# A Generalization of Caristi’s Fixed Point Theorem in the Variable Exponent Weighted Formal Power Series Space

Authors: Awad A. Bakery; M. H. El Dewaik
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Supposepn be sequence of positive reals. By Hwpn, we represent the space of all formal power series ∑n=0∞anzn equipped with ∑n=0∞λan/n+1pn<∞, for some λ>0. Various topological and geometric behavior of Hwpn and the prequasi ideal constructs by s-numbers and Hwpn have been considered. The upper bounds for s-numbers of infinite series of the weighted n-th power forward shift operator on Hwpn with applications to some entire functions are granted. Moreover, we investigate an extrapolation of Caristi’s fixed point theorem in Hwpn.


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## Body

## 1. Introduction

While a statement of fixed Lebesgue spaces, variable exponent Lebesgue spaces go back many years, and in successive centuries, variable Lebesgue and Sobolev spaces have been systematically examined. As then, many variable exponent real function spaces and complex function spaces have presented, for example, Hardy spaces, Besov spaces, Bessel potential spaces, Trieble-Lizorkin spaces, Morrey spaces, Herz-Morrey spaces, Fock spaces, Bergman spaces, and Herz spaces with variable exponents. For three centuries, variable exponent function spaces have been widely applied in approximation theory, image processing, and differential equations. The learn about of the variable exponent Lebesgue spaces obtained in addition impetus from the mathematical description of the hydrodynamics of non-Newtonian fluids [1, 2]. Applications of non-Newtonian fluids moreover regarded as electrorheological, vary from their use in army science to civil engineering and orthopedics. We will use the next conventions during the article; if others are used, we will state them.Conventions 1.
ℕ=0,1,2,⋯.ℂ:The space of all complex numbers
ℝℕ:The space of all sequences of real numbers
ℓ∞:The space of bounded sequences of real numbers
ℓr:The space of r-absolutely summable sequences of real numbers
c0:The space of null sequences of real numbers
el=0,0,⋯,1,0,0,⋯,:As 1 lies at the lth coordinate, for all l∈N
F:The space of each sequences with finite nonzero coordinates
cardG:The number of elements of G
miZ:The space of all monotonic increasing sequences of positive reals
L:The ideal of all bounded linear mappings between any arbitrary Banach spaces
F:The ideal of finite rank mappings between any arbitrary Banach spaces
Λ:The ideal of approximable mappings between any arbitrary Banach spaces
Lc:The ideal of compact mappings between any arbitrary Banach spaces
LX,Y:The space of all bounded linear mappings from a Banach space X into a Banach space Y
LX:The space of all bounded linear mappings from a Banach space X into itself
FX,Y:The space of finite rank mappings from a Banach space X into a Banach space Y
FX:The space of finite rank mappings from a Banach space X into itself
ΛX,Y:The space of approximable mappings from a Banach space X into a Banach space Y
ΛX:The space of approximable mappings from a Banach space X into itself
LcX,Y:The space of compact mappings from a Banach space X into a Banach space Y
LcX:The space of compact mappings from a Banach space X into itself
saGa∈ℕ:The sequence of s-numbers of the bounded linear operator G
αaGa∈ℕ:The sequence of approximation numbers of the bounded linear operator G
saGa∈ℕ:The sequence of Kolmogorov numbers of the bounded linear operator G
SV:The operator ideals constructed by the sequence of s-numbers in any sequence space V
SVapp:The operator ideals constructed by the sequence of approximation numbers in any sequence space V
SVKol:The operator ideals constructed by the sequence of Kolmogorov numbers in any sequence space V.The theory of operator ideals has many activities in fixed point theorems, eigenvalue distributions, geometry of Banach spaces, spectral theorems, etc. Some of operator ideals in the class of Banach spaces or Hilbert spaces are generated by sequence of real numbers. For example,Lc is constructed by daGa∈ℕ and c0. Pietsch [3] examined the quasi ideals Sℓtapp for 0<t<∞. He proved that the ideals of Hilbert Schmidt operators and of nuclear operators between Hilbert spaces are generated by ℓ2 and ℓ1, respectively. Also, he examined that F¯=Sℓt, for 1≤t<∞, and Sℓt is simple Banach space. Pietsch [4] showed that Sℓt, where 0<t<∞, is small. Makarov and Faried [5] investigated that for any Banach spaces X and Y with dimX=dimY=∞. They have for all r>t>0 that SℓtappX,YÞSℓrappX,YÜLX,Y. Faried and Bakery [6] generalized the concept of quasi ideal by introducing the concept of prequasi ideal. They introduced some geometric and topological structure of the spaces Scest and SℓM. Yaying et al. [7] suggested the sequence space, χrt, with its r-Cesàro matrix in ℓt, with r∈0,1 and 1≤t≤∞. They examined the quasi Banach ideal of type χrt, for r∈0,1 and 1<t<∞. They found its Schauder basis, α−, β−, and γ− duals and committed to certain matrix classes linked to this sequence space. Mursaleen and Noman [8, 9] investigated the compact operators on some difference sequence spaces. The multiplication maps on Cesàro sequence spaces with the Luxemburg norm explored by Komal et al. [10]. İlkhan et al. [11] affected the multiplication maps on Cesàro second order function spaces. Recently, many authors in the literature have investigated some nonabsolute kind sequence spaces and brought recent splendid papers; for example, Mursaleen and Noman [12] defined the sequence spaces ℓpλ and ℓ∞λ of nonabsolute type and proved that the spaces ℓpλ and ℓpλ are linearly isomorphic for 0<p≤∞, ℓpλ is a p-normed space, and a BK-space in the cases for 0<p<1 and 1≤p≤∞ and modeled the basis for the space ℓpλ for 1≤p<∞. In [13], they considered the α−, β−, and γ− duals of ℓpλ and ℓ∞λ of nonabsolute type, for 1≤p<∞. They were given a picture of some related matrix classes and shown the characterizations of some other classes by means of a given basic lemma. On Cesàro summable sequences, Mursaleen and Basar [14] defined some spaces of double sequences whose Cesàro transforms are bounded, convergent in the Pringsheim’s sense, null in the Pringsheim’s sense, both convergent in the Pringsheim’s sense and bounded, regularly convergent, and absolutely q-summable, respectively, and investigated some topological properties of those sequence spaces. The Banach fixed point theorem [15] gave many mathematicians the way to examine many generalizations for the contraction operators defined on the space or on the space itself. Kannan [16] examined an example of a class of operators with the identical fixed point actions as contractions but fails to be continuous. Ghoncheh [17] was the only one who investigated Kannan operators in modular vector spaces. He showed that the existence of a fixed point of Kannan mapping in complete modular spaces that have Fatou property. Bakery and Mohamed [18] explored the concept of the prequasi norm on Nakano sequence space so as to its variable exponent in 0,1. They investigated the sufficient conditions on it equipped with the definite prequasi norm to generate prequasi Banach and closed space and approved the Fatou property of distinct prequasi norms on it. Moreover, they showed that the existence of a fixed point of Kannan prequasi norm contraction maps on it and on the prequasi Banach operator ideal generated by s-numbers which belong to this sequence space. According to the spectral decomposition theorem [3], for A∈LcH, where H is a Hilbert space, then Ay=∑a=0∞αa≺y,ra≻wa, where ra and wa are orthonormal families in H. Assume taa∈ℕ∈ℝℕ be decreasing and D:ηa⟶taηa be the diagonal operator on ℓp with p≥1. Hence, saD=ta. Shields [19] gave an illustration to the weighted shift operators as formal power series in unilateral shifts and formal Laurent series in bilateral shifts. Hedayatian [20] explored the space of formal power series with power r, Hrba, where ba is a sequence of positive numbers with b0=1 and r>0. By the space Hpba, he thought that the set of all formal power series ∑a=0∞fa^za with ∑a=0∞bafa^p<∞. He investigated cyclic vectors for the forward shift operator and super cyclic vectors for the backward shift operator on the space Hpba. However, Emamirad and Heshmati [21], studied the idea of functions evident on the Bargmann space by fz=∑a=0∞caza/a! with f=∑a=0∞ca2<∞, where za/a!:a∈ℕ is an orthonormal basis. Faried et al. [22] examined the upper bounds for s-numbers of infinite series of the weighted n-th power forward shift operator on Hrba, for 1≤r<∞, with some applications to some entire functions.The aim of this paper is organized as follows: In Section3, we introduce the definition of the space Hwpn under the function ρ. We offer the enough setup on Hwpn to become premodular special space of formal power series which implies that Hwpn is a prequasi normed space. In Section 4, firstly, the operator ideals generated by s-numbers and Hwpn so as to SHwpn constructs an operator ideal are presented. Secondly, we offer the sufficient conditions (not necessary) on Hwpnρ, such that F is dense in SHwpnρ. This presents the nonlinearity of s-type Hwpnρ spaces which offers an answer of Rhoades [23] open problem. Thirdly, we examine the setup on Hwpnρ so that the elements of prequasi ideal SHwpnρ are complete and closed. Fourthly, we study the enough setup on Hwpnρ such that SHwpnρ is strictly contained for distinct powers. We investigate the smallness of SHwpnρ. Fifthly, we explain the simpleness of SHwpnρ. Sixthly, we offer the enough conditions on Hwpnρ so as to the class L with its sequence of eigenvalues in Hwpnρ is strictly contained in SHwpnρ. In Section 5, we evaluate the upper bounds for s-numbers of infinite series of the weighted n-th power forward shift operator on Hwpn with applications to some entire functions. Finally, in Section 6, we appraise a generalization of Caristi’s fixed point theorem in Hwpnρ.


## 2. Definitions and Preliminaries

Definition 2 [24].
A maps:LX,Y⟶0,∞ℕ is named an s-number, if the sequence sbBa=0∞, for every B∈LX,Y, verifies the next conditions:
(a)
ssumeB∈LX,Y, thenB=s0B≥s1B≥s2B≥⋯≥0(b)
sb+aB1+B2≤sbB1+saB2, for all B1,B2∈LX,Y, b, a∈ℕ(c)
The inequalitysaABD≤AsaBD satisfies, if D∈LX0,X, B∈LX,Y, and A∈LY,Y0, where X0 and Y0 are any two Banach spaces(d)
IfA∈LX,Y and λ∈ℝ, then saλA=λsaA(e)
AssumerankA≤b then sbA=0, whenever A∈LX,Y(f)
SupposeIa denotes the unit map on the a-dimensional Hilbert space ℓ2a, then sr≥aIa=0 or sr<aIa=1
We introduce a few examples ofs-numbers as indicated:
(i)
Then-th approximation number, αbA, where(1)αbA=infA−B:B∈LX,YandrankB≤b.(ii)
Then-th Kolmogorov number, dbA, where(2)dbA=infdimY≤bsup∥u∥≤1infv∈YAu−v.Remark 3 [24].
AssumeA∈LcH, where H be a Hilbert space, then all the s-numbers equal the eigenvalues of A, where A=A∗A.Lemma 4 [3].
SupposeA∈LX,Y and A∉ΛX,Y, then there are maps D∈LX and M∈LY with MADeb=eb, for all b∈ℕ.Definition 5 [3].
A Banach spaceY is called simple if LY has a unique nontrivial closed ideal.Theorem 6 [3].
AssumeD be a Banach space with dimD=∞, one has
(3)FD⫋ΛD⫋LcD⫋LD.Definition 7 [3].
A classU⊆L is called an operator ideal if each element UX,Y=U∩LX,Y verifies the next conditions:
(4)F⊆U.(i)
UX,Y is linear space on ℝ(ii)
SupposeD∈LX0,X, B∈UX,Y, and A∈LY,Y0 then, ABD∈UX0,Y0Definition 8 [6].
A mapg:U⟶0,∞ is named a prequasi norm on the ideal U if it verifies the next conditions:
(1)
AssumeA∈LX,Y, gA≥0, and gA=0⇔A=0(2)
We haveM≥1 with gμA≤MμgA, for each μ∈ℝ and A∈UX,Y(3)
We findK≥1 with gA1+A2≤KgA1+gA2, for every A1,A2∈UX,Y(4)
We obtainC≥1 so that if A∈LX0,X, B∈UX,Y, and D∈LY,Y0 then gDBA≤CDgBA, where X0 and Y0 are normed spacesTheorem 9 [6].
Ifg is a quasi norm on the ideal U, then g is a prequasi norm on the ideal U.Theorem 10 [25].
Ifs-type Vυ≔f=srT∈ℝℕ:T∈LX,Yandυf<∞. Assume SVυ be an operator ideal, one has
(1)
F⊂s− type Vυ(2)
SupposesrT1r=0∞∈s-type Vυ and srT2r=0∞∈s-type Vυ, then srT1+T2r=0∞∈s-type Vυ(3)
Assumeλ∈ℝ and srTr=0∞∈s-type Vυ, then λsrTr=0∞∈s-type Vυ(4)
The sequence spaceVυ is solid; i.e., if srGr=0∞∈s-type Vυ and srT≤srG, for all r∈ℕ and T,G∈LX,Y, then srTr=0∞∈s-type VυLemma 11 [26].
Assumeξii∈Ψ be a bounded family of ℝ. Then
(5)supcardG=a+1infi∈Gξi=infcardG=asupi∉Gξi.
The next inequality in [27] will be used in the sequel. For ra,ta∈ℝℕ, and qa∈0,∞ℕ, one has
(6)ra+taqa≤Kraqa+taqa,
whereK=max1,2ϖq−1 and ϖq=max1,supaqa.


## 3. Main Results

### 3.1. The Space of FunctionsHwpnρ

In this section, we current the definition of the spaceHwpnρ equipped with the function ρ. We introduce the sufficient setup on Hwpnρ to form premodular special space of formal power series (ssfps). This gives that Hwpnρ is a prequasi normed (ssfps).Assumep=pnn∈ℕ∈0,∞ℕ, we define the following space:
(7)Hwpnρ=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂandρλf<∞,forsomeλ>0,where
(8)ρf=∑n=0∞fn^n+1pn.Whenpn∈ℓ∞, we have
(9)Hwpnρ=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂandρλf<∞,forsomeλ>0=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂand∑n=0∞λfn^n+1pn<∞,forsomeλ>0=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂandinfnλpn∑n=0∞fn^n+1pn<∞,forsomeλ>0=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂand∑n=0∞fn^n+1pn<∞=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂandρλf<∞,foranyλ>0.Definition 12.
The linear space of formal power series,(10)H=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂ,is named a (ssfps) when:
(1)
em∈H, for every m∈ℕ, where emz=∑n=0∞enm^zn=zm(2)
Supposeg∈H and fn^≤gn^, for all n∈ℕ, then f∈H(3)
Letf∈H, then f.∈H, where f.z=∑b=0∞fb/2^zb and b/2 indicates the integral part of b/2
ByF, we indicate the space of finite formal power series, i.e, for f∈F; then, there is l∈ℕ so that fz=∑n=0lfn^zn.Definition 13.
A subspaceHρ of the (ssfps) is called a premodular (ssfps), if there is a map ρ:H⟶0,∞ satisfies the following setup:
(i)
Assumef∈H, one has ρf≥0 and f=θ⇔ρf=0, where θ is the zero function of H(ii)
Iff∈H and λ∈ℝ, one has l≥1 with ρλf≤λlρf(iii)
Iff,g∈H, one has K≥1 so as to ρf+g≤Kρf+ρg(iv)
Supposefb^≤gb^, for every b∈ℕ, then ρf≤ρg(v)
One hasK0≥1 such that ρf≤ρf.≤K0ρf(11)F¯=Hρ.(vi)
We haveξ>0 with ρλe0≥ξλρe0, where λ∈ℝ
Remark that the continuity ofρf at θ follows from part (ii). The part (1) in Definition 12 and part (vi) in Definition 13 explain that emm∈ℕ is a Schauder basis of Hρ.
The (ssfps)Hρ is named a prequasi normed (ssfps) if ρ satisfies the parts (i)–(iii) of Definition 13, and if the space H is complete equipped with ρ, then Hρ is named a prequasi Banach (ssfps).Theorem 14.
Every premodular (ssfps)Hρ is a prequasi normed (ssfps).Theorem 15.
Ifpn∈miZ∩ℓ∞ with p0>0, then Hwpnρ is a premodular Banach (ssfps).Proof.
(1-i) Assumef,g∈Hwpn. Hence, fz=∑n=0∞fn^zn and gz=∑n=0∞gn^zn converge for any z∈ℂ. One has f+gz=∑n=0∞fn^+gn^zn converges for any z∈ℂ. As pn∈ℓ∞, one obtains
(12)∑n=0∞fn^+gn^n+1pn≤K∑n=0∞fn^n+1pn+∑n=0∞gn^n+1pn<∞,
thenf+g∈Hwpn.
(1-ii) Supposeλ∈ℝ and f∈Hwpn. Hence, fz=∑n=0∞fn^zn converges for any z∈ℂ. One has λfz=∑n=0∞λfn^zn converges for any z∈ℂ. As pn∈ℓ∞, one can see
(13)∑n=0∞λfn^n+1pn≤supnλpn∑n=0∞fn^n+1pn<∞.
Soλf∈Hwpn. Hence, from parts (1-i) and (1-ii), one gets that the space Hwpn is linear. Also em∈Hwpn, for every m∈ℕ, where emz=∑n=0∞enm^zn=zm and ∑n=0∞enm^/n+1pn=1/m+1pm.
(2) Supposefn^≤gn^, for every n∈ℕ and g∈Hwpn. Hence, gz=∑n=0∞gn^zn converges for any z∈ℂ. We have
(14)∑n=0∞fn^n+1pn≤∑n=0∞gn^n+1pn<∞.
Therefore,fz=∑n=0∞fn^zn converges for any z∈ℂ and ρf<∞. So, f∈Hwpn.
(3) Iff∈Hwpn and pn∈miZ∩ℓ∞ with p0>0. Hence, fz=∑n=0∞fn^zn converges for any z∈ℂ and ρf<∞. We obtain
(15)ρf.=∑n=0∞fn/2^n+1pn=∑n=0∞fn^2n+1p2n+∑n=0∞fn^2n+2p2n+1≤2∑n=0∞fn^n+1pn=2ρf.
Therefore,f.z=∑n=0∞fn/2^zn converges for any z∈ℂ and ρf.<∞. Then, f.∈Hwpn.
(i)
Definitely, for allf∈Hwpn, then ρf≥0 and ρf=0⇔f=θ(ii)
One hasl=max1,supnηpn−1≥1, for every η∈ℝ\0 and l≥1, for η=0 so that(16)ρηf=∑n=0∞ηbnfn^pn≤supnηpn∑n=0∞bnfn^pn≤lηρf,for all f∈Hwpn.
(iii)
One gets someK=max1,2supnpn−1≥1 satisfy the inequality(17)ρf+g=∑n=0∞fn^+gn^n+1pn≤Kρf+ρg,for every f,g∈Hwpn.
(iv)
Definitely from the proof part (2)(v)
From the proof part (3), one hasK0=2≥1(vi)
DefinitelyF¯=Hwpn(vii)
We haveζ with 0<ζ≤ηp0−1 so that ρηe0≥ζηρe0, for all η≠0 and ζ>0, when η=0
So, the spaceHwpnρ is premodular (ssfps). To prove that Hwpnρ is a premodular Banach (ssfps), let fi is a Cauchy sequence in Hwpnρ; hence, for all ε∈0,1, one has i0∈ℕ so that for every i,j≥i0, we have
(18)ρfi−fj=∑n=0∞fni^−fnj^n+1pn<εϖp.
Fori,j≥i0, and k∈ℕ, one gets
(19)fki^−fkj^<ε.
Therefore,fkj^ is a Cauchy sequence in ℝ, for constant k∈ℕ; hence, limj⟶∞fkj^=fk0^, for constant k∈ℕ. So, ρfi−f0<εϖp, for all i≥i0. Finally, to investigate that f0∈Hwpn, one has
(20)ρf0=ρf0−fn+fn≤Kρfn−f0+ρfn<∞.
Therefore,f0∈Hwpn. So, the space Hwpnρ is a premodular Banach (ssfps).
By using Theorem14, we offer the next theorem.☐Theorem 16.
Assumepn∈miZ∩ℓ∞ with p0>0, then the space Hwpnρ be a prequasi Banach (ssfps), where
(21)ρf=∑n=0∞fn^n+1pn,forallf∈Hwpn.Theorem 17.
Ifpn∈miZ∩ℓ∞ with p0>0, then the space Hwpnρ is a prequasi closed (ssfps), where
(22)ρf=∑n=0∞fn^n+1pn,forallf∈Hwpn.Proof.
From Theorem16, the space Hwpnρ is a prequasi normed (ssfps). To prove that Hwpnρ is a prequasi closed (ssfps), suppose fii=0∞∈Hwpnρ and limi⟶∞ρfi−f0=0; then, for each ε∈0,1, one has i0∈ℕ so that for every i≥i0, we have
(23)ε>ρfi−f0=∑a=0∞fai^−fa0^n+1pa1/ϖp.
Therefore, fori≥i0 and a∈ℕ, one obtains fai^−fa0^<ε. Hence, fai^ is a convergent sequence in ℝ, for constant a∈ℕ. Hence, limi⟶∞fai^=fa0^, for constant a∈ℕ. Finally, to show that f0∈Hwpnρ, one has
(24)ρf0=ρf0−fi+fi≤ρfi−f0+ρfi<∞,so f0∈Hwpnρ. This explains that Hwpnρ is a prequasi closed (ssfps).☐


## 3.1. The Space of FunctionsHwpnρ

In this section, we current the definition of the spaceHwpnρ equipped with the function ρ. We introduce the sufficient setup on Hwpnρ to form premodular special space of formal power series (ssfps). This gives that Hwpnρ is a prequasi normed (ssfps).Assumep=pnn∈ℕ∈0,∞ℕ, we define the following space:
(7)Hwpnρ=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂandρλf<∞,forsomeλ>0,where
(8)ρf=∑n=0∞fn^n+1pn.Whenpn∈ℓ∞, we have
(9)Hwpnρ=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂandρλf<∞,forsomeλ>0=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂand∑n=0∞λfn^n+1pn<∞,forsomeλ>0=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂandinfnλpn∑n=0∞fn^n+1pn<∞,forsomeλ>0=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂand∑n=0∞fn^n+1pn<∞=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂandρλf<∞,foranyλ>0.Definition 12.
The linear space of formal power series,(10)H=f:fz=∑n=0∞fn^znconvergesforanyz∈ℂ,is named a (ssfps) when:
(1)
em∈H, for every m∈ℕ, where emz=∑n=0∞enm^zn=zm(2)
Supposeg∈H and fn^≤gn^, for all n∈ℕ, then f∈H(3)
Letf∈H, then f.∈H, where f.z=∑b=0∞fb/2^zb and b/2 indicates the integral part of b/2
ByF, we indicate the space of finite formal power series, i.e, for f∈F; then, there is l∈ℕ so that fz=∑n=0lfn^zn.Definition 13.
A subspaceHρ of the (ssfps) is called a premodular (ssfps), if there is a map ρ:H⟶0,∞ satisfies the following setup:
(i)
Assumef∈H, one has ρf≥0 and f=θ⇔ρf=0, where θ is the zero function of H(ii)
Iff∈H and λ∈ℝ, one has l≥1 with ρλf≤λlρf(iii)
Iff,g∈H, one has K≥1 so as to ρf+g≤Kρf+ρg(iv)
Supposefb^≤gb^, for every b∈ℕ, then ρf≤ρg(v)
One hasK0≥1 such that ρf≤ρf.≤K0ρf(11)F¯=Hρ.(vi)
We haveξ>0 with ρλe0≥ξλρe0, where λ∈ℝ
Remark that the continuity ofρf at θ follows from part (ii). The part (1) in Definition 12 and part (vi) in Definition 13 explain that emm∈ℕ is a Schauder basis of Hρ.
The (ssfps)Hρ is named a prequasi normed (ssfps) if ρ satisfies the parts (i)–(iii) of Definition 13, and if the space H is complete equipped with ρ, then Hρ is named a prequasi Banach (ssfps).Theorem 14.
Every premodular (ssfps)Hρ is a prequasi normed (ssfps).Theorem 15.
Ifpn∈miZ∩ℓ∞ with p0>0, then Hwpnρ is a premodular Banach (ssfps).Proof.
(1-i) Assumef,g∈Hwpn. Hence, fz=∑n=0∞fn^zn and gz=∑n=0∞gn^zn converge for any z∈ℂ. One has f+gz=∑n=0∞fn^+gn^zn converges for any z∈ℂ. As pn∈ℓ∞, one obtains
(12)∑n=0∞fn^+gn^n+1pn≤K∑n=0∞fn^n+1pn+∑n=0∞gn^n+1pn<∞,
thenf+g∈Hwpn.
(1-ii) Supposeλ∈ℝ and f∈Hwpn. Hence, fz=∑n=0∞fn^zn converges for any z∈ℂ. One has λfz=∑n=0∞λfn^zn converges for any z∈ℂ. As pn∈ℓ∞, one can see
(13)∑n=0∞λfn^n+1pn≤supnλpn∑n=0∞fn^n+1pn<∞.
Soλf∈Hwpn. Hence, from parts (1-i) and (1-ii), one gets that the space Hwpn is linear. Also em∈Hwpn, for every m∈ℕ, where emz=∑n=0∞enm^zn=zm and ∑n=0∞enm^/n+1pn=1/m+1pm.
(2) Supposefn^≤gn^, for every n∈ℕ and g∈Hwpn. Hence, gz=∑n=0∞gn^zn converges for any z∈ℂ. We have
(14)∑n=0∞fn^n+1pn≤∑n=0∞gn^n+1pn<∞.
Therefore,fz=∑n=0∞fn^zn converges for any z∈ℂ and ρf<∞. So, f∈Hwpn.
(3) Iff∈Hwpn and pn∈miZ∩ℓ∞ with p0>0. Hence, fz=∑n=0∞fn^zn converges for any z∈ℂ and ρf<∞. We obtain
(15)ρf.=∑n=0∞fn/2^n+1pn=∑n=0∞fn^2n+1p2n+∑n=0∞fn^2n+2p2n+1≤2∑n=0∞fn^n+1pn=2ρf.
Therefore,f.z=∑n=0∞fn/2^zn converges for any z∈ℂ and ρf.<∞. Then, f.∈Hwpn.
(i)
Definitely, for allf∈Hwpn, then ρf≥0 and ρf=0⇔f=θ(ii)
One hasl=max1,supnηpn−1≥1, for every η∈ℝ\0 and l≥1, for η=0 so that(16)ρηf=∑n=0∞ηbnfn^pn≤supnηpn∑n=0∞bnfn^pn≤lηρf,for all f∈Hwpn.
(iii)
One gets someK=max1,2supnpn−1≥1 satisfy the inequality(17)ρf+g=∑n=0∞fn^+gn^n+1pn≤Kρf+ρg,for every f,g∈Hwpn.
(iv)
Definitely from the proof part (2)(v)
From the proof part (3), one hasK0=2≥1(vi)
DefinitelyF¯=Hwpn(vii)
We haveζ with 0<ζ≤ηp0−1 so that ρηe0≥ζηρe0, for all η≠0 and ζ>0, when η=0
So, the spaceHwpnρ is premodular (ssfps). To prove that Hwpnρ is a premodular Banach (ssfps), let fi is a Cauchy sequence in Hwpnρ; hence, for all ε∈0,1, one has i0∈ℕ so that for every i,j≥i0, we have
(18)ρfi−fj=∑n=0∞fni^−fnj^n+1pn<εϖp.
Fori,j≥i0, and k∈ℕ, one gets
(19)fki^−fkj^<ε.
Therefore,fkj^ is a Cauchy sequence in ℝ, for constant k∈ℕ; hence, limj⟶∞fkj^=fk0^, for constant k∈ℕ. So, ρfi−f0<εϖp, for all i≥i0. Finally, to investigate that f0∈Hwpn, one has
(20)ρf0=ρf0−fn+fn≤Kρfn−f0+ρfn<∞.
Therefore,f0∈Hwpn. So, the space Hwpnρ is a premodular Banach (ssfps).
By using Theorem14, we offer the next theorem.☐Theorem 16.
Assumepn∈miZ∩ℓ∞ with p0>0, then the space Hwpnρ be a prequasi Banach (ssfps), where
(21)ρf=∑n=0∞fn^n+1pn,forallf∈Hwpn.Theorem 17.
Ifpn∈miZ∩ℓ∞ with p0>0, then the space Hwpnρ is a prequasi closed (ssfps), where
(22)ρf=∑n=0∞fn^n+1pn,forallf∈Hwpn.Proof.
From Theorem16, the space Hwpnρ is a prequasi normed (ssfps). To prove that Hwpnρ is a prequasi closed (ssfps), suppose fii=0∞∈Hwpnρ and limi⟶∞ρfi−f0=0; then, for each ε∈0,1, one has i0∈ℕ so that for every i≥i0, we have
(23)ε>ρfi−f0=∑a=0∞fai^−fa0^n+1pa1/ϖp.
Therefore, fori≥i0 and a∈ℕ, one obtains fai^−fa0^<ε. Hence, fai^ is a convergent sequence in ℝ, for constant a∈ℕ. Hence, limi⟶∞fai^=fa0^, for constant a∈ℕ. Finally, to show that f0∈Hwpnρ, one has
(24)ρf0=ρf0−fi+fi≤ρfi−f0+ρfi<∞,so f0∈Hwpnρ. This explains that Hwpnρ is a prequasi closed (ssfps).☐


## 4. Properties of Operator Ideal

In this section, we offer some geometric and topological structure of the operator ideals generated bys-numbers and Hwpnρ.


### 4.1. Prequasi Ideal

Notations 18.
(25)SH≔SHX,Y;XandYareBanachSpaces,whereSHX,Y≔P∈LX,Y:fs∈H,where,fsz=∑n=0∞snPznconvergesforanyz∈ℂ.SHapp≔SHappX,Y;XandYareBanachSpaces,whereSHappX,Y≔P∈LX,Y:fapp∈H,where,fappz=∑n=0∞αnPznconvergesforanyz∈ℂ.SHKol≔SHKolX,Y;XandYareBanachSpaces,whereSHKolX,Y≔P∈LX,Y:fKol∈H,where,fKolz=∑n=0∞dnPznconvergesforanyz∈ℂTheorem 19.
SH is an operator ideal, whenever H is a (ssfps).Proof.
To proveSH is an operator ideal:
(i)
AssumeB∈FX,Y and rankB=n, for all n∈ℕ, as ek∈H, for every k∈ℕ and H is a linear space, then siBi=0∞=s0B,s1B,⋯,sn−1B,0,0,0,⋯. Hence(26)fsz=∑k=0∞skBekz=∑k=0n−1skBekz=∑k=0n−1skBzk,converges for any z∈ℂ. So fs∈H and hence B∈SHX,Y which gives FX,Y⊆SHX,Y.
(ii)
SupposeP1,P2∈SHX,Y and b1,b2∈ℝ. Hence fs,gs∈H so that fsz=∑n=0∞snP1zn and gsz=∑n=0∞snP2zn converge for any z∈ℂ. From Definition 12 part (3), we have f.,g.∈H, where f.z=∑n=0∞sn/2P1zn and g.z=∑n=0∞sn/2P2zn converge for any z∈ℂ. Since n≥2n/2, by using the definition of s-numbers and snP is a decreasing sequence, one has
snb1P1+b2P2≤s2n/2b1P1+b2P2≤sn/2b1P1+sn/2b2P2=b1sn/2P1+b2sn/2P2, for every n∈ℕ. As from Definition 12 part (2) and H is a linear space, one gets hs∈H, where hsz≕∑n=0∞snb1P1+b2P2zn converges for any z∈ℂ. Hence, b1P1+b2P2∈SHX,Y.
(iii)
AssumeP∈LX0,X, T∈SHX,Y, and R∈LY,Y0. Hence, fs∈H, where fsz=∑n=0∞snTzn converges for any z∈ℂ. As snRTP≤RsnTP, from Definition 12 parts (1) and (2), one obtains hs∈H, where hsz≕∑n=0∞snRTPzn converges for any z∈ℂ. This gives RTP∈SHX0,Y0. ☐Corollary 20.
Assumepn∈mi↗∩ℓ∞ with p0>0, then SHwpn be an operator ideal.


### 4.2. Ideal of Finite Rank Operators

In this section, the sufficient conditions (not necessary) onHwpnρ such that F is dense in SHwpnρ are presented. This investigates the nonlinearity of the s-type Hwpnρ spaces (Rhoades open problem [23]).Theorem 21.
FX,Y¯=SHwpnρX,Y, where
(27)ρf=∑n=0∞fn^n+1pn,forallf∈Hwpn,whenever pn∈miZ∩ℓ∞ with p0>0.Proof.
Evidently,FX,Y¯⊂SHwpnρX,Y, as the space SHwpnρ is an operator ideal. Hence, we have to show that SHwpnρX,Y⊆FX,Y¯. By choosing T∈SHwpnρX,Y, hence, fs∈Hwpnρ, with fsz=∑n=0∞snTzn converges for any z∈ℂ. Therefore, ρfs<∞; let ε∈0,1, one has m∈ℕ−0 so that ρfs−∑n=0m−1en<ε/4. Since snTn∈ℕ is decreasing, one obtains
(28)∑n=m+12ms2mTn+1pn≤∑n=m+12msnTn+1pn≤∑n=m∞snTn+1pn<ε4.
So, one hasA∈F2mX,Y, rankA≤2m, and
(29)∑n=2m+13mT−An+1pn≤∑n=m+12mT−An+1pn<ε4.
Becausepn∈ℓ∞, so we have
(30)∑n=0mT−An+1pn<ε4.
AsT−A∈SHwpnρX,Y, hence hs∈Hwpnρ, where hsz≔∑n=0∞snT−Azn converges for any z∈ℂ. In view of pn is increasing also the inequalities (2)–(4) explain that
(31)dT,A=ρhs=∑n=03m−1snT−An+1pn+∑n=3m∞snT−An+1pn≤∑n=03mT−An+1pn+∑n=m∞sn+2mT−An+2m+1pn+2m≤3∑n=0mT−An+1pn+∑n=m∞snTn+1pn<ε.
AsI7∈SH0ρX,Y but the condition p0>0 is not satisfied which implies a negative example of the converse statement. This completes the proof. ☐From Theorem21, we can say that if pn∈miZ∩ℓ∞ with p0>0, then each compact operators can be approximated by finite rank operators, and the converse is not always true.


### 4.3. Banach and Closed Prequasi Ideal

For which spaceHρ are the elements of prequasi operator ideal SHρ complete?Theorem 22.
IfHρ is a premodular (ssfps), then the function gP=ρfs is a prequasi norm on SHρ, where fsz=∑n=0∞snPzn converges for any z∈ℂ.Proof.
LetHρ be a premodular (ssfps), so g satisfies the following conditions:
(1)
SupposeP∈SHρX,Y, gP=ρfs≥0, and gP=ρfs=0, if and only if, snP=0, for all n∈N, if and only if, P=0(2)
One hasl≥1 with gλP=ρλfs≤lλρfs=lλgP, for all P∈SHρX,Y and λ∈ℝ(3)
We haveKK0≥1 so that for P1,P2∈SHρX,Y. Hence, f1sz=∑n=0∞snP1zn and f2sz=∑n=0∞snP2zn converge for any z∈ℂ. Hence, for hsz≕∑n=0∞snP1+P2zn, we have(32)gP1+P2=ρhs≤ρf1s.+f2s.≤Kρf1s.+ρf2s.≤KK0gP1+gP2.(4)
There areC≥1, let A∈LX0,X, B∈SHρX,Y, and D∈LY,Y0. Hence, fsz=∑n=0∞snBzn converges for all z∈ℂ. Then, for hsz≕∑n=0∞snDBAzn, one has(33)gDBA=ρhs≤ρADfs≤CAgBD.
☐Theorem 23.
SupposeX and Y be Banach spaces, and Hρ be a premodular (ssfps), then SHρ,g be a prequasi Banach operator ideal, where gP=ρfs and fsz=∑n=0∞snPzn converge for any z∈ℂ.Proof.
SinceHρ is a premodular (ssfps), then the function gP=ρfs is a prequasi norm on SHρ. Assume Pm be a Cauchy sequence in SHρX,Y. Hence, fsm∈Hρ and fsmz=∑n=0∞snPmzn converge for any z∈ℂ. Let hsz≕∑n=0∞snPi−Pjzn, then by using conditions (iv) and (vii) of Definition 13 and as LX,Y⊇SHρX,Y, one has
(34)gPi−Pj=ρhs≥ρs0Pi−Pje0=ρPi−Pje0≥ξPi−Pjρe0,hence, Pmm∈ℕ is a Cauchy sequence in LX,Y. But the space LX,Y is a Banach space, so one has P∈LX,Y with limm⟶∞Pm−P=0 and since fsm∈Hρ, for all m∈ℕ. Therefore, from Theorem 22 and the continuously of ρ at θ, one gets
(35)gP=gP−Pm+Pm≤KK0gPm−P+gPm=KK0ρPm−P∑m=0∞em+KK0ρfsm<ε,then fs∈Hρ, this gives P∈SHρX,Y. ☐Corollary 24.
SupposeX and Y be Banach spaces, and pn∈miZ∩ℓ∞ with p0>0, then SHwpn be a prequasi Banach operator ideal.Theorem 25.
AssumeX and Y be Banach spaces and Hρ be a premodular (ssfps), then SHρ,g is a prequasi closed operator ideal, where gP=ρfs, where fsz=∑n=0∞snPzn converges for any z∈ℂ.Proof.
SinceHρ is a premodular (ssfps), then the function gP=ρfs is a prequasi norm on SHρ. Assume Pm∈SHρX,Y, with m∈ℕ and limm⟶∞gPm−P=0. Hence, fsm∈Hρ and fsmz=∑n=0∞snPmzn converge for any z∈ℂ. Let hsz≕∑n=0∞snPi−Pjzn, then by using conditions (iv) and (vii) of Definition 13 and as LX,Y⊇SHρX,Y, one obtains
(36)gP−Pj=ρhs≥ρs0P−Pje0=ρP−Pje0≥ξP−Pjρe0,hence, Pmm∈ℕ is a convergent sequence in LX,Y. But the space LX,Y is a Banach space, so one gets P∈LX,Y with limm⟶∞Pm−P=0 and since fsm∈Hρ, for all m∈ℕ, so from Theorem 22 and the continuously of ρ at θ, we have
(37)gP=gP−Pm+Pm≤KK0gPm−P+gPm=KK0ρPm−P∑m=0∞em+KK0ρfsm<ε,
thenfs∈Hρ, this implies P∈SHρX,Y. ☐Corollary 26.
SupposeX and Y be Banach spaces and pn∈mi↗∩ℓ∞ with p0>0, then SHwpn be a prequasi closed operator ideal.In view of Theorem10, we explain the next properties of the s-type Hwpnρ.Theorem 27.
Fors-type Hwpnρ≔snT∈ℝℕ:T∈SHwpnρX,Y. The following conditions are satisfied:
(1)
One hass-type Hwpnρ⊃F(2)
AssumesrT1r=0∞∈s-type Hwpnρ and srT2r=0∞∈s-type Hwpnρ, then srT1+T2r=0∞∈s-type Hwpnρ(3)
We haveλ∈ℝ and srTr=0∞∈s-type Hwpnρ, then λsrTr=0∞∈s- ype Hwpnρ(4)
Thes-type Hwpnρ is solid


### 4.4. Small Prequasi Banach Ideal

We offer here a few inclusion relations for the spaceSHwpnρ for distinct pn.Theorem 28.
SupposeX and Y be Banach spaces with dimX=dimY=∞ and pn,qn∈mi↗∩ℓ∞ with p0>0 and pn<qn, for every n∈ℕ, one has
(38)SHwpnρX,Y⫋SHwqnρX,YÜLX,Y.Proof.
LetT∈SHwpnρX,Y. Hence, fs∈Hwpnρ and fsz=∑n=0∞snTzn converge for any z∈ℂ. We have
(39)∑n=0∞snTn+1qn<∑n=0∞snTn+1pn<∞,so T∈SHwqnρX,Y. After, by choosing T with snT=n+11−1/pn, hence T∉SHwpnρX,Y and T∈SHwqnρX,Y. Obviously, SHwqnρX,Y⊂LX,Y. Next, by taking T with snT=n+11−1/qn, then T∉SHwqnρX,Y and T∈LX,Y. This completes the proof. ☐
In this part, we explore the conditions for whichSHwpnρapp is small. ☐Theorem 29.
AssumeX and Y be Banach spaces with dimX=dimY=∞. Suppose pn∈mi↗∩0,1ℕ, then SHwpnρapp be small.Proof.
Evidently, the spaceSHwpnρapp,g constructs a prequasi Banach operator ideal, with gT=∑j=0∞αjT/j+1pj. Assume SHwpnρappX,Y=LX,Y. Therefore, one has C>0 with gT≤CT, for all T∈LX,Y. From Dvoretzky’s Theorem [28] with r∈ℕ, we have quotient spaces X/λr and subspaces ηr of Y that transformed onto ℓ2r by isomorphisms Dr and Br with DrDr−1≤2 and BrBr−1≤2. Assume Ir be the identity operator on ℓ2r, ζr be the quotient operator from X onto X/λr, and Jr be the natural embedding operator from ηr into Y. Let ha be the Bernstein numbers [29], one obtains
(40)1=haIr=haBrBr−1IrDrDr−1≤BrhaBr−1IrDrDr−1=BrhaJrBr−1IrDrDr−1≤BrdaJrBr−1IrDrDr−1=BrdaJrBr−1IrDrζrDr−1≤BrαaJrBr−1IrDrζrDr−1,for 0≤j≤r. Hence, for l≥1, we have
(41)1j+1pj≤BrDr−1pjαjJrBr−1IrDrζrj+1pj⇒1j+1pj≤lBrαjJrBr−1IrDrζrj+1pjDr−1⇒∑j=0r1j+1pj≤lBrDr−1∑j=0rαjJrBr−1IrDrζrj+1pj⇒∑j=0r1j+1pj≤lBrDr−1gJrBr−1IrDrζr⇒∑j=0r1j+1pj≤lCBrDr−1∥JrBr−1IrDrζr∥⇒∑j=0r1j+1pj≤lCBrDr−1∥JrBr−1∥∥Ir∥∥Drζr∥=lCBrDr−1Br−1IrDr⇒∑j=0r1j+1pj≤4lC.
Sincer⟶∞, then ∑j=0∞1/j+1pj<∞. As ∑j=0∞1/j+1pj≥∑j=0∞1/j+1=∞. Therefore, the space SHwpnρapp is small. ☐Theorem 30.
SupposeX and Y be Banach spaces with dimX=dimY=∞. Let pn∈mi↗∩0,1ℕ, then SHwpnρKol be small.


### 4.5. Simple Prequasi Ideal

We declare an answer of the following investigation; for whichHwpnρ is the space SHwpnρ simple?Theorem 31.
Supposepn,qn∈mi↗∩ℓ∞ with 1≤pn<qn, for all n∈ℕ, hence
(42)LSHwqnρ,SHwpnρ=ΛSHwqnρ,SHwpnρ.Proof.
Assume we haveT∈LSHwqnρ,SHwpnρ and T∉ΛSHwqnρ,SHwpnρ. From Lemma 4, there are G∈LSHwqnρ and B∈LSHwpnρ with BTGIm=Im. For all m∈ℕ, we have
(43)ImSHwpnρ=∑n=0∞snImn+1pn1/suppn=∑n=0m−11n+1pn1/suppn≤BTGImSHwqnρ≤∑n=0∞snImn+1qn1/supqn=∑n=0m−11n+1qn1/supqn.
This defies Theorem28. ☐Corollary 32.
Assumepn,qn∈mi↗∩ℓ∞ with 1≤pn<qn, for all n∈ℕ, hence
(44)LSHwqnρ,SHwpnρ=LCSHwqnρ,SHwpnρ.Proof.
Definitely fromΛ⊆Lc. ☐Theorem 33.
Supposepn∈mi↗∩ℓ∞ with p0≥1, hence SHwpnρ is simple.Proof.
LetT∈LCSHwpnρ and T∉ΛSHwpnρ. By using Lemma 4, there are G,B∈LSHwpnρ such that BTGIk=Ik. One obtains ISHwpnρ∈LCSHwpnρ. Hence, LSHwpnρ=LCSHwpnρ. This gives a unique nontrivial closed ideal ΛSHwpnρ in LSHwpnρ. ☐


### 4.6. Spectrum of Prequasi Ideal

In this part, we give the sufficient conditions onHwpnρ such that the class L with sequence of eigenvalues in Hwpnρ is strictly contained in SHwpnρ.Notations 34.
(45)SHρλ≔SHρλX,Y;XandYareBanachSpaces,whereSHρλX,Y≔T∈LX,Y:fλ∈Hρ,where,fλz=∑n=0∞λnTznconvergesforanyz∈ℂandT−λlTI=0,foreveryl∈ℕ.Theorem 35.
Pick up any Banach spacesX and Y with dimX=dimY=∞. Assume pn∈mi↗∩ℓ∞ with p0>0, one has
(46)SHwpnρλX,YÜSHwpnρX,Y.Proof.
SupposeT∈SHwpnρλX,Y, therefore fλ∈Hwpnρ, where fλz=∑n=0∞λnTzn converges for every z∈ℂ with ρfλ=∑n=0∞λnT/n+1pn<∞, and T−λlTI=0, for every l∈ℕ. One can see T=λlTI, with l∈ℕ, hence slT=slλlTI=λlT, with l∈ℕ. As a result, fs∈Hwpnρ, hence T∈SHwpnρX,Y. Finally, by taking snT=n+11−2/pn and λnT=n+11−1/pn. Therefore, T∉SHwpnρλX,Y and T∈SHwpnρX,Y. This finishes the proof. ☐


## 4.1. Prequasi Ideal

Notations 18.
(25)SH≔SHX,Y;XandYareBanachSpaces,whereSHX,Y≔P∈LX,Y:fs∈H,where,fsz=∑n=0∞snPznconvergesforanyz∈ℂ.SHapp≔SHappX,Y;XandYareBanachSpaces,whereSHappX,Y≔P∈LX,Y:fapp∈H,where,fappz=∑n=0∞αnPznconvergesforanyz∈ℂ.SHKol≔SHKolX,Y;XandYareBanachSpaces,whereSHKolX,Y≔P∈LX,Y:fKol∈H,where,fKolz=∑n=0∞dnPznconvergesforanyz∈ℂTheorem 19.
SH is an operator ideal, whenever H is a (ssfps).Proof.
To proveSH is an operator ideal:
(i)
AssumeB∈FX,Y and rankB=n, for all n∈ℕ, as ek∈H, for every k∈ℕ and H is a linear space, then siBi=0∞=s0B,s1B,⋯,sn−1B,0,0,0,⋯. Hence(26)fsz=∑k=0∞skBekz=∑k=0n−1skBekz=∑k=0n−1skBzk,converges for any z∈ℂ. So fs∈H and hence B∈SHX,Y which gives FX,Y⊆SHX,Y.
(ii)
SupposeP1,P2∈SHX,Y and b1,b2∈ℝ. Hence fs,gs∈H so that fsz=∑n=0∞snP1zn and gsz=∑n=0∞snP2zn converge for any z∈ℂ. From Definition 12 part (3), we have f.,g.∈H, where f.z=∑n=0∞sn/2P1zn and g.z=∑n=0∞sn/2P2zn converge for any z∈ℂ. Since n≥2n/2, by using the definition of s-numbers and snP is a decreasing sequence, one has
snb1P1+b2P2≤s2n/2b1P1+b2P2≤sn/2b1P1+sn/2b2P2=b1sn/2P1+b2sn/2P2, for every n∈ℕ. As from Definition 12 part (2) and H is a linear space, one gets hs∈H, where hsz≕∑n=0∞snb1P1+b2P2zn converges for any z∈ℂ. Hence, b1P1+b2P2∈SHX,Y.
(iii)
AssumeP∈LX0,X, T∈SHX,Y, and R∈LY,Y0. Hence, fs∈H, where fsz=∑n=0∞snTzn converges for any z∈ℂ. As snRTP≤RsnTP, from Definition 12 parts (1) and (2), one obtains hs∈H, where hsz≕∑n=0∞snRTPzn converges for any z∈ℂ. This gives RTP∈SHX0,Y0. ☐Corollary 20.
Assumepn∈mi↗∩ℓ∞ with p0>0, then SHwpn be an operator ideal.


## 4.2. Ideal of Finite Rank Operators

In this section, the sufficient conditions (not necessary) onHwpnρ such that F is dense in SHwpnρ are presented. This investigates the nonlinearity of the s-type Hwpnρ spaces (Rhoades open problem [23]).Theorem 21.
FX,Y¯=SHwpnρX,Y, where
(27)ρf=∑n=0∞fn^n+1pn,forallf∈Hwpn,whenever pn∈miZ∩ℓ∞ with p0>0.Proof.
Evidently,FX,Y¯⊂SHwpnρX,Y, as the space SHwpnρ is an operator ideal. Hence, we have to show that SHwpnρX,Y⊆FX,Y¯. By choosing T∈SHwpnρX,Y, hence, fs∈Hwpnρ, with fsz=∑n=0∞snTzn converges for any z∈ℂ. Therefore, ρfs<∞; let ε∈0,1, one has m∈ℕ−0 so that ρfs−∑n=0m−1en<ε/4. Since snTn∈ℕ is decreasing, one obtains
(28)∑n=m+12ms2mTn+1pn≤∑n=m+12msnTn+1pn≤∑n=m∞snTn+1pn<ε4.
So, one hasA∈F2mX,Y, rankA≤2m, and
(29)∑n=2m+13mT−An+1pn≤∑n=m+12mT−An+1pn<ε4.
Becausepn∈ℓ∞, so we have
(30)∑n=0mT−An+1pn<ε4.
AsT−A∈SHwpnρX,Y, hence hs∈Hwpnρ, where hsz≔∑n=0∞snT−Azn converges for any z∈ℂ. In view of pn is increasing also the inequalities (2)–(4) explain that
(31)dT,A=ρhs=∑n=03m−1snT−An+1pn+∑n=3m∞snT−An+1pn≤∑n=03mT−An+1pn+∑n=m∞sn+2mT−An+2m+1pn+2m≤3∑n=0mT−An+1pn+∑n=m∞snTn+1pn<ε.
AsI7∈SH0ρX,Y but the condition p0>0 is not satisfied which implies a negative example of the converse statement. This completes the proof. ☐From Theorem21, we can say that if pn∈miZ∩ℓ∞ with p0>0, then each compact operators can be approximated by finite rank operators, and the converse is not always true.


## 4.3. Banach and Closed Prequasi Ideal

For which spaceHρ are the elements of prequasi operator ideal SHρ complete?Theorem 22.
IfHρ is a premodular (ssfps), then the function gP=ρfs is a prequasi norm on SHρ, where fsz=∑n=0∞snPzn converges for any z∈ℂ.Proof.
LetHρ be a premodular (ssfps), so g satisfies the following conditions:
(1)
SupposeP∈SHρX,Y, gP=ρfs≥0, and gP=ρfs=0, if and only if, snP=0, for all n∈N, if and only if, P=0(2)
One hasl≥1 with gλP=ρλfs≤lλρfs=lλgP, for all P∈SHρX,Y and λ∈ℝ(3)
We haveKK0≥1 so that for P1,P2∈SHρX,Y. Hence, f1sz=∑n=0∞snP1zn and f2sz=∑n=0∞snP2zn converge for any z∈ℂ. Hence, for hsz≕∑n=0∞snP1+P2zn, we have(32)gP1+P2=ρhs≤ρf1s.+f2s.≤Kρf1s.+ρf2s.≤KK0gP1+gP2.(4)
There areC≥1, let A∈LX0,X, B∈SHρX,Y, and D∈LY,Y0. Hence, fsz=∑n=0∞snBzn converges for all z∈ℂ. Then, for hsz≕∑n=0∞snDBAzn, one has(33)gDBA=ρhs≤ρADfs≤CAgBD.
☐Theorem 23.
SupposeX and Y be Banach spaces, and Hρ be a premodular (ssfps), then SHρ,g be a prequasi Banach operator ideal, where gP=ρfs and fsz=∑n=0∞snPzn converge for any z∈ℂ.Proof.
SinceHρ is a premodular (ssfps), then the function gP=ρfs is a prequasi norm on SHρ. Assume Pm be a Cauchy sequence in SHρX,Y. Hence, fsm∈Hρ and fsmz=∑n=0∞snPmzn converge for any z∈ℂ. Let hsz≕∑n=0∞snPi−Pjzn, then by using conditions (iv) and (vii) of Definition 13 and as LX,Y⊇SHρX,Y, one has
(34)gPi−Pj=ρhs≥ρs0Pi−Pje0=ρPi−Pje0≥ξPi−Pjρe0,hence, Pmm∈ℕ is a Cauchy sequence in LX,Y. But the space LX,Y is a Banach space, so one has P∈LX,Y with limm⟶∞Pm−P=0 and since fsm∈Hρ, for all m∈ℕ. Therefore, from Theorem 22 and the continuously of ρ at θ, one gets
(35)gP=gP−Pm+Pm≤KK0gPm−P+gPm=KK0ρPm−P∑m=0∞em+KK0ρfsm<ε,then fs∈Hρ, this gives P∈SHρX,Y. ☐Corollary 24.
SupposeX and Y be Banach spaces, and pn∈miZ∩ℓ∞ with p0>0, then SHwpn be a prequasi Banach operator ideal.Theorem 25.
AssumeX and Y be Banach spaces and Hρ be a premodular (ssfps), then SHρ,g is a prequasi closed operator ideal, where gP=ρfs, where fsz=∑n=0∞snPzn converges for any z∈ℂ.Proof.
SinceHρ is a premodular (ssfps), then the function gP=ρfs is a prequasi norm on SHρ. Assume Pm∈SHρX,Y, with m∈ℕ and limm⟶∞gPm−P=0. Hence, fsm∈Hρ and fsmz=∑n=0∞snPmzn converge for any z∈ℂ. Let hsz≕∑n=0∞snPi−Pjzn, then by using conditions (iv) and (vii) of Definition 13 and as LX,Y⊇SHρX,Y, one obtains
(36)gP−Pj=ρhs≥ρs0P−Pje0=ρP−Pje0≥ξP−Pjρe0,hence, Pmm∈ℕ is a convergent sequence in LX,Y. But the space LX,Y is a Banach space, so one gets P∈LX,Y with limm⟶∞Pm−P=0 and since fsm∈Hρ, for all m∈ℕ, so from Theorem 22 and the continuously of ρ at θ, we have
(37)gP=gP−Pm+Pm≤KK0gPm−P+gPm=KK0ρPm−P∑m=0∞em+KK0ρfsm<ε,
thenfs∈Hρ, this implies P∈SHρX,Y. ☐Corollary 26.
SupposeX and Y be Banach spaces and pn∈mi↗∩ℓ∞ with p0>0, then SHwpn be a prequasi closed operator ideal.In view of Theorem10, we explain the next properties of the s-type Hwpnρ.Theorem 27.
Fors-type Hwpnρ≔snT∈ℝℕ:T∈SHwpnρX,Y. The following conditions are satisfied:
(1)
One hass-type Hwpnρ⊃F(2)
AssumesrT1r=0∞∈s-type Hwpnρ and srT2r=0∞∈s-type Hwpnρ, then srT1+T2r=0∞∈s-type Hwpnρ(3)
We haveλ∈ℝ and srTr=0∞∈s-type Hwpnρ, then λsrTr=0∞∈s- ype Hwpnρ(4)
Thes-type Hwpnρ is solid


## 4.4. Small Prequasi Banach Ideal

We offer here a few inclusion relations for the spaceSHwpnρ for distinct pn.Theorem 28.
SupposeX and Y be Banach spaces with dimX=dimY=∞ and pn,qn∈mi↗∩ℓ∞ with p0>0 and pn<qn, for every n∈ℕ, one has
(38)SHwpnρX,Y⫋SHwqnρX,YÜLX,Y.Proof.
LetT∈SHwpnρX,Y. Hence, fs∈Hwpnρ and fsz=∑n=0∞snTzn converge for any z∈ℂ. We have
(39)∑n=0∞snTn+1qn<∑n=0∞snTn+1pn<∞,so T∈SHwqnρX,Y. After, by choosing T with snT=n+11−1/pn, hence T∉SHwpnρX,Y and T∈SHwqnρX,Y. Obviously, SHwqnρX,Y⊂LX,Y. Next, by taking T with snT=n+11−1/qn, then T∉SHwqnρX,Y and T∈LX,Y. This completes the proof. ☐
In this part, we explore the conditions for whichSHwpnρapp is small. ☐Theorem 29.
AssumeX and Y be Banach spaces with dimX=dimY=∞. Suppose pn∈mi↗∩0,1ℕ, then SHwpnρapp be small.Proof.
Evidently, the spaceSHwpnρapp,g constructs a prequasi Banach operator ideal, with gT=∑j=0∞αjT/j+1pj. Assume SHwpnρappX,Y=LX,Y. Therefore, one has C>0 with gT≤CT, for all T∈LX,Y. From Dvoretzky’s Theorem [28] with r∈ℕ, we have quotient spaces X/λr and subspaces ηr of Y that transformed onto ℓ2r by isomorphisms Dr and Br with DrDr−1≤2 and BrBr−1≤2. Assume Ir be the identity operator on ℓ2r, ζr be the quotient operator from X onto X/λr, and Jr be the natural embedding operator from ηr into Y. Let ha be the Bernstein numbers [29], one obtains
(40)1=haIr=haBrBr−1IrDrDr−1≤BrhaBr−1IrDrDr−1=BrhaJrBr−1IrDrDr−1≤BrdaJrBr−1IrDrDr−1=BrdaJrBr−1IrDrζrDr−1≤BrαaJrBr−1IrDrζrDr−1,for 0≤j≤r. Hence, for l≥1, we have
(41)1j+1pj≤BrDr−1pjαjJrBr−1IrDrζrj+1pj⇒1j+1pj≤lBrαjJrBr−1IrDrζrj+1pjDr−1⇒∑j=0r1j+1pj≤lBrDr−1∑j=0rαjJrBr−1IrDrζrj+1pj⇒∑j=0r1j+1pj≤lBrDr−1gJrBr−1IrDrζr⇒∑j=0r1j+1pj≤lCBrDr−1∥JrBr−1IrDrζr∥⇒∑j=0r1j+1pj≤lCBrDr−1∥JrBr−1∥∥Ir∥∥Drζr∥=lCBrDr−1Br−1IrDr⇒∑j=0r1j+1pj≤4lC.
Sincer⟶∞, then ∑j=0∞1/j+1pj<∞. As ∑j=0∞1/j+1pj≥∑j=0∞1/j+1=∞. Therefore, the space SHwpnρapp is small. ☐Theorem 30.
SupposeX and Y be Banach spaces with dimX=dimY=∞. Let pn∈mi↗∩0,1ℕ, then SHwpnρKol be small.


## 4.5. Simple Prequasi Ideal

We declare an answer of the following investigation; for whichHwpnρ is the space SHwpnρ simple?Theorem 31.
Supposepn,qn∈mi↗∩ℓ∞ with 1≤pn<qn, for all n∈ℕ, hence
(42)LSHwqnρ,SHwpnρ=ΛSHwqnρ,SHwpnρ.Proof.
Assume we haveT∈LSHwqnρ,SHwpnρ and T∉ΛSHwqnρ,SHwpnρ. From Lemma 4, there are G∈LSHwqnρ and B∈LSHwpnρ with BTGIm=Im. For all m∈ℕ, we have
(43)ImSHwpnρ=∑n=0∞snImn+1pn1/suppn=∑n=0m−11n+1pn1/suppn≤BTGImSHwqnρ≤∑n=0∞snImn+1qn1/supqn=∑n=0m−11n+1qn1/supqn.
This defies Theorem28. ☐Corollary 32.
Assumepn,qn∈mi↗∩ℓ∞ with 1≤pn<qn, for all n∈ℕ, hence
(44)LSHwqnρ,SHwpnρ=LCSHwqnρ,SHwpnρ.Proof.
Definitely fromΛ⊆Lc. ☐Theorem 33.
Supposepn∈mi↗∩ℓ∞ with p0≥1, hence SHwpnρ is simple.Proof.
LetT∈LCSHwpnρ and T∉ΛSHwpnρ. By using Lemma 4, there are G,B∈LSHwpnρ such that BTGIk=Ik. One obtains ISHwpnρ∈LCSHwpnρ. Hence, LSHwpnρ=LCSHwpnρ. This gives a unique nontrivial closed ideal ΛSHwpnρ in LSHwpnρ. ☐


## 4.6. Spectrum of Prequasi Ideal

In this part, we give the sufficient conditions onHwpnρ such that the class L with sequence of eigenvalues in Hwpnρ is strictly contained in SHwpnρ.Notations 34.
(45)SHρλ≔SHρλX,Y;XandYareBanachSpaces,whereSHρλX,Y≔T∈LX,Y:fλ∈Hρ,where,fλz=∑n=0∞λnTznconvergesforanyz∈ℂandT−λlTI=0,foreveryl∈ℕ.Theorem 35.
Pick up any Banach spacesX and Y with dimX=dimY=∞. Assume pn∈mi↗∩ℓ∞ with p0>0, one has
(46)SHwpnρλX,YÜSHwpnρX,Y.Proof.
SupposeT∈SHwpnρλX,Y, therefore fλ∈Hwpnρ, where fλz=∑n=0∞λnTzn converges for every z∈ℂ with ρfλ=∑n=0∞λnT/n+1pn<∞, and T−λlTI=0, for every l∈ℕ. One can see T=λlTI, with l∈ℕ, hence slT=slλlTI=λlT, with l∈ℕ. As a result, fs∈Hwpnρ, hence T∈SHwpnρX,Y. Finally, by taking snT=n+11−2/pn and λnT=n+11−1/pn. Therefore, T∉SHwpnρλX,Y and T∈SHwpnρX,Y. This finishes the proof. ☐


## 5. Weighted Shift Operators onHwpnρ

In this section, we offer the upper bounds ofs-numbers for infinite series of the weighted n-th power forward shift operator on Hwpn with applications to some entire functions.Definition 36.
IfHρ is a prequasi normed (ssfps). An operator Vz:Hρ⟶Hρ is named forward shift operator if Vzf=zf, for every f∈Hρ, where Vzfz=∑n=0∞fn^zn+1 converges for any z∈ℂ and ρVzf<∞.Definition 37.
IfHρ is a prequasi normed (ssfps). An operator Bz:Hρ⟶Hρ is named backward shift operator if Bzfz=fz−f0/z, for every f∈Hρ, where Bzfz=∑n=0∞fn+1^zn converges for all z∈ℂ and ρBzf<∞.Theorem 38.
Ifpn∈mi↗∩ℓ∞ with p0>0, hence Vz∈LHwpnρ with
(47)suprr+1prr+2pr+11/ϖp≤Vz≤suprr+1r+2pr/ϖp,where ρf=∑r=0∞fr^/r+1pr1/ϖp, for each f∈Hwpnρ.Proof.
Assume the setup be verified. Forf∈Hwpnρ. As pn∈mi↗∩ℓ∞ with p0>0, hence
(48)ρVzf=ρzf=∑r=0∞fr^r+2pr+11/ϖp≤∑r=0∞fr^r+2pr1/ϖp≤suprr+1r+2pr/ϖp∑r=0∞fr^r+1pr1/ϖp=suprr+1r+2pr/ϖpρf.
Then,Vz∈LHwpnρ with Vz≤suprr+1/r+2pr/ϖp. As Vz∈LHwpnρ. Hence, one has A>0 so that ρVzf≤Aρf, for every f∈Hwpnρ. Therefore, ρVzer≤Aρer, then suprr+1pr/r+2pr+11/ϖp≤Vz. This finishes the proof. ☐Theorem 39.
Ifpn∈mi↗∩ℓ∞ with p0>0, hence Bz∈LHwpnρ with
(49)suprr+2pr+1r+1pr1/ϖp≤Bz≤suprr+2r+1pr/ϖp,where ρf=∑r=0∞fr^/r+1pr1/ϖp, for any f∈Hwpnρ.Proof.
Assume the setup be verified. Forf∈Hwpnρ. As pn∈miZ∩ℓ∞ with p0>0, hence
(50)ρBzf=∑r=0∞fr+1^r+1pr1/ϖp≤suprr+2r+1pr/ϖp∑r=0∞fr+1^r+2pr1/ϖp≤suprr+2r+1pr/ϖp∑r=0∞fr^r+1pr1/ϖp=suprr+2r+1pr/ϖpρf.
Then,Bz∈LHwpnρ with Bz≤suprr+2/r+1pr/ϖp. As Bz∈LHwpnρ. Hence, one has A>0 so that ρBzf≤Aρf, for every f∈Hwpnρ. Therefore, ρBzer+1≤Aρer+1, so suprr+2pr+1/r+1pr1/ϖp≤Bz. This finishes the proof. ☐Theorem 40.
Ifpn∈mi↗∩ℓ∞ with p0≥1. Assume limsupn⟶∞1/n+1pn/n=1, hence all function in Hwpnρ is analytic on the open unit disc D. More, the convergence in Hwpnρ gives the uniform convergence on compact subsets of D, where ρf=∑r=0∞fr^/r+1pr1/ϖp, for any f∈Hwpnρ.Proof.
Assumelimsupn⟶∞1/n+1pn/n=1, and f∈Hwpnρ. Hence, fz=∑n=0∞fn^zn converges for any z∈ℂ and ρf=∑n=0∞fn^/n+1pn1/ϖp<∞. So, limsupn⟶∞fn^/n+1pnn<1. One has
(51)limsupn⟶∞fn^pnn<1limsupn⟶∞1/n+1pn/n=1.
Aspn∈miZ∩ℓ∞ with p0≥1, one gets limsupn⟶∞fn^nz<z<1, for every z∈D. Then, fz=∑n=0∞fn^zn converges for any complex value of z∈D. Suppose A be a compact subset of D, and fkz∈A, for every k∈ℕ. Assume fk converges to f∈Hwpnρ, one obtains
(52)fkz−fz=∑n=0∞fnk^−fn^zn≤∑n=0∞fnk^−fn^zn≤∑n=0∞fnk^−fn^n+1pn1/ϖp∑n=0∞n+1qnznqn1/ϖp=∑n=0∞n+1qnznqn1/ϖpρfk−f,
whereqn∈miZ∩ℓ∞ with q0≥1 and 1/pn+1/qn=1, for every n∈ℕ. Obviously, limsupn⟶∞n+1qn/nzqn<1, hence ∑n=0∞n+1qnznqn1/ϖq<∞. Then, limk⟶∞fkz=fz∈A. ☐Theorem 41.
SupposeVz be the forward shift operator on Hwpnρ, with ρf=∑r=0∞fr^/r+1pr1/ϖp, for every f∈Hwpnρ. One has
(53)supcardξ=r+1infk∈ξk+1k+n+1pk+n/ϖp1An≤srVzn≤supcardξ=r+1infk∈ξk+1k+n+1pk+n/ϖp,where An=∑k=0∞fk^/k+1pk1/ϖp/∑k∈ξfk^/k+1pk+n1/ϖp.Proof.
Assume cardξ=r+1 and since Vznf∈Hwpnρ, for every f∈Hwpnρ, where fz=∑k=0∞fk^zk converges for any z∈ℂ and ρf=∑k=0∞fk^/k+1pk1/ϖp<∞. Therefore, Vznfz=∑k=0∞fk^zk+n and ρVznf=∑k=0∞fk^/k+n+1pk+n1/ϖp<∞.
SupposePξ be an operator on Hwpnρ with rank Pξ=r+1 evident by
(54)Pξgz=Pξ∑k=0∞fk^zk+n=∑k∈ξ∞fk^zk+n.
AsρPξg=∑k∈ξfk^/k+n+1pk+n1/ϖp≤∑k=0∞fk^/k+n+1pk+n1/ϖp=ρg. This implies Pξ≤1. Define an operator Szn by Sznhz=Szn∑k∈ξfk^zk+n=∑k=0∞fk^zk, one gets
(55)ρSznh=∑k=0∞fk^k+1pk1/ϖp≤Un∑k∈ξfk^k+n+1pk+n1/ϖp=Unρg.
Therefore,Szn≤Un, where 1≤Un=∑k=0∞fk^/k+1pk1/ϖp/∑k∈ξfk^/k+n+1pk+n1/ϖp<∞. Hence, the identity operator will be Ir+1=PξVznSzn, in view of the definition of s-numbers, one has
(56)srIr+1=1≤PξsrVznSzn≤srVznSzn⇒srVzn≥1Szn≥1Un=∑k∈ξfk^/k+n+1pk+n1/ϖp∑k=0∞fk^/k+1pk1/ϖp≥infk∈ξk+1k+n+1pk+n/ϖp1An.
This inequality is confirmed for all cardξ=r+1, we have
(57)srVzn≥supcardξ=r+1infk∈ξk+1k+n+1pk+n/ϖp1An.
On the other hand, supposeξ be a subset of ℕ with card ξ=r. Define the finite rank operator Rzn by Rznvz=Rzn∑k=0∞fk^zk=∑k∈ξfk^zk+n. From the definition of approximation numbers, one gets
(58)srVzn≤αrVzn≤Vzn−Rzn≤supfz≠0Vzn−Rznfzfz=supfz≠0∑k∉ξfk^zk+nfz≤supfz≠0∑k∉ξfk^/k+n+1pk+n1/ϖpfz≤supk∉ξk+1k+n+1pk+n/ϖp.
This inequality is satisfied for any cardξ=r and from Lemma 11, we have
(59)supcardξ=r+1infk∈ξk+1k+n+1pk+n/ϖp1An≤srVzn≤infcardξ=rsupk∉ξk+1k+n+1pk+n/ϖp=supcardξ=r+1infk∈ξk+1k+n+1pk+n/ϖp.
This finishes the proof. ☐Here, the upper and lower bounds of norm∑m=0∞cmVzm on the space Hwpnρ have been introduced.Theorem 42.
Acting∑m=0∞cmVzm on the space Hwpnρ, where ρf=∑r=0∞fr^/r+1pr1/ϖp, for every f∈Hwpnρ, cmm=0∞∈ℓpm/ϖp, and pn∈mi↗∩ℓ∞ with p0≥1, one has
(60)supk∑m=0∞cmpm+kk+1pkm+k+1pm+k1/ϖp≤∑m=0∞cmVzm≤supm,kk+1m+k+1pm+k/ϖp∑m=0∞cmpm/ϖp.Proof.
Letf∈Hwpnρ, one gets ∑m=0∞cmVzmfz=∑k=0∞∑m=0∞cmfk^zk+m. We have
(61)∑m=0∞cmVzm≥ρ∑m=0∞cmVzmekρek=∑m=0∞cm/m+k+1pm+k1/k+1pk1/ϖp≥supk∑m=0∞cmpm+kk+1pkm+k+1pm+k1/ϖp.
Asρ verifies the triangle inequality, one can see
(62)∑m=0∞cmVzm=supρf≠0ρ∑m=0∞cmVzmfρf≤supρf≠0∑m=0∞∑k=0∞cmfk^/m+k+1pm+k1/ϖp∑k=0∞fk^/k+1pk1/ϖp≤supm,kk+1m+k+1pm+k/ϖp∑m=0∞∑k=0∞cmfk^/k+1pm+k1/ϖp∑k=0∞fk^/k+1pk1/ϖp≤supm,kk+1m+k+1pm+k/ϖp∑m=0∞cmpm/ϖp.
The next theorem investigates an upper estimation to thes-numbers of ∑m=0∞cmVzm on the space Hwpnρ. ☐Theorem 43.
Acting∑m=0∞cmVzm on the space Hwpnρ, where ρf=∑r=0∞fr^/r+1pr1/ϖp, for everyf∈Hwpnρ, the s−numbers of this operator are supposed by
(63)sr∑m=0∞cmVzm≤supcardξ=r+1infk∈ξsupmk+1m+k+1pm+k/ϖp∑m=0∞cmpm/ϖp,for all cmm=0∞∈ℓpm/ϖp and pn∈miZ∩ℓ∞ with p0≥1.Proof.
Assumeξ be a subset of ℕ and card ξ=r. From the definition of s-numbers. Define the finite rank operator R by Rfz=R∑k=0∞fk^zk=∑k∈ξ∑m=0kcmfk−m^zk. From the definition of approximation numbers and as ρ verifies the triangle inequality, one has
(64)sr∑m=0∞cmVzm≤αr∑m=0∞cmVzm≤∑m=0∞cmVzm−R≤supρf≠0ρ∑m=0∞cmVzmf−Rfρf≤supρf≠0∑m=0∞∑k∉ξcmfk^/k+m+1pk+mρf≤supk∉ξ,mk+1m+k+1pm+k/ϖp∑m=0∞cmpm/ϖp.
This inequality is satisfied for all cardξ=r, we have
(65)sr∑m=0∞cmVzm≤infcardξ=rsupk∉ξ,mk+1m+k+1pm+k/ϖp∑m=0∞cmpm/ϖp=supcardξ=r+1infk∈ξsupmk+1m+k+1pm+k/ϖp∑m=0∞cmpm/ϖp.
This performs the proof. ☐Definition 44.
By using the power series of an entire functiongz=∑m=0∞amzm, the shift operator Vgz is defined as:
(66)Vgzfz=∑m=0∞amVzmfz.
The next theorem implies a direct actions of the last theorem for some entire functions, for example, the exponential and the sine functions.Corollary 45.
Assumepn∈mi↗∩ℓ∞ with p0≥1. Suppose Vez be shift operator on Hwpnρ, where ρf=∑r=0∞fr^/r+1pr1/ϖp, for every f∈Hwpnρ and ez=∑m=0∞zm/m!. The upper estimation of the s-numbers of Vez is pretended by
(67)srVez≤supcardξ=r+1infk∈ξsupmk+1m+k+1pm+k/ϖp∑m=0∞1m!pm/ϖp.Proof.
Clearly, from Theorem43.☐Corollary 46.
Supposepn∈mi↗∩ℓ∞ with p0≥1. Assume Vsinz be shift operator on Hwpnρ, where ρf=∑r=0∞fr^/r+1pr1/ϖp, for every f∈Hwpnρ and sinz=∑m=0∞−1mz2m+1/2m+1!. The upper estimation of the s-numbers of Vsinz is presented by
(68)srVsinz≤supcardξ=r+1infk∈ξsupmk+1m+k+1pm+k/ϖp∑m=0∞12m+1!pm/ϖp.Proof.
Obviously, from Theorem43. ☐


## 6. Caristi’s Fixed Point Theorem inHwpnρ

The Ekeland variational principle (EVP) can not examine in modular spaces as the modular does not satisfy the triangle inequality. In this section in view of Farkas [30], we offer a development of Caristi’s fixed point theorem in Hwpnρ.Definition 47.
(a)
The prequasi normed (ssfps)ρ on Hwpnρ is named ρ-convex, if ρωv+1−ωt≤ωρv+1−ωρt, for each ω∈0,1 and v,t∈Hwpnρ(b)
vaa∈ℕ⊆Hwpnρ is ρ-convergent to v∈Hwpnρ, if and only if, lima⟶∞ρva−v=0. If the ρ-limit exists, then it is unique(c)
vaa∈ℕ⊆Hwpnρ is ρ-Cauchy, if lima,b⟶∞ρva−vb=0(d)
Y⊂Hwpnρ is ρ-closed, if for every ρ-converging uaa∈ℕ⊂Y to u, then u∈Y(e)
Y⊂Hwpnρ is ρ-bounded, if δρY=supρv−t:v,t∈Y<∞(f)
Theρ-ball of radius d≥0 and center v, for all v∈Hwpnρ, is defined as:(69)Bρv,d=t∈Hwpnρ:ρv−t≤d.(g)
A prequasi normed (ssfps)ρ on Hwpn verifies the Fatou property, if for every sequence ta⊆Hwpnρ with lima⟶∞ρta−t=0 and every v∈Hwpnρ hence(70)ρv−t≤supjinfa≥jρv−ta.
Contemplate that the Fatou property gives theρ-closedness of the ρ-balls.Theorem 48.
The functionρf=∑r=0∞fr^/r+1pr1/ϖp, for all f∈Hwpnρ, verifies the Fatou property when pn∈mi↗∩ℓ∞ with p0>0.Proof.
Letfi⊆Hwpnρ with limi⟶∞ρfi−f=0. As the space Hwpnρ is a prequasi closed space, hence f∈Hwpnρ. So for all g∈Hwpnρ, we have
(71)ρg−f=∑a=0∞ga^−fa^a+1pa1/ϖp≤∑a=0∞ga^−fai^a+1pa1/ϖp+∑a=0∞fai^−fa^a+1pa1/ϖp≤supjinfi≥jρg−fi.
☐Theorem 49.
The functionρf=∑r=0∞fr^/r+1pr, for all f∈Hwpnρ, does not verify the Fatou property, when pn∈mi↗∩ℓ∞ with p0>1.Proof.
Supposefi⊆Hwpnρ with limi⟶∞ρfi−f=0. As the space Hwpnρ is a prequasi closed space, hence f∈Hwpnρ. So for every g∈Hwpnρ, one gets
(72)ρg−f=∑a=0∞ga^−fa^a+1pa≤2supapa−1∑a=0∞ga^−fai^a+1pa+∑a=0∞fai^−fa^a+1pa≤2supapa−1supjinfi≥jρg−fi.
Therefore,ρ does not verify the Fatou property. ☐Example 50.
The functionρf=∑r=0∞fr^/r+1pr1/ϖp, for every f∈Hwpnρ, is a prequasi normed (ssfps), not quasi normed (ssfps), and not a normed (ssfps), for all f∈Hwpnρ.Example 51.
The functionρf=∑r=0∞fr^/r+1p1/p, for every f∈Hbn,pρ, is a prequasi normed (ssfps), quasi normed (ssfps), and not a normed (ssfps) on Hwpρ, for 0<p<1.Example 52.
The functionρf=infκ>0:∑r=0∞fr^/κr+1pr≤1, is a prequasi normed (ssfps), a quasi normed (ssfps), and a normed (ssfps) on Hwpn.Definition 53.
Supposeg:Hwpnρ⟶−∞,∞ be a function. The function g is named lower semicontinuous at f0∈Hwpnρ if liminff⟶f0gf=gf0, whereliminff⟶f0gf=supV∈Vf0inff∈Vgf, where Vf0 is a neighborhood system of f0.Definition 54.
Assumeg:Hwpnρ⟶−∞,∞ be a function. The function g is named proper, if
(73)Dg=f∈Hwpnρ:gf<∞≠∅.Theorem 55.
Pick upY≠∅ and Y be a ρ-closed subset of Hwpaρ, with ρf=∑a=0∞fa^/a+1pa1/ϖp, for every f∈Hwpaρ, and g:Y⟶−∞,∞ be a proper, ρ-lower semicontinuous function with inff∈Ygf>−∞. Suppose γ>0, βa⊂0,∞, and f0∈Y with gf0≤inff∈Ygf+γ. Therefore, one has fa∈Y which ρ-converges to some fγ, with
(i)
ρfγ−fa≤γ/2aβ0, for all a∈ℕ(74)gfγ+∑a=0∞βaρfγ−fa≤gf0,(ii)
iff≠fγ, one has
(75)gfγ+∑a=0∞βaρfγ−fa<gf+∑a=0∞βaρf−fa.Proof.
PlaceSf0=f∈Y:gf+β0ρf−f0≤gf0. As f0∈Sf0, hence Sf0≠∅. Since g is ρ-lower semicontinuous, ρ verifies the Fatou property, and Y is ρ-closed, one has Sf0 is ρ-closed. Choose f1∈Sf0 with
(76)gf1+β0ρf1−f0≤inff∈Sf0gf+β0ρf−f0+γβ12β0.
After place(77)Sf1=f∈Sf0:gf+∑j=01βjρf−fj≤gf1+β0ρf1−f0.
Similar toSf0, one has Sf1≠∅ and ρ-closed. Assume that one has built f0,f1,f2,,fa and Sf0,Sf1,Sf2,,Sfa. Next that, choose fa+1∈Sfa with
(78)gfa+1+∑j=0aβjρfa+1−fj≤inff∈Sfagf+∑j=0aβjρf−fj+γβa2aβ0.
Let(79)Sfa+1≔f∈Sfa:gf+∑j=0a+1βjρf−fj≤gfa+1+∑j=0aβjρfa+1−fj.
Hence, we form the sequencesfa and Sfa by induction. For fixed a∈ℕ. Suppose z∈Sfa. We have
(80)gz+∑j=0aβjρz−fj≤gfa+∑j=0a−1βjρfa−fj,hence
(81)βaρz−fa≤gfa+∑j=0a−1βjρfa−fj−gz+∑j=0a−1βjρz−fj≤gfa+∑j=0a−1βjρfa−fj−inff∈Sfa−1gf+∑j=0a−1βjρf−fj≤γβa2aβ0.
AsSfa is decreasing with fa∈Sfa, for every n∈ℕ, we obtain
(82)ρfa+h−fa≤γ2aβ0,for all a,h∈ℕ. This implies fa is ρ-Cauchy. As Hwpaρ is ρ-Banach space. Hence, fa has ρ-limits fγ, and ⋂a∈ℕSfa=fγ verifies. Since fa+1∈Sfa, we can see
(83)gfa+1+∑j=0aβjρfa+1−fj≤gfa+∑j=0a−1βjρfa−fj,this gives gfa+∑j=0a−1βjρfa−fj is decreasing. Next, suppose f≠fγ. Hence, we have m∈ℕ with f∉Sfa, for all a≥m, i.e.,
(84)gfa+∑j=0a−1βjρfa−fj<gf+∑j=0aβjρf−fj.
Sincefγ∈Sfa, with a≥m, we get
(85)gfγ+∑j=0aβjρfγ−fj≤gfa+∑j=0a−1βjρfa−fj≤gfm+∑j=0m−1βjρfm−fj.
Puta⟶∞ in the previous inequality, we have
(86)gfγ+∑j=0∞βjρfγ−fj≤gxm+∑j=0m−1βjρfm−fj<gf+∑j=0mβjρf−fj≤gf+∑j=0∞βjρf−fj.
This gives(87)gfγ+∑a=0∞βaρfγ−fa<gf+∑a=0∞βaρf−fa.This completes the proof. ☐In view of Theorem55, we present a generalization of Caristi’s fixed point theorem in Hwpaρ.Theorem 56.
SupposeY≠∅ and Y be a ρ-closed subset ofHwpaρ, with ρf=∑a=0∞fa^/a+1pa1/ϖp, for every f∈Hwpaρ. Let γ>0 and βn with 0<χ=∑n=0∞βn<∞. Assume T:Y⟶Y be a mapping and one has a function g:Y⟶−∞,∞ which is a proper and ρ-lower semicontinuous with inff∈Ygf>−∞ and
(1)
ρTf−h−ρf−h≤ρTf−f, for any f,h∈Y(2)
ρTf−f≤gf−gTf, for any f∈Y
Therefore,T has a fixed point in Y.Proof.
As0<χ=∑n=0∞βn<∞, one has g∗≔χg is also proper, ρ-lower semicontinuous, and bounded from below. Suppose f∈Y, we have
(88)χρTf−f≤g∗f−g∗Tf.
Sinceinff∈Yg∗f>−∞, one obtains f0∈Y with g∗f0<inff∈Yg∗f+γ. From Theorem 55, there is fa which ρ-converges to some fγ∈Y, with
(89)g∗fγ+∑a=0∞βaρfγ−fa<g∗f+∑a=0∞βaρf−fa,for every f≠fγ. Assume that Tfγ≠fγ, we have
(90)g∗fγ+∑a=0∞βaρfγ−fa<g∗Tfγ+∑a=0∞βaρTfγ−fa,then
(91)g∗fγ−g∗Tfγ<∑a=0∞βaρTfγ−fa−∑a=0∞βaρfγ−fa=∑a=0∞βaρTfγ−fa−ρfγ−fa.
By using setting (6), we get
(92)g∗fγ−g∗Tfγ<∑a=0∞βaρTfγ−fγ=χρTfγ−fγ.
The inequality 5 implies that(93)χρTfγ−fγ≤g∗fγ−g∗Tfγ<χρTfγ−fγ.
This is a disagreement. Therefore,Tfγ=fγ. This finishes the proof. ☐



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*Source: 9919420-2021-06-07.xml*