces on using (65) to$$\begin{array}{}\text{(66)}& \underset{m⟶\infty }{\lim }{\Lambda }_{e_0^{a,b}p,q,{\ell }_1}^m\le {T_{\Lambda }}_{\chi }\le 4\underset{m⟶\infty }{\lim }{\Lambda }_{e_0^{a,b}p,q,{\ell }_1}^m,\end{array}$$

  as desired.

(e) Notice that$$\begin{array}{}\text{(67)}& {{\displaystyle \sum _{l=0}^r{\Lambda }_l}}_{e_0^{a,b}p,q}^{†}={{\displaystyle \sum _{l=0}^r{\Omega }_l}}_{c_0}^{†}={{\displaystyle \sum _{l=0}^r{\Omega }_l}}_{{\ell }_1}={\displaystyle \sum _{v=0}^{\infty }{\displaystyle \sum _{l=0}^r{\omega }_{lv}}},\end{array}$$

  which on using (a) of Lemma 10 yields$$\begin{array}{}\text{(68)}& {T_{\Lambda }}_{\chi }=\underset{r⟶\infty }{\lim \sup }{\displaystyle \sum _{v=0}^{\infty }{\displaystyle \sum _{l=0}^r{\omega }_{lv}}}.\end{array}$$

(f) We have$$\begin{array}{}\text{(69)}& {{\displaystyle \sum _{l=0}^r{\Omega }_l}-\tilde{\omega }}_{c_0}^{†}={{\displaystyle \sum _{l=0}^r{\Omega }_l}-\tilde{\omega }}_{{\ell }_1}={\displaystyle \sum _{v=0}^{\infty }{\displaystyle \sum _{l=0}^r{\omega }_{lv}}-{\tilde{\omega }}_v},\end{array}$$

  for each $r\in {\mathbb{N}}_0$. Let $\mathrm{\Lambda }\in e_0^{a,b}p,q,cs$. Then, Lemma 11 implies that $\mathrm{\Omega }\in c_0,cs$. Thus, in view of (b) of Lemma 10, we deduce that$$\begin{array}{}\text{(70)}& \frac{1}{2}\underset{r⟶\infty }{\lim \sup }{{\displaystyle \sum _{l=0}^r{\Omega }_l-{\tilde{\omega }}_v}}_{c_0}^{†}\le {T_{\Lambda }}_{\chi }\le \underset{r⟶\infty }{\lim \sup }{{\displaystyle \sum _{l=0}^r{\Omega }_l-{\tilde{\omega }}_l}}_{c_0}^{†},\end{array}$$

  which on using (69) yields$$\begin{array}{}\text{(71)}& \frac{1}{2}\underset{r⟶\infty }{\lim \sup }{\displaystyle \sum _{v=0}{\displaystyle \sum _{l=0}^r{\omega }_{lv}-{\tilde{\omega }}_v}}\le {T_{\Lambda }}_{\chi }\le \underset{r⟶\infty }{\lim \sup }{\displaystyle \sum _{v=0}^{\infty }{\displaystyle \sum _{l=0}^r{\omega }_{lv}-{\tilde{\omega }}_v}},\end{array}$$

  as desired.

(g) This proof is similar to the proof of (e). Hence, the details are excluded.

Now, we have the following corollaries.

Corollary 6.

The following statements hold:

(a) If $\mathrm{\Lambda }\in e_0^{a,b}p,q,c_0$, then  $T_{\mathrm{\Lambda }}$ is compact iff  ${\lim }_{r⟶\infty }{\displaystyle {\sum }_{v=0}^{\infty }{\omega }_{rv}}=0$.

(b) If $\mathrm{\Lambda }\in e_0^{a,b}p,q,c$, then  $T_{\mathrm{\Lambda }}$ is compact iff  ${\lim }_{r⟶\infty }{\displaystyle {\sum }_{v=0}^{\infty }{\tilde{\omega }}_{ij}-{\tilde{\omega }}_j}=0.$

(c) If $\mathrm{\Lambda }\in e_0^{a,b}p,q,{\ell }_{\infty }$, then  $T_{\mathrm{\Lambda }}$ is compact if  ${\lim }_{r⟶\infty }{\displaystyle {\sum }_{v=0}^{\infty }{\omega }_{rv}}=0$.

(d) If $\mathrm{\Lambda }\in e_0^{a,b}p,q,{\ell }_1$, then  $T_{\mathrm{\Lambda }}$ is compact iff  ${\lim }_{r⟶\infty }\underset{R\in {\mathrm{ℛ}}_m}{\text{sup}}{\displaystyle {\sum }_{v=0}^{\infty }{\displaystyle {\sum }_{r\in R}{\omega }_{rv}}}=0$.

(e) If $\mathrm{\Lambda }\in e_0^{a,b}p,q,c{s}_0$, then  $T_{\mathrm{\Lambda }}$ is compact iff  ${\lim }_{r⟶\infty }{\displaystyle {\sum }_{v=0}^{\infty }{\displaystyle {\sum }_{l=0}^r{\omega }_{lv}}}=0$.

(f) If $\mathrm{\Lambda }\in e_0^{a,b}p,q,cs$, then  $T_{\mathrm{\Lambda }}$ is compact iff  ${\lim }_{r⟶\infty }{\displaystyle {\sum }_{v=0}^{\infty }{\displaystyle {\sum }_{l=0}^r{\omega }_{lv}-\tilde{\omega }}}=0$.

(g) If $\mathrm{\Lambda }\in e_0^{a,b}p,q,bs$, then  $T_{\mathrm{\Lambda }}$ is compact if ${\lim }_{r⟶\infty }{\displaystyle {\sum }_{v=0}^{\infty }{\displaystyle {\sum }_{l=0}^r{\omega }_{lv}}}=0$.

6. Mapping Ideals

Here, we define $s$-type mapping ideals on ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$. The family of all bounded linear operators between any two Banach spaces is denoted by $\mathrm{ℬ}$. Specifically, $\mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$ indicates the family of all bounded linear mappings from Banach space $\mathfrak{X}$ to Banach space $\mathfrak{Y}$. Before proceeding to our results, we list the following notations and definitions.

Definition 6 (see [36]).

Assume ${\omega }^{+}$ is the set of nonnegative real sequences and $ℂ$ is the set of complex numbers. Therefore, $s$-number is a mapping $s:\mathrm{ℬ}\mathfrak{X},\mathfrak{Y}⟶{\omega }^{+}$ satisfying the following conditions:

(i) $\varphi =s_0\varphi \ge s_1\varphi \ge s_2\varphi \ge \dots \ge 0$, $\forall \varphi \in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$.

(ii) $s_{a+b-1}\varphi +\psi \le s_a\varphi +s_b\psi$, for all $\varphi ,\psi \in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$ and $a,b\in {\mathbb{N}}_0$.

(iii) $s_a\varphi \theta \psi \le \varphi s_a\theta \psi$, for each $\varphi \in \mathrm{ℬ}{\mathfrak{X}}_0,\mathfrak{X}$, $\theta \in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$, and $\psi \in \mathrm{ℬ}\mathfrak{Y},{\mathfrak{Y}}_0$, where ${\mathfrak{X}}_0$ and ${\mathfrak{Y}}_0$ are any two Banach sequence spaces.

(iv) Let $\varphi \in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$ and $\nu \in ℂ$. Then, $s_a\nu \varphi =\nu s_a\varphi$.

(v) If $\text{rank}\varphi \le a$, then $s_a\varphi =0$ for all $\varphi \in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$.

(vi) $s_v{\Im }_a=0$ for $v\ge a$ or $s_v{\Im }_a=1$ for $v<a$, where ${\Im }_a$ indicates the identity mapping on $a$-dimensional Hilbert space ${\ell }_2^a$.

We indicate the following parameters using an assortment of $s$-numbers:

(1) The $a$-th Kolmogorov number, indicated by $d_{a}X$, is defined by $d_{a}X={\inf }_{\text{dim}J\le a}{\text{sup}}_{f\le 1}{\inf }_{g\in J}Xf-g$.

(2) The $a$-th approximation number, indicated by ${\alpha }_{a}X$, is defined by ${\alpha }_{a}X=\inf X-Y:Y\in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}\text{and\hspace{0.17em}rank}Y\le a$.

Definition 7 (see [37]).

Suppose $\mathcal{W}\subset \mathrm{ℬ}$ and denote $\mathcal{W}\mathfrak{X},\mathfrak{Y}=\mathcal{W}\cap \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$. Then, $\mathcal{W}$ is called a mapping ideal if the following conditions are satisfied:

(i) ${\Im }_{\mathfrak{D}}\in \mathcal{W}$, where $\mathfrak{D}$ is the one-dimensional Banach sequence space.

(ii) $\mathcal{W}\mathfrak{X},\mathfrak{Y}$ is linear over $ℂ$.

(iii) If $\psi \in \mathrm{ℬ}{\mathfrak{X}}_0,\mathfrak{X}$, $\theta \in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$ and $\varphi \in \mathrm{ℬ}{\mathfrak{Y}}_0,\mathfrak{Y}$, then $\varphi \theta \psi \in \mathrm{ℬ}{\mathfrak{X}}_0,{\mathfrak{Y}}_0$, where ${\mathfrak{X}}_0$ and ${\mathfrak{Y}}_0$ are any two normed spaces.

Definition 8 (see [38]).

A pre-quasi-norm on the ideal $\mathcal{W}$ is a mapping $\mu :\mathcal{W}⟶{\omega }^{+}$ that satisfies the following conditions:

(i) $\mu \varphi \ge 0$ and $\mu \varphi =0$ iff $\varphi =0$, $\forall \varphi \in \mathcal{W}\mathfrak{X},\mathfrak{Y}$.

(ii) There is $m_0\ge 1$ so that $\mu \varsigma \varphi \le m_0\varsigma \mu \varphi$, for all $\varphi \in \mathcal{W}\mathfrak{X},\mathfrak{Y}$.

(iii) There is $n_0\ge 1$ so that $\mu \varphi +\psi \le n_0\mu \varphi +\mu \psi$, for all $\varphi ,\psi \in \mathcal{W}\mathfrak{X},\mathfrak{Y}$.

(iv) One has $p_0\ge 1$ so that $\mu \varphi \theta \psi \le p_0\varphi \mu \theta \psi$, if $\psi \in \mathrm{ℬ}{\mathfrak{X}}_0,\mathfrak{X}$, $\theta \in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$, and $\varphi \in \mathrm{ℬ}{\mathfrak{Y}}_0,\mathfrak{Y}$.

Definition 9 (see [38]).

The subspace $\mathrm{ℨ}\subset \omega$ is called a private sequence space (or in short $pss$) if following statements are satisfied:

(i) ${\mathfrak{e}}_v\in \mathrm{ℨ}$, for all $v\in {\mathbb{N}}_0$, where ${\mathfrak{e}}_v$ indicates the sequence with 1 in the $v^{\text{th}}$ position and 0 elsewhere.

(ii) If $\mathfrak{g}={\mathfrak{g}}_v\in \omega$, $\mathfrak{h}={\mathfrak{h}}_v\in \mathrm{ℨ}$, and ${\mathfrak{g}}_v\le {\mathfrak{h}}_v$, for $v\in {\mathbb{N}}_0$, then $\mathfrak{g}\in \mathrm{ℨ}$.

(iii) $g_{v/2}\in \mathrm{ℨ}$, if $g_v\in \mathrm{ℨ}$, where $v/2$ indicates the integral part of $v/2$.

Definition 10 (see [39]).

A subspace of the $pss$ is called a pre-modular $pss$, if there is a function $\upsilon :\mathrm{ℨ}⟶0,\infty$ satisfying the following statements:

(i) For all $j\in \mathrm{ℨ}$, $j=0\iff \upsilon j=0$, and $\upsilon j\ge 0$, with $0$ being the zero vector of $\mathrm{ℨ}$.

(ii) If $j\in \mathrm{ℨ}$ and $\rho \in ℂ$, then there is $E_0\ge 1$ with $\upsilon \rho j\le \rho E_0\upsilon j$.

(iii) $\upsilon h+j\le G_0\upsilon h+\upsilon j$ holds for some $G_0\ge 1$, with $f,g\in \mathrm{ℨ}$.

(iv) Suppose $x\in {\mathbb{N}}_0$, $h_x\le j_x$; then, $\upsilon h_x\le \upsilon j_x$.

(v) The inequality $\upsilon j_x\le \upsilon j_{x/2}\le D_0\upsilon j_x$ holds, for $D_0\ge 1$.

(vi) $\overline{\mathrm{ℭ}}={\mathrm{ℨ}}_{\upsilon }$, where $\overline{\mathrm{ℭ}}$ indicates the closure of the space of all sequences with infinite zero coordinates.

(vii) One has $\eta >0$ such that $\upsilon \rho ,\mathrm{0,0,0},\dots \ge \eta \rho \upsilon \mathrm{1,0,0,0},\dots$, with $\rho \in ℂ$.

Definition 11 (see [39]).

The $pss{\mathrm{ℨ}}_{\upsilon }$ is called a pre-quasi-normed $pss$, if $\upsilon$ satisfies conditions (i)–(iii) in Definition 10. If $\mathrm{ℨ}$ is complete equipped with $\upsilon$, then ${\mathrm{ℨ}}_{\upsilon }$ is said to be a pre-quasi-Banach $pss$.

Lemma 12 (see [38]).

Every pre-modular $pss$ is a pre-quasi-normed $pss$.

Definition 12.

We introduce the next sequence space:$$\begin{array}{}\text{(72)}& {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}≔\mathfrak{g}={\mathfrak{g}}_v\in \omega :\underset{r⟶\infty }{\lim }\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{\mathfrak{g}}_v}=0,\end{array}$$where $\delta ={\delta }_v\in {\omega }^{+}$, $\mathrm{\varrho }\mathfrak{g}={\text{sup}}_{r\in {\mathbb{N}}_0}1/{a+b}_{p,q}^r{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{\mathfrak{g}}_v}$, and $Ar,v={\begin{array}{c}r\\ v\end{array}}_{p,q}{p}^{\begin{array}{c}r-v\\ 2\end{array}}{q}^{\begin{array}{c}v\\ 2\end{array}}{a}^v{b}^{r-v}$.

By ${\Im }_{\nearrow }$ and ${\Im }_{\searrow }$, we indicate the space of all monotonic increasing and decreasing sequences of positive reals, respectively.

Theorem 10.

$e_0^{a,b}p,q,\delta$is a$pss$, if${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$or,${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$and there is$C\ge 1$so that${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$.

Proof.

(i) Suppose $\mathfrak{g},\mathfrak{h}\in e_0^{a,b}p,q,\delta$, and one has$$\begin{array}{c}\text{(73)}& {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{\mathfrak{g}}_v+{\mathfrak{h}}_v}\le {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{\mathfrak{g}}_v}+{\text{sup}}_{r\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{\mathfrak{h}}_v}<\infty .\end{array}$$

  Hence, $\mathfrak{g}+\mathfrak{h}\in e_0^{a,b}p,q,\delta$.

  Let $\varsigma \in ℂ$ and $\mathfrak{g}\in e_0^{a,b}p,q,\delta$. So, one has$$\begin{array}{}\text{(74)}& {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v\varsigma {\mathfrak{g}}_v}=\varsigma {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{\mathfrak{g}}_v}<\infty .\end{array}$$

  This implies $e_0^{a,b}p,q,\delta$ is a linear space. Next$$\begin{array}{}\text{(75)}& {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{{\mathfrak{e}}_t}_v}={\delta }_t{\text{sup}}_{r\in {\mathbb{N}}_0}\frac{Ar,t}{{a+b}_{p,q}^r}<\infty .\end{array}$$

  This gives ${\mathfrak{e}}_t\in e_0^{a,b}p,q,\delta$, for all $t\in {\mathbb{N}}_0$.

(ii) Assume that ${\mathfrak{g}}_v\le {\mathfrak{h}}_v$, for all $v\in {\mathbb{N}}_0$ and $\mathfrak{h}\in e_0^{a,b}p,q,\delta$. Then, we have$$\begin{array}{}\text{(76)}& {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{\mathfrak{g}}_v}\le {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{\mathfrak{h}}_v}<\infty .\end{array}$$

  This implies that $\mathfrak{g}\in e_0^{a,b}p,q,\delta$.

(iii) Let ${\mathfrak{g}}_v\in e_0^{a,b}p,q,\delta$, ${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$ and $\exists C\ge 1$ so that ${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$. So, one has$$\begin{array}{c}\text{(77)}& {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v}{\mathfrak{g}}_{\frac{v}{2}}\le {\text{sup}}_{m\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^{2m}}{\displaystyle \sum _{v=0}^{2m}{\delta }_{v}A2m,v}{\mathfrak{g}}_{\frac{v}{2}}+{\text{sup}}_{m\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^{2m+1}}{\displaystyle \sum _{v=0}^{2m+1}{\delta }_{v}A2m+1,v}{\mathfrak{g}}_{\frac{v}{2}}\\ \le {\text{sup}}_{m\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^{2m}}{\delta }_{2m}A2m,2m{\mathfrak{g}}_m+{\displaystyle \sum _{v=0}^m{\delta }_{2v}A2m,2v{\mathfrak{g}}_v+{\delta }_{2v+1}A2m,2v+1{\mathfrak{g}}_v}\\ +{\text{sup}}_{m\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^{2m+1}}{\displaystyle \sum _{v=0}^m{\delta }_{2v}A2m+\mathrm{1,2}v{\mathfrak{g}}_v+{\delta }_{2v+1}A2m+\mathrm{1,2}v+1{\mathfrak{g}}_v}\\ \le {\text{sup}}_{m\in {\mathbb{N}}_0}\frac{C}{{a+b}_{p,q}^{2m}}{\displaystyle \sum _{v=0}^m{\delta }_{v}A2m,v{\mathfrak{g}}_v}+{\text{sup}}_{m\in {\mathbb{N}}_0}\frac{2C}{{a+b}_{p,q}^{2m}}{\displaystyle \sum _{v=0}^m{\delta }_{v}A2m,v{\mathfrak{g}}_v}\\ +{\text{sup}}_{m\in {\mathbb{N}}_0}\frac{2C}{{a+b}_{p,q}^{2m+1}}{\displaystyle \sum _{v=0}^m{\delta }_{v}A2m+1,v{\mathfrak{g}}_v}\\ \le 5C{\text{sup}}_{m\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^m}{\displaystyle \sum _{v=0}^m{\delta }_{v}Am,v{\mathfrak{g}}_v}<\infty .\end{array}$$

Thus, ${\mathfrak{g}}_{v/2}\in e_0^{a,b}p,q,\delta$.

This finishes the proof.

Define the sets ${\mathrm{ℬ}}_{\mathrm{ℨ}}^s\mathfrak{X},\mathfrak{Y}$, ${\mathrm{ℬ}}_{\mathrm{ℨ}}^{\alpha }\mathfrak{X},\mathfrak{Y}$, and ${\mathrm{ℬ}}_{\mathrm{ℨ}}^d\mathfrak{X},\mathfrak{Y}$ by$$\begin{array}{c}\text{(78)}& {\mathrm{ℬ}}_{\mathrm{ℨ}}^s\mathfrak{X},\mathfrak{Y}≔\varphi \in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}:s_a\varphi \in \mathrm{ℨ},{\mathrm{ℬ}}_{\mathrm{ℨ}}^{\alpha }\mathfrak{X},\mathfrak{Y}≔\varphi \in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}:{\alpha }_a\varphi \in \mathrm{ℨ},\\ {\mathrm{ℬ}}_{\mathrm{ℨ}}^d\mathfrak{X},\mathfrak{Y}≔\varphi \in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}:d_a\varphi \in \mathrm{ℨ},\end{array}$$where $\mathfrak{X}$ and $\mathfrak{Y}$ are any two Banach sequence spaces. We denote ${\mathrm{ℬ}}_{\mathrm{ℨ}}^s≔{\mathrm{ℬ}}_{\mathrm{ℨ}}^s\mathfrak{X},\mathfrak{Y}$, ${\mathrm{ℬ}}_{\mathrm{ℨ}}^{\alpha }≔{\mathrm{ℬ}}_{\mathrm{ℨ}}^{\alpha }\mathfrak{X},\mathfrak{Y}$, and ${\mathrm{ℬ}}_{\mathrm{ℨ}}^d≔{\mathrm{ℬ}}_{\mathrm{ℨ}}^d\mathfrak{X},\mathfrak{Y}$, respectively.

Lemma 13 (see [37]).

Suppose the linear sequence space $\mathrm{ℨ}$ is a $pss$. Then, ${\mathrm{ℬ}}_{\mathrm{ℨ}}^s$is a mapping ideal.

Theorem 11.

Suppose${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$or,${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$and there is$C\ge 1$so that${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$. Then, ${\mathrm{ℬ}}_{e_0^{a,b}p,q,\delta }^s$is a mapping ideal.

Proof.

It comes directly from Lemma 13.

Theorem 12.

Let${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$or,${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$and there exists$C\ge 1$such that${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$. Then,${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$is a pre-modular$pss$.

Proof.

(i) Obviously, for every $\mathfrak{g}\in {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$ that $\mathrm{\varrho }\mathfrak{g}\ge 0$ and $\mathrm{\varrho }\mathfrak{g}=0$, if and only if, $\mathfrak{g}=0$.

(ii) For all $ϵ\ge 1$. Then, $\mathrm{\varrho }\alpha \mathfrak{g}\le ϵ\alpha \mathrm{\varrho }\mathfrak{g}$, for each $\mathfrak{g}\in e_0^{a,b}p,q,\delta$ and $\alpha \in ℂ$.

(iii) Note that $\mathrm{\varrho }\mathfrak{g}+\mathfrak{h}\le \mathrm{\varrho }\mathfrak{g}+\mathrm{\varrho }\mathfrak{h}$, for every $\mathfrak{g},\mathfrak{h}\in e_0^{a,b}p,q,\delta$.

(iv) One has $\mathrm{\varrho }{\mathfrak{g}}_k\le \mathrm{\varrho }{\mathfrak{h}}_k$, if ${\mathfrak{g}}_k\le {\mathfrak{h}}_k$ (see proof of (ii) in Theorem 10).

(v) From proof of (iii) in Theorem 10, we have that $\mathrm{\varrho }{\mathfrak{g}}_k\le \mathrm{\varrho }{\mathfrak{g}}_{k/2}\le \eta \mathrm{\varrho }{\mathfrak{g}}_k$ with $\eta =5C$.

(vi) $\overline{\mathrm{ℭ}}=e_0^{a,b}p,q,\delta$.

(vii) One has, if $\alpha \ne 0$ then $0<\gamma \le 1$, for $\mathrm{\varrho }\alpha ,\mathrm{0,0},\dots \ge \gamma \alpha \mathrm{\varrho }\mathrm{1,0,0},\dots$ and when $\alpha =0$ then $\gamma >0$.

This finishes the proof.

Theorem 13.

Suppose that${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$or,${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$and there is$C\ge 1$so that${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$. Then,${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$is a pre-quasi-Banach$pss$.

Proof.

Keeping in mind Theorem 12 and Lemma 12, it is enough to show that every Cauchy sequence in ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$ is convergent in ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$. We suppose that ${\mathfrak{g}}^m={\mathfrak{g}}_k^m$ is a Cauchy sequence in ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$. Then, for every $\epsilon \in \mathrm{0,1}$, there is $n_0\in {\mathbb{N}}_0$ so that$$\begin{array}{}\text{(79)}& \mathrm{\varrho }{\mathfrak{g}}^m-{\mathfrak{g}}^n={\text{sup}}_{r\in {\mathbb{N}}_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{\mathfrak{g}}_v^m-{\mathfrak{g}}_v^n}<\epsilon ,\end{array}$$for each $m,n\ge n_0$. This gives that ${\mathfrak{g}}_v^m-{\mathfrak{g}}_v^n<\epsilon$, for fixed $v$ and for every $m,n\ge n_0$. Hence, ${\mathfrak{g}}_v^m$ is a Cauchy sequence in $ℂ$. As $ℂ$ is complete, ${\lim }_{m⟶\infty }{\mathfrak{g}}_v^m={\mathfrak{g}}_v$, for a fixed $v\in {\mathbb{N}}_0$. This gives, by using (81), that $\mathrm{\varrho }{\mathfrak{g}}^m-\mathfrak{g}<\epsilon$, for every $m\ge n_0$. Also, one has $\mathrm{\varrho }\mathfrak{g}\le \mathrm{\varrho }{\mathfrak{g}}^m-\mathfrak{g}+\mathrm{\varrho }{\mathfrak{g}}^m<\infty$. This implies that $\mathfrak{g}\in {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$. So, ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$ is a pre-quasi-Banach $pss$.

Theorem 14 (see [40]).

Let $s-$type ${\mathcal{E}}_{\mathrm{\varrho }}≔h=s_{x}H\in {\omega }^{+}:H\in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}\text{and}\mathrm{\varrho }h<\infty$. If ${\mathbb{B}}_{{\mathcal{E}}_{\mathrm{\varrho }}}^s$ is a mapping ideal, then the following conditions are satisfied:

(1) $\mathrm{ℭ}\subset s-$type${\mathcal{E}}_{\mathrm{\varrho }}$.

(2) If${s_x{H}_1}_{x=0}^{\infty }\in s-$type${\mathcal{E}}_{\mathrm{\varrho }}$and${s_x{H}_2}_{x=0}^{\infty }\in s-$type${\mathcal{E}}_{\mathrm{\varrho }}$, then${s_x{H}_1+H_2}_{x=0}^{\infty }\in s-$type${\mathcal{E}}_{\mathrm{\varrho }}$.

(3) If$\lambda \in ℂ$and${s_{x}H}_{x=0}^{\infty }\in s-$type${\mathcal{E}}_{\mathrm{\varrho }}$, then$\lambda {s_{x}H}_{x=0}^{\infty }\in s-$type${\mathcal{E}}_{\mathrm{\varrho }}$.

(4) The sequence space${\mathcal{E}}_{\mathrm{\varrho }}$is solid, i.e., if${s_{x}J}_{x=0}^{\infty }\in s-$type${\mathcal{E}}_{\mathrm{\varrho }}$and$s_{x}H\le s_{x}J$, for every$x\in {\mathbb{N}}_0$and$H,J\in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$, then${s_{x}H}_{x=0}^{\infty }\in s-$type${\mathcal{E}}_{\mathrm{\varrho }}$.

By using Theorem 14, we introduce the following properties of the $s-$type ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$.

Theorem 15.

Assume$s-$type${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}≔f=s_{n}X\in {\omega }^{+}:X\in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}\text{and}\mathrm{\varrho }f<\infty$. The following settings are satisfied:

(1) We have$s-$type${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}\supset \mathrm{ℭ}$.

(2) Let${s_r{X}_1}_{r=0}^{\infty }\in s-$type${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$and${s_r{X}_2}_{r=0}^{\infty }\in s-$type${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$; then, ${s_r{X}_1+X_2}_{r=0}^{\infty }\in s-$type${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$.

(3) Suppose$\lambda \in ℂ$and${s_{r}X}_{r=0}^{\infty }\in s-$type${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$; then, $\lambda {s_{r}X}_{r=0}^{\infty }\in s-$type${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$.

(4) The$s-$type${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$is solid.

7. Properties of the Pre-Quasi-Ideal

The following notations and definitions are essential for our investigation. Let $\mathfrak{X}$ and $\mathfrak{Y}$ be two Banach sequence spaces.

Convention 1.

$$\begin{array}{c}\text{(80)}& \mathbb{F}:\text{the\hspace{0.17em}ideal\hspace{0.17em}of\hspace{0.17em}finite\hspace{0.17em}rank\hspace{0.17em}mappings\hspace{0.17em}between\hspace{0.17em}any\hspace{0.17em}arbitrary\hspace{0.17em}Banach\hspace{0.17em}spaces}\text{.}\mathcal{A}:\text{the\hspace{0.17em}ideal\hspace{0.17em}of\hspace{0.17em}approximable\hspace{0.17em}mappings\hspace{0.17em}between\hspace{0.17em}any\hspace{0.17em}arbitrary\hspace{0.17em}Banach\hspace{0.17em}spaces}.\\ \mathcal{K}:\text{the\hspace{0.17em}ideal\hspace{0.17em}of\hspace{0.17em}compact\hspace{0.17em}mappings\hspace{0.17em}between\hspace{0.17em}any\hspace{0.17em}arbitrary\hspace{0.17em}Banach\hspace{0.17em}spaces}.\\ \mathbb{F}\mathfrak{X},\mathfrak{Y}:\text{the\hspace{0.17em}space\hspace{0.17em}of\hspace{0.17em}finite\hspace{0.17em}rank\hspace{0.17em}mappings\hspace{0.17em}from\hspace{0.17em}}\mathfrak{X}\text{into\hspace{0.17em}}\mathfrak{Y}.\\ \mathbb{F}\mathfrak{X}:\text{the\hspace{0.17em}space\hspace{0.17em}of\hspace{0.17em}finite\hspace{0.17em}rank\hspace{0.17em}mappings\hspace{0.17em}from\hspace{0.17em}}\mathfrak{X}\text{into\hspace{0.17em}itself}\text{.}\\ \mathcal{A}\mathfrak{X},\mathfrak{Y}:\text{the\hspace{0.17em}space\hspace{0.17em}of\hspace{0.17em}approximable\hspace{0.17em}mappings\hspace{0.17em}from\hspace{0.17em}}\mathfrak{X}\text{into\hspace{0.17em}}\mathfrak{Y}.\\ \mathcal{A}\mathfrak{X}:\text{the\hspace{0.17em}space\hspace{0.17em}of\hspace{0.17em}approximable\hspace{0.17em}mappings\hspace{0.17em}from\hspace{0.17em}}\mathfrak{X}\text{into\hspace{0.17em}itself}.\\ \mathcal{K}\mathfrak{X},\mathfrak{Y}:\text{the\hspace{0.17em}space\hspace{0.17em}of\hspace{0.17em}compact\hspace{0.17em}mappings\hspace{0.17em}from\hspace{0.17em}a\hspace{0.17em}}\mathfrak{X}\text{into\hspace{0.17em}}\mathfrak{Y}.\\ \mathcal{K}\mathfrak{X}:\text{the\hspace{0.17em}space\hspace{0.17em}of\hspace{0.17em}compact\hspace{0.17em}mappings\hspace{0.17em}from\hspace{0.17em}}\mathfrak{X}\text{into\hspace{0.17em}itself}\text{.}\end{array}$$

Lemma 14 (see [41]).

Assume $M\in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$ and $M\notin \mathcal{A}\mathfrak{X},\mathfrak{Y}$; then, there exist operators $Q\in \mathrm{ℬ}\mathfrak{X}$ and $L\in \mathrm{ℬ}\mathfrak{Y}$ such that $LMQ{e}_x=e_x$, for every $x\in {\mathbb{N}}_0$.

Definition 13 (see [41]).

A Banach space $\mathcal{E}$ is said to be simple if the algebra $\mathrm{ℬ}\mathcal{E}$ contains one and only one nontrivial closed ideal.

Theorem 16 (see [41]).

If $\mathcal{E}$ is a Banach space with $\text{dim}\mathcal{E}=\infty$, then$$\begin{array}{}\text{(81)}& \mathbb{F}\mathcal{E}\subset \mathcal{A}\mathcal{E}\subset \mathcal{K}\mathcal{E}\subset \mathrm{ℬ}\mathcal{E}.\end{array}$$

In this section, firstly, we obtain sufficient conditions (not necessary) on ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$ so that $\overline{\mathbb{F}}={\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s$. This provides a negative answer of Rhoades [42] open problem about the linearity of $s-$type ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$ space. Secondly, for which conditions on ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$, is ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s$ closed and complete? Thirdly, we explain enough setups on ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$ such that ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s$ is strictly contained for different weights, and we obtain conditions so that the Banach pre-quasi-ideal ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s$ is simple Banach space. Fourthly, we obtain enough conditions on ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$ such that the space of all bounded linear operators which sequence of eigenvalues in ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$ equals ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s$.

7.1. Finite Rank Pre-Quasi-Ideal

Theorem 17.

${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}=\overline{\mathbb{F}\mathfrak{X},\mathfrak{Y}}$, if the condition${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$or${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$and there is$C\ge 1$so that${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$hold. But the converse is not necessarily true.

Proof.

To explain that $\overline{\mathbb{F}\mathfrak{X},\mathfrak{Y}}\subseteq {\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$, as $e_l\in {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$, for all $l\in {\mathbb{N}}_0$ and ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$is a linear space. Assume $Z\in \mathbb{F}\mathfrak{X},\mathfrak{Y}$, and one gets ${s_{l}Z}_{l=0}^{\infty }\in \mathrm{ℭ}$. To show that ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}\subseteq \overline{\mathbb{F}\mathfrak{X},\mathfrak{Y}}$, assume $Z\in {\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$, and one has ${s_{l}Z}_{l=0}^{\infty }\in {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$. As $\mathrm{\varrho }{s_{r}Z}_{r=0}^{\infty }<\infty$, suppose $\rho \in \mathrm{0,1}$; then, there is $r_0\in {\mathbb{N}}_0-0$ with $\mathrm{\varrho }{s_{r}Z}_{r=r_0}^{\infty }<\rho /16d$, for some $d\ge 1$. Since $s_{r}Z\in {\Im }_{\searrow }$, we have$$\begin{array}{c}\text{(82)}& {\text{sup}}_{r=r_0+1}^{2r_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{s}_{2r_0}Z}\le {\text{sup}}_{r=r_0+1}^{2r_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{s}_{v}Z}\le {\text{sup}}_{r=r_0}^{\infty }\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{s}_{v}Z}<\frac{\rho }{16d}.\end{array}$$

Therefore, there exists $Y\in {\mathbb{F}}_{2r_0}\mathfrak{X},\mathfrak{Y}$ such that rank $Y\le 2r_0$ and$$\begin{array}{}\text{(83)}& {\text{sup}}_{r=2r_0+1}^{3r_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,vZ-Y}\le {\text{sup}}_{r=r_0+1}^{2r_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,vZ-Y}<\frac{\rho }{16d}.\end{array}$$

One has$$\begin{array}{}\text{(84)}& \frac{{\displaystyle {\sum }_{v=0}^{r_0}{\delta }_{v}A{r}_0,vZ-Y}}{{a+b}_{p,q}^{r_0}}<\frac{\rho }{16}.\end{array}$$

Hence, we have$$\begin{array}{}\text{(85)}& {\text{sup}}_{r=0}^{r_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,vZ-Y}<\frac{\rho }{16d}.\end{array}$$

According to inequalities (82)–(85) and ${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$, we obtain$$\begin{array}{c}\text{(86)}& dZ,Y=\mathrm{\varrho }{s_{r}Z-Y}_{r=0}^{\infty }\le {\text{sup}}_{r=0}^{3r_0-1}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{s}_{v}Z-Y}+{\text{sup}}_{r=3r_0}^{\infty }\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{s}_{v}Z-Y}\\ \le {\text{sup}}_{r=0}^{3r_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{s}_{v}Z-Y}+{\text{sup}}_{r=r_0}^{\infty }\frac{1}{{a+b}_{p,q}^{r+2r_0}}{\displaystyle \sum _{v=0}^{r+2r_0}{\delta }_{v}Ar+2r_0,v{s}_{v}Z-Y}\\ \le {\text{sup}}_{r=0}^{3r_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,v{s}_{v}Z-Y}+{\text{sup}}_{r=r_0}^{\infty }\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^{r+2r_0}{\delta }_{v}Ar,v{s}_{v}Z-Y}\\ \le 3{\text{sup}}_{r=0}^{r_0}\frac{1}{{a+b}_{p,q}^r}{\displaystyle \sum _{v=0}^r{\delta }_{v}Ar,vZ-Y}\\ +{\text{sup}}_{r=r_0}^{\infty }\frac{{\displaystyle {\sum }_{v=0}^{2r_0-1}{\delta }_{v}Ar,v{s}_{v}Z-Y}+{\displaystyle {\sum }_{v=2r_0}^{r+2r_0}{\delta }_{v}Ar,v{s}_{v}Z-Y}}{{a+b}_{p,q}^r}\\ \le 3{\text{sup}}_{r=0}^{r_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,vZ-Y}}{{a+b}_{p,q}^r}+{\text{sup}}_{r=r_0}^{\infty }\frac{{\displaystyle {\sum }_{v=0}^{2r_0-1}{\delta }_{v}Ar,v{s}_{v}Z-Y}}{{a+b}_{p,q}^r}\\ +{\text{sup}}_{r=r_0}^{\infty }\frac{{\displaystyle {\sum }_{v=2r_0}^{r+2r_0}{\delta }_{v}Ar,v{s}_{v}Z-Y}}{{a+b}_{p,q}^r}\\ \le 3{\text{sup}}_{r=0}^{r_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,vZ-Y}}{{a+b}_{p,q}^r}+{\text{sup}}_{r=r_0}^{\infty }\frac{{\displaystyle {\sum }_{v=0}^{2r_0-1}{\delta }_{v}Ar,vZ-Y}}{{a+b}_{p,q}^r}\\ +{\text{sup}}_{r=r_0}^{\infty }\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v+2r_0}Ar,v+2r_0{s}_{v+2r_0}Z-Y}}{{a+b}_{p,q}^r}\\ \le 3{\text{sup}}_{r=0}^{r_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,vZ-Y}}{{a+b}_{p,q}^r}+\frac{2{\displaystyle {\sum }_{v=0}^{r_0}{\delta }_{v}A{r}_0,vZ-Y}}{{a+b}_{p,q}^{r_0}}\\ +{\text{sup}}_{r=r_0}^{\infty }\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{s}_{v}Z-Y}}{{a+b}_{p,q}^r}<\rho .\end{array}$$

On the other hand, one has a negative example as $I_2\in {\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$, where ${{\delta }_{v}Ar,v}_{v\in {\mathbb{N}}_0}=\mathrm{0,0,1,0,1,0},\dots$. This completes the proof.

7.2. Banach and Closed Pre-Quasi-Ideal

Theorem 18 (see [38]).

The function $\mathrm{\Psi }$ is a pre-quasi-norm on ${\mathrm{ℬ}}_{{\mathcal{E}}_{\mathrm{\varrho }}}^s$, where $\mathrm{\Psi }Y=\mathrm{\varrho }{s_{b}Y}_{b=0}^{\infty }$, for all $Y\in {\mathrm{ℬ}}_{{\mathcal{E}}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$, when ${\mathcal{E}}_{\mathrm{\varrho }}$ is a pre-modular $pss$.

Theorem 19.

Assume the condition${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$or${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$and there is$C\ge 1$so that${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$hold; then, ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s,\mathrm{\Psi }$is a pre-quasi-Banach ideal, where$\psi X=\mathrm{\varrho }{s_{l}X}_{l=0}^{\infty }$.

Proof.

Since ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$ is a pre-modular $pss$, by Theorem 18, $\mathrm{\Psi }$ is a pre-quasi-norm on ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s$. Let ${X_j}_{j\in {\mathbb{N}}_0}$ be a Cauchy sequence in ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$. Since $\mathrm{ℬ}\mathfrak{X},\mathfrak{Y}\supseteq {\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$, we have$$\begin{array}{}\text{(87)}& \Psi X_i-X_j={\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{s}_v{X}_i-X_j}}{{a+b}_{p,q}^r}\ge {\delta }_0{X}_i-X_j.\end{array}$$

Then, ${X_j}_{j\in {\mathbb{N}}_0}$ is a Cauchy sequence in $\mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$. As $\mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$ is a Banach space, then there exists $X\in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$ with ${\lim }_{j⟶\infty }{X}_j-X=0$. Since ${s_l{X}_j}_{l=0}^{\infty }\in {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$, for all $j\in {\mathbb{N}}_0$, in view of (ii), (iii), and (v) in Definition 10, we have$$\begin{array}{c}\text{(88)}& \Psi X={\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{s}_{v}X}}{{a+b}_{p,q}^r}\le {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{s}_{v/2}X-X_j}}{{a+b}_{p,q}^r}+{\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{s}_{v/2}{X}_j}}{{a+b}_{p,q}^r}\\ \le {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,vX-X_j}}{{a+b}_{p,q}^r}+D_0{\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{s}_v{X}_j}}{{a+b}_{p,q}^r}<\infty .\end{array}$$

Hence, ${s_{l}X}_{l=0}^{\infty }\in {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$, so $X\in {\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$.

Theorem 20.

Suppose$\mathfrak{X}$,$\mathfrak{Y}$are normed spaces, and the condition${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$or${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$and there is$C\ge 1$so that${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$is satisfied; then, ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s,\mathrm{\Psi }$is a pre-quasi-closed ideal, where$\mathrm{\Psi }X=\mathrm{\varrho }{s_{l}X}_{l=0}^{\infty }$.

Proof.

Since ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$ is a pre-modular $pss$, from Theorem 0.1, we have that $\mathrm{\Psi }$ is a pre-quasi-norm on ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s$. Let $X_j\in {\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$, for all $j\in {\mathbb{N}}_0$ and ${\lim }_{j⟶\infty }\mathrm{\Psi }{X}_j-X=0$. As $\mathrm{ℬ}\mathfrak{X},\mathfrak{Y}\supseteq {\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$, one has$$\begin{array}{}\text{(89)}& \Psi X-X_j={\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{s}_{v}X-X_j}}{{a+b}_{p,q}^r}\ge {\delta }_{0}X-X_j.\end{array}$$

Then, ${X_j}_{j\in {\mathbb{N}}_0}$ is a convergent sequence in $\mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$. As ${s_l{X}_j}_{l=0}^{\infty }\in {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$, for every $j\in {\mathbb{N}}_0$, in view of (ii), (iii), and (v) in Definition 10, we have$$\begin{array}{c}\text{(90)}& \Psi X={\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{s}_{v}X}}{{a+b}_{p,q}^r}\le {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{s}_{v/2}X-X_j}}{{a+b}_{p,q}^r}+{\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{s}_{v/2}{X}_j}}{{a+b}_{p,q}^r}\\ \le {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,vX-X_j}}{{a+b}_{p,q}^r}+D_0{\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{s}_v{X}_j}}{{a+b}_{p,q}^r}<\infty .\end{array}$$

We get ${s_{l}X}_{l=0}^{\infty }\in {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$, so $X\in {\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$.

7.3. Simple Banach Pre-Quasi-Ideal

Theorem 21.

Assume$\mathfrak{X}$and$\mathfrak{Y}$are Banach spaces with$\text{dim}\mathfrak{X}=\text{dim}\mathfrak{Y}=\infty$, and the condition${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$or${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$and there is$C\ge 1$so that${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$are satisfied with${\delta }_l^2\le {\delta }_l^1$, for every$l\in {\mathbb{N}}_0$; then,$$\begin{array}{}\text{(91)}& {\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}\subset {\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}\subset \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}.\end{array}$$

Proof.

Suppose $Z\in {\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$; then, $s_{l}Z\in {e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}$. We have$$\begin{array}{}\text{(92)}& {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_v^{2}Ar,v{s}_{v}X}}{{a+b}_{p,q}^r}\le {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_v^{1}Ar,v{s}_{v}X}}{{a+b}_{p,q}^r}<\infty .\end{array}$$

Then, $Z\in {\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$. Next, if we put ${s_{r}Z}_{r=0}^{\infty }$ with ${\displaystyle {\sum }_{v=0}^r{\delta }_v^{1}Ar,v{s}_{v}X}={a+b}_{p,q}^r$ and ${\displaystyle {\sum }_{v=0}^r{\delta }_v^{2}Ar,v{s}_{v}X}={a+b}_{p,q}^r/r+1$, we obtain $Z\in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$ so that $Z\notin {\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$ and $Z\in {\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$.

Evidently, ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}\subset \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$. Next, if we put ${s_{r}Z}_{r=0}^{\infty }$ so that ${\displaystyle {\sum }_{v=0}^r{\delta }_v^{2}Ar,v{s}_{v}X}={a+b}_{p,q}^r$, we have $Z\in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}$ such that $Z\notin {\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$. This finishes the proof.

Theorem 22.

Let$\mathfrak{X}$and$\mathfrak{Y}$be Banach spaces with$\text{dim}\mathfrak{X}=\text{dim}\mathfrak{Y}=\infty$, and the condition${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$or${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$and there is$C\ge 1$such that${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$are satisfied with${\delta }_l^2\le {\delta }_l^1$, for every$l\in {\mathbb{N}}_0$; then,$$\begin{array}{}\text{(93)}& \mathrm{ℬ}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y},{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}=\mathcal{A}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y},{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}.\end{array}$$

Proof.

Suppose $X\in \mathrm{ℬ}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y},{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$ and $X\notin \mathcal{A}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y},{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$. From Lemma 14, there are $Y\in \mathrm{ℬ}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$ and $Z\in \mathrm{ℬ}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$ with $ZXY{I}_j=I_j$. Hence, for all $j\in {\mathbb{N}}_0$, one has$$\begin{array}{c}\text{(94)}& {I_j}_{{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}}={\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_v^{1}Ar,v{s}_v{I}_j}}{{a+b}_{p,q}^r}\le ZXY{I_j}_{{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}}\\ \le {\text{sup}}_{r\in {\mathbb{N}}_0}\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_v^{2}Ar,v{s}_v{I}_j}}{{a+b}_{p,q}^r}.\end{array}$$

This contradicts Theorem 21. Therefore, $X\in \mathcal{A}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y},{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$, as desired.

Corollary 7.

Let$\mathfrak{X}$and$\mathfrak{Y}$be Banach spaces with$\text{dim}\mathfrak{X}=\text{dim}\mathfrak{Y}=\infty$, and the condition${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$or${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$and there is$C\ge 1$so that${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$are satisfied with${\delta }_l^2\le {\delta }_l^1$, for every$l\in {\mathbb{N}}_0$; then,$$\begin{array}{}\text{(95)}& \mathrm{ℬ}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y},{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}=\mathcal{K}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^2}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y},{\mathrm{ℬ}}_{{e_0^{a,b}p,q,{\delta }_l^1}_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}.\end{array}$$

Proof.

The result is immediate since $\mathcal{A}\subset \mathcal{K}$.

Theorem 23.

Suppose$\mathfrak{X}$and$\mathfrak{Y}$are Banach spaces with$\text{dim}\mathfrak{X}=\text{dim}\mathfrak{Y}=\infty$, and the condition${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$or${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$and there is$C\ge 1$so that${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$are satisfied; then, ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s$is simple.

Proof.

Let the closed ideal $\mathcal{K}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$ contain an operator $X\notin \mathcal{A}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$. From Lemma 14, one has $Y,Z\in \mathrm{ℬ}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$ with $ZXY{I}_j=I_j$. This implies that $I_{{\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}}\in \mathcal{K}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$. Then, $\mathrm{ℬ}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}=\mathcal{K}{\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$. Therefore, ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s$ is simple Banach space.

7.4. Eigenvalues of s-Type Operators

Conventions 2.

$$\begin{array}{c}\text{(96)}& {{\mathrm{ℬ}}_{\mathcal{E}}^s}^{\rho }≔{{\mathrm{ℬ}}_{\mathcal{E}}^s}^{\rho }\mathfrak{X},\mathfrak{Y};\mathfrak{X}\text{\hspace{0.17em}and\hspace{0.17em}}\mathfrak{Y}\text{\hspace{0.17em}are\hspace{0.17em}Banach\hspace{0.17em}Spaces},\text{where}{{\mathrm{ℬ}}_{\mathcal{E}}^s}^{\rho }\mathfrak{X},\mathfrak{Y}≔X\in \mathrm{ℬ}\mathfrak{X},\mathfrak{Y}:{{\rho }_{l}X}_{n=0}^{\infty }\in \mathcal{E}\text{\hspace{0.17em}and\hspace{0.17em}}\text{X}-{\rho }_{\text{l}}\text{X}\text{I\hspace{0.17em}is\hspace{0.17em}not\hspace{0.17em}invertible},\text{foral}l\text{\hspace{0.17em}l}\in {\mathbb{N}}_{\text{0}}.\end{array}$$

Theorem 24.

Assume$\mathfrak{X}$and$\mathfrak{Y}$are Banach spaces with$\text{dim}\mathfrak{X}=\text{dim}\mathfrak{Y}=\infty$, and the condition${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\searrow }$or${{\delta }_{v}Ar,v}_{v=0}^{\infty }\in {\Im }_{\nearrow }\cap {\ell }_{\infty }$and there is$C\ge 1$so that${\delta }_{2v+1}Ar,2v+1\le C{\delta }_{v}Ar,v$are satisfied with$\underset{r\in {\mathbb{N}}_0}{\inf }{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v}/{a+b}_{p,q}^r>0$; then,$$\begin{array}{}\text{(97)}& {{\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s}^{\rho }\mathfrak{X},\mathfrak{Y}={\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}.\end{array}$$

Proof.

Suppose $X\in {{\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s}^{\rho }\mathfrak{X},\mathfrak{Y}$; then, ${{\rho }_{l}X}_{l=0}^{\infty }\in {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$ and $X-{\rho }_{l}XI=0$, for every $l\in {\mathbb{N}}_0$. One has $X={\rho }_{l}XI$, for every $l\in {\mathbb{N}}_0$, so $s_{l}X=s_l{\rho }_{l}XI={\rho }_{l}X$, for all $l\in {\mathbb{N}}_0$. This implies ${s_{l}X}_{l=0}^{\infty }\in {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$; then, $X\in {\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$.

Secondly, assume $X\in {\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}$. Then, ${s_{l}X}_{l=0}^{\infty }\in {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$. Therefore, one gets$$\begin{array}{}\text{(98)}& \underset{r⟶\infty }{\lim }\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v{s}_{v}X}}{{a+b}_{p,q}^r}\ge \underset{r\in {\mathbb{N}}_0}{\inf }\frac{{\displaystyle {\sum }_{v=0}^r{\delta }_{v}Ar,v}}{{a+b}_{p,q}^r}\underset{l⟶\infty }{\lim }{s}_{l}X.\end{array}$$

Hence, ${\lim }_{l⟶\infty }{s}_{l}X=0$. Suppose ${X-s_{l}XI}^{-1}$ exists, for all $l\in {\mathbb{N}}_0$. Therefore, ${X-s_{l}XI}^{-1}$ exists and is bounded, for all $l\in {\mathbb{N}}_0$. Hence, ${\lim }_{l⟶\infty }{X-s_{l}XI}^{-1}=X^{-1}$ exists and is bounded. As ${\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s,\mathrm{\Psi }$ is a pre-quasi-operator ideal, one has$$\begin{array}{}\text{(99)}& I=X{X}^{-1}\in {\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s\mathfrak{X},\mathfrak{Y}\Rightarrow {s_{l}I}_{l=0}^{\infty }\in {e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}\Rightarrow \underset{l⟶\infty }{\lim }{s}_{l}I=0.\end{array}$$

Therefore, one gets a contradiction, as ${\lim }_{l⟶\infty }{s}_{l}I=1$. This implies $X-s_{l}XI=0$, for all $l\in {\mathbb{N}}_0$. So, $X\in {{\mathrm{ℬ}}_{{e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}}^s}^{\rho }\mathfrak{X},\mathfrak{Y}$. This completes the proof.

8. Conclusion

The development of sequence spaces using post-quantum analog of well-known matrices is a new study in the field of sequence spaces. We have seen some studies concerning construction of sequence spaces using the quantum analog of special matrices [6, 26]. The approach towards constructing post-quantum analog of well-known sequence spaces is another achievement in the field of sequence spaces. In the current article, we presented new Euler sequence space obtained by the domain of generalized $p,q$-Euler matrix $E^{a,b}p,q$. Finally, we obtained certain geometric and topological properties of ${e_0^{a,b}p,q,\delta }_{\mathrm{\varrho }}$ as well as the eigenvalue distribution of mapping ideals generated by this space and $s$-numbers. Thus, our result advances some of the well-known spaces in the literature including binomial sequence spaces [22] and Euler sequence spaces [1].

The application of $p,q$ or $q$-calculus in the field of the sequence spaces is a new study and has the tendency to attract more researchers in this domain. Several $p,q$-analogs of well-known operators, for instance, $p,q$-Cesàro mean [38], $p,q$-difference operator [43], and so on, can be found in the literature. One may use these special matrices to construct$p,q$-sequence spaces in the similar line to our study. Moreover, for future scope, by extending this study, one may construct the domain of the $p,q$-Euler matrix $E^{a,b}p,q$ in Maddox’s spaces.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors have read and approved the final manuscript.

Acknowledgments

This work was funded by the University of Jeddah, Saudi Arabia, under grant no. (UJ-22-DR-7). The authors, therefore, acknowledge with thanks the University technical and financial support.

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