# Mappings Ideal of Type the Domain of Generalizedp,q-Euler Matrix in c0 and c

Authors: Taja Yaying; Bipan Hazarika; S. A. Mohiuddine; OM Kalthum S. K. Mohamed; Awad A. Bakery
Journal: Journal of Mathematics
 (2023)

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## Abstract

We develop new Banach sequence spacese0a,bp,q and eca,bp,q derived by the domain of generalized p,q-Euler matrix Ea,bp,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 α, β, and γ-duals of the spaces e0a,bp,q and eca,bp,q. Certain classes of matrix transformations are characterized from e0a,bp,q and eca,bp,q to Z∈ℓ∞,c,c0,ℓ1,ℓk. We obtain essential conditions of compactness of operators from e0a,bp,q and eca,bp,q to Z∈ℓ∞,c,c0,ℓ1,bs,cs,cs0. Finally, under a definite functional ϱ and a weighted sequence of positive reals δ, we define a new sequence space e0a,bp,q,δϱ. 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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## Body

## 1. Introduction and Preliminaries

Throughout this paper, the set of all real-valued sequences is symbolised bys. A sequence space is a linear subspace of s. Few examples of classical sequence spaces which are used frequently in this study are(1)c=z=zr∈s:limr⟶∞zrexists,c0=z=zr∈s:limr⟶∞zr=0,ℓ∞=z=zr∈s:supr∈ℕ0zr<∞,ℓk=z=zr∈s:∑r=0∞zrk<∞,1≤k<∞,cs=z=zr∈s:limr⟶∞∑v=0rzvexists,bs=z=zr∈s:supr∈ℕ0∑v=0rzv<∞andcs0=z=zr∈s:limr⟶∞∑v=0rzv=0.Here and in what follows,ℕ0=0,1,2,…. A Banach space is a complete normed space. Furthermore, a Banach space having continuous coordinates is called a BK-space. The sequence spaces c0 and c are BK spaces endowed with supremum norm defined by zc0=zc=supr∈ℕ0zr.LetΛ=λrv be an infinite matrix consisting of real entries and Z and W be any two sequence spaces. We denote the sequence in the rth row of the matrix Λ by Λr. The notation Λz=Λzr=∑v=0∞λrvzv is called Λ-transform of z=zr, given that ∑v=0∞λrvzv converges r∈ℕ0. Furthermore, if for each z∈Z, Λz∈W, then Λ is called a matrix transformation from Z to W. We use the symbol Z,W to mean the family of matrix transformation from Z to W. Additionally, Λ=λrv is said to be a triangle if λrv=0 for v>r and λrr≠0.The domain ofΛ in Z, denoted by ZΛ, is given by(2)ZΛ=z∈Z:Λz∈Z.The theory of matrix domain is very crucial in constructing sequence spaces. It is known that the matrix domainZΛ is itself a sequence space. Moreover, if Z is a BK space and Λ is a triangle, then we enjoy the celebrated property that ZΛ is also a BK-space endowed with the norm zZΛ=ΛzZ. 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

Thep,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 inp,q-Calculus

Definition 1 (see [10]).
Thep,q-number 0<q<p≤1 is given by(3)vp,q=pv−qvp−q,v=1,2,3,…,0,v=0.It is well known that whenp=1, vp,q reduces to its q version vq (cf. [8]).Definition 2 (see [9]).
Thep,q-binomial coefficient is given by(4)rvp,q=rp,q!r−vp,q!vp,q!,r≥v,0,v>r.
Here the symbolvp,q! is the p,q-factorial of v which is expressed by(5)vp,q!=vp,qv−1p,q…2p,q1p,q.Lemma 1 (see [11]).
Thep,q-binomial formula is expressed as(6)a+bp,qr=a+bqa+pbq2a+p2b…qr−1a+pr−1b,r=1,2,3,…,1,r=0,=∑v=0rrvp,qpr−v2qv2avbr−v.Clearly whenp=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

Let0<a<1; then, the Euler mean Ea=erva is defined by(7)erva=rv1−ar−vbv,0≤v≤r,0,v>r.Let0<a<1 and a+b≠1; then, the binomial matrix Ea,b=erva,b is given by(8)erva,b=1a+brrvarbr−v,0≤v≤r,0,v>r.It is clear that whena+b=1, then Ea,b reduces to Ea. A good number of publications can be found in the literature concerning construction of sequence spaces using Euler mean Ea and binomial matrix Ea,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 ΔB=δrvB, ΔBm=δrvBm, ΔBq=δrvBq, Bm=brvm, and Bnm=brvm,n that are used in the table:(9)δrvB=−1r−v,r−1≤v≤r,0,v>r.δrvBm=−1r−vmr−v,max0,r−m≤v≤r,0,0≤v<max0,r−morv>r.brvm=mr−vam−r+vbr−v,max0,r−m≤v≤r,0,0≤v≤max0,r−morv>r,brvm,n=mr−vam−r+vbr−vnv,max0,r−m≤v≤r,0,0≤v≤max0,r−morv>r,δrvBq=−1r−vΓq+1r−v!Γq−r+v+1,max0≤v≤r,0,v>r,where m∈ℕ0, q∈ℕ, and n=nr is any fixed sequence of real numbers.Table 1
Euler spaces.
Euler spacesReferencese0a=c0Ea, eca=cEa[1]eka=ℓkEa, e∞a=ℓ∞Ea[2, 4]e0aΔB=c0EaΔB, ecaΔB=cEaΔB[13]e0aΔBm=c0EaΔBm, ecaΔBm=cEaΔBm, e∞aΔBm=ℓ∞EaΔBm[14]ekaΔBq=ℓkEaΔBq, e0aΔBq=c0EaΔBq, ecaΔBq=cEaΔBq, e∞rΔBq=ℓ∞EaΔBq[15]eak=ℓkEa[16]e0aΔB,k=c0EaΔB, ecaΔB,k=cEaΔB and e∞aΔB,k=ℓ∞EaΔB[17]e0aΔBm,k=c0kEaΔBm, ecrΔBm,k=ckEaΔBm, e∞aΔBm,k=ℓ∞kEaΔBm[18]f0Ea=f0Ea, fEa=fEa[19]e0aBm=c0EaBm, ecaBm=cEaBm, e∞aBm=ℓ∞EaBm[20]e0aBnm=c0EaBnm, ecaBnm=cEaBnm[21]eka,b=ℓkEa,b, e0a,b=c0Ea,b, eca,b=cEa,b, e∞a,b=ℓ∞Ea,b[22, 23]e0a,bBm=c0Ea,bBm, eca,bBm=cEa,bBm, e∞a,bBm=ℓ∞Ea,bBm[24]e0a,bΔBq=c0Ea,bΔBq, eca,bΔBq=cEa,bΔBq, e∞a,bΔBq=ℓ∞Ea,bΔBq[25]eka,bΔBq=ℓkEa,bΔBq[5]


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

For0<q<1, q-Cesàro matrix Cq=crvq [12] is defined by(10)crvq=qvr+1q,0≤v≤r,0,v>r.The development of sequence spaces using post-quantum (orp,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 X0q and Xcq are studied by Demiriz and Şahin [26] generated by q-Cesàro matrix in c0 and c, respectively. As a natural continuation, Yaying et al. [6] developed its domains Xkq and X∞q in ℓk and ℓ∞, respectively, and studied associated operator ideals. More recently, Yaying et al. [27] constructed p,q-Euler sequence spaces eka,bp,q=ℓkEa,bp,q and e∞a,bp,q=ℓ∞Ea,bp,q and studied certain geometric properties of these spaces. Yaying et al. [28] introduced q-Euler sequence spaces e0a,bq=c0Ea,bq and eca,bq=cEa,bq.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 c0. The main objective of this paper is to study generalized Euler spaces e0a,bp,q and eca,bp,q due to p,q-Euler matrix Ea,bp,q in c0 and c, respectively. We obtain some topological properties and inclusion nature and determine bases for the spaces e0a,bp,q and eca,bp,q. In Section 3, we determine α, β, and γ-duals of e0a,bp,q and eca,bp,q. In Section 4, certain matrix transformations are characterized from e0a,bp,q and eca,bp,q to μ∈ℓ∞,c,c0,ℓ1,cs,cs0,bs. Section 5 is devoted to the computation of essential conditions of compactness for operators on e0a,bp,q. Finally, in Sections 6 and 7, under a definite functional ϱ and a weighted sequence of positive reals δ, we define new sequence space e0a,bp,q,δϱ. We exhibit some geometric and topological natures of this space and eigenvalue distribution of mapping ideals constructed by this space and s-numbers.


## 1.1.p,q-Calculus

Thep,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 inp,q-Calculus

Definition 1 (see [10]).
Thep,q-number 0<q<p≤1 is given by(3)vp,q=pv−qvp−q,v=1,2,3,…,0,v=0.It is well known that whenp=1, vp,q reduces to its q version vq (cf. [8]).Definition 2 (see [9]).
Thep,q-binomial coefficient is given by(4)rvp,q=rp,q!r−vp,q!vp,q!,r≥v,0,v>r.
Here the symbolvp,q! is the p,q-factorial of v which is expressed by(5)vp,q!=vp,qv−1p,q…2p,q1p,q.Lemma 1 (see [11]).
Thep,q-binomial formula is expressed as(6)a+bp,qr=a+bqa+pbq2a+p2b…qr−1a+pr−1b,r=1,2,3,…,1,r=0,=∑v=0rrvp,qpr−v2qv2avbr−v.Clearly whenp=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.


## 1.1.1. Relevant Notions inp,q-Calculus

Definition 1 (see [10]).
Thep,q-number 0<q<p≤1 is given by(3)vp,q=pv−qvp−q,v=1,2,3,…,0,v=0.It is well known that whenp=1, vp,q reduces to its q version vq (cf. [8]).Definition 2 (see [9]).
Thep,q-binomial coefficient is given by(4)rvp,q=rp,q!r−vp,q!vp,q!,r≥v,0,v>r.
Here the symbolvp,q! is the p,q-factorial of v which is expressed by(5)vp,q!=vp,qv−1p,q…2p,q1p,q.Lemma 1 (see [11]).
Thep,q-binomial formula is expressed as(6)a+bp,qr=a+bqa+pbq2a+p2b…qr−1a+pr−1b,r=1,2,3,…,1,r=0,=∑v=0rrvp,qpr−v2qv2avbr−v.Clearly whenp=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

Let0<a<1; then, the Euler mean Ea=erva is defined by(7)erva=rv1−ar−vbv,0≤v≤r,0,v>r.Let0<a<1 and a+b≠1; then, the binomial matrix Ea,b=erva,b is given by(8)erva,b=1a+brrvarbr−v,0≤v≤r,0,v>r.It is clear that whena+b=1, then Ea,b reduces to Ea. A good number of publications can be found in the literature concerning construction of sequence spaces using Euler mean Ea and binomial matrix Ea,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 ΔB=δrvB, ΔBm=δrvBm, ΔBq=δrvBq, Bm=brvm, and Bnm=brvm,n that are used in the table:(9)δrvB=−1r−v,r−1≤v≤r,0,v>r.δrvBm=−1r−vmr−v,max0,r−m≤v≤r,0,0≤v<max0,r−morv>r.brvm=mr−vam−r+vbr−v,max0,r−m≤v≤r,0,0≤v≤max0,r−morv>r,brvm,n=mr−vam−r+vbr−vnv,max0,r−m≤v≤r,0,0≤v≤max0,r−morv>r,δrvBq=−1r−vΓq+1r−v!Γq−r+v+1,max0≤v≤r,0,v>r,where m∈ℕ0, q∈ℕ, and n=nr is any fixed sequence of real numbers.Table 1
Euler spaces.
Euler spacesReferencese0a=c0Ea, eca=cEa[1]eka=ℓkEa, e∞a=ℓ∞Ea[2, 4]e0aΔB=c0EaΔB, ecaΔB=cEaΔB[13]e0aΔBm=c0EaΔBm, ecaΔBm=cEaΔBm, e∞aΔBm=ℓ∞EaΔBm[14]ekaΔBq=ℓkEaΔBq, e0aΔBq=c0EaΔBq, ecaΔBq=cEaΔBq, e∞rΔBq=ℓ∞EaΔBq[15]eak=ℓkEa[16]e0aΔB,k=c0EaΔB, ecaΔB,k=cEaΔB and e∞aΔB,k=ℓ∞EaΔB[17]e0aΔBm,k=c0kEaΔBm, ecrΔBm,k=ckEaΔBm, e∞aΔBm,k=ℓ∞kEaΔBm[18]f0Ea=f0Ea, fEa=fEa[19]e0aBm=c0EaBm, ecaBm=cEaBm, e∞aBm=ℓ∞EaBm[20]e0aBnm=c0EaBnm, ecaBnm=cEaBnm[21]eka,b=ℓkEa,b, e0a,b=c0Ea,b, eca,b=cEa,b, e∞a,b=ℓ∞Ea,b[22, 23]e0a,bBm=c0Ea,bBm, eca,bBm=cEa,bBm, e∞a,bBm=ℓ∞Ea,bBm[24]e0a,bΔBq=c0Ea,bΔBq, eca,bΔBq=cEa,bΔBq, e∞a,bΔBq=ℓ∞Ea,bΔBq[25]eka,bΔBq=ℓkEa,bΔBq[5]


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

For0<q<1, q-Cesàro matrix Cq=crvq [12] is defined by(10)crvq=qvr+1q,0≤v≤r,0,v>r.The development of sequence spaces using post-quantum (orp,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 X0q and Xcq are studied by Demiriz and Şahin [26] generated by q-Cesàro matrix in c0 and c, respectively. As a natural continuation, Yaying et al. [6] developed its domains Xkq and X∞q in ℓk and ℓ∞, respectively, and studied associated operator ideals. More recently, Yaying et al. [27] constructed p,q-Euler sequence spaces eka,bp,q=ℓkEa,bp,q and e∞a,bp,q=ℓ∞Ea,bp,q and studied certain geometric properties of these spaces. Yaying et al. [28] introduced q-Euler sequence spaces e0a,bq=c0Ea,bq and eca,bq=cEa,bq.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 c0. The main objective of this paper is to study generalized Euler spaces e0a,bp,q and eca,bp,q due to p,q-Euler matrix Ea,bp,q in c0 and c, respectively. We obtain some topological properties and inclusion nature and determine bases for the spaces e0a,bp,q and eca,bp,q. In Section 3, we determine α, β, and γ-duals of e0a,bp,q and eca,bp,q. In Section 4, certain matrix transformations are characterized from e0a,bp,q and eca,bp,q to μ∈ℓ∞,c,c0,ℓ1,cs,cs0,bs. Section 5 is devoted to the computation of essential conditions of compactness for operators on e0a,bp,q. Finally, in Sections 6 and 7, under a definite functional ϱ and a weighted sequence of positive reals δ, we define new sequence space e0a,bp,q,δϱ. 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 Spacese0a,bp,q and eca,bp,q

The generalizedp,q-Euler matrix Ea,bp,q=erva,b of order a,b is defined by (cf. [27])(11)erva,b=1a+bp,qrrvp,qpr−v2qv2avbr−v,0≤v≤r,0,v>r,where 0<q<p≤1 and a and b are nonnegative real numbers. Equivalently,(12)Ea,bp,q=1000…ba+bp,qaa+bp,q00…pb2a+bp,q2p+qaba+bp,q2qa2a+bp,q20…p3b3a+bp,q3pp2+pq+q2ab2a+bp,q3qp2+pq+q2a2ba+bp,q3q3a3a+bp,q3…⋮⋮⋮⋮⋱.We notice thatlimr⟶∞erva,b=0 and ∑v=0rerva,b=1r∈ℕ0. Thus, Ea,bp,q is regular. Furthermore, we may term Ea,bp,q as the p,q analog of the binomial matrix Ea,b. The definition so provided is justifiable as one may see that the matrix Ea,bp,q reduces to binomial matrix Ea,b when p=1 and q approaches 1. It is worth mentioning that Ea,bp,q reduces to the triangle Ea,bq when p=1 with entries rvqqv2avbr−v/a+bqr. We call Ea,bq as the q-analog of the binomial matrix Ea,b. Moreover, when b=1−a, then the matrix Ea,bp,q reduces to triangle Eap,q with entries 1/a+1−ap,qrrvp,qpr−v2qv2av1−ar−v.The generalizedp,q-Euler spaces e0a,bp,q and eca,bp,q are defined by(13)e0a,bp,q=z=zv∈s:limr⟶∞1a+bp,qr∑v=0rrvp,qpr−v2qv2avbr−vzv=0,eca,bp,q=z=zv∈s:limr⟶∞1a+bp,qr∑v=0rrvp,qpr−v2qv2avbr−vzvexists.In view of (2), we noticed that e0a,bp,q and eca,bp,q can also be defined in the form(14)e0a,bp,q=c0Ea,bp,q,eca,bp,q=cEa,bp,q.We emphasize here thate0a,bp,q and eca,bp,q reduce to the below given spaces depending on the choice of p,q and a,b:(1)
Ifp=1, e0a,bp,q and eca,bp,q reduce to q-binomial spaces e0a,bq=c0Ea,bq and eca,bq=cEa,bq, respectively, due to Yaying et al. [28].(2)
Ifp=1 and q⟶1−, e0a,bp,q and eca,bp,q reduce to binomial spaces e0a,b and eca,b, respectively, due to Bişgin [22].(3)
Ifp=1 and b=1−a, e0a,bp,q and eca,bp,q reduce to q-Euler space e0aq=c0Eaq and ecaq=cEaq, respectively.(4)
Ifp=1 and q⟶1−, e0a,bp,q and eca,bp,q reduce to Euler spaces e0a and eca, respectively, due to Altay and Başar [1].(5)
Ifb=1−a, e0a,bp,q and eca,bp,q reduce to p,q-Euler spaces e0ap,q=c0Eap,q and ecap,q=cEap,q.Consider the sequencew=wv defined in terms of the sequence z=zv by(15)wr=Ea,bp,qzr=1a+bp,qr∑v=0rrvp,qpr−v2qv2avbr−vzv,r∈ℕ0.Then,w=wr is called Ea,bp,q-transform of z=zv. Due to (15), we get that(16)zr=∑v=0r−1r−vrvp,qqr−v2br−va+bp,qvarqr2wvr∈ℕ0.Theorem 1.
The spacese0a,bp,qandeca,bp,qareBKspaces because(17)ze0a,bp,q=zeca,bp,q=supr∈ℕ01a+bp,qr∑v=0rrvp,qpr−v2qv2avbr−vzv.Proof.
This is a routine work.Remark 1.
(i)
In the casep=1, e0a,bq and eca,bq are BKspaces because(18)ze0a,bq=zeca,bq=supr∈ℕ01a+bqr∑v=0rrvqqv2avbr−vzv,as obtained by Yaying et al. [28].(ii)
In the casep=1 and q⟶1, e0a,b and eca,b are BK because(19)ze0a,b=zeca,b=supr∈ℕ01a+br∑v=0rrvavbr−vzv,as deduced by Bişgin [22].Theorem 2.
The spacese0a,bp,q≅c0andeca,bp,q≅c.Proof.
Since the proof of both spaces is similar, we provide the proof of the first case only. Note that the mappingπ:e0a,bp,q⟶c0 defined by πz=Ea,bp,qz for all z∈e0a,bp,q is linear and 1−1. Let w=wr be any sequence in c0 and z=zr be as defined in (16). Then, one obtains(20)limr⟶∞1a+bp,qr∑v=0rrvp,qpr−v2qv2avbr−vzv=limr⟶∞1a+bp,qr∑v=0rrvp,qpr−v2qv2avbr−v∑l=0v−1v−lvlp,qqv−l2tv−la+bp,qlavqv2wl=limr⟶∞wr=0.
Thus,z∈e0a,bp,q and π:e0a,bp,q⟶c0 is onto. This proves that the space e0a,bp,q≅c0.Remark 2.
(i)
In the casep=1, the spacese0a,bq≅c0andeca,bq≅cas obtained by Yaying et al. [28].(ii)
In the casep=1andq⟶1−, the spacese0a,b≅c0andeca,b≅cas obtained by Bişgin [22].Theorem 3.
The following inclusions strictly hold:(a)
e0a,bp,q⊂eca,bp,q.(b)
c0⊂e0a,bp,qandc⊂eca,bp,q.Proof.
The inclusiona is obvious. Hence, we present the proof for inclusion b. We know that the matrix Ea,bp,q is regular; therefore, the inclusion c0⊂e0a,bp,q holds. Now, let us consider the sequence z=1,−1,1,−1,…; then, Ea,bp,q− transform of z, that is, Ea,bp,q=−a+br/a+bp,qrr∈ℕ0, is a sequence in the space c0, although z is not a sequence in c0. This proves that the inclusion c0⊂e0a,bp,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 domainZΛ has a basis iff Z has basis, where Λ is a triangle (see [29]). In view of this fact, one obtains the following result.Theorem 4.
Letϵvp,q=ϵrvp,q∈e0a,bp,qv∈ℕ0defined by(21)ϵrvp,q=−1r−vrvp,qqr−v2br−va+bp,qvarqr2,v≤r,0,v>r.Then, the set(a)
ϵ0p,q,ϵ1p,q,ϵ2p,q,… forms a basis of e0a,bp,q and every z∈e0p,qp,q has a unique representation z=∑v=0∞wvzvp,q.(b)
e,ϵ0p,q,ϵ1p,q,ϵ2p,q,… forms a basis of eca,bp,q, where e=1,1,1,…, and every z∈eca,bp,q has a unique representation z=ξe+∑v=0∞wv−ξϵvp,q, where ξ=limv⟶∞wv=limv⟶∞Ea,bp,qzv.


## 3.α, β, and γ-Duals

In the present section, we computeα, β, and γ-duals of e0a,bp,q and eca,bp,q. Because the techniques of the proof for both spaces e0a,bp,q and eca,bp,q are the same, we present the proof for the space e0a,bp,q only.Definition 3.
Theα, β, and γ−duals of Z⊂s are given by(22)Zα=ς=ςr∈s:ςz=ςrzr∈ℓ1∀z∈Z,Zβ=ς=ςr∈s:ςz=ςrzr∈cs∀z∈Z,Zγ=ς=ςr∈s:ςz=ςrzr∈bs∀z∈Z,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, ℛ represents the family of all finite subsets of ℕ0.Lemma 2.
Λ=λrv∈c0,ℓ1iff(23)supR∈ℛ∑v=0∞∑r∈Rλrv<∞.Lemma 3.
Λ=λrv∈c0,ciff(24)supr∈ℕ0∑v=0rλrv<∞,(25)limr⟶∞λrvexists for each v∈ℕ0.Lemma 4.
Λ=λrv∈c0,ℓ∞iff (24) holds.Theorem 5.
Define the setd1p,qby(26)d1p,q=ς=ςr∈s:supR∈ℛ∑v=0∞∑r∈R−1r−vrvp,qqr−v2br−va+bp,qvarqr2ςr<∞.
Then,e0a,bp,qα=eca,bp,qα=d1p,q.Proof.
Letςr∈s. Then, by using equality (16), we have(27)ςrzr=∑v=0r−1r−vrvp,qqr−v2br−va+bp,qvarqr2ςrwv=Φa,bp,qwr,r∈ℕ0,where Φa,bp,q=ϕrva,bp,q is a triangle defined by(28)ϕrva,bp,q=−1r−vrvp,qqr−v2br−va+bp,qvarqr2ςr,0≤v≤r,0,v>r.
Thus, we observe on using (27) that ςz=ςrzr∈ℓ1 whenever z∈e0a,bp.q iff Φa,bp,qw∈ℓ1 whenever w∈c0. Thus, we realize that ς=ςr∈e0p,qp,qα iff Ea,bp,q∈c0,ℓ1. Thus, using Lemma 2, we conclude that e0a,bp,qα=d1p,q.Theorem 6.
Define the setsd2p,q,d3p,q, andd4p,qby(29)d2p,q=ς=ςr∈s:∑l=v∞−1l−vlvp,qql−v2bl−va+bp,qvalql2ςlexists∀v∈ℕ0,d3p,q=ς=ςr∈s:supr∈ℕ0∑v=0r∑l=vr−1l−vlvp,qql−v2bl−va+bp,qvalql2ςl<∞,d4p,q=ς=ςr∈s:limr⟶∞∑v=0r∑l=vr−1l−vlvp,qql−v2bl−va+bp,qvalql2ςlexists.Then,e0a,bp,qβ=d2p,q∩d3p,q and eca,bp,qβ=d2p,q∩d3p,q∩d4p,q.Proof.
Letςr∈s and w=wr be the Ea,bp,q-transform of the sequence z=zr. Then, using equality (16), we get(30)∑v=0rςvzv=∑v=0r∑l=0v−1v−lvlp,qqv−l2bv−la+bp,qlavqv2wlςv=∑v=0r∑l=vr−1l−vlvp,qql−v2bl−va+bp,qvalql2ςlwv(31)=Ψa,bp,qwr,r∈ℕ0,where Ψa,bp,q=ψrva,bp,q is given by(32)ψrva,bp,q=∑l=vr−1l−vlvp,qql−v2bl−va+bp,qvalql2ςl,0≤v≤r,0,v>r.
In the light of (31), one obtains that ςz=ςrzr∈cs whenever z=zr∈e0a,bp,q iff Ψa,bp,qw∈c whenever w=wv∈c0. Thus, ς=ςr∈e0a,bp,qβ iff Ψa,bp,q∈c0,c. Thus, one obtains on using Lemma 3 that(33)supr∈ℕ0∑v=0rψrva,bp,q<∞andlimr⟶∞ψrva,bp,qexists∀v∈ℕ0.
Thus,e0a,bp,q=d2p,q∩d3p,q.Theorem 7.
e0a,bp,qγ=eca,bp,qγ=d3p,q.Proof.
The proof is similar to the previous theorem except that Lemma4 is used in place of Lemma 3.


## 4. Matrix Transformations

Now we characterize certain class of matrix transformations frome0a,bp,q and eca,bp,q to W∈ℓ∞,c,c0,ℓ1,ℓk.Theorem 8.
LetZ⊂sbe arbitrary. Then, Λ=λrv∈e0a,bp,q,Z(or, respectively, eca,bp,q,Z) if and only ifΩr=ωmvr∈c0,c(or, respectively, c,c) for eachr∈ℕ0, andΩ=ωrv∈c0,Z(or, respectively, c,Z), where(34)ωmvr=0,v>m,∑l=vm−1l−vlvp,qql−v2bl−va+bp,qvalql2λrl,0≤v≤m,ωrv=∑l=v∞−1l−vlvp,qql−v2bl−va+bp,qvalql2λrl.for all r,v∈ℕ0.Proof.
This is straightforward from ([3], Theorem 8).Now, in the light of Theorem8 and recalling the celebrated results of Stielglitz and Tietz [30], we get the following results.Corollary 1.
The following statements hold:(1)
Λ∈e0a,bp,q,ℓ∞iff(35)supm∈ℕ0∑v=0∞ωmvr<∞,(36)limm⟶∞ωmvrexists for allv∈ℕ0,(37)supr∈ℕ0∑v=0∞ωrv<∞.(2)
Λ∈e0a,bp,q,ciff (35) and (36) hold, and(38)supr∈ℕ0∑v=0∞ωrv<∞,(39)limr⟶∞ωrvexists for all v∈ℕ0.(3)
Λ∈e0a,bp,q,c0iff (35)–(37) hold, and(40)limr⟶∞ωrv=0for all v∈ℕ0.(4)
Λ∈e0a,bp,q,ℓ1iff (35) and (36) hold, and(41)supR∈ℛ∑v=0∞∑r∈Rωrv<∞.(5)
Λ∈e0a,bp,q,ℓkiff (35) and (36) hold, and(42)supV∈ℛ∑r=0∞∑v∈Vωrvk<∞k>1.Corollary 2.
The following statements hold:(1)
Λ∈eca,bp,q,ℓ∞iff (35), (36) and (38) hold, and(43)limm⟶∞∑v=0∞ωmvr.(2)
Λ∈eca,bp,q,ciff (35)–(37), (39), and (43) hold, and(44)limr⟶∞∑v=0rωrvexists.(3)
Λ∈eca,bp,q,c0iff (35)–(37), (39), and (43) hold.(45)limr⟶∞∑v=0rωrv=0.(4)
Λ∈eca,bp,q,ℓ1iff (35), (36), (41), and (43) hold.(5)
Λ∈eca,bp,q,ℓkiff (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]).
LetΛ be an infinite matrix, Ω be a triangle, and Z and W be any two sequence spaces. Then, Λ∈Z,WΩif and only ifΩΛ∈Z,W.LetΛ=λrv 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 matrixCα=crvαby(46)crvα=∑m=0rα+r−m−1r−mα+rrλmvα>1,for all r,v∈ℕ0. Then, the necessary and sufficient condition that Λ is in any one of the families e0a,bp,q,Ckα, e0a,bp,q,C∞α, eca,bp,q,Ckα, and eca,bp,q,C∞α is determined from the respective ones in Corollaries 1 and 2, by replacing the elements of Λ with those of Cα, where Ckα and C∞α are generalized Cesàro sequence spaces of order α given by Roopaei et al. [32].Corollary 4.
LetΛ=λrvbe an infinite matrix and define the matrixC˜=Crvby(47)C˜rv=∑m=0rCmCr−mCr+1λmvm,v∈ℕ0,whereCrrepresent sequence of Catalan numbers. Then, the necessary and sufficient condition thatΛbelongs to one of the classese0a,bp,q,c0C˜,e0a,bp,q,cC˜,eca,bp,q,c0C˜, andeca,bp,q,cC˜is determined from the respective ones inCorollaries 1 and 2by replacing the elements ofΛwith those ofC˜, wherecC˜andc0C˜are Catalan sequence spaces given by İlkhan [33].Corollary 5.
Defineλ˜=λ˜rvby(48)λ˜rv=∑m=0rλmv,r,v∈ℕ0.Then, the necessary and sufficient condition thatΛ belongs to one of the families e0a,bp,q,bs, e0a,bp,q,cs, e0a,bp,q,cs0, eca,bp,q,bs, eca,bp,q,cs, and eca,bp,q,cs0 is determined by replacing the elements of Λ with those of λ˜ from the respective ones in Corollaries 1 and 2.


## 5. Compactness by Hmnc

LetZ and W be any two Banach spaces. We represent the set of all bounded linear operators T:Z⟶W by BZ,W, which itself is a Banach space endowed with the norm T=supz∈BZTz, where the notation BZ denotes open ball in Z. Define ςZ†=supz∈BZ∑v=0∞ςvzv. In this case, we realize that ς=ςv∈Zβ, 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 operatorT:Z⟶W is compact if the domain of Z is all of Z and the sequence Tzr has a convergent subsequence in W for every bounded sequence zr in Z.Definition 5.
LetJ be a bounded subset of a metric space Z. Then, Hmnc of J is defined by(49)χJ=infε>0:J⊂∪v=0rBzv,av,zv∈Z,av<εv=0,1,2,…,r,r∈ℕ0,where Bzv,av is an open sphere with centre at zv and radius av.The Hmnc plays a very crucial role in determining the compactness of operators between theBK spaces. An operator T:Z⟶W is compact iff Tχ=0, where Tχ represents Hmnc of T given by Tχ=χTBZ. We refer the readers to [25, 34, 35].LetZ and W be any two BK-spaces. Then, Λ∈Z,W corresponds a linear operator TΛ∈BZ,W, where TΛz=Λz∀z∈Z (see ([7], Theorem 3.2.4(a))). Additionally, if Z⊃σ, then TΛ=ΛZ,W=supr∈ℕ0|ΛrZ†<∞ (see ([34], Theorem 1.23)), where σ is the set of sequences that end in zeroes.Lemma 6.
ℓ∞β=cβ=c0β=ℓ1. Furthermore, ifZ∈ℓ∞,c,c0, thenzZ†=zℓ1.Lemma 7 (see ([34], Theorem 2.15)).
LetJ⊂c0be bounded and define the operatorπr:c0⟶c0byπrz0,z1,z2…=z0,z1,z2…,zr,0,0,…∀z=zr∈c0; then,(50)χJ=limr⟶∞supr∈JI−πrz,where I is the identity operator on c0.Lemma 8 (see ([35], Theorem 7)).
LetZ⊃σbe aBK−space. Then, one has the following statements:(a)
IfΛ∈Z,c0, thenTΛχ=limsupr⟶∞ΛrZ†andTΛis compact ifflimr⟶∞ΛrZ†=0.(b)
IfZhasAKandΛ∈Z,c, then(51)12limsupr⟶∞Λr−λZ†≤TΛχ≤limsupr⟶∞Λr−λZ†,andTΛis compact ifflimr⟶∞Λr−λZ†=0,whereλ=λrwithλr=limr⟶∞λrv∀v∈ℕ0.(c)
IfΛ∈Z,ℓ∞, then0≤TΛχ≤limsupr⟶∞ΛrZ†andTΛis compact iflimr⟶∞ΛrZ†=0.We use the symbolℛm to indicate the subfamily of ℛ with elements that are greater than m.Lemma 9 (see ([35], Theorem 3.11)).
LetZ⊃σbe aBK−space. IfΛ∈Z,ℓ1, then(52)limm⟶∞supR∈ℛm∑r∈RΛrZ†≤TΛχ≤4⋅limm⟶∞supR∈ℛm∑r∈RΛrZ†.
TΛ is compact iff limm⟶∞supR∈ℛm∑r∈RΛrZ†=0.Lemma 10 (see ([35], Theorem 4.4, Corollary 3)).
LetZ⊃σbe aBK−space and let(53)Λbsr=∑v=0rΛvZ†.
Then, the following statements hold:(a)
IfΛ∈Z,cs0, then TΛχ=limsupr⟶∞ΛZ,bsr and TΛ is compact iff limr⟶∞ΛZ,bsr=0.(b)
IfZ has AK and Λ∈Z,cs, then(54)12limsupr⟶∞∑v=0rΛv−λ˜Z†≤TΛχ≤limsupr⟶∞∑v=0rΛv−λ˜Z†.TΛ is compact iff limsupr⟶∞∑v=0rΛv−λ˜Z†=0, where λ˜=λ˜v with λ˜v=limr⟶∞∑m=0rλmv for all v∈ℕ0.(c)
IfΛ∈Z,bs, then 0≤TΛχ≤limsupr⟶∞ΛZ,bsr and TΛ is compact if limr⟶∞ΛZ,bsr=0.Lemma 11.
IfΛ∈e0a,bp,q,Z, thenΩ∈c0,ZandΛz=Ωwfor allz∈e0a,bp,q, where the matrixΩ=ωrvis defined in (34).Proof.
LetΛ∈e0a,bp,q,Z and z∈e0a,bp,q. Then, Λr=λrvv∈ℕ0∈e0a,bp,qβ∀r∈ℕ0. One obtains(55)Ωwv=∑v=0∞ωrvwv=∑v=0∞∑m=v∞−1m−vmvp,qqm−v2bm−va+bp,qvamqm/2λrm⋅1a+bp,qv∑l=0vvlp,qpv−l2ql2albv−lzl=∑v=0∞λrvzv=Λzvv∈ℕ0.
Thus,Ωr∈ℓ1 for each r∈ℕ0 and Ωw∈Z. Consequently, Ω∈c0,Z.Theorem 9.
The following statements hold:(a)
IfΛ∈e0a,bp,q,c0, thenTΛχ=limsupr⟶∞∑v=0∞ωrv.(b)
IfΛ∈e0a,bp,q,c, then(56)12limsupr⟶∞∑v=0∞ωrv−ω≤TΛχ≤limsupr⟶∞∑v=0∞ωrv−ω,whereω=ωv and ωv=limr⟶∞ωrv∀v∈ℕ0.(c)
IfΛ∈e0a,bp,q,ℓ∞, then0≤TΛχ≤limsupr⟶∞∑v=0∞ωrv.(d)
IfΛ∈e0a,bp,q,ℓ1, then(57)limm⟶∞Λe0a,bp,q,ℓ1m≤TΛχ≤4limm⟶∞Λe0a,bp,q,ℓ1m,whereΛe0a,bp,q,ℓ1m=supR∈ℛm∑v=0∞∑r∈Rωrv.(e)
IfΛ∈e0a,bp,q,cs0, thenTΛχ=limsupr⟶∞∑v=0∞∑l=0rωlv.(f)
IfΩ∈e0a,bp,q,cs, then(58)12limsupr⟶∞∑v=0∞∑l=0rωlv−ω˜≤TΛχ≤limsupr⟶∞∑v=0∞∑l=0rωlv−ω˜,whereω˜=ω˜v with ω˜v=limr⟶∞∑l=0rωlv∀v∈ℕ0.(g)
IfΛ∈e0a,bp,q,bs, then0≤TΛχ≤limsupr⟶∞∑v=0∞∑l=0rωlv.Proof.
(a)
LetΛ∈e0a,bp,q,c0. We observe that(59)Λre0a,bp,q†=Ωrc0†=Ωrℓ1=∑v=0∞ωrv,r∈ℕ0.One obtains on engaging Lemma8 (a) that(60)TΛχ=limsupr⟶∞∑v=0∞ωrv.(b)
We observe that(61)Ωr−ωc0†=Ωr−ωℓ1=∑v=0rωrv−ωvr∈ℕ0.Now, letΛ∈e0a,bp,q,c; then, by using Lemma 11, one obtains that Ω∈c0,c. By engaging Lemma 8 (b), we deduce that(62)12limsupr⟶∞Ωr−ωc0†≤TΛχ≤limsupr⟶∞Ωr−ωc0†,which in the light of (62) yields(63)12limsupr⟶∞∑v=0∞ωrv−ωv≤TΛχ≤limsupr⟶∞∑v=0∞ωrv−ωv,which is the required result.(c)
The proof is similar to the proof of (a). Hence, the details are excluded.(d)
We have(64)∑r∈RΩrc0†=∑r∈RΩrℓ1=∑v=0∞∑r∈Rωrv.LetΛ∈e0a,bp,q,ℓ1. Then, Lemma 11 implies that Ω∈c0,ℓ1. Hence, by using Lemma 9, we get(65)limm⟶∞supR∈ℛm∑r∈RΩrc0†≤TΛχ≤4⋅limm⟶∞supR∈ℛm∑r∈RΩrc0†,which further reduces on using (65) to(66)limm⟶∞Λe0a,bp,q,ℓ1m≤TΛχ≤4limm⟶∞Λe0a,bp,q,ℓ1m,as desired.(e)
Notice that(67)∑l=0rΛle0a,bp,q†=∑l=0rΩlc0†=∑l=0rΩlℓ1=∑v=0∞∑l=0rωlv,which on using (a) of Lemma10 yields(68)TΛχ=limsupr⟶∞∑v=0∞∑l=0rωlv.(f)
We have(69)∑l=0rΩl−ω˜c0†=∑l=0rΩl−ω˜ℓ1=∑v=0∞∑l=0rωlv−ω˜v,for eachr∈ℕ0. Let Λ∈e0a,bp,q,cs. Then, Lemma 11 implies that Ω∈c0,cs. Thus, in view of (b) of Lemma 10, we deduce that(70)12limsupr⟶∞∑l=0rΩl−ω˜vc0†≤TΛχ≤limsupr⟶∞∑l=0rΩl−ω˜lc0†,which on using (69) yields(71)12limsupr⟶∞∑v=0∑l=0rωlv−ω˜v≤TΛχ≤limsupr⟶∞∑v=0∞∑l=0rωlv−ω˜v,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)
IfΛ∈e0a,bp,q,c0, then  TΛ is compact iff  limr⟶∞∑v=0∞ωrv=0.(b)
IfΛ∈e0a,bp,q,c, then  TΛ is compact iff  limr⟶∞∑v=0∞ω˜ij−ω˜j=0.(c)
IfΛ∈e0a,bp,q,ℓ∞, then  TΛ is compact if  limr⟶∞∑v=0∞ωrv=0.(d)
IfΛ∈e0a,bp,q,ℓ1, then  TΛ is compact iff  limr⟶∞supR∈ℛm∑v=0∞∑r∈Rωrv=0.(e)
IfΛ∈e0a,bp,q,cs0, then  TΛ is compact iff  limr⟶∞∑v=0∞∑l=0rωlv=0.(f)
IfΛ∈e0a,bp,q,cs, then  TΛ is compact iff  limr⟶∞∑v=0∞∑l=0rωlv−ω˜=0.(g)
IfΛ∈e0a,bp,q,bs, then  TΛ is compact if limr⟶∞∑v=0∞∑l=0rωlv=0.


## 6. Mapping Ideals

Here, we defines-type mapping ideals on e0a,bp,q,δϱ. The family of all bounded linear operators between any two Banach spaces is denoted by ℬ. Specifically, ℬX,Y indicates the family of all bounded linear mappings from Banach space X to Banach space Y. Before proceeding to our results, we list the following notations and definitions.Definition 6 (see [36]).
Assumeω+ is the set of nonnegative real sequences and ℂ is the set of complex numbers. Therefore, s-number is a mapping s:ℬX,Y⟶ω+ satisfying the following conditions:(i)
ϕ=s0ϕ≥s1ϕ≥s2ϕ≥…≥0, ∀ϕ∈ℬX,Y.(ii)
sa+b−1ϕ+ψ≤saϕ+sbψ, for all ϕ,ψ∈ℬX,Y and a,b∈ℕ0.(iii)
saϕθψ≤ϕsaθψ, for each ϕ∈ℬX0,X, θ∈ℬX,Y, and ψ∈ℬY,Y0, where X0 and Y0 are any two Banach sequence spaces.(iv)
Letϕ∈ℬX,Y and ν∈ℂ. Then, saνϕ=νsaϕ.(v)
Ifrankϕ≤a, then saϕ=0 for all ϕ∈ℬX,Y.(vi)
svℑa=0 for v≥a or svℑa=1 for v<a, where ℑa indicates the identity mapping on a-dimensional Hilbert space ℓ2a.We indicate the following parameters using an assortment ofs-numbers:(1)
Thea-th Kolmogorov number, indicated by daX, is defined by daX=infdimJ≤asupf≤1infg∈JXf−g.(2)
Thea-th approximation number, indicated by αaX, is defined by αaX=infX−Y:Y∈ℬX,Yand rankY≤a.Definition 7 (see [37]).
SupposeW⊂ℬ and denote WX,Y=W∩ℬX,Y. Then, W is called a mapping ideal if the following conditions are satisfied:(i)
ℑD∈W, where D is the one-dimensional Banach sequence space.(ii)
WX,Y is linear over ℂ.(iii)
Ifψ∈ℬX0,X, θ∈ℬX,Y and ϕ∈ℬY0,Y, then ϕθψ∈ℬX0,Y0, where X0 and Y0 are any two normed spaces.Definition 8 (see [38]).
A pre-quasi-norm on the idealW is a mapping μ:W⟶ω+ that satisfies the following conditions:(i)
μϕ≥0 and μϕ=0 iff ϕ=0, ∀ϕ∈WX,Y.(ii)
There ism0≥1 so that μςϕ≤m0ςμϕ, for all ϕ∈WX,Y.(iii)
There isn0≥1 so that μϕ+ψ≤n0μϕ+μψ, for all ϕ,ψ∈WX,Y.(iv)
One hasp0≥1 so that μϕθψ≤p0ϕμθψ, if ψ∈ℬX0,X, θ∈ℬX,Y, and ϕ∈ℬY0,Y.Definition 9 (see [38]).
The subspaceℨ⊂ω is called a private sequence space (or in short pss) if following statements are satisfied:(i)
ev∈ℨ, for all v∈ℕ0, where ev indicates the sequence with 1 in the vth position and 0 elsewhere.(ii)
Ifg=gv∈ω, h=hv∈ℨ, and gv≤hv, for v∈ℕ0, then g∈ℨ.(iii)
gv/2∈ℨ, if gv∈ℨ, where v/2 indicates the integral part of v/2.Definition 10 (see [39]).
A subspace of thepss is called a pre-modular pss, if there is a function υ:ℨ⟶0,∞ satisfying the following statements:(i)
For allj∈ℨ, j=0⇔υj=0, and υj≥0, with 0 being the zero vector of ℨ.(ii)
Ifj∈ℨ and ρ∈ℂ, then there is E0≥1 with υρj≤ρE0υj.(iii)
υh+j≤G0υh+υj holds for some G0≥1, with f,g∈ℨ.(iv)
Supposex∈ℕ0, hx≤jx; then, υhx≤υjx.(v)
The inequalityυjx≤υjx/2≤D0υjx holds, for D0≥1.(vi)
ℭ¯=ℨυ, where ℭ¯ indicates the closure of the space of all sequences with infinite zero coordinates.(vii)
One hasη>0 such that υρ,0,0,0,…≥ηρυ1,0,0,0,…, with ρ∈ℂ.Definition 11 (see [39]).
Thepssℨυ is called a pre-quasi-normed pss, if υ satisfies conditions (i)–(iii) in Definition 10. If ℨ is complete equipped with υ, then ℨυ is said to be a pre-quasi-Banach pss.Lemma 12 (see [38]).
Every pre-modularpss is a pre-quasi-normed pss.Definition 12.
We introduce the next sequence space:(72)e0a,bp,q,δϱ≔g=gv∈ω:limr⟶∞1a+bp,qr∑v=0rδvAr,vgv=0,where δ=δv∈ω+, ϱg=supr∈ℕ01/a+bp,qr∑v=0rδvAr,vgv, and Ar,v=rvp,qpr−v2qv2avbr−v.Byℑ↗ and ℑ↘, we indicate the space of all monotonic increasing and decreasing sequences of positive reals, respectively.Theorem 10.
e0a,bp,q,δis apss, ifδvAr,vv=0∞∈ℑ↘or,δvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,v.Proof.
(i)
Supposeg,h∈e0a,bp,q,δ, and one has(73)supr∈ℕ01a+bp,qr∑v=0rδvAr,vgv+hv≤supr∈ℕ01a+bp,qr∑v=0rδvAr,vgv+supr∈ℕ01a+bp,qr∑v=0rδvAr,vhv<∞.Hence,g+h∈e0a,bp,q,δ.Letς∈ℂ and g∈e0a,bp,q,δ. So, one has(74)supr∈ℕ01a+bp,qr∑v=0rδvAr,vςgv=ςsupr∈ℕ01a+bp,qr∑v=0rδvAr,vgv<∞.This impliese0a,bp,q,δ is a linear space. Next(75)supr∈ℕ01a+bp,qr∑v=0rδvAr,vetv=δtsupr∈ℕ0Ar,ta+bp,qr<∞.This giveset∈e0a,bp,q,δ, for all t∈ℕ0.(ii)
Assume thatgv≤hv, for all v∈ℕ0 and h∈e0a,bp,q,δ. Then, we have(76)supr∈ℕ01a+bp,qr∑v=0rδvAr,vgv≤supr∈ℕ01a+bp,qr∑v=0rδvAr,vhv<∞.This implies thatg∈e0a,bp,q,δ.(iii)
Letgv∈e0a,bp,q,δ, δvAr,vv=0∞∈ℑ↗∩ℓ∞ and ∃C≥1 so that δ2v+1Ar,2v+1≤CδvAr,v. So, one has(77)supr∈ℕ01a+bp,qr∑v=0rδvAr,vgv2≤supm∈ℕ01a+bp,q2m∑v=02mδvA2m,vgv2+supm∈ℕ01a+bp,q2m+1∑v=02m+1δvA2m+1,vgv2≤supm∈ℕ01a+bp,q2mδ2mA2m,2mgm+∑v=0mδ2vA2m,2vgv+δ2v+1A2m,2v+1gv+supm∈ℕ01a+bp,q2m+1∑v=0mδ2vA2m+1,2vgv+δ2v+1A2m+1,2v+1gv≤supm∈ℕ0Ca+bp,q2m∑v=0mδvA2m,vgv+supm∈ℕ02Ca+bp,q2m∑v=0mδvA2m,vgv+supm∈ℕ02Ca+bp,q2m+1∑v=0mδvA2m+1,vgv≤5Csupm∈ℕ01a+bp,qm∑v=0mδvAm,vgv<∞.
Thus,gv/2∈e0a,bp,q,δ.
This finishes the proof.Define the setsℬℨsX,Y, ℬℨαX,Y, and ℬℨdX,Y by(78)ℬℨsX,Y≔ϕ∈ℬX,Y:saϕ∈ℨ,ℬℨαX,Y≔ϕ∈ℬX,Y:αaϕ∈ℨ,ℬℨdX,Y≔ϕ∈ℬX,Y:daϕ∈ℨ,where X and Y are any two Banach sequence spaces. We denote ℬℨs≔ℬℨsX,Y, ℬℨα≔ℬℨαX,Y, and ℬℨd≔ℬℨdX,Y, respectively.Lemma 13 (see [37]).
Suppose the linear sequence spaceℨ is a pss. Then, ℬℨsis a mapping ideal.Theorem 11.
SupposeδvAr,vv=0∞∈ℑ↘or,δvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,v. Then, ℬe0a,bp,q,δsis a mapping ideal.Proof.
It comes directly from Lemma13.Theorem 12.
LetδvAr,vv=0∞∈ℑ↘or,δvAr,vv=0∞∈ℑ↗∩ℓ∞and there existsC≥1such thatδ2v+1Ar,2v+1≤CδvAr,v. Then,e0a,bp,q,δϱis a pre-modularpss.Proof.
(i)
Obviously, for everyg∈e0a,bp,q,δϱ that ϱg≥0 and ϱg=0, if and only if, g=0.(ii)
For allϵ≥1. Then, ϱαg≤ϵαϱg, for each g∈e0a,bp,q,δ and α∈ℂ.(iii)
Note thatϱg+h≤ϱg+ϱh, for every g,h∈e0a,bp,q,δ.(iv)
One hasϱgk≤ϱhk, if gk≤hk (see proof of (ii) in Theorem 10).(v)
From proof of (iii) in Theorem10, we have that ϱgk≤ϱgk/2≤ηϱgk with η=5C.(vi)
ℭ¯=e0a,bp,q,δ.(vii)
One has, ifα≠0 then 0<γ≤1, for ϱα,0,0,…≥γαϱ1,0,0,… and when α=0 then γ>0.
This finishes the proof.Theorem 13.
Suppose thatδvAr,vv=0∞∈ℑ↘or,δvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,v. Then,e0a,bp,q,δϱis a pre-quasi-Banachpss.Proof.
Keeping in mind Theorem12 and Lemma 12, it is enough to show that every Cauchy sequence in e0a,bp,q,δϱ is convergent in e0a,bp,q,δϱ. We suppose that gm=gkm is a Cauchy sequence in e0a,bp,q,δϱ. Then, for every ε∈0,1, there is n0∈ℕ0 so that(79)ϱgm−gn=supr∈ℕ01a+bp,qr∑v=0rδvAr,vgvm−gvn<ε,for each m,n≥n0. This gives that gvm−gvn<ε, for fixed v and for every m,n≥n0. Hence, gvm is a Cauchy sequence in ℂ. As ℂ is complete, limm⟶∞gvm=gv, for a fixed v∈ℕ0. This gives, by using (81), that ϱgm−g<ε, for every m≥n0. Also, one has ϱg≤ϱgm−g+ϱgm<∞. This implies that g∈e0a,bp,q,δϱ. So, e0a,bp,q,δϱ is a pre-quasi-Banach pss.Theorem 14 (see [40]).
Lets−type Eϱ≔h=sxH∈ω+:H∈ℬX,Yandϱh<∞. If BEϱs is a mapping ideal, then the following conditions are satisfied:(1)
ℭ⊂s−typeEϱ.(2)
IfsxH1x=0∞∈s−typeEϱandsxH2x=0∞∈s−typeEϱ, thensxH1+H2x=0∞∈s−typeEϱ.(3)
Ifλ∈ℂandsxHx=0∞∈s−typeEϱ, thenλsxHx=0∞∈s−typeEϱ.(4)
The sequence spaceEϱis solid, i.e., ifsxJx=0∞∈s−typeEϱandsxH≤sxJ, for everyx∈ℕ0andH,J∈ℬX,Y, thensxHx=0∞∈s−typeEϱ.By using Theorem14, we introduce the following properties of the s−type e0a,bp,q,δϱ.Theorem 15.
Assumes−typee0a,bp,q,δϱ≔f=snX∈ω+:X∈ℬX,Yandϱf<∞. The following settings are satisfied:(1)
We haves−typee0a,bp,q,δϱ⊃ℭ.(2)
LetsrX1r=0∞∈s−typee0a,bp,q,δϱandsrX2r=0∞∈s−typee0a,bp,q,δϱ; then, srX1+X2r=0∞∈s−typee0a,bp,q,δϱ.(3)
Supposeλ∈ℂandsrXr=0∞∈s−typee0a,bp,q,δϱ; then, λsrXr=0∞∈s−typee0a,bp,q,δϱ.(4)
Thes−typee0a,bp,q,δϱis solid.


## 7. Properties of the Pre-Quasi-Ideal

The following notations and definitions are essential for our investigation. LetX and Y be two Banach sequence spaces.Convention 1.
(80)F:the ideal of finite rank mappings between any arbitrary Banach spaces.A:the ideal of approximable mappings between any arbitrary Banach spaces.K:the ideal of compact mappings between any arbitrary Banach spaces.FX,Y:the space of finite rank mappings fromXintoY.FX:the space of finite rank mappings fromXinto itself.AX,Y:the space of approximable mappings fromXintoY.AX:the space of approximable mappings fromXinto itself.KX,Y:the space of compact mappings from aXintoY.KX:the space of compact mappings fromXinto itself.Lemma 14 (see [41]).
AssumeM∈ℬX,Y and M∉AX,Y; then, there exist operators Q∈ℬX and L∈ℬY such that LMQex=ex, for every x∈ℕ0.Definition 13 (see [41]).
A Banach spaceE is said to be simple if the algebra ℬE contains one and only one nontrivial closed ideal.Theorem 16 (see [41]).
IfE is a Banach space with dimE=∞, then(81)FE⊂AE⊂KE⊂ℬE.In this section, firstly, we obtain sufficient conditions (not necessary) one0a,bp,q,δϱ so that F¯=ℬe0a,bp,q,δϱs. This provides a negative answer of Rhoades [42] open problem about the linearity of s−type e0a,bp,q,δϱ space. Secondly, for which conditions on e0a,bp,q,δϱ, is ℬe0a,bp,q,δϱs closed and complete? Thirdly, we explain enough setups on e0a,bp,q,δϱ such that ℬe0a,bp,q,δϱs is strictly contained for different weights, and we obtain conditions so that the Banach pre-quasi-ideal ℬe0a,bp,q,δϱs is simple Banach space. Fourthly, we obtain enough conditions on e0a,bp,q,δϱ such that the space of all bounded linear operators which sequence of eigenvalues in e0a,bp,q,δϱ equals ℬe0a,bp,q,δϱs.


### 7.1. Finite Rank Pre-Quasi-Ideal

Theorem 17.
ℬe0a,bp,q,δϱsX,Y=FX,Y¯, if theconditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vhold. But the converse is not necessarily true.Proof.
To explain thatFX,Y¯⊆ℬe0a,bp,q,δϱsX,Y, as el∈e0a,bp,q,δϱ, for all l∈ℕ0 and e0a,bp,q,δϱis a linear space. Assume Z∈FX,Y, and one gets slZl=0∞∈ℭ. To show that ℬe0a,bp,q,δϱsX,Y⊆FX,Y¯, assume Z∈ℬe0a,bp,q,δϱsX,Y, and one has slZl=0∞∈e0a,bp,q,δϱ. As ϱsrZr=0∞<∞, suppose ρ∈0,1; then, there is r0∈ℕ0−0 with ϱsrZr=r0∞<ρ/16d, for some d≥1. Since srZ∈ℑ↘, we have(82)supr=r0+12r01a+bp,qr∑v=0rδvAr,vs2r0Z≤supr=r0+12r01a+bp,qr∑v=0rδvAr,vsvZ≤supr=r0∞1a+bp,qr∑v=0rδvAr,vsvZ<ρ16d.
Therefore, there existsY∈F2r0X,Y such that rank Y≤2r0 and(83)supr=2r0+13r01a+bp,qr∑v=0rδvAr,vZ−Y≤supr=r0+12r01a+bp,qr∑v=0rδvAr,vZ−Y<ρ16d.
One has(84)∑v=0r0δvAr0,vZ−Ya+bp,qr0<ρ16.
Hence, we have(85)supr=0r01a+bp,qr∑v=0rδvAr,vZ−Y<ρ16d.
According to inequalities (82)–(85) and δvAr,vv=0∞∈ℑ↘, we obtain(86)dZ,Y=ϱsrZ−Yr=0∞≤supr=03r0−11a+bp,qr∑v=0rδvAr,vsvZ−Y+supr=3r0∞1a+bp,qr∑v=0rδvAr,vsvZ−Y≤supr=03r01a+bp,qr∑v=0rδvAr,vsvZ−Y+supr=r0∞1a+bp,qr+2r0∑v=0r+2r0δvAr+2r0,vsvZ−Y≤supr=03r01a+bp,qr∑v=0rδvAr,vsvZ−Y+supr=r0∞1a+bp,qr∑v=0r+2r0δvAr,vsvZ−Y≤3supr=0r01a+bp,qr∑v=0rδvAr,vZ−Y+supr=r0∞∑v=02r0−1δvAr,vsvZ−Y+∑v=2r0r+2r0δvAr,vsvZ−Ya+bp,qr≤3supr=0r0∑v=0rδvAr,vZ−Ya+bp,qr+supr=r0∞∑v=02r0−1δvAr,vsvZ−Ya+bp,qr+supr=r0∞∑v=2r0r+2r0δvAr,vsvZ−Ya+bp,qr≤3supr=0r0∑v=0rδvAr,vZ−Ya+bp,qr+supr=r0∞∑v=02r0−1δvAr,vZ−Ya+bp,qr+supr=r0∞∑v=0rδv+2r0Ar,v+2r0sv+2r0Z−Ya+bp,qr≤3supr=0r0∑v=0rδvAr,vZ−Ya+bp,qr+2∑v=0r0δvAr0,vZ−Ya+bp,qr0+supr=r0∞∑v=0rδvAr,vsvZ−Ya+bp,qr<ρ.
On the other hand, one has a negative example asI2∈ℬe0a,bp,q,δϱsX,Y, where δvAr,vv∈ℕ0=0,0,1,0,1,0,…. This completes the proof.


### 7.2. Banach and Closed Pre-Quasi-Ideal

Theorem 18 (see [38]).
The functionΨ is a pre-quasi-norm on ℬEϱs, where ΨY=ϱsbYb=0∞, for all Y∈ℬEϱsX,Y, when Eϱ is a pre-modular pss.Theorem 19.
Assume the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vhold; then, ℬe0a,bp,q,δϱs,Ψis a pre-quasi-Banach ideal, whereψX=ϱslXl=0∞.Proof.
Sincee0a,bp,q,δϱ is a pre-modular pss, by Theorem 18, Ψ is a pre-quasi-norm on ℬe0a,bp,q,δϱs. Let Xjj∈ℕ0 be a Cauchy sequence in ℬe0a,bp,q,δϱsX,Y. Since ℬX,Y⊇ℬe0a,bp,q,δϱsX,Y, we have(87)ΨXi−Xj=supr∈ℕ0∑v=0rδvAr,vsvXi−Xja+bp,qr≥δ0Xi−Xj.
Then,Xjj∈ℕ0 is a Cauchy sequence in ℬX,Y. As ℬX,Y is a Banach space, then there exists X∈ℬX,Y with limj⟶∞Xj−X=0. Since slXjl=0∞∈e0a,bp,q,δϱ, for all j∈ℕ0, in view of (ii), (iii), and (v) in Definition 10, we have(88)ΨX=supr∈ℕ0∑v=0rδvAr,vsvXa+bp,qr≤supr∈ℕ0∑v=0rδvAr,vsv/2X−Xja+bp,qr+supr∈ℕ0∑v=0rδvAr,vsv/2Xja+bp,qr≤supr∈ℕ0∑v=0rδvAr,vX−Xja+bp,qr+D0supr∈ℕ0∑v=0rδvAr,vsvXja+bp,qr<∞.
Hence,slXl=0∞∈e0a,bp,q,δϱ, so X∈ℬe0a,bp,q,δϱsX,Y.Theorem 20.
SupposeX,Yare normed spaces, and the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vis satisfied; then, ℬe0a,bp,q,δϱs,Ψis a pre-quasi-closed ideal, whereΨX=ϱslXl=0∞.Proof.
Sincee0a,bp,q,δϱ is a pre-modular pss, from Theorem 0.1, we have that Ψ is a pre-quasi-norm on ℬe0a,bp,q,δϱs. Let Xj∈ℬe0a,bp,q,δϱsX,Y, for all j∈ℕ0 and limj⟶∞ΨXj−X=0. As ℬX,Y⊇ℬe0a,bp,q,δϱsX,Y, one has(89)ΨX−Xj=supr∈ℕ0∑v=0rδvAr,vsvX−Xja+bp,qr≥δ0X−Xj.
Then,Xjj∈ℕ0 is a convergent sequence in ℬX,Y. As slXjl=0∞∈e0a,bp,q,δϱ, for every j∈ℕ0, in view of (ii), (iii), and (v) in Definition 10, we have(90)ΨX=supr∈ℕ0∑v=0rδvAr,vsvXa+bp,qr≤supr∈ℕ0∑v=0rδvAr,vsv/2X−Xja+bp,qr+supr∈ℕ0∑v=0rδvAr,vsv/2Xja+bp,qr≤supr∈ℕ0∑v=0rδvAr,vX−Xja+bp,qr+D0supr∈ℕ0∑v=0rδvAr,vsvXja+bp,qr<∞.
We getslXl=0∞∈e0a,bp,q,δϱ, so X∈ℬe0a,bp,q,δϱsX,Y.


### 7.3. Simple Banach Pre-Quasi-Ideal

Theorem 21.
AssumeXandYare Banach spaces withdimX=dimY=∞, and the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vare satisfied withδl2≤δl1, for everyl∈ℕ0; then,(91)ℬe0a,bp,q,δl1ϱsX,Y⊂ℬe0a,bp,q,δl2ϱsX,Y⊂ℬX,Y.Proof.
SupposeZ∈ℬe0a,bp,q,δl1ϱsX,Y; then, slZ∈e0a,bp,q,δl1ϱ. We have(92)supr∈ℕ0∑v=0rδv2Ar,vsvXa+bp,qr≤supr∈ℕ0∑v=0rδv1Ar,vsvXa+bp,qr<∞.
Then,Z∈ℬe0a,bp,q,δl2ϱsX,Y. Next, if we put srZr=0∞ with ∑v=0rδv1Ar,vsvX=a+bp,qr and ∑v=0rδv2Ar,vsvX=a+bp,qr/r+1, we obtain Z∈ℬX,Y so that Z∉ℬe0a,bp,q,δl1ϱsX,Y and Z∈ℬe0a,bp,q,δl2ϱsX,Y.
Evidently,ℬe0a,bp,q,δl2ϱsX,Y⊂ℬX,Y. Next, if we put srZr=0∞ so that ∑v=0rδv2Ar,vsvX=a+bp,qr, we have Z∈ℬX,Y such that Z∉ℬe0a,bp,q,δl2ϱsX,Y. This finishes the proof.Theorem 22.
LetXandYbe Banach spaces withdimX=dimY=∞, and the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1such thatδ2v+1Ar,2v+1≤CδvAr,vare satisfied withδl2≤δl1, for everyl∈ℕ0; then,(93)ℬℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y=Aℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y.Proof.
SupposeX∈ℬℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y and X∉Aℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y. From Lemma 14, there are Y∈ℬℬe0a,bp,q,δl2ϱsX,Y and Z∈ℬℬe0a,bp,q,δl1ϱsX,Y with ZXYIj=Ij. Hence, for all j∈ℕ0, one has(94)Ijℬe0a,bp,q,δl1ϱsX,Y=supr∈ℕ0∑v=0rδv1Ar,vsvIja+bp,qr≤ZXYIjℬe0a,bp,q,δl2ϱsX,Y≤supr∈ℕ0∑v=0rδv2Ar,vsvIja+bp,qr.
This contradicts Theorem21. Therefore, X∈Aℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y, as desired.Corollary 7.
LetXandYbe Banach spaces withdimX=dimY=∞, and the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vare satisfied withδl2≤δl1, for everyl∈ℕ0; then,(95)ℬℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y=Kℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y.Proof.
The result is immediate sinceA⊂K.Theorem 23.
SupposeXandYare Banach spaces withdimX=dimY=∞, and the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vare satisfied; then, ℬe0a,bp,q,δϱsis simple.Proof.
Let the closed idealKℬe0a,bp,q,δϱsX,Y contain an operator X∉Aℬe0a,bp,q,δϱsX,Y. From Lemma 14, one has Y,Z∈ℬℬe0a,bp,q,δϱsX,Y with ZXYIj=Ij. This implies that Iℬe0a,bp,q,δϱsX,Y∈Kℬe0a,bp,q,δϱsX,Y. Then, ℬℬe0a,bp,q,δϱsX,Y=Kℬe0a,bp,q,δϱsX,Y. Therefore, ℬe0a,bp,q,δϱs is simple Banach space.


### 7.4. Eigenvalues of s-Type Operators

Conventions 2.
(96)ℬEsρ≔ℬEsρX,Y;XandYare Banach Spaces,whereℬEsρX,Y≔X∈ℬX,Y:ρlXn=0∞∈EandX-ρlXI is not invertible,foralll∈ℕ0.Theorem 24.
AssumeXandYare Banach spaces withdimX=dimY=∞, and the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vare satisfied withinfr∈ℕ0∑v=0rδvAr,v/a+bp,qr>0; then,(97)ℬe0a,bp,q,δϱsρX,Y=ℬe0a,bp,q,δϱsX,Y.Proof.
SupposeX∈ℬe0a,bp,q,δϱsρX,Y; then, ρlXl=0∞∈e0a,bp,q,δϱ and X−ρlXI=0, for every l∈ℕ0. One has X=ρlXI, for every l∈ℕ0, so slX=slρlXI=ρlX, for all l∈ℕ0. This implies slXl=0∞∈e0a,bp,q,δϱ; then, X∈ℬe0a,bp,q,δϱsX,Y.
Secondly, assumeX∈ℬe0a,bp,q,δϱsX,Y. Then, slXl=0∞∈e0a,bp,q,δϱ. Therefore, one gets(98)limr⟶∞∑v=0rδvAr,vsvXa+bp,qr≥infr∈ℕ0∑v=0rδvAr,va+bp,qrliml⟶∞slX.
Hence,liml⟶∞slX=0. Suppose X−slXI−1 exists, for all l∈ℕ0. Therefore, X−slXI−1 exists and is bounded, for all l∈ℕ0. Hence, liml⟶∞X−slXI−1=X−1 exists and is bounded. As ℬe0a,bp,q,δϱs,Ψ is a pre-quasi-operator ideal, one has(99)I=XX−1∈ℬe0a,bp,q,δϱsX,Y⇒slIl=0∞∈e0a,bp,q,δϱ⇒liml⟶∞slI=0.
Therefore, one gets a contradiction, asliml⟶∞slI=1. This implies X−slXI=0, for all l∈ℕ0. So, X∈ℬe0a,bp,q,δϱsρX,Y. This completes the proof.


## 7.1. Finite Rank Pre-Quasi-Ideal

Theorem 17.
ℬe0a,bp,q,δϱsX,Y=FX,Y¯, if theconditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vhold. But the converse is not necessarily true.Proof.
To explain thatFX,Y¯⊆ℬe0a,bp,q,δϱsX,Y, as el∈e0a,bp,q,δϱ, for all l∈ℕ0 and e0a,bp,q,δϱis a linear space. Assume Z∈FX,Y, and one gets slZl=0∞∈ℭ. To show that ℬe0a,bp,q,δϱsX,Y⊆FX,Y¯, assume Z∈ℬe0a,bp,q,δϱsX,Y, and one has slZl=0∞∈e0a,bp,q,δϱ. As ϱsrZr=0∞<∞, suppose ρ∈0,1; then, there is r0∈ℕ0−0 with ϱsrZr=r0∞<ρ/16d, for some d≥1. Since srZ∈ℑ↘, we have(82)supr=r0+12r01a+bp,qr∑v=0rδvAr,vs2r0Z≤supr=r0+12r01a+bp,qr∑v=0rδvAr,vsvZ≤supr=r0∞1a+bp,qr∑v=0rδvAr,vsvZ<ρ16d.
Therefore, there existsY∈F2r0X,Y such that rank Y≤2r0 and(83)supr=2r0+13r01a+bp,qr∑v=0rδvAr,vZ−Y≤supr=r0+12r01a+bp,qr∑v=0rδvAr,vZ−Y<ρ16d.
One has(84)∑v=0r0δvAr0,vZ−Ya+bp,qr0<ρ16.
Hence, we have(85)supr=0r01a+bp,qr∑v=0rδvAr,vZ−Y<ρ16d.
According to inequalities (82)–(85) and δvAr,vv=0∞∈ℑ↘, we obtain(86)dZ,Y=ϱsrZ−Yr=0∞≤supr=03r0−11a+bp,qr∑v=0rδvAr,vsvZ−Y+supr=3r0∞1a+bp,qr∑v=0rδvAr,vsvZ−Y≤supr=03r01a+bp,qr∑v=0rδvAr,vsvZ−Y+supr=r0∞1a+bp,qr+2r0∑v=0r+2r0δvAr+2r0,vsvZ−Y≤supr=03r01a+bp,qr∑v=0rδvAr,vsvZ−Y+supr=r0∞1a+bp,qr∑v=0r+2r0δvAr,vsvZ−Y≤3supr=0r01a+bp,qr∑v=0rδvAr,vZ−Y+supr=r0∞∑v=02r0−1δvAr,vsvZ−Y+∑v=2r0r+2r0δvAr,vsvZ−Ya+bp,qr≤3supr=0r0∑v=0rδvAr,vZ−Ya+bp,qr+supr=r0∞∑v=02r0−1δvAr,vsvZ−Ya+bp,qr+supr=r0∞∑v=2r0r+2r0δvAr,vsvZ−Ya+bp,qr≤3supr=0r0∑v=0rδvAr,vZ−Ya+bp,qr+supr=r0∞∑v=02r0−1δvAr,vZ−Ya+bp,qr+supr=r0∞∑v=0rδv+2r0Ar,v+2r0sv+2r0Z−Ya+bp,qr≤3supr=0r0∑v=0rδvAr,vZ−Ya+bp,qr+2∑v=0r0δvAr0,vZ−Ya+bp,qr0+supr=r0∞∑v=0rδvAr,vsvZ−Ya+bp,qr<ρ.
On the other hand, one has a negative example asI2∈ℬe0a,bp,q,δϱsX,Y, where δvAr,vv∈ℕ0=0,0,1,0,1,0,…. This completes the proof.


## 7.2. Banach and Closed Pre-Quasi-Ideal

Theorem 18 (see [38]).
The functionΨ is a pre-quasi-norm on ℬEϱs, where ΨY=ϱsbYb=0∞, for all Y∈ℬEϱsX,Y, when Eϱ is a pre-modular pss.Theorem 19.
Assume the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vhold; then, ℬe0a,bp,q,δϱs,Ψis a pre-quasi-Banach ideal, whereψX=ϱslXl=0∞.Proof.
Sincee0a,bp,q,δϱ is a pre-modular pss, by Theorem 18, Ψ is a pre-quasi-norm on ℬe0a,bp,q,δϱs. Let Xjj∈ℕ0 be a Cauchy sequence in ℬe0a,bp,q,δϱsX,Y. Since ℬX,Y⊇ℬe0a,bp,q,δϱsX,Y, we have(87)ΨXi−Xj=supr∈ℕ0∑v=0rδvAr,vsvXi−Xja+bp,qr≥δ0Xi−Xj.
Then,Xjj∈ℕ0 is a Cauchy sequence in ℬX,Y. As ℬX,Y is a Banach space, then there exists X∈ℬX,Y with limj⟶∞Xj−X=0. Since slXjl=0∞∈e0a,bp,q,δϱ, for all j∈ℕ0, in view of (ii), (iii), and (v) in Definition 10, we have(88)ΨX=supr∈ℕ0∑v=0rδvAr,vsvXa+bp,qr≤supr∈ℕ0∑v=0rδvAr,vsv/2X−Xja+bp,qr+supr∈ℕ0∑v=0rδvAr,vsv/2Xja+bp,qr≤supr∈ℕ0∑v=0rδvAr,vX−Xja+bp,qr+D0supr∈ℕ0∑v=0rδvAr,vsvXja+bp,qr<∞.
Hence,slXl=0∞∈e0a,bp,q,δϱ, so X∈ℬe0a,bp,q,δϱsX,Y.Theorem 20.
SupposeX,Yare normed spaces, and the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vis satisfied; then, ℬe0a,bp,q,δϱs,Ψis a pre-quasi-closed ideal, whereΨX=ϱslXl=0∞.Proof.
Sincee0a,bp,q,δϱ is a pre-modular pss, from Theorem 0.1, we have that Ψ is a pre-quasi-norm on ℬe0a,bp,q,δϱs. Let Xj∈ℬe0a,bp,q,δϱsX,Y, for all j∈ℕ0 and limj⟶∞ΨXj−X=0. As ℬX,Y⊇ℬe0a,bp,q,δϱsX,Y, one has(89)ΨX−Xj=supr∈ℕ0∑v=0rδvAr,vsvX−Xja+bp,qr≥δ0X−Xj.
Then,Xjj∈ℕ0 is a convergent sequence in ℬX,Y. As slXjl=0∞∈e0a,bp,q,δϱ, for every j∈ℕ0, in view of (ii), (iii), and (v) in Definition 10, we have(90)ΨX=supr∈ℕ0∑v=0rδvAr,vsvXa+bp,qr≤supr∈ℕ0∑v=0rδvAr,vsv/2X−Xja+bp,qr+supr∈ℕ0∑v=0rδvAr,vsv/2Xja+bp,qr≤supr∈ℕ0∑v=0rδvAr,vX−Xja+bp,qr+D0supr∈ℕ0∑v=0rδvAr,vsvXja+bp,qr<∞.
We getslXl=0∞∈e0a,bp,q,δϱ, so X∈ℬe0a,bp,q,δϱsX,Y.


## 7.3. Simple Banach Pre-Quasi-Ideal

Theorem 21.
AssumeXandYare Banach spaces withdimX=dimY=∞, and the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vare satisfied withδl2≤δl1, for everyl∈ℕ0; then,(91)ℬe0a,bp,q,δl1ϱsX,Y⊂ℬe0a,bp,q,δl2ϱsX,Y⊂ℬX,Y.Proof.
SupposeZ∈ℬe0a,bp,q,δl1ϱsX,Y; then, slZ∈e0a,bp,q,δl1ϱ. We have(92)supr∈ℕ0∑v=0rδv2Ar,vsvXa+bp,qr≤supr∈ℕ0∑v=0rδv1Ar,vsvXa+bp,qr<∞.
Then,Z∈ℬe0a,bp,q,δl2ϱsX,Y. Next, if we put srZr=0∞ with ∑v=0rδv1Ar,vsvX=a+bp,qr and ∑v=0rδv2Ar,vsvX=a+bp,qr/r+1, we obtain Z∈ℬX,Y so that Z∉ℬe0a,bp,q,δl1ϱsX,Y and Z∈ℬe0a,bp,q,δl2ϱsX,Y.
Evidently,ℬe0a,bp,q,δl2ϱsX,Y⊂ℬX,Y. Next, if we put srZr=0∞ so that ∑v=0rδv2Ar,vsvX=a+bp,qr, we have Z∈ℬX,Y such that Z∉ℬe0a,bp,q,δl2ϱsX,Y. This finishes the proof.Theorem 22.
LetXandYbe Banach spaces withdimX=dimY=∞, and the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1such thatδ2v+1Ar,2v+1≤CδvAr,vare satisfied withδl2≤δl1, for everyl∈ℕ0; then,(93)ℬℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y=Aℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y.Proof.
SupposeX∈ℬℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y and X∉Aℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y. From Lemma 14, there are Y∈ℬℬe0a,bp,q,δl2ϱsX,Y and Z∈ℬℬe0a,bp,q,δl1ϱsX,Y with ZXYIj=Ij. Hence, for all j∈ℕ0, one has(94)Ijℬe0a,bp,q,δl1ϱsX,Y=supr∈ℕ0∑v=0rδv1Ar,vsvIja+bp,qr≤ZXYIjℬe0a,bp,q,δl2ϱsX,Y≤supr∈ℕ0∑v=0rδv2Ar,vsvIja+bp,qr.
This contradicts Theorem21. Therefore, X∈Aℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y, as desired.Corollary 7.
LetXandYbe Banach spaces withdimX=dimY=∞, and the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vare satisfied withδl2≤δl1, for everyl∈ℕ0; then,(95)ℬℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y=Kℬe0a,bp,q,δl2ϱsX,Y,ℬe0a,bp,q,δl1ϱsX,Y.Proof.
The result is immediate sinceA⊂K.Theorem 23.
SupposeXandYare Banach spaces withdimX=dimY=∞, and the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vare satisfied; then, ℬe0a,bp,q,δϱsis simple.Proof.
Let the closed idealKℬe0a,bp,q,δϱsX,Y contain an operator X∉Aℬe0a,bp,q,δϱsX,Y. From Lemma 14, one has Y,Z∈ℬℬe0a,bp,q,δϱsX,Y with ZXYIj=Ij. This implies that Iℬe0a,bp,q,δϱsX,Y∈Kℬe0a,bp,q,δϱsX,Y. Then, ℬℬe0a,bp,q,δϱsX,Y=Kℬe0a,bp,q,δϱsX,Y. Therefore, ℬe0a,bp,q,δϱs is simple Banach space.


## 7.4. Eigenvalues of s-Type Operators

Conventions 2.
(96)ℬEsρ≔ℬEsρX,Y;XandYare Banach Spaces,whereℬEsρX,Y≔X∈ℬX,Y:ρlXn=0∞∈EandX-ρlXI is not invertible,foralll∈ℕ0.Theorem 24.
AssumeXandYare Banach spaces withdimX=dimY=∞, and the conditionδvAr,vv=0∞∈ℑ↘orδvAr,vv=0∞∈ℑ↗∩ℓ∞and there isC≥1so thatδ2v+1Ar,2v+1≤CδvAr,vare satisfied withinfr∈ℕ0∑v=0rδvAr,v/a+bp,qr>0; then,(97)ℬe0a,bp,q,δϱsρX,Y=ℬe0a,bp,q,δϱsX,Y.Proof.
SupposeX∈ℬe0a,bp,q,δϱsρX,Y; then, ρlXl=0∞∈e0a,bp,q,δϱ and X−ρlXI=0, for every l∈ℕ0. One has X=ρlXI, for every l∈ℕ0, so slX=slρlXI=ρlX, for all l∈ℕ0. This implies slXl=0∞∈e0a,bp,q,δϱ; then, X∈ℬe0a,bp,q,δϱsX,Y.
Secondly, assumeX∈ℬe0a,bp,q,δϱsX,Y. Then, slXl=0∞∈e0a,bp,q,δϱ. Therefore, one gets(98)limr⟶∞∑v=0rδvAr,vsvXa+bp,qr≥infr∈ℕ0∑v=0rδvAr,va+bp,qrliml⟶∞slX.
Hence,liml⟶∞slX=0. Suppose X−slXI−1 exists, for all l∈ℕ0. Therefore, X−slXI−1 exists and is bounded, for all l∈ℕ0. Hence, liml⟶∞X−slXI−1=X−1 exists and is bounded. As ℬe0a,bp,q,δϱs,Ψ is a pre-quasi-operator ideal, one has(99)I=XX−1∈ℬe0a,bp,q,δϱsX,Y⇒slIl=0∞∈e0a,bp,q,δϱ⇒liml⟶∞slI=0.
Therefore, one gets a contradiction, asliml⟶∞slI=1. This implies X−slXI=0, for all l∈ℕ0. So, X∈ℬe0a,bp,q,δϱsρX,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 Ea,bp,q. Finally, we obtained certain geometric and topological properties of e0a,bp,q,δϱ 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 ofp,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 constructp,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 Ea,bp,q in Maddox’s spaces.



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