On Some Sequence Spaces via q-Pascal Matrix and Its Geometric Properties

Abstract

We develop some new sequence spaces ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$ by using q-Pascal matrix $\mathcal{P}\left(q\right).$ We discuss some topological properties of the newly defined spaces, obtain the Schauder basis for the space ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ and determine the Alpha-($\alpha$-), Beta-($\beta$-) and Gamma-($\gamma$-) duals of the newly defined spaces. We characterize a certain class $({𝓁}_p\left(\mathcal{P}\left(q\right)\right),X)$ of infinite matrices, where $X\in \{{𝓁}_{\infty },c,c_0\}$. Furthermore, utilizing the proposed results, we characterize certain other classes of infinite matrices. We also examine some geometric properties, like the approximation property, Dunford–Pettis property, Hahn–Banach extension property, and Banach–Saks-type p property of the spaces ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$.

1. Introduction and Preliminaries

The theory of the q-analogue, as the nomenclature suggests, deals with expanding the classical mathematical concept to a more generalized one by utilizing a new parameter q. The generalized expression, i.e., q-analogue reduces to the original expression as $q\to 1^{-}.$ Although the history of the q-analogue dates back to Euler, the true application of the q-analogue in developing q-differentiation and q-integration was architected by Jackson [1]. This concept of generalization is widely accepted by the mathematical community, and as a result, several applications of q-analogues in different branches of mathematics, like algebra, combinatorics, integro-differential equations, approximation theory, special functions, hypergeometric functions, etc., are witnessed in the literature. Indeed, the application of q-theory has also been spotted in the field of summability and sequence spaces. For instances, Demiriz and Şahin [2] and Yaying et al. [3] developed the q-analogue of the well-known Cesàro sequence spaces. Mursaleen et al. [4] discussed a regular summability method by using q-statistical convergence. They further obtained a condition for q-statistical convergent sequences to be Cesàro summable.

We now turn to certain basic definitions in q-theory:

Definition 1. 

The q-integer ${\left[v\right]}_q$ $(q\in (0,1\left)\right)$ is defined by

$$\begin{array}{c}\hfill {\left[v\right]}_q=\left\{\begin{array}{ccc}\sum _{i=0}^{v-1}{q}^i& ,& v\in \mathbb{N},\\ 0& ,& v=0.\end{array}\right.\end{array}$$

Here and onwards, $\mathbb{N}=\{1,2,3,\dots \}$ and ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$. Apparently, ${\left[v\right]}_q=v$ as $q\to 1^{-}$.

Definition 2. 

The notation ${\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q$ defined by

$$\begin{array}{c}\hfill {\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q=\left\{\begin{array}{ccc}{\displaystyle \frac{{\left[n\right]}_q!}{{[n-v]}_q!{\left[v\right]}_q!}}& ,& n\ge v,\\ 0& ,& n<v,\end{array}\right.\end{array}$$

is the natural q-analogue of the binomial coefficient $\left(\genfrac{}{}{0pt}{}{n}{v}\right).$ Here, ${\left[v\right]}_q!={\prod }_{i=1}^v{\left[i\right]}_q$ is the natural q-analogue of $v!.$

Apparently, the relations ${\left(\genfrac{}{}{0pt}{}{0}{0}\right)}_q={\left(\genfrac{}{}{0pt}{}{v}{0}\right)}_q={\left(\genfrac{}{}{0pt}{}{v}{v}\right)}_q=1$ and ${\left(\genfrac{}{}{0pt}{}{v}{v-k}\right)}_q={\left(\genfrac{}{}{0pt}{}{v}{k}\right)}_q$ hold true for the q-binomial coefficient ${\left(\genfrac{}{}{0pt}{}{v}{k}\right)}_q.$ We recommend the monograph [5] for a basic idea of q-theory.

1.1. Sequence Space

The set $\omega$ of all real- or complex-valued sequences forms a vector space with the algebraic operations addition $(+)$ and scalar multiplication $(·)$ defined for the vectors $y=\left(y_n\right),z=\left(z_n\right)\in \omega$ and real or complex scalar $\alpha$ by

$$\begin{array}{ccc}\hfill y+z& =& \left(y_n\right)+\left(z_n\right)=(y_n+z_n),\hfill \\ \hfill \alpha ·y& =& \alpha ·\left(y_n\right)=\left(\alpha y_n\right).\hfill \end{array}$$

Any vector subspace of $\omega$ is known as a sequence space. The sets ${𝓁}_p$ $(0<p<\infty )$ of all absolutely p-summable sequences, $c_0$ of all null sequences, c of all convergent sequences, and ${𝓁}_{\infty }$ of all bounded sequences are some examples of basic sequence spaces. Let

$$\begin{array}{ccc}\hfill bs& =& \left\{u=\left(u_n\right)\in \omega :\underset{n\in {\mathbb{N}}_0}{sup}\left|\sum _{v=0}^n{u}_v\right|<\infty \right\};\hfill \\ \hfill cs& =& \left\{u=\left(u_n\right)\in \omega :\underset{n\to \infty }{lim}\sum _{v=0}^n{u}_v=\beta \mathrm{for}\mathrm{some}\beta \in \mathbb{C}\right\};\hfill \end{array}$$

be the spaces of all sequences with bounded partial sums and summable sequences, respectively.

A Banach sequence space with continuous coordinates is called a $BK$-space. The spaces ${𝓁}_p$ $(1\le p<\infty )$ and ${𝓁}_{\infty }$ are $BK$-spaces normed by

$${∥y∥}_p={\left(\sum _{v=0}^{\infty }{\left|y_v\right|}^p\right)}^{1/p}\mathrm{and}{∥y∥}_{\infty }=\underset{v\in {\mathbb{N}}_0}{sup}\left|y_v\right|,$$

(1)

respectively. For $0<p<1,$ the space ${𝓁}_p$ is a complete p-normed space due to p-norm ${∥y∥}_p={\sum }_{v=0}^{\infty }{\left|y_v\right|}^p.$

Let $G=\left(g_{n,v}\right)$ be any infinite matrix with real entries. Let $G_n={\left(g_{n,v}\right)}_{v=0}^{\infty }$, i.e., $G_n$ denotes the $n^{\mathrm{th}}$ row of the matrix $G.$ For any $y=\left(y_v\right)\in \omega ,$ the sequence

$$Gy=\left\{{\left(Gy\right)}_n\right\}={\left(\sum _{v=0}^{\infty }{g}_{n,v}{y}_v\right)}_{n\in {\mathbb{N}}_0}$$

is called G-transform of $y,$ where we presume that the infinite sum exists for each $n\in {\mathbb{N}}_0$. Let $X,Y\subset \omega$. Then, G corresponds a matrix mapping from X to Y if $Gy\in Y$ for all $y\in X.$ The family of all such matrices from X to Y is denoted by $(X,Y).$ Further, the domain $X_G$ of the matrix G in a sequence space X is given by $X_G=\{y\in \omega :Gy\in X\}$ and is a sequence space. In addition, if G is a triangle matrix and X is a $BK$-space, then $X_G$ is a $BK$-space normed by ${∥y∥}_{X_G}={∥Gy∥}_X.$ The monographs [6,7] are worth mentioning for detailed studies concerning the domain of notable triangles in classical sequence spaces. We also mention the papers [8,9] so that readers may glean brief insight into some interesting types of summable sequence spaces.

Lately, the scholarly publications have seen an emergence of the q-analogue of well-known sequence spaces. For instances, the domain $X\left(C\left(q\right)\right):=X_{C\left(q\right)}$ for $X\in \{{𝓁}_p,c_0,c,{𝓁}_{\infty }\}$ are discussed by Demiriz and Şahin [2] and Yaying et al. [3], defined via the q-Cesàro matrix $C\left(q\right)={\left(c_{nv}^q\right)}_{n,v\in {\mathbb{N}}_0}$ given by

$$c_{nv}^q=\left\{\begin{array}{cc}\frac{q^v}{{[n+1]}_q}\hfill & (0\le v\le n),\hfill \\ 0\hfill & (v>n).\hfill \end{array}\right.$$

Quite recently, some topological and geometric properties of the spaces ${\left({𝓁}_p\right)}_{C\left(q\right)}$ and ${\left({𝓁}_{\infty }\right)}_{C\left(q\right)}$ have been studied by Yılmaz and Akdemir [10]. Alotaibi et al. [11] engineered the sequence spaces ${𝓁}_p\left({\nabla }_q^2\right):={\left({𝓁}_p\right)}_{{\nabla }_q^2}$ and ${𝓁}_{\infty }\left({\nabla }_q^2\right):={\left({𝓁}_{\infty }\right)}_{{\nabla }_q^2}$ of the operator ${\nabla }_q^2$ in ${𝓁}_p$ and ${𝓁}_{\infty },$ respectively. We recall that a sequence space X exhibits symmetry (as defined in [12]) if $y_{\pi \left(n\right)}$ belongs to X for any $\left(y_n\right)$ in X, where $\pi \left(n\right)$ represents a permutation on ${\mathbb{N}}_0$. Alotaibi et al. [11] proved that the space ${𝓁}_{\infty }\left({\nabla }_q^2\right)$ does not exhibit symmetricity.

The theory of q-sequence spaces is further strengthened by the introduction of q-Catalan sequence space $X\left(\mathcal{C}\left(q\right)\right)=X_{C\left(q\right)}$ by Yaying et al. [13] for $X\in \{c,c_0\}$, and $\mathcal{C}\left(q\right)={\left({\tilde{c}}_{nv}^q\right)}_{n,v\in {\mathbb{N}}_0}$ is defined by

$$\begin{array}{c}\hfill {\tilde{c}}_{nv}\left(q\right)=\left\{\begin{array}{ccc}{q}^v{\displaystyle \frac{c_v\left(q\right)c_{n-v}\left(q\right)}{c_{n+1}\left(q\right)}}& ,& 0\le v\le n,\\ 0& ,& v>n,\end{array}\right.\end{array}$$

where $c\left(q\right)={\left(c_v\left(q\right)\right)}_{v\in {\mathbb{N}}_0}$ is a sequence of q-Catalan numbers.

1.2. Pascal Matrix and Motivation

The Pascal matrix $\mathcal{P}={\left({\tilde{p}}_{n,v}\right)}_{n,v\in {\mathbb{N}}_0}$ is defined by (see [14,15])

$$\begin{array}{c}\hfill {\tilde{p}}_{n,v}=\left\{\begin{array}{ccc}\left(\genfrac{}{}{0pt}{}{n}{n-v}\right)& ,& 0\le v\le n,\\ 0& ,& v>n.\end{array}\right.\end{array}$$

The matrix domains $c_0\left(\mathcal{P}\right)={\left(c_0\right)}_{\mathcal{P}}$ and $c\left(\mathcal{P}\right)=c_{\mathcal{P}}$ were investigated by Polat [16]. These domains were further generalized by Aydın and Polat [14] by introducing Pascal difference sequence spaces $c_0(P\nabla )={\left(c_0\right)}_{P\nabla }$ and $c(P\nabla )=c_{P\nabla },$ where ∇ is the first-order backward difference operator. Yaying and Başar [17] recently studied the sequence space $X\left(G\right)$, where G is the product of the Lambda $\left(\mathrm{\Lambda }\right)$ matrix and Pascal matrix, i.e., $G=\mathrm{\Lambda }\mathcal{P}$ and X is any of the spaces ${𝓁}_p,c_0,c$ or ${𝓁}_{\infty }.$

Let $q\in (0,1).$ Then, the q-Pascal matrix $\mathcal{P}\left(q\right)=\left({\tilde{p}}_{n,v}^q\right)$ is defined by

$$\begin{array}{c}\hfill {\tilde{p}}_{n,v}^q=\left\{\begin{array}{ccc}{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q& ,& 0\le v\le n,\\ 0& ,& v>n\end{array}\right.\end{array}$$

for all $n,v\in {\mathbb{N}}_0$ (see [18]). By using the q-Pascal matrix $\mathcal{P}\left(q\right),$ Yaying et al. [19] defined q-Pascal sequence spaces $c_0\left(\mathcal{P}\left(q\right)\right)={\left(c_0\right)}_{\mathcal{P}\left(q\right)}$ and $c\left(\mathcal{P}\left(q\right)\right)=c_{\mathcal{P}\left(q\right)}$. Further, the authors informed that

$$\sum _{v=0}^n{\tilde{p}}_{n,v}^q=G_n\left(q\right)\mathrm{and}\underset{n\to \infty }{lim}{\tilde{p}}_{n,v}^q=\frac{1}{{\left[v\right]}_q!{(1-q)}^v},$$

where $G_n\left(q\right)$ is the $n^{\mathrm{th}}$-Galois number defined by the recurrence relation $G_{n+1}\left(q\right)=2G_n\left(q\right)+(q^n-1)G_{n-1}\left(q\right)$ with $G_0\left(q\right)=1$ and $G_1\left(q\right)=2$ (see [5] for more details). Apparently, the matrix $\mathcal{P}\left(q\right)$ is a conservative matrix.

Inspired by the above studies, we develop q-Pascal sequence spaces ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$ and explore several properties of these spaces.

2. The Spaces ${𝓁}_{\mathit{p}}\left(\mathcal{P}\left(\mathit{q}\right)\right)$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(\mathit{q}\right)\right)$

Define the spaces ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$ by

$$\begin{array}{ccc}\hfill {𝓁}_p\left(\mathcal{P}\left(q\right)\right)& :=& \left\{y=\left(y_n\right)\in \omega :\sum _{n=0}^{\infty }{\left|\sum _{v=0}^n{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q{y}_v\right|}^p<\infty \right\};\hfill \\ \hfill {𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)& :=& \left\{y=\left(y_n\right)\in \omega :\underset{n\in {\mathbb{N}}_0}{sup}\left|\sum _{v=0}^n{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q{y}_v\right|<\infty \right\};\hfill \end{array}$$

Here, the sequence

$$z=\left(z_n\right)=\left\{\sum _{v=0}^n{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q{y}_v\right\}$$

is termed the $\mathcal{P}\left(q\right)$-transform of $y=\left(y_n\right)$. Thus, an equivalent definition of the spaces ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$ may be given as

$${𝓁}_p\left(\mathcal{P}\left(q\right)\right)={\left({𝓁}_p\right)}_{\mathcal{P}\left(q\right)}\mathrm{and}{𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)={\left({𝓁}_{\infty }\right)}_{\mathcal{P}\left(q\right)}.$$

It is easy to observe that when $q\to 1^{-}$, the spaces ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$ are reduced to the spaces ${𝓁}_p\left(\mathcal{P}\right)={\left({𝓁}_p\right)}_{\mathcal{P}}$ and ${𝓁}_{\infty }\left(\mathcal{P}\right)={\left({𝓁}_{\infty }\right)}_{\mathcal{P}}.$

Definition 3 

([19]). The inverse of the q-Pascal matrix $\mathcal{P}\left(q\right)$ is given by the matrix $\mathcal{Q}\left(q\right)={\left({\left\{{\tilde{p}}^q\right\}}_{n,v}^{-1}\right)}_{n,v\in {\mathbb{N}}_0}$ as follows:

$$\begin{array}{c}\hfill {\left\{{\tilde{p}}^q\right\}}_{n,v}^{-1}=\left\{\begin{array}{ccc}{(-1)}^{n-v}{q}^{\left(\genfrac{}{}{0pt}{}{n-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q& ,& 0\le v\le n,\\ 0& ,& v>n.\end{array}\right.\end{array}$$

The $\mathcal{Q}\left(q\right)$-transform or ${\left\{\mathcal{P}\left(q\right)\right\}}^{-1}$-transform of the sequence $z=\left(z_n\right)$ is given by the sequence $y=\left(y_n\right)$, where

$$\begin{array}{c}\hfill y_n=\sum _{v=0}^n{(-1)}^{n-v}{q}^{\left(\genfrac{}{}{0pt}{}{n-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q{z}_v\end{array}$$

(2)

for each $n\in {\mathbb{N}}_0$. Throughout the article, unless stated otherwise, we keep in mind that the sequence z is the $\mathcal{P}\left(q\right)$-transform of the sequence y, or equivalently, the sequence y is the ${\left\{\mathcal{P}\left(q\right)\right\}}^{-1}$-transform of the sequence z.

Theorem 1. 

Each of the following statements holds true:

1. 

The space ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ is a complete p-normed space for $0<p<1$ due to the p-norm

$${∥y∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}={∥\mathcal{P}\left(q\right)y∥}_p=\sum _{n=0}^{\infty }{\left|\sum _{v=0}^n{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q{y}_v\right|}^p.$$

2. 

The space ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ is a $BK$-space for $1\le p<\infty$ normed by

$${∥y∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}={∥\mathcal{P}\left(q\right)y∥}_p={\left(\sum _{n=0}^{\infty }{\left|\sum _{v=0}^n{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q{y}_v\right|}^p\right)}^{1/p}.$$

3. 

The space ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$ is a $BK$-space normed by

$${∥y∥}_{{𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)}={∥\mathcal{P}\left(q\right)y∥}_{\infty }=\underset{n\in {\mathbb{N}}_0}{sup}\left|\sum _{v=0}^n{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q{y}_v\right|.$$

Proof. 

It is known that the spaces ${𝓁}_p$ for $1<p<\infty$ and ${𝓁}_{\infty }$ are $BK$-spaces equipped by their natural norms defined in (1) and $\mathcal{P}\left(q\right)$ is a triangle. Thus, Parts (2) and (3) follow immediately by using Wilansky’s work [20] (Theorem 4.3.2).

A similar argument holds for the complete p-normed space ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ $(0<p\le 1).$ □

Let X denote either the space ${𝓁}_p$ or ${𝓁}_{\infty }.$ Consider the mapping $\mathcal{L}$ defined by

$$\begin{array}{c}\hfill \begin{array}{ccccc}\mathcal{L}& :& X\left(\mathcal{P}\right(q\left)\right)& ⟶& X\\ & & y& ⟼& \mathcal{L}y=z=\mathcal{P}\left(q\right)y.\end{array}\end{array}$$

It is noted that $\mathcal{P}\left(q\right)$ is the matrix representation of the operator $\mathcal{L}$. Additionally, $\mathcal{P}\left(q\right)$ is a triangle. As a result of this fact, the mapping $\mathcal{L}$ is the norm or p-norm, preserving the linear bijection. Thus, $X\left(\mathcal{P}\right(q\left)\right)$ is linearly isomorphic to $X.$ That is, ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)\cong {𝓁}_p$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)\cong {𝓁}_{\infty }.$

Definition 4. 

A sequence $s=\left(s_k\right)\in (X,∥·∥)$ is termed a Schauder basis of X if for each $y\in X$, there corresponds a unique sequence of scalars $\left({\alpha }_k\right)$ such that

$$\underset{n\to \infty }{lim}∥y-\sum _{v=0}^n{\alpha }_k{s}_k∥=0.$$

It is known from Theorem 2.3 of Jarrah and Malkowsky [21] that if G is a triangle matrix, then a normed linear space $X_G$ has a basis iff X has a basis. As a result of this fact, we have the following corollary:

Corollary 1. 

Define the sequence $s^{\left(v\right)}\left(q\right)=\left(s_n^{\left(v\right)}\left(q\right)\right)$ for each fixed $v\in {\mathbb{N}}_0$ by

$$\begin{array}{c}\hfill s_n^{\left(v\right)}\left(q\right)=\left\{\begin{array}{ccc}{(-1)}^{n-v}{q}^{\left(\genfrac{}{}{0pt}{}{n-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q& ,& 0\le v\le n,\\ 0& ,& v>n.\end{array}\right.\end{array}$$

Then, $\left\{s^{\left(0\right)}\left(q\right),s^{\left(1\right)}\left(q\right),s^{\left(2\right)}\left(q\right),\dots \right\}$ is the basis of the space ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ and each $y\in {𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ is uniquely given by

$$y=\sum _{v=0}^{\infty }{z}_k{s}^{\left(k\right)}\left(q\right).$$

Remark 1. 

There is no basis for the space ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right).$

3. Some Duals of the Spaces ${𝓁}_{\mathit{p}}\left(\mathcal{P}\left(\mathit{q}\right)\right)$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(\mathit{q}\right)\right)$

Definition 5. 

The α-, β- and γ-duals $X^{\alpha }$, $X^{\beta }$ and $X^{\gamma }$ of any $X\subset \omega$ are defined by

$$\begin{array}{cc}\hfill & X^{\alpha }:=\{c=\left(c_n\right)\in \omega :cy=\left(c_n{y}_n\right)\in {𝓁}_1\mathit{for}\mathit{all}y\in X\},\hfill \\ \hfill & X^{\beta }:=\{c=\left(c_n\right)\in \omega :cy=\left(c_n{y}_n\right)\in cs\mathit{for}\mathit{all}y\in X\}and\hfill \\ \hfill & X^{\gamma }:=\{c=\left(c_n\right)\in \omega :cy=\left(c_n{y}_n\right)\in bs\mathit{for}\mathit{all}y\in X\},\hfill \end{array}$$

respectively.

Let $\mathfrak{N}$ denote the family of all finite subsets of ${\mathbb{N}}_0$, and $p^{\prime }$ be the complement of p, i.e., $1/p+1/p^{\prime }=1.$ We recall certain lemmas that are important for determining the duals.

Lemma 1 

([22,23,24]). Let $G=\left(g_{n,v}\right)$ be an infinite matrix. Then, each of the following statements holds true:

(1) 

$G=\left(g_{n,v}\right)\in ({𝓁}_{\infty },{𝓁}_{\infty })$ iff

$$\begin{array}{ccc}\hfill & & \underset{n\in {\mathbb{N}}_0}{sup}\sum _{v=0}^{\infty }\left|g_{n,v}\right|<\infty .\hfill \end{array}$$

(3)

(2) 

$G=\left(g_{n,v}\right)\in ({𝓁}_{\infty },c)$ iff

$$\begin{array}{ccc}\hfill & & \exists g_v\in \mathbb{C}\ni \underset{n\to \infty }{lim}{g}_{n,v}=g_v\mathrm{for} \mathrm{all}v\in {\mathbb{N}}_0,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill & & \underset{n\to \infty }{lim}\sum _{v=0}^{\infty }\left|g_{n,v}\right|=\sum _{v=0}^{\infty }\left|\underset{n\to \infty }{lim}{g}_{n,v}\right|.\hfill \end{array}$$

(5)

(3) 

$G=\left(g_{n,v}\right)\in ({𝓁}_{\infty },{𝓁}_1)$ iff

$$\begin{array}{c}\hfill \underset{N\in \mathfrak{N}}{sup}\sum _{v=0}^{\infty }\left|\sum _{n\in N}{g}_{n,v}\right|<\infty .\end{array}$$

(6)

(4) 

$G=\left(g_{n,v}\right)\in ({𝓁}_p,{𝓁}_{\infty })$ iff

$$\begin{array}{ccc}& & \underset{n\in {\mathbb{N}}_0}{sup}\sum _{v=0}^{\infty }{\left|g_{n,v}\right|}^{p^{\prime }}<\infty ,(1<p<\infty ).\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill & & \underset{n,v\in {\mathbb{N}}_0}{sup}{\left|g_{n,v}\right|}^p<\infty ,(0<p\le 1).\hfill \end{array}$$

(8)

(5) 

(a)  For $1<p<\infty$, $G=\left(g_{n,v}\right)\in ({𝓁}_p,c)$ iff (4) and (7) hold.

(b)  For $0<p\le 1$, $G=\left(g_{n,v}\right)\in ({𝓁}_p,c)$ iff (4) and (8) hold.

(6) 

$G=\left(g_{n,v}\right)\in ({𝓁}_p,{𝓁}_1)$ iff

$$\begin{array}{ccc}\hfill & & \underset{N\in \mathfrak{N}}{sup}\underset{v\in {\mathbb{N}}_0}{sup}{\left|\sum _{n\in N}{g}_{n,v}\right|}^p<\infty ,(0<p\le 1).\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & \underset{N\in \mathcal{N}}{sup}\sum _{v=0}^{\infty }{\left|\sum _{n\in N}{g}_{n,v}\right|}^{p^{\prime }}<\infty ,(1<p<\infty ).\hfill \end{array}$$

(10)

Theorem 2. 

Define the sets ${\lambda }_1,$ ${\lambda }_2,$ and ${\lambda }_3$ by

$$\begin{array}{ccc}\hfill {\lambda }_1& :=& \left\{c=\left(c_n\right)\in \omega :\underset{N\in \mathfrak{N}}{sup}\underset{v\in {\mathbb{N}}_0}{sup}{\left|\sum _{n\in N}{(-1)}^{n-v}{q}^{\left(\genfrac{}{}{0pt}{}{n-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q{c}_n\right|}^p<\infty \right\},(0<p\le 1),\hfill \\ \hfill {\lambda }_2& :=& \left\{c=\left(c_n\right)\in \omega :\underset{N\in \mathfrak{N}}{sup}\sum _{v=0}^{\infty }{\left|\sum _{n\in N}{(-1)}^{n-v}{q}^{\left(\genfrac{}{}{0pt}{}{n-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q{c}_n\right|}^{p^{\prime }}<\infty \right\},(1<p<\infty ),\hfill \\ \hfill {\lambda }_3& :=& \left\{c=\left(c_n\right)\in \omega :\underset{N\in \mathfrak{N}}{sup}\sum _{v=0}^{\infty }\left|\sum _{n\in N}{(-1)}^{n-v}{q}^{\left(\genfrac{}{}{0pt}{}{n-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q{c}_n\right|<\infty \right\}.\hfill \end{array}$$

Then,

(i) 

${\left[{𝓁}_p\left(\mathcal{P}\left(q\right)\right)\right]}^{\alpha }=\left\{\begin{array}{ccc}{\lambda }_1& ,& 0<p\le 1,\\ {\lambda }_2& ,& 1<p<\infty .\end{array}\right.$

(ii) 

${\left[{𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)\right]}^{\alpha }={\lambda }_3$.

Proof. 

Consider the triangle $D={\left(d_{n,v}\right)}_{n,v\in {\mathbb{N}}_0}$ defined by

$$\begin{array}{c}\hfill d_{n,v}=\left\{\begin{array}{ccc}{(-1)}^{n-v}{q}^{\left(\genfrac{}{}{0pt}{}{n-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q{c}_n& ,& 0\le v\le n,\\ 0& ,& n>v.\end{array}\right.\end{array}$$

Then, the following relation holds true:

$$\begin{array}{cc}\hfill c_n{y}_n& =\sum _{v=0}^n{(-1)}^{n-v}{q}^{\left(\genfrac{}{}{0pt}{}{n-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{n}{v}\right)}_q{c}_n{z}_v\hfill \\ \hfill & ={\left(Dz\right)}_n,(n\in {\mathbb{N}}_0).\hfill \end{array}$$

This clearly indicates that $cy=\left(c_n{y}_n\right)\in {𝓁}_1$ whenever $y\in {𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ iff $Dz\in {𝓁}_1$ whenever $z\in {𝓁}_p.$ This means that $c=\left(c_n\right)\in {\left[{𝓁}_p\left(\mathcal{P}\left(q\right)\right)\right]}^{\alpha }$ iff $D\in ({𝓁}_p,{𝓁}_1).$ As a result of this fact and together with Lemma 1/(6), we obtain that

$${\left[{𝓁}_p\left(\mathcal{P}\left(q\right)\right)\right]}^{\alpha }=\left\{\begin{array}{ccc}{\lambda }_1& ,& 0<p\le 1,\\ {\lambda }_2& ,& 1<p<\infty .\end{array}\right.$$

The $\alpha$-dual of ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$ is given in a similar manner by utilizing Lemma 1/(3) in place of Lemma 1/(6). This ends the proof. □

Theorem 3. 

Define the sets ${\mu }_i$ $(i=1,2,3,4)$ by

$$\begin{array}{cc}\hfill & {\mu }_1=\left\{c=\left(c_n\right)\in \omega :\underset{n\to \infty }{lim}\sum _{u=v}^n{(-1)}^{u-v}{q}^{\left(\genfrac{}{}{0pt}{}{u-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{u}{v}\right)}_q{c}_u\mathrm{exists} \mathrm{for} \mathrm{each} v\in {\mathbb{N}}_0\right\},\hfill \\ \hfill & {\mu }_2=\left\{c=\left(c_n\right)\in \omega :\underset{n,v\in {\mathbb{N}}_0}{sup}{\left|\sum _{u=v}^n{(-1)}^{u-v}{q}^{\left(\genfrac{}{}{0pt}{}{u-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{u}{v}\right)}_q{c}_u\right|}^p<\infty \right\},(0<p\le 1),\hfill \\ \hfill & {\mu }_3=\left\{c=\left(c_n\right)\in \omega :\underset{n\in {\mathbb{N}}_0}{sup}\sum _{v=0}^n{\left|\sum _{u=v}^n{(-1)}^{u-v}{q}^{\left(\genfrac{}{}{0pt}{}{u-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{u}{v}\right)}_q{c}_u\right|}^{p^{\prime }}<\infty \right\},\hfill \\ \hfill & {\mu }_4=\left\{c=\left(c_n\right)\in \omega :\underset{n\to \infty }{lim}\sum _{v=0}^{\infty }\left|\sum _{u=v}^n{(-1)}^{u-v}{q}^{\left(\genfrac{}{}{0pt}{}{u-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{u}{v}\right)}_q{c}_u\right|=\sum _{v=0}^{\infty }\left|\underset{n\to \infty }{lim}\sum _{u=v}^n{(-1)}^{u-v}{q}^{\left(\genfrac{}{}{0pt}{}{u-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{u}{v}\right)}_q{c}_u\right|\right\}.\hfill \end{array}$$

Then,

(i) 

${\left[{𝓁}_p\left(\mathcal{P}\left(q\right)\right)\right]}^{\beta }=\left\{\begin{array}{ccc}{\mu }_1\cap {\mu }_2& ,& 0<p\le 1,\\ {\mu }_1\cap {\mu }_3& ,& 1<p<\infty .\end{array}\right.$

(ii) 

${\left[{𝓁}_{\infty }\mathcal{P}\left(q\right)\right]}^{\beta }={\lambda }_3\cap {\mu }_4$.

Proof. 

Consider the triangle $E={\left(e_{n,v}\right)}_{n,v\in {\mathbb{N}}_0}$ defined by

$$\begin{array}{c}\hfill e_{n,v}=\left\{\begin{array}{ccc}\sum _{u=v}^n{(-1)}^{n-v}{q}^{\left(\genfrac{}{}{0pt}{}{u-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{u}{v}\right)}_q{c}_u& ,& 0\le v\le n,\\ 0& ,& v>n.\end{array}\right.\end{array}$$

Then, the following relation holds true:

$$\begin{array}{ccc}\hfill \sum _{v=0}^n{c}_v{y}_v& =& \sum _{v=0}^n\left[\sum _{u=0}^v{(-1)}^{v-u}{q}^{\left(\genfrac{}{}{0pt}{}{v-u}{2}\right)}{\left(\genfrac{}{}{0pt}{}{v}{u}\right)}_q{z}_u\right]c_v\hfill \\ & =& \sum _{v=0}^n\left[\sum _{u=v}^n{(-1)}^{u-v}{q}^{\left(\genfrac{}{}{0pt}{}{u-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{u}{v}\right)}_q{c}_u\right]z_v\hfill \\ & =& {\left(Ez\right)}_n\hfill \end{array}$$

for each $n\in {\mathbb{N}}_0.$ This indicates that $cy=\left(c_n{y}_n\right)\in cs$ whenever $y=\left(y_n\right)\in {𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ iff $Ez\in c$ whenever $z=\left(z_n\right)\in {𝓁}_p$, which means that $c=\left(c_n\right)\in {\left[{𝓁}_p\left(\mathcal{P}\left(q\right)\right)\right]}^{\beta }$ iff $E\in ({𝓁}_p,c).$ Thus, by using Lemma 1/(5), one obtains that

$${\left[{𝓁}_p\left(\mathcal{P}\left(q\right)\right)\right]}^{\beta }=\left\{\begin{array}{ccc}{\mu }_1\cap {\mu }_2& ,& 0<p\le 1,\\ {\mu }_1\cap {\mu }_3& ,& 1<p<\infty .\end{array}\right.$$

The $\beta$-dual of ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$ is obtained in a similar manner by utilizing Lemma 1/(1) in place of Lemma 1/(5) in the above proof. □

By proceeding in the way similar to the above given proof of the Beta-dual, we observe that the sequence $cy=\left(c_n{y}_n\right)\in bs$ whenever $y=\left(y_n\right)\in {𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ or ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$ iff $Ez\in {𝓁}_{\infty }$ whenever $z=\left(z_n\right)\in {𝓁}_p$ or ${𝓁}_{\infty }$, which means that $c=\left(c_n\right)\in {\left[{𝓁}_p\left(\mathcal{P}\left(q\right)\right)\right]}^{\beta }$ or ${\left[{𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)\right]}^{\beta }$ iff $E\in ({𝓁}_p,{𝓁}_{\infty })$ or $({𝓁}_{\infty },{𝓁}_{\infty }).$ As a consequence of this fact and together with Lemma 1/(4) and Lemma 1/(1), the $\gamma$-dual of the spaces ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$ may be given as follows:

Theorem 4. 

The given results hold true:

(i) 

${\left[{𝓁}_p\left(\mathcal{P}\left(q\right)\right)\right]}^{\gamma }=\left\{\begin{array}{ccc}{\mu }_2& ,& 0<p\le 1,\\ {\mu }_3& ,& 1<p<\infty .\end{array}\right.$

(ii) 

${\left[{𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)\right]}^{\gamma }={\mu }_3$ with $p^{\prime }=1$.

4. Matrix Transformation on the Space ${𝓁}_{\mathit{p}}\left(\mathcal{P}\left(\mathit{q}\right)\right)$

In this section, certain classes of infinite matrices $({𝓁}_p\left(\mathcal{P}\left(q\right)\right),{𝓁}_{\infty }),$ $({𝓁}_p\left(\mathcal{P}\left(q\right)\right),c)$ and $({𝓁}_p\left(\mathcal{P}\left(q\right)\right),c_0)$ are characterized. As a consequence, we also give a characterization of the other classes of infinite matrices by means of a theorem.

For all $n,v\in {\mathbb{N}}_0$, define the matrix $\tilde{G}={\tilde{g}}_{n,v}$ by

$${\tilde{g}}_{n,v}=\sum _{u=v}^{\infty }{(-1)}^{u-v}{q}^{\left(\genfrac{}{}{0pt}{}{u-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{u}{v}\right)}_q{g}_{nu}.$$

Let $G={\left(g_{n,v}\right)}_{n,v\in {\mathbb{N}}_0}$ be an infinite matrix over the field of complex numbers.

Theorem 5. 

Let $1<p<\infty .$ Then, $G=\left(g_{n,v}\right)\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),{𝓁}_{\infty })$ iff both of the given conditions hold true:

$$\begin{array}{ccc}& & G_n={\left(g_{n,v}\right)}_{v\in {\mathbb{N}}_0}\in {\mu }_3,\mathit{for}\mathit{each}n\in {\mathbb{N}}_0;\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & \underset{n\in {\mathbb{N}}_0}{sup}\sum _{v=0}^{\infty }{\left|{\tilde{g}}_{n,v}\right|}^{p^{\prime }}<\infty .\hfill \end{array}$$

(12)

Proof. 

Let $1<p<\infty$ and $G\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),{𝓁}_{\infty }).$ Then, for all $y\in {𝓁}_p\left(\mathcal{P}\left(q\right)\right)$, $\mathcal{P}\left(q\right)y$ exists and is contained in the space ${𝓁}_{\infty }.$ This implies that $G_n\in {\left\{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)\right\}}^{\beta }$, $(n\in {\mathbb{N}}_0),$ which proves the necessity of the relation (11).

Now, consider the following equality for all $n\in {\mathbb{N}}_0$, obtained by using the relation (2)

$$\begin{array}{ccc}\hfill \sum _{v=0}^{\infty }{g}_{n,v}{y}_v& =& \sum _{v=0}^{\infty }{g}_{n,v}\left\{\sum _{u=0}^v{(-1)}^{v-u}{q}^{\left(\genfrac{}{}{0pt}{}{v-u}{2}\right)}{\left(\genfrac{}{}{0pt}{}{v}{u}\right)}_q{z}_u\right\}\hfill \\ & =& \sum _{v=0}^{\infty }{g}_{n,v}\left\{\sum _{u=v}^{\infty }{(-1)}^{u-v}{q}^{\left(\genfrac{}{}{0pt}{}{u-v}{2}\right)}{\left(\genfrac{}{}{0pt}{}{u}{v}\right)}_q{g}_{n,u}\right\}{z}_v\hfill \\ & =& \sum _{v=0}^{\infty }{\tilde{g}}_{n,v}{z}_v.\hfill \end{array}$$

(13)

We recall that the matrix mappings between $BK$-spaces are continuous and the fact that the spaces ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ and ${𝓁}_{\infty }$ are $BK$-spaces. It follows that there exists some $M>0$ such that

$${∥Gy∥}_{{𝓁}_{\infty }}\le M{∥y∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}\forall y\in {𝓁}_p\left(\mathcal{P}\left(q\right)\right).$$

Thus, by using Hölder’s inequality together with the relation (13), we obtain that

$$\begin{array}{ccc}\hfill \frac{{∥Gy∥}_{{𝓁}_{\infty }}}{{∥z∥}_{{𝓁}_p}}& =& \underset{n\in {\mathbb{N}}_0}{sup}\frac{\left|{\sum }_{v=0}^{\infty }{\tilde{g}}_{n,v}{z}_v\right|}{{\left|z\right|}_{{𝓁}_p}}\hfill \\ & \le & \underset{n\in {\mathbb{N}}_0}{sup}\frac{{\left({\sum }_{v=0}^{\infty }{\left|{\tilde{g}}_{n,v}\right|}^{p^{\prime }}\right)}^{1/p^{\prime }}{\left({\sum }_{v=0}^{\infty }{\left|z_v\right|}^p\right)}^{1/p^{\prime }}}{{∥z∥}_{{𝓁}_p}}\hfill \\ & =& \underset{n\in {\mathbb{N}}_0}{sup}{\left(\sum _{v=0}^{\infty }{\left|{\tilde{g}}_{n,v}\right|}^{p^{\prime }}\right)}^{1/p^{\prime }}<\infty \hfill \end{array}$$

as desired.

Conversely, assume that the given conditions hold true and choose any $y=\left(y_v\right)\in {𝓁}_p\left(\mathcal{P}\left(q\right)\right)$. The condition (11) means that the sequence $G_n\in {\left\{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)\right\}}^{\beta }$, which implies that $Gy$ exists. We recall that the space ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ is linearly isomorphic to the space ${𝓁}_p.$ In view of this fact, we may write that the sequence $z=\left(z_n\right)\in {𝓁}_p.$ We again apply Hölder’s inequality to the relation (13) and obtain that

$${∥Gy∥}_{{𝓁}_{\infty }}=\underset{n\in {\mathbb{N}}_0}{sup}\left|\sum _{v=0}^{\infty }{\tilde{g}}_{n,v}{z}_v\right|\le \underset{n\in {\mathbb{N}}_0}{sup}{\left({\left|\sum _{v=0}^{\infty }{\tilde{g}}_{n,v}\right|}^{p^{\prime }}\right)}^{1/p^{\prime }}{\left(\sum _{v=0}^{\infty }{\left|z_v\right|}^p\right)}^{1/p}<\infty .$$

This proves that $Gy\in {𝓁}_{\infty }$ for all $y\in {𝓁}_p\left(\mathcal{P}\left(q\right)\right)$. That is, $G\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),{𝓁}_{\infty }).$ □

Theorem 6. 

Let $1<p<\infty .$ Then, $G\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),c)$ iff (11) and (12) hold true and

$$\exists {\alpha }_v\in \mathbb{C}\ni \underset{n\to \infty }{lim}{\tilde{g}}_{n,v}={\alpha }_v,(v\in {\mathbb{N}}_0).$$

(14)

Proof. 

Let $G\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),c)$. Then, for all $y\in {𝓁}_p\left(\mathcal{P}\left(q\right)\right)$, the sequence $Gy$ exists and belongs to the space $c.$ Since $c\subset {𝓁}_{\infty },$ the necessity of the conditions (11) and (12) follows immediately from Theorem 5.

Let $e^{\left(v\right)}$ be a sequence with 1 in the $v^{\mathrm{th}}$ position and 0 elsewhere. Then, for $y=\mathcal{Q}\left(q\right)e^{\left(v\right)}$, we obtain by using the relation (13) that

$$Gy=G\left(\mathcal{Q}\left(q\right)e^{\left(v\right)}\right)=\tilde{G}\left(\mathcal{P}\left(q\right)\left(\mathcal{Q}\left(q\right)e^{\left(v\right)}\right)\right)=\tilde{G}{e}^{\left(v\right)}={\left({\tilde{g}}_{n,v}\right)}_{n\in {\mathbb{N}}_0}$$

for each fixed $v\in {\mathbb{N}}_0.$ This shows the necessity of the condition (14).

Conversely, assume that the conditions (11), (12) and (14) hold true, and choose any $y=\left(y_n\right)\in {𝓁}_p\left(\mathcal{P}\left(q\right)\right).$ Then, $G_n={\left(g_{n,v}\right)}_{v\in {\mathbb{N}}_0}\in {\left\{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)\right\}}^{\beta }$ for each $n\in {\mathbb{N}}_0,$ and so $Gy$ exists. Therefore, we again obtain the relation (13). Thus, the conditions (4) and (7) are satisfied by the matrix $\tilde{G},$ that is to say that $\tilde{G}z\in c.$ Thus, by using (13), we conclude that $Gy\in c$ as desired. □

Theorem 7. 

Let $1<p<\infty .$ Then, $G\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),c_0)$ iff (11) and (12) hold true, and (14) also holds true with ${\alpha }_v=0$ for all $v\in {\mathbb{N}}_0.$

Proof. 

We leave the proof as it follows immediately from Theorem 6. □

Theorem 8. 

Let $X\subset \omega$, $\mu \in \left\{{𝓁}_{\infty },c,c_0\right\}$ and G be any infinite matrix over the complex field $\mathbb{C}.$ Let R be the product matrix of infinite matrices $\mathcal{P}\left(q\right)$ and G, i.e., $R=\mathcal{P}\left(q\right)G=\left(r_{nv}\right)$ is given by

$$r_{nv}=\sum _{u=0}^n{\left(\genfrac{}{}{0pt}{}{n}{u}\right)}_q{g}_{uv}$$

(15)

for all $n,v\in {\mathbb{N}}_0.$ Then, $G\in (X,Y_{\mathcal{P}\left(q\right)})$ iff $R\in (X,Y).$

Proof. 

Choose any sequence $y=\left(y_n\right)\in X.$ Then, it immediately follows from the relation (15) that

$$\mathcal{P}\left(q\right)\left(Gy\right)=\left(\mathcal{P}\right(q\left)G\right)y=Ry.$$

(16)

As a result, one has $Gy\in Y_{\mathcal{P}\left(q\right)}$ whenever $y\in X$ iff $Ry\in Y$ whenever $y\in X$, which means that $G\in (X,Y_{\mathcal{P}\left(q\right)})$ iff $R\in (X,Y).$ □

The following corollaries are some of the many consequences of Theorem 8.

Corollary 2. 

Let $G=\left(g_{n,v}\right)$ be an infinite matrix over the complex field $\mathbb{C}$, $R=\left(r_{n,v}\right)$ is defined as in (15), and the spaces $c_0\left(\mathcal{P}\left(q\right)\right)$ and $c\left(\mathcal{P}\right(q\left)\right)$ are defined as in the work of Yaying et al. [19]. Then, each of the following assertions holds true:

1. 

$G=\left(g_{n,v}\right)\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),{𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right))$ iff each of the conditions (11) and (12) holds true with $r_{n,v}$ in place of $g_{n,v}$ for all $n,v\in {\mathbb{N}}_0.$

2. 

$G=\left(g_{n,v}\right)\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),c\left(\mathcal{P}\left(q\right)\right))$ iff each of the conditions (11), (12), and (14) holds true with $r_{n,v}$ in place of $g_{n,v}$ for all $n,v\in {\mathbb{N}}_0.$

3. 

$G=\left(g_{n,v}\right)\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),c_0\left(\mathcal{P}\left(q\right)\right))$ iff each of the conditions (11) and (12) holds true, and the condition (14) also holds true with ${\alpha }_v=0$ for all $v\in {\mathbb{N}}_0$ and $r_{n,v}$ in place of $g_{n,v}$ for all $n,v\in {\mathbb{N}}_0.$

Corollary 3. 

Let $G=\left(g_{n,v}\right)$ be an infinite matrix over the complex field $\mathbb{C}$, $g(n,v)={\sum }_{u=0}^n{g}_{u,v}$ for all $n,v\in {\mathbb{N}}_0$, and $c{s}_0$ is the space consisting of series converging to zero. Then, each of the following assertions holds true:

1. 

$G=\left(g_{n,v}\right)\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),bs)$ iff each of the conditions (11) and (12) holds true with $g(n,v)$ in place of $g_{n,v}$ for all $n,v\in {\mathbb{N}}_0.$

2. 

$G=\left(g_{n,v}\right)\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),cs)$ iff each of the conditions (11), (12) and (14) holds true with $g(n,v)$ in place of $g_{n,v}$ for all $n,v\in {\mathbb{N}}_0.$

3. 

$G=\left(g_{n,v}\right)\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),c{s}_0)$ iff each of the conditions (11) and (12) holds true, and the condition (14) also holds true with ${\alpha }_v=0$ for all $v\in {\mathbb{N}}_0$ and $g(n,v)$ in place of $g_{n,v}$ for all $n,v\in {\mathbb{N}}_0.$

Consider an increasing sequence $\lambda =\left({\lambda }_n\right)$ of positive real numbers with ${\lambda }_n\to \infty$ as $n\to \infty .$ Then, the matrix $\mathrm{\Lambda }=\left({\lambda }_{n,v}\right)$ is defined by (see [25])

$$\begin{array}{c}\hfill {\lambda }_{n,v}=\left\{\begin{array}{ccc}\frac{{\lambda }_v-{\lambda }_{v-1}}{{\lambda }_n}& ,& 0\le v\le n,\\ 0& ,& v>n,\end{array}\right.\end{array}$$

for $n,v\in {\mathbb{N}}_0.$ Now, in view of Theorem 8, define the matrix $\tilde{\mathrm{\Lambda }}=\left({\tilde{\lambda }}_{n,v}\right)$ by

$${\tilde{\lambda }}_{n,v}=\sum _{u=0}^n\frac{{\lambda }_u-{\lambda }_{u-1}}{{\lambda }_n}{g}_{u,v}$$

for all $n,v\in {\mathbb{N}}_0.$

Corollary 4. 

Let $G=\left(g_{n,v}\right)$ be an infinite matrix over the complex field $\mathbb{C}$ and the spaces ${𝓁}_{\infty }^{\lambda },$ $c^{\lambda }$ and $c_0^{\lambda }$ are defined as in [25]. Then, each of the following assertions holds true:

1. 

$G=\left(g_{n,v}\right)\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),{𝓁}_{\infty }^{\lambda })$ iff each of the conditions (11) and (12) holds true with ${\tilde{\lambda }}_{n,v}$ in place of $g_{n,v}$ for all $n,v\in {\mathbb{N}}_0.$

2. 

$G=\left(g_{n,v}\right)\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),c^{\lambda })$ iff each of the conditions (11), (12) and (14) holds true with ${\tilde{\lambda }}_{n,v}$ in place of $g_{n,v}$ for all $n,v\in {\mathbb{N}}_0.$

3. 

$G=\left(g_{n,v}\right)\in ({𝓁}_p\left(\mathcal{P}\left(q\right)\right),c_0^{\lambda })$ iff each of the conditions (11) and (12) holds true, and the condition (14) also holds true with ${\alpha }_v=0$ for all $v\in {\mathbb{N}}_0$ and ${\tilde{\lambda }}_{n,v}$ in place of $g_{n,v}$ for all $n,v\in {\mathbb{N}}_0.$

5. Some Geometric Properties of the Spaces ${𝓁}_{\mathit{p}}\left(\mathcal{P}\left(\mathit{q}\right)\right)$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(\mathit{q}\right)\right)$

In this section, we examine some geometric properties like the approximation property, Dunford–Pettis property, Hahn–Banach extension property and Banach–Saks-type p property of the spaces ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ $(1\le p<\infty )$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$.

Let X and Y be two Banach spaces. Then, a linear operator $\mathcal{L}:X\to Y$ is compact if whenever Q is a bounded subset of $X,$ $L\left(Q\right)$ is a relatively compact subset of Y [26] (Definition 3.4.1). Throughout, we use the following notation:

$$\begin{array}{cc}\hfill & C(X,Y):=\left\{\mathcal{L}:X\to Y\mathrm{such}\mathrm{that}\mathcal{L}\mathrm{is}\mathrm{a}\mathrm{compact}\mathrm{operator}\right\};\hfill \\ \hfill & B(X,Y):=\left\{\mathcal{L}:X\to Y\mathrm{such}\mathrm{that}\mathcal{L}\mathrm{is}\mathrm{a}\mathrm{bounded}\mathrm{linear}\mathrm{operator}\right\}.\hfill \end{array}$$

Apparently, $C(X,Y)$ is a closed subspace of $B(X,Y),$ for any two Banach spaces X and Y.

We recall that an operator between any two Banach spaces is said to have a finite rank if the range of the operator is finitely dimensional. Moreover, the range of a compact linear operator between any two Banach spaces is closed iff the operator has a finite rank [26] (Proposition 3.4.6).

Definition 6 

([26] (Definition 3.4.26)). A Banach space X is said to have approximation property if the set of finite rank members of $B(Y,X)$ is dense in $C(Y,X)$ for any Banach space Y.

Theorem 9 

([26] (Theorem 3.4.27)). The space ${𝓁}_p$ $(1\le p<\infty )$ has the approximation property.

Definition 7 

([26] (Definition 3.5.1)). Let X and Y be any two Banach spaces. Then, a linear operator $\mathcal{L}:X\to Y$ is said to be weakly compact if whenever Q is a bounded subset of $X,$ $\mathcal{L}\left(Q\right)$ is a relatively weakly compact subset of Y.

Definition 8 

([26] (Definition 3.5.15)). Let X and Y be any two Banach spaces. Then, a linear operator $\mathcal{L}:X\to Y$ is said to be completely continuous (or Dunford–Pettis operator) if $\mathcal{L}\left(Q\right)$ is a compact subset of Y whenever Q is a weakly compact subset of X. In this case, X is said to have a Dunford–Pettis property (in short, D-P property) if each weakly compact linear operator from X to Y is completely continuous.

Lemma 2 

([26] (Exercise 3.50, p. 339)). Let $\mathcal{L}$ be a continuous linear operator from a normed space X to a normed space $Y.$ Then, $\mathcal{L}\left(Q\right)$ is weakly compact in Y whenever Q is weakly compact in $X.$

Theorem 10 

([26] (Art 1.9.6, p. 75) (Hahn–Banach extension theorem)). Let V be any subspace of a normed linear space X and ${\mathcal{L}}_0$ be a bounded linear functional on V. Then, $L_0$ may be extended to a bounded linear functional $\mathcal{L}$ on X such that $∥{\mathcal{L}}_0∥=∥\mathcal{L}∥$.

Theorem 11 

([27]). Let V be a linear subspace of a Banach space X and ${\mathcal{L}}_0:V\to {𝓁}_{\infty }$ be a bounded linear operator. Then, ${\mathcal{L}}_0$ may be extended to a bounded linear operator $\mathcal{L}:X\to {𝓁}_{\infty }$ such that $∥{\mathcal{L}}_0∥=∥\mathcal{L}∥.$

In this case, the space ${𝓁}_{\infty }$ is said to possess the Hahn–Banach extension property.

Definition 9 

([28]). A Banach space X is said to possess a weak Banach–Saks property if every weakly null sequence $\left(y_n\right)\in X$ contains a subsequence $\left(y_{n_v}\right)$ whose Cesàro transformation is norm-convergent to zero, that is,

$$\underset{n\to \infty }{lim}∥\frac{1}{n+1}\sum _{v=0}^n{y}_{n_v}∥=0.$$

Additionally, X is said to possess the Banach–Saks property if every bounded sequence in X contains a subsequence whose Cesàro transformation is norm-convergent in X.

Definition 10 

([29]). A Banach space X is said to possess Banach–Saks-type p if every weakly null sequence $\left(y_n\right)$ has a subsequence $\left(y_{n_v}\right)$ such that, for some $C>0,$

$$∥\sum _{v=0}^n{y}_{n_v}∥\le C{(n+1)}^{1/p},$$

for all $n\in {\mathbb{N}}_0.$.

Now we turn to the main results of this section:

Theorem 12. 

For $1\le p<\infty ,$ the space ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ has the approximation property.

Proof. 

Let $\mathcal{L}:X\to {𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ be a compact linear operator for any Banach space $X.$ This means that for each bounded sequence $y=\left(y_n\right)\in X,$ the sequence $\left(\mathcal{L}{y}_n\right)$ has a convergent sub-sequence $\left(\mathcal{L}{y}_{n_v}\right)$ in ${𝓁}_p\left(\mathcal{P}\left(q\right)\right).$ That is,

$${∥\mathcal{L}{y}_{n_u}-\mathcal{L}{y}_{n_v}∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}^p={∥\mathcal{L}\left(y_{n_u}-y_{n_v}\right)∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}^p={∥\left(\mathcal{P}\left(q\right)\mathcal{L}\right)\left(y_{n_u}-y_{n_v}\right)∥}_{{𝓁}_p}^p\to 0$$

as $u,v\to \infty .$ Thus, the operator $\mathcal{P}\left(q\right)\mathcal{L}:X\to {𝓁}_p$ is well defined and compact. Now, we turn our attention to the space ${𝓁}_p$ that has the approximation property. It follows that there exists a sequence $\left({\mathcal{T}}_n\right)$ of finite rank bounded linear operators from X to ${𝓁}_p$ such that

$$∥\mathcal{P}\left(q\right)\mathcal{L}-{\mathcal{T}}_n∥\to 0$$

as $n\to \infty .$ As a consequence of this fact, we realize that the sequence $\left({\mathcal{P}\left(q\right)}^{-1}{\mathcal{T}}_n\right)$ of bounded linear operators from X to ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ is the required sequence of a finite rank. Additionally,

$$\begin{array}{ccc}\hfill ∥\mathcal{L}-{\mathcal{P}\left(q\right)}^{-1}{\mathcal{T}}_n∥& =& \underset{∥y∥=1}{sup}{∥\left(\mathcal{L}-{\mathcal{P}\left(q\right)}^{-1}{\mathcal{T}}_n\right)y∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}^p\hfill \\ & =& \underset{∥y∥=1}{sup}{∥\mathcal{L}y-\left({\mathcal{P}\left(q\right)}^{-1}{\mathcal{T}}_n\right)y∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}^p\hfill \\ & =& \underset{∥y∥=1}{sup}{∥\mathcal{P}\left(q\right)\mathcal{L}y-{\mathcal{T}}_{n}y∥}_{{𝓁}_p}^p\hfill \\ & =& \underset{∥y∥=1}{sup}{∥\left(\mathcal{P}\left(q\right)\mathcal{L}-{\mathcal{T}}_n\right)y∥}_{{𝓁}_p}^p\hfill \\ & \to & 0\mathrm{as}n\to \infty .\hfill \end{array}$$

This completes the proof. □

Theorem 13. 

The space ${𝓁}_1\left(\mathcal{P}\left(q\right)\right)$ has the D-P property.

Proof. 

Let $\mathcal{L}$ be a weakly compact operator from the Banach space ${𝓁}_1\left(\mathcal{P}\left(q\right)\right)$ to a space $X.$ Then, $\mathcal{L}{\left\{\mathcal{P}\left(q\right)\right\}}^{-1}$ is a bounded linear operator from ${𝓁}_1$ to $X.$ We intend to show that $\mathcal{L}$ is completely continuous.

Let Q be a bounded set in ${𝓁}_1.$ Then, it follows that ${\left\{\mathcal{P}\left(q\right)\right\}}^{-1}Q$ is a bounded set in ${𝓁}_1\left(\mathcal{P}\left(q\right)\right)$. Since $\mathcal{L}$ is weakly compact, it follows that the set

$$\mathcal{L}\left({\left\{\mathcal{P}\left(q\right)\right\}}^{-1}Q\right)=\left(\mathcal{L}{\left\{\mathcal{P}\left(q\right)\right\}}^{-1}\right)Q$$

is relatively weakly compact in $X.$ Therefore, we observe that $\mathcal{L}{\left\{\mathcal{P}\left(q\right)\right\}}^{-1}$ is a weakly compact operator from ${𝓁}_1$ to X. Now, since the space ${𝓁}_1$ has the D-P property, it follows that the operator $\mathcal{L}{\left\{\mathcal{P}\left(q\right)\right\}}^{-1}$ is completely continuous. Suppose that C is a weakly compact subset of ${𝓁}_1\left(\mathcal{P}\left(q\right)\right)$. Then, it follows from Lemma 2 that $\mathcal{P}\left(q\right)C$ is a weakly compact subset of ${𝓁}_1.$ Since $\mathcal{L}{\left\{\mathcal{P}\left(q\right)\right\}}^{-1}$ is completely continuous, therefore $\mathcal{L}{\left\{\mathcal{P}\left(q\right)\right\}}^{-1}\left(\mathcal{P}\left(q\right)C\right)=\mathcal{L}\left(C\right)$ is a compact set in $Y.$ This concludes that $\mathcal{L}$ is completely continuous as desired. □

In the next result, we show that the space ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)$ has the Hahn–Banach extension property.

Theorem 14. 

Let V be a linear subspace of a Banach space X and ${\mathcal{L}}_0\in B(V,{𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)).$ Then, ${\mathcal{L}}_0$ may be extended to a bounded linear operator $\mathcal{L}\in B(X,{𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right))$ such that $∥{\mathcal{L}}_0∥=∥\mathcal{L}∥.$

Proof. 

Let ${\mathcal{L}}_0\in B(V,{𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)).$ Then, $\mathcal{P}\left(q\right){\mathcal{L}}_0\in B(V,{𝓁}_{\infty }).$ Now, since the space ${𝓁}_{\infty }$ has the Hahn–Banach extension property by Theorem 11, it follows that the operator $\mathcal{P}\left(q\right){\mathcal{L}}_0$ may be extended to the operator $\mathcal{T}\in B(X,{𝓁}_{\infty })$ such that $∥\mathcal{P}\left(q\right){\mathcal{L}}_0∥=∥\mathcal{T}∥.$ Choose the operator $\mathcal{L}={\left\{\mathcal{P}\left(q\right)\right\}}^{-1}\mathcal{T}.$ Then, it is clear that $\mathcal{L}\in B(X,{𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right)).$ Further, for any $y\in V,$ we observe that

$$\mathcal{L}y=\left({\left\{\mathcal{P}\left(q\right)\right\}}^{-1}\mathcal{T}\right)y={\left\{\mathcal{P}\left(q\right)\right\}}^{-1}\left(\mathcal{T}y\right)={\left\{\mathcal{P}\left(q\right)\right\}}^{-1}\left(\left(\mathcal{P}\left(q\right){\mathcal{L}}_0\right)y\right)={\mathcal{L}}_{0}y.$$

Additionally

$$∥\mathcal{L}∥=∥{\left\{\mathcal{P}\left(q\right)\right\}}^{-1}\mathcal{T}∥=∥{\left\{\mathcal{P}\left(q\right)\right\}}^{-1}\left(\mathcal{P}\left(q\right){\mathcal{L}}_0\right)∥=∥{\mathcal{L}}_0∥,$$

as desired. □

Theorem 15. 

The space ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ possesses the Banach–Saks-type p property.

Proof. 

Let $\left({ϵ}_n\right)$ be a sequence of positive reals with ${\displaystyle \sum _{n=0}^{\infty }}{ϵ}_n\le \frac{1}{2}.$ Let $\left(y_n\right)$ be any weakly null sequence in $B(𝓁(\mathcal{P}\left(q\right)\left)\right).$ Choose $u_0=y_0=0$ and $u_1=y_{n_1}=y_1.$ Then, there exists $c_1\in {\mathbb{N}}_0$ such that

$${∥\sum _{k=c_1+1}^{\infty }{u}_1\left(k\right)e^{\left(k\right)}∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}<{ϵ}_1.$$

As per the hypothesis, $\left(y_n\right)$ is a weakly null sequence. This means that $y_n\to 0$ coordinate wise. This implies the existence of an $n_2\in {\mathbb{N}}_0$ such that

$${∥\sum _{k=0}^{c_1}{y}_n\left(k\right)e^{\left(k\right)}∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}<{ϵ}_1,$$

when $n\ge n_2.$ Again, choose $u_2=y_{n_2}.$ Then, there exists $c_2>c_1$ such that

$${∥\sum _{k=c_2+1}^{\infty }{u}_2\left(k\right)e^{\left(k\right)}∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}<{ϵ}_2.$$

Since $y_n\to 0$ coordinate wise, it follows that there exists $n_3>n_2$ such that

$${∥\sum _{k=0}^{c_2}{y}_n\left(k\right)e^{\left(k\right)}∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}<{ϵ}_2,$$

when $n\ge n_3.$

Proceeding in a similar way to the above technique will yield two increasing sequences $\left(n_k\right)$ and $\left(c_k\right)$ such that

$${∥\sum _{k=0}^{c_m}{y}_n\left(k\right)e^{\left(k\right)}∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}<{ϵ}_m,$$

for all $n\ge n_{k+1}$ and

$${∥\sum _{k=c_m+1}^{\infty }{u}_m\left(k\right)e^{\left(k\right)}∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}<{ϵ}_m.$$

where $u_m=y_{n_m}.$ Thus

$$\begin{array}{cc}\hfill {∥\sum _{m=0}^n{u}_m∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}& ={∥\sum _{m=0}^n\left(\sum _{k=0}^{c_{m-1}}{u}_m\left(k\right)e^{\left(k\right)}+\sum _{k=c_{m-1}+1}^{c_m}{u}_m\left(k\right)e^{\left(k\right)}+\sum _{k=c_m}^{\infty }{u}_m\left(k\right)e^{\left(k\right)}\right)∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}\hfill \\ \hfill & \le {∥\sum _{m=0}^n\left(\sum _{k=c_{m-1}+1}^{c_m}{u}_m\left(k\right)e^{\left(k\right)}\right)∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}+2\sum _{m=0}^n{ϵ}_m.\hfill \end{array}$$

Now, since $y_n\in B\left({𝓁}_p\left(\mathcal{P}\left(q\right)\right)\right)$ and ${∥y∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}={\displaystyle \sum _{n=0}^{\infty }}\left|{\sum }_{k=0}^n{\tilde{p}}_{n,k}{y}_k\right|,$ it follows that ${∥y∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}\le 1.$ Therefore, we have

$$\begin{array}{cc}\hfill {∥\sum _{m=0}^n\left(\sum _{k=c_{m-1}+1}^{c_m}{u}_m\left(k\right)e^{\left(k\right)}\right)∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}^p& =\sum _{m=0}^n\sum _{k=c_{m-1}+1}^{c_m}{\left|\sum _{v=0}^k{\tilde{p}}_{k,v}{u}_m\left(v\right)\right|}^p\hfill \\ \hfill & \le \sum _{m=0}^n\sum _{k=0}^{\infty }{\left|\sum _{v=0}^k{\tilde{p}}_{k,v}{u}_m\left(v\right)\right|}^p\le n+1.\hfill \end{array}$$

Now, using the fact that $1\le {(n+1)}^{1/p}$ for all $n\in {\mathbb{N}}_0$ and $1\le p<\infty ,$ we obtain

$$\begin{array}{c}\hfill {∥\sum _{m=0}^n{u}_m∥}_{{𝓁}_p\left(\mathcal{P}\left(q\right)\right)}\le {(n+1)}^{1/p}+1\le 2{(n+1)}^{1/p}.\end{array}$$

Thus, we conclude that ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ is of the Banach–Saks-type $p.$ □

6. Conclusions

In this article, we progressed the exploration pertaining to the advancement of “sequence spaces via specific q-matrices”. This advancement was accomplished through the utilization of the q-Pascal matrix $\mathcal{P}\left(q\right)$. Additionally, we introduced and conducted an analysis on the sequence spaces ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right).$ Furthermore, we examined a set of outcomes related to the Schauder basis, the $\alpha$-, $\beta$-, and $\gamma$-duals, along with matrix transformations related to these spaces. Within the concluding section, a comprehensive investigation into various geometric properties, namely the approximation property, Dunford–Pettis property, Hahn–Banach extension property, and the Banach–Saks-type p property, was undertaken for the spaces ${𝓁}_p\left(\mathcal{P}\left(q\right)\right)$ and ${𝓁}_{\infty }\left(\mathcal{P}\left(q\right)\right).$

As a prospect for future research, the exploration of the domain $X_{\mathcal{P}\left(q\right)}$ of the q-Pascal matrix $\mathcal{P}\left(q\right)$ in Maddox’s space $X\in \{𝓁\left(p\right),c_0\left(p\right),c\left(p\right),{𝓁}_{\infty }\left(p\right)\}$ holds promise.