# The Spectrum of Mapping Ideals of Type Variable Exponent Function Space of Complex Variables with Some Applications

Authors: Awad A. Bakery; Mustafa M. Mohammed
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The topological and geometric behaviors of the variable exponent formal power series space, as well as the prequasi-ideal construction bys-numbers and this function space of complex variables, are investigated in this article. Upper bounds for s-numbers of infinite series of the weighted nth power forward and backward shift operator on this function space are being investigated, with applications to some entire functions.


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## Body

## 1. Introduction

Operator ideal theory has various applications in the geometry of Banach spaces, xed point theory, spectral theory, and other areas of mathematics, among other areas of knowledge. Throughout the article, we will adhere to the etymological conventions listed below. If any other sources are used, we will make a note of them.


### 1.1. Conventions 1.1

ℕ=0,1,2,⋯.ℂ: complex number spaceℝℕ: the space of all real sequencesℓ∞: the space of bounded real sequencesℓr: the space of r-absolutely summable real sequencesc0: the space of null real sequencesel=0,0,⋯,1,0,0,⋯, as 1 lies at the lth coordinate, for all l∈ℕF: the space of each sequence with finite nonzero coordinatescardG: the number of elements of the set Gmi↗: the space of all monotonic increasing sequences of positive realsL: the ideal of all bounded linear operators between any arbitrary Banach spacesF: the ideal of finite rank operators between any arbitrary Banach spacesΛ: the ideal of approximable operators between any arbitrary Banach spacesLc: the ideal of compact operators between any arbitrary Banach spacesLX,Y: the space of all bounded linear operators from a Banach space X into a Banach space YLX : the space of all bounded linear operators from a Banach space X into itselfFX,Y: the space of finite rank operators from a Banach space X into a Banach space YFX: the space of finite rank operators from a Banach space X into itselfΛX,Y : the space of approximable operators from a Banach space X into a Banach space YΛX: the space of approximable operators from a Banach space X into itselfLcX,Y: the space of compact operators from a Banach space X into a Banach space YLcX: the space of compact operators from a Banach space X into itselfsaGa∈ℕ: the sequence of s-numbers of the bounded linear operator GαaGa∈ℕ: the sequence of approximation numbers of the bounded linear operator GsaGa∈ℕ: the sequence of Kolmogorov numbers of the bounded linear operator GSv: the operator ideals formed by the sequence of s-numbers in any sequence space VSVapp: the operator ideals formed by the sequence of approximation numbers in any sequence space VSVKol: the operator ideals formed by the sequence of Kolmogorov numbers in any sequence space V


### 1.2. Notations 1.2 (see [1])

SH≔SHX,Y;XandYareBanachSpaces, where
(1)SHX,Y≔P∈LX,Y:fs∈H,wherefsz=∑n=0∞snPznconvergesforanyz∈ℂSHapp≔SHappX,Y;XandYareBanachSpaces, where
(2)SHappX,Y≔P∈LX,Y:fapp∈H,wherefappz=∑n=0∞αnPznconvergesforanyz∈ℂSHKol;SHKolX,Y;XandYareBanachSpaces, where
(3)SHKolX,Y≔P∈LX,Y:fKol∈H,wherefKolz=∑n=0∞dnPznconvergesforanyz∈ℂSHρλ≔SHρλX,Y;XandYareBanachSpaces, where
(4)SHρλX,Y≔T∈LX,Y:fλ∈Hρ,wherefλz=∑n=0∞λnTznconvergesforanyz∈ℂandT−λlTI=0,foreveryl∈ℕSeveral operator ideals in the class of Banach or Hilbert spaces are defined by sequences of real numbers.Lc, for example, is produced by daGa∈ℕ and c0. Pietsch [2] looked into the quasi-ideals Sℓtapp, for 0<t<∞. He demonstrated how ℓ2 and ℓ1 yield the ideals of Hilbert Schmidt operators and nuclear operators between Hilbert spaces, respectively. In addition, he proved that F¯=Sℓt, for 1≤t<∞, and Sℓt is a simple Banach space. Pietsch [3] explained that Sℓt, where 0<t<∞, is small. Makarov and Faried [4] showed that for any Banach spaces X and Y with dimX=dimY=∞, then for every r>t>0, one has SℓtappX,Y⊂≠SℓrappX,Y⊂≠LX,Y. The concept of prequasi-ideal was developed by Faried and Bakery [5], who elaborated on the concept of quasi-ideal. They investigated some geometric and topological properties of the spaces Scest and SℓM. According to the spectral decomposition theorem [2], for A∈LcH, where H is a Hilbert space, one has Ay=∑a=0∞αa≺y,ra≻wa, where ra and wa are orthonormal families in H. Suppose taa∈ℕ∈ℝℕ be decreasing and D:ηa⟶taηa be the diagonal operator on ℓp with p≥1. Therefore, saD=ta. Shields [6] investigated an indication to the weighted shift operators as formal power series in unilateral shifts and formal Laurent series in bilateral shifts. Hedayatian [7] offered 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 meant that the set of all formal power series ∑a=0∞fa^za with ∑a=0∞bafa∧p<∞. He studied cyclic vectors for the H forward shift operator and supercyclic vectors for the backward shift operator on the space Hpba.However, Emamirad and Heshmati [8] explored 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. [9] introduced the upper bounds for s-numbers of infinite series of the weighted nth power forward shift operator on Hrba, for 1≤r<∞, with some applications to some entire functions.The paper is arranged as follows. In Section3, we offer the definition of the space Hp. with definite function ρ. We introduce the sufficient conditions on Hp. to generate premodular special space of formal power series. This gives that Hp. is a prequasinormed space. In Section 4, firstly, we give the sufficient conditions on Hp. such that the class SHp. generates an operator ideal. Secondly, we explain enough settings (not necessary) on Hp.ρ, so that F¯=SHp.ρ. This shows the nonlinearity of s-type Hp.ρ spaces which gives an answer of Rhoades [10] open problem. Thirdly, we investigate the conditions on Hp.ρ such that the prequasi-ideal SHp.ρ are Banach and closed. Fourthly, we examine the sufficient conditions on Hp.ρ such that SHp.ρ is strictly contained for different powers. We show the smallness of SHp.ρ. Fifthly, we investigate the simpleness of SHp.ρ. Sixthly, we present the enough setup on Hp.ρ such that the class L with its sequence of eigenvalues in Hp.ρ equals SHp.ρ. In Section 5, we estimate the upper bounds for s-numbers of infinite series of the weighted nth power forward and backward shift operator on Hp. with approaches to some entire functions.


## 1.1. Conventions 1.1

ℕ=0,1,2,⋯.ℂ: complex number spaceℝℕ: the space of all real sequencesℓ∞: the space of bounded real sequencesℓr: the space of r-absolutely summable real sequencesc0: the space of null real sequencesel=0,0,⋯,1,0,0,⋯, as 1 lies at the lth coordinate, for all l∈ℕF: the space of each sequence with finite nonzero coordinatescardG: the number of elements of the set Gmi↗: the space of all monotonic increasing sequences of positive realsL: the ideal of all bounded linear operators between any arbitrary Banach spacesF: the ideal of finite rank operators between any arbitrary Banach spacesΛ: the ideal of approximable operators between any arbitrary Banach spacesLc: the ideal of compact operators between any arbitrary Banach spacesLX,Y: the space of all bounded linear operators from a Banach space X into a Banach space YLX : the space of all bounded linear operators from a Banach space X into itselfFX,Y: the space of finite rank operators from a Banach space X into a Banach space YFX: the space of finite rank operators from a Banach space X into itselfΛX,Y : the space of approximable operators from a Banach space X into a Banach space YΛX: the space of approximable operators from a Banach space X into itselfLcX,Y: the space of compact operators from a Banach space X into a Banach space YLcX: the space of compact operators from a Banach space X into itselfsaGa∈ℕ: the sequence of s-numbers of the bounded linear operator GαaGa∈ℕ: the sequence of approximation numbers of the bounded linear operator GsaGa∈ℕ: the sequence of Kolmogorov numbers of the bounded linear operator GSv: the operator ideals formed by the sequence of s-numbers in any sequence space VSVapp: the operator ideals formed by the sequence of approximation numbers in any sequence space VSVKol: the operator ideals formed by the sequence of Kolmogorov numbers in any sequence space V


## 1.2. Notations 1.2 (see [1])

SH≔SHX,Y;XandYareBanachSpaces, where
(1)SHX,Y≔P∈LX,Y:fs∈H,wherefsz=∑n=0∞snPznconvergesforanyz∈ℂSHapp≔SHappX,Y;XandYareBanachSpaces, where
(2)SHappX,Y≔P∈LX,Y:fapp∈H,wherefappz=∑n=0∞αnPznconvergesforanyz∈ℂSHKol;SHKolX,Y;XandYareBanachSpaces, where
(3)SHKolX,Y≔P∈LX,Y:fKol∈H,wherefKolz=∑n=0∞dnPznconvergesforanyz∈ℂSHρλ≔SHρλX,Y;XandYareBanachSpaces, where
(4)SHρλX,Y≔T∈LX,Y:fλ∈Hρ,wherefλz=∑n=0∞λnTznconvergesforanyz∈ℂandT−λlTI=0,foreveryl∈ℕSeveral operator ideals in the class of Banach or Hilbert spaces are defined by sequences of real numbers.Lc, for example, is produced by daGa∈ℕ and c0. Pietsch [2] looked into the quasi-ideals Sℓtapp, for 0<t<∞. He demonstrated how ℓ2 and ℓ1 yield the ideals of Hilbert Schmidt operators and nuclear operators between Hilbert spaces, respectively. In addition, he proved that F¯=Sℓt, for 1≤t<∞, and Sℓt is a simple Banach space. Pietsch [3] explained that Sℓt, where 0<t<∞, is small. Makarov and Faried [4] showed that for any Banach spaces X and Y with dimX=dimY=∞, then for every r>t>0, one has SℓtappX,Y⊂≠SℓrappX,Y⊂≠LX,Y. The concept of prequasi-ideal was developed by Faried and Bakery [5], who elaborated on the concept of quasi-ideal. They investigated some geometric and topological properties of the spaces Scest and SℓM. According to the spectral decomposition theorem [2], for A∈LcH, where H is a Hilbert space, one has Ay=∑a=0∞αa≺y,ra≻wa, where ra and wa are orthonormal families in H. Suppose taa∈ℕ∈ℝℕ be decreasing and D:ηa⟶taηa be the diagonal operator on ℓp with p≥1. Therefore, saD=ta. Shields [6] investigated an indication to the weighted shift operators as formal power series in unilateral shifts and formal Laurent series in bilateral shifts. Hedayatian [7] offered 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 meant that the set of all formal power series ∑a=0∞fa^za with ∑a=0∞bafa∧p<∞. He studied cyclic vectors for the H forward shift operator and supercyclic vectors for the backward shift operator on the space Hpba.However, Emamirad and Heshmati [8] explored 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. [9] introduced the upper bounds for s-numbers of infinite series of the weighted nth power forward shift operator on Hrba, for 1≤r<∞, with some applications to some entire functions.The paper is arranged as follows. In Section3, we offer the definition of the space Hp. with definite function ρ. We introduce the sufficient conditions on Hp. to generate premodular special space of formal power series. This gives that Hp. is a prequasinormed space. In Section 4, firstly, we give the sufficient conditions on Hp. such that the class SHp. generates an operator ideal. Secondly, we explain enough settings (not necessary) on Hp.ρ, so that F¯=SHp.ρ. This shows the nonlinearity of s-type Hp.ρ spaces which gives an answer of Rhoades [10] open problem. Thirdly, we investigate the conditions on Hp.ρ such that the prequasi-ideal SHp.ρ are Banach and closed. Fourthly, we examine the sufficient conditions on Hp.ρ such that SHp.ρ is strictly contained for different powers. We show the smallness of SHp.ρ. Fifthly, we investigate the simpleness of SHp.ρ. Sixthly, we present the enough setup on Hp.ρ such that the class L with its sequence of eigenvalues in Hp.ρ equals SHp.ρ. In Section 5, we estimate the upper bounds for s-numbers of infinite series of the weighted nth power forward and backward shift operator on Hp. with approaches to some entire functions.


## 2. Definitions and Preliminaries

Definition 1 (see [11]).
A functions:LX,Y⟶0,∞ℕ is called an s-number, if the sequence sbBa=0∞, for all B∈LX,Y, shows the following settings:
(a)
IfB∈LX,Y, then B=s0B≥s1B≥s2B≥⋯≥0(b)
sb+a−1B1+B2≤sbB1+saB2, for every B1,B2∈LX,Y,b,a∈ℕ(c)
The inequalitysaABD≤AsaBD holds, if D∈LX0,X,B∈LX,Y and A∈LY,Y0, where X0 and Y0 are arbitrary Banach spaces(d)
SupposeA∈LX0,X and λ∈ℝ, then saλA=λsaA(e)
LetrankA≤b then sbA=0, whenever A∈LX0,X(f)
AssumeIλ indicates the identity operator on the λ-dimensional Hilbert space ℓ2λ, then sr≥λIλ=0 or sr<λIλ=1
Consider the following examples ofs-numbers:
(i)
Thebth approximation number, αbA, where
(5)αbA=infA−B:B∈LX,YandrankB≤b(ii)
Thebth Kolmogorov number, dbA, where
(6)dbA=infdimY≤bsupu≤1infv∈YAu−v.Remark 2 (see [11]).
IfB∈LcH, where H be a Hilbert space, then all the s-numbers equal the eigenvalues of ∣B∣, where B=B∗B.Lemma 3 (see [2]).
IfB∈LX0,X and B∉ΛX0,X, then D∈LX and M∈LY with MBDeb=eb, for each b∈ℕ.Definition 4 (see [2]).
A Banach spaceY is said to be simple if LY has one and only one nontrivial closed ideal.Theorem 5 (see [2]).
IfD is a Banach space with dimD=∞, then
(7)FD⊂≠ΛD⊂≠LcD⊂≠LD.Definition 6. (see [2]).
A classU⊆L is said to be an operator ideal if every vector UX,Y=U∩LX,Y shows the following settings:
(i)
F⊆U(ii)
UX,Y is linear space on ℝ(iii)
IfD∈LX0,X,B∈UX,Y and A∈LY,Y0 then, ABD∈UX0,Y0Definition 7 (see [5]).
A functiong:U⟶0,∞ is called a prequasinorm on the ideal U if it shows the next settings:
(1)
For eachA∈LX,Y,gA≥0 and gA=0⇔A=0(2)
One hasM≥1 with gβA≤M∣β∣gA, for all β∈ℝ and A∈UX,Y(3)
One hasK≥1 with gA1+A2≤KgA1+gA2, for every A1,A2∈UX,Y(4)
There existsC≥1 so that if A∈LX0,X,B∈UX,Y and D∈LY,Y0 then gDBA≤C∥D∥gB∥A∥, where X0 and Y0 are normed spacesTheorem 8 (see [5]).
Supposeg is a quasinorm on the ideal U, then g is a prequasinorm on the ideal U.Theorem 9 (see [12]).
Assumes‐typeVυ≔f=srT∈ℝℕ:T∈LX,Yandυf<∞. If Svυ is an operator ideal, then we have
(1)
F⊂s-type Vυ(2)
AssumesrT1r=0∞∈s‐typeVvandsrT2r=0∞∈s‐typeVv,thensrT1+T2r=0∞∈s‐typeVv(3)
Supposeλ∈ℝ and srTr=0∞∈s‐typeVv,thenλsrTr=0∞∈s‐typeVv(4)
The sequence spaceVv is solid. i.e., when srGr=0∞∈s‐typeVvandsrT≤srG,foreveryr∈ℕandT,G∈LX,Y,thensrTr=0∞∈s‐typeVvLemma 10 (see [13]).
Ifξii∈Ψ is a bounded family of ℝ. We have
(8)supcardG=a+1infξii∈G=infcardG=asupξii∉G.Lemma 11 (see [14]).
Ifra,ta∈ℝℕandqa∈0,∞ℕ,withK=max1,2ϖq−1andϖq=max1,supaqa, then
(9)ra+taqa≤Kraqa+taqa.Definition 12 (see [1]).
The linear space of formal power series(10)H=f:fz=∑n=0∞f⌢nznconvergesforanyz∈ℂ,,is called a special space of formal power series (or in short (ssfps)), if it shows the following settings:
(1)
em∈H, for all m∈ℕ, where emz=∑n=0∞enm⌢zn=zm(2)
Ifg∈H and f⌢n≤g⌢n,foralln∈ℕ,thenf∈H(3)
Supposef∈H, then f.∈H, where f.z=∑b=0∞f⌢b/2zbandb/2 marks the integral part of b/2Theorem 13 (see [1]).
IfH is a (ssfps), then SH is an operator ideal.
ByF, we explain the space of finite formal power series, i.e, for f∈F, one has l∈ℕ with fz=∑n=0lf⌢nzn.Definition 14 (see [1]).
A subspaceHρ of the (ssfps) is called a premodular (ssfps), if there is a function ρ:H⟶0,∞ verifies the next conditions:
(i)
Forf∈H, we have ρf≥0 and f=θ⇔ρf=0, where θ is the zero function of H(ii)
Supposef∈H and λ∈ℝ, then there is l≥1 with ρλf≤λlρf(iii)
Letf,g∈H, then there is K≥1 such that ρf+g≤Kρf+ρg(iv)
Supposef⌢b≤g⌢b, for every b∈ℕ, then ρf≤ρg(v)
There isK0≥1 so that ρf≤ρf.≤K0ρf(vi)
F¯=Hρ(vii)
one hasξ>0 with ρλe0≥ξλρe0, where λ∈ℝ
Note that the continuity ofρf at θ comes from condition (ii). Condition (1) in Definition 12 and condition (vi) in Definition 14 investigate that emm∈ℕ is a Schauder basis of Hρ.
The (ssfps)Hρ is called a prequasinormed (ssfps) if ρ shows the conditions (i)–(iii) of Definition 14, and if the space H is complete under ρ, then Hρ is called a prequasi-Banach (ssfps).Theorem 15 (see [1]).
Every premodular (ssfps)Hρ is a prequasinormed (ssfps).Definition 16 (see [1]).
AssumeHρ is a prequasinormed (ssfps). An operator Vz:Hρ⟶Hρ is called forward shift, if Vzf=zf, for all f∈Hρ, where Vzfz=∑n=0∞f⌢nzn+1 converges for every z∈ℂ and ρVzf<∞.Definition 17 (see [1]).
SupposeHρ is a prequasinormed (ssfps). An operator BzHρ⟶Hρ is called backward shift, if Bzfz=fz−f0/z, for all f∈Hρ, where Bzfz=∑n=0∞f⌢n+1zn converges for every z∈ℂ and ρBzf<∞.Definition 18 (see [9]).
By using the power series of an entire functiongz=∑m=0∞amzm, the shift operator Vgz is defined as
(11)Vgzfz=∑m=0∞amVzmfz.Definition 19 (see [9]).
By using the power series of an entire functiongz=∑m=0∞amzm, the shift operator Bgz is defined as
(12)Bgzfz=∑m=0∞amBzmfz.


## 3. Main Results

### 3.1. The Space of FunctionsHp.ρ

We define in this section the spaceHp.ρ under the function ρ and give enough conditions on it to create pre-modular (ssfps) which implies that is a prequasi-Banach (ssfps).Ifp=pvv∈ℕ0,∞ℕ, we define the new space of functions:
(13)Hp.ρ=f:fz=∑v=0∞f⌢vzvconvergesforanyz∈ℂandρζf<∞,forsomeζ>0,where
(14)ρf=∑v=0∞1pvf⌢vpv.Ifpv∈ℓ∞, one has
(15)Hp.ρ=f:fz=∑v=0∞f⌢vzvconvergesforanyz∈ℂandρζf<∞,forsomeζ>0=f:fz=∑v=0∞f⌢vzvconvergesforanyz∈ℂand∑v=0∞1pvζf⌢vpv<∞,forsomeζ>0=f:fz=∑v=0∞f⌢vzvconvergesforanyz∈ℂandinfvζpv∑v=0∞1pvζf⌢vpv<∞,forsomeζ>0=f:fz=∑v=0∞f⌢vzvconvergesforanyz∈ℂand∑v=0∞1pvf⌢vpv<∞=f:fz=∑v=0∞f⌢vzvconvergesforanyz∈ℂandρζf<∞,foranyζ>0.Theorem 20.
Considerpv∈mi↗∩ℓ∞with p0>0, one has Hp.ρ is a premodular Banach (ssfps).Proof (1-i).
Letf,g∈Hp.. Therefore, fz=∑v=0∞f⌢vzvandgz=∑v=0∞g⌢vzv converge for any z∈ℂ. Then, f+gz=∑v=0∞f⌢v+g⌢vzvconvergesforanyz∈ℂ. From pv∈ℓ∞, we have∑v=0∞1/pvf⌢v+g⌢vpv≤K∑v=0∞1/pvf⌢vpv+∑v=0∞1/pvg⌢vpv<∞,so f+g∈Hp..
(1-ii) Letλ∈ℝandf∈Hp.. Therefore, fz=∑v=0∞f⌢vzv converges for any z∈ℂ. Then, λfz=∑v=0∞λf⌢vzv converges for any z∈ℂ. From pv∈ℓ∞, we have∑v=0∞1/pvλf⌢vpv≤supvλpv∑v=0∞1/pvf⌢vpv<∞.
Soλf∈Hp.. Therefore, from conditions (1-i) and (1-ii), the space Hp. is linear. To prove em∈Hp.,
for allm∈ℕ, where emz=∑v=0∞evm⌢zv=zmand∑v=0∞1/pvevm⌢pv=1/pm(2)
Assumefv^≤gv^, for all v∈ℕ and g∈Hp.. Then,converges for any z∈ℂ
One has(16)∑v=0∞1pvf∧vpv≤∑v=0∞1pvg∧vpv<∞.
So,fz=∑v=0∞f^vzv and z∈ℂ and ρf<∞. Hence, f∈Hp⋅(3)
Letf∈Hp⋅ and pv∈mi↗∩ℓ∞ with p0>0. Then, fz=∑v=0∞f^vzv converges for any z∈ℂand ρf<∞. One has
(17)ρf⋅=∑v=0∞1pvfv/2∧pv=∑v=0∞1p2vfv∧p2v+∑v=0∞1p2v+1fv∧p2v+1≤2∑v=0∞1pvfv∧pv=2ρf
Hence,f⋅z=∑v=0∞fv/2^zv converges for any z∈ℂ and ρf⋅<∞. Then f⋅∈Hp⋅.
(i)
Obviously, iff∈Hp., one gets ρf≥0 and ρf=0⇔f=θ(ii)
There isl=max1,supvηpv−1≥1, for all η∈ℝ\0 and l≥1, for η=0 so that
(18)ρηf=∑v=0∞1pvηfv∧pv≤supvηpv∑v=0∞1pvfv∧pv≤lηρf,for all f∈Hp.(iii)
There isK=max1,2supvpv−1≥1 so that
(19)ρηf=∑v=0∞1pvηfv^≤supvηpv∑v=0∞1pvfv∧pv≤lηρf,for every f, g∈Hp.(iv)
Obviously from the proof part (2).(v)
From the proof part (3), one hasK0=2≥1(vi)
Clearly,F¯=Hp.(vii)
One hasζ with 0<ζ≤ηp0−1 with ρηe0≥ζηρe0, for each η≠0 and ζ>0, when η=0. Therefore, the space Hp.ρ is a premodular (ssfps). To show that Hp.ρ is a premodular Banach (ssfps), we suppose fi to be a Cauchy sequence in Hp.ρ, then for every ε∈0,1, there is i0∈N such that for all i,j≥i0, one gets
(20)ρfi−fj=∑v=0∞1pvfvi∧−fvj∧pv<εϖp
Fori,j≥i0 and v∈ℕ, we have
(21)fvi^−fvj^<ε.
So,fvj^ is a Cauchy sequence in ℝ, for fixed v∈ℕ, hence limj⟶∞fvj^=fv0^, for fixed v∈ℕ.
Therefore,ρfi−f0<εϖp, for every i≥i0. Finally, to show that f0∈Hp., we have
(22)ρf0=ρf0−fi+fi≤Kρfi−f0+ρfi<∞.
Hence,f0∈Hp.. Then, the space Hp.ρ is a premodular Banach (ssfps).
In view of Theorems15 and 20, we conclude the following theorem.Theorem 21.
Ifpv∈mi↗∩ℓ∞ with p0>0, then the space Hp.ρ is a prequasi-Banach (ssfps), where
ρf=∑v=0∞1/pvfv∧pv,forallf∈Hp⋅.Theorem 22.
Supposepv∈mi↗∩ℓ∞ with p0>0, one has Hp.ρ is a prequasiclosed (ssfps), where
(23)ρf=∑v=0∞1pvfv∧pv,forallf∈Hp⋅.Proof.
According to Theorem21, the space Hp.ρ is a prequasinormed (ssfps). To explain that Hp.ρ is a prequasiclosed (ssfps), let fii=0∞∈Hp⋅ρ and limi⟶∞ρfi−f0=0, we have for all ε∈0,1, there is i0∈ℕ such that for all i≥i0, one gets
(24)ε>ρfi−f0=∑a=0∞1pafai∧−fa0∧pa1/ϖp.
So, fori≥i0 and a∈ℕ, we have fai^−fa0^<ε.Therefore, fai^ is a convergent sequence in ℝ, for fixed a∈ℕ. Then, limi⟶∞fai^=fa0^ for fixed a∈ℕ. Finally to prove that f0∈Hp.ρ, we have
(25)ρf0=ρf0−fi+fi≤ρfi−f0+ρfi<∞,this gives f0∈Hp.ρ which shows that Hp.ρ is a prequasiclosed (ssfps).


## 3.1. The Space of FunctionsHp.ρ

We define in this section the spaceHp.ρ under the function ρ and give enough conditions on it to create pre-modular (ssfps) which implies that is a prequasi-Banach (ssfps).Ifp=pvv∈ℕ0,∞ℕ, we define the new space of functions:
(13)Hp.ρ=f:fz=∑v=0∞f⌢vzvconvergesforanyz∈ℂandρζf<∞,forsomeζ>0,where
(14)ρf=∑v=0∞1pvf⌢vpv.Ifpv∈ℓ∞, one has
(15)Hp.ρ=f:fz=∑v=0∞f⌢vzvconvergesforanyz∈ℂandρζf<∞,forsomeζ>0=f:fz=∑v=0∞f⌢vzvconvergesforanyz∈ℂand∑v=0∞1pvζf⌢vpv<∞,forsomeζ>0=f:fz=∑v=0∞f⌢vzvconvergesforanyz∈ℂandinfvζpv∑v=0∞1pvζf⌢vpv<∞,forsomeζ>0=f:fz=∑v=0∞f⌢vzvconvergesforanyz∈ℂand∑v=0∞1pvf⌢vpv<∞=f:fz=∑v=0∞f⌢vzvconvergesforanyz∈ℂandρζf<∞,foranyζ>0.Theorem 20.
Considerpv∈mi↗∩ℓ∞with p0>0, one has Hp.ρ is a premodular Banach (ssfps).Proof (1-i).
Letf,g∈Hp.. Therefore, fz=∑v=0∞f⌢vzvandgz=∑v=0∞g⌢vzv converge for any z∈ℂ. Then, f+gz=∑v=0∞f⌢v+g⌢vzvconvergesforanyz∈ℂ. From pv∈ℓ∞, we have∑v=0∞1/pvf⌢v+g⌢vpv≤K∑v=0∞1/pvf⌢vpv+∑v=0∞1/pvg⌢vpv<∞,so f+g∈Hp..
(1-ii) Letλ∈ℝandf∈Hp.. Therefore, fz=∑v=0∞f⌢vzv converges for any z∈ℂ. Then, λfz=∑v=0∞λf⌢vzv converges for any z∈ℂ. From pv∈ℓ∞, we have∑v=0∞1/pvλf⌢vpv≤supvλpv∑v=0∞1/pvf⌢vpv<∞.
Soλf∈Hp.. Therefore, from conditions (1-i) and (1-ii), the space Hp. is linear. To prove em∈Hp.,
for allm∈ℕ, where emz=∑v=0∞evm⌢zv=zmand∑v=0∞1/pvevm⌢pv=1/pm(2)
Assumefv^≤gv^, for all v∈ℕ and g∈Hp.. Then,converges for any z∈ℂ
One has(16)∑v=0∞1pvf∧vpv≤∑v=0∞1pvg∧vpv<∞.
So,fz=∑v=0∞f^vzv and z∈ℂ and ρf<∞. Hence, f∈Hp⋅(3)
Letf∈Hp⋅ and pv∈mi↗∩ℓ∞ with p0>0. Then, fz=∑v=0∞f^vzv converges for any z∈ℂand ρf<∞. One has
(17)ρf⋅=∑v=0∞1pvfv/2∧pv=∑v=0∞1p2vfv∧p2v+∑v=0∞1p2v+1fv∧p2v+1≤2∑v=0∞1pvfv∧pv=2ρf
Hence,f⋅z=∑v=0∞fv/2^zv converges for any z∈ℂ and ρf⋅<∞. Then f⋅∈Hp⋅.
(i)
Obviously, iff∈Hp., one gets ρf≥0 and ρf=0⇔f=θ(ii)
There isl=max1,supvηpv−1≥1, for all η∈ℝ\0 and l≥1, for η=0 so that
(18)ρηf=∑v=0∞1pvηfv∧pv≤supvηpv∑v=0∞1pvfv∧pv≤lηρf,for all f∈Hp.(iii)
There isK=max1,2supvpv−1≥1 so that
(19)ρηf=∑v=0∞1pvηfv^≤supvηpv∑v=0∞1pvfv∧pv≤lηρf,for every f, g∈Hp.(iv)
Obviously from the proof part (2).(v)
From the proof part (3), one hasK0=2≥1(vi)
Clearly,F¯=Hp.(vii)
One hasζ with 0<ζ≤ηp0−1 with ρηe0≥ζηρe0, for each η≠0 and ζ>0, when η=0. Therefore, the space Hp.ρ is a premodular (ssfps). To show that Hp.ρ is a premodular Banach (ssfps), we suppose fi to be a Cauchy sequence in Hp.ρ, then for every ε∈0,1, there is i0∈N such that for all i,j≥i0, one gets
(20)ρfi−fj=∑v=0∞1pvfvi∧−fvj∧pv<εϖp
Fori,j≥i0 and v∈ℕ, we have
(21)fvi^−fvj^<ε.
So,fvj^ is a Cauchy sequence in ℝ, for fixed v∈ℕ, hence limj⟶∞fvj^=fv0^, for fixed v∈ℕ.
Therefore,ρfi−f0<εϖp, for every i≥i0. Finally, to show that f0∈Hp., we have
(22)ρf0=ρf0−fi+fi≤Kρfi−f0+ρfi<∞.
Hence,f0∈Hp.. Then, the space Hp.ρ is a premodular Banach (ssfps).
In view of Theorems15 and 20, we conclude the following theorem.Theorem 21.
Ifpv∈mi↗∩ℓ∞ with p0>0, then the space Hp.ρ is a prequasi-Banach (ssfps), where
ρf=∑v=0∞1/pvfv∧pv,forallf∈Hp⋅.Theorem 22.
Supposepv∈mi↗∩ℓ∞ with p0>0, one has Hp.ρ is a prequasiclosed (ssfps), where
(23)ρf=∑v=0∞1pvfv∧pv,forallf∈Hp⋅.Proof.
According to Theorem21, the space Hp.ρ is a prequasinormed (ssfps). To explain that Hp.ρ is a prequasiclosed (ssfps), let fii=0∞∈Hp⋅ρ and limi⟶∞ρfi−f0=0, we have for all ε∈0,1, there is i0∈ℕ such that for all i≥i0, one gets
(24)ε>ρfi−f0=∑a=0∞1pafai∧−fa0∧pa1/ϖp.
So, fori≥i0 and a∈ℕ, we have fai^−fa0^<ε.Therefore, fai^ is a convergent sequence in ℝ, for fixed a∈ℕ. Then, limi⟶∞fai^=fa0^ for fixed a∈ℕ. Finally to prove that f0∈Hp.ρ, we have
(25)ρf0=ρf0−fi+fi≤ρfi−f0+ρfi<∞,this gives f0∈Hp.ρ which shows that Hp.ρ is a prequasiclosed (ssfps).


## 4. Properties of Operator Ideal

Throughout this section, some geometric and topological properties of the prequasi-ideals formed bys-numbers and Hp.ρ are presented.


### 4.1. Ideal of Finite Rank Operators

In this part, enough settings (not necessary) onHp.ρ so that F¯=SHp.ρ are given. This explains the nonlinearity of the s-type Hp.ρ spaces (Rhoades open problem [10]).In view of Theorems13 and 20, we conclude the next theorem.Theorem 23.
Considerpv∈mi↗∩ℓ∞ with p0>0, then SHp. is an operator ideal.Theorem 24.
Ifpv∈mi↗∩ℓ∞ with p0>0, then F¯=SHp.ρ, where
(26)ρf=∑v=0∞1pvfv^,foreveryf∈Hp⋅.Proof.
Clearly,FX,Y¯⊂SHp.ρX,Y, since the space SHp.ρ is an operator ideal. Therefore, we have to show that SHp.ρX,Y⊆FX,Y¯. By letting T∈SHp.ρX,Y, then, fs∈Hp.ρ, with fsz=∑v=0∞svTzv converges for any z∈ℂ. So, ρfs<∞, fix ε∈0,1, we have m∈ℕ−0 with ρfs−∑v=0m−1ev<ε/4. As svTv∈ℕ is decreasing, we have
(27)∑v=m+12m1pvs2mTpv≤∑v=m+12m1pvsvTpv≤∑v=m∞1pvsvTpv<ε4.
Therefore, we haveA∈F2mX,Y, rank A≤2m and
(28)∑v=2m+13m1pvT−Apv≤∑v=m+12m1pvT−Apv<ε4.
Aspv∈ℓ∞, then
(29)∑v=0m1pvT−Apv<ε4.
SinceT−A∈SHp.pX,Y, then hs∈Hp.p, where hsz≔∑v=0∞svT−Azv converges for any z∈ℂ. Because pv is increasing and from the inequalities (27)–(29), we get
(30)dT,A=ρhs=∑v=03m−11pvsvT−Apv+∑v=3m∞1pvsvT−Apv≤∑v=03m1pvT−Apv+∑v=m∞1pv+2msv+2mT−Apv+2m≤3∑v=0m1pvT−Apv+∑v=m∞1pvsvTpv<ε.
SinceI2∈SH1/n+1pX,Y but the condition pv∈mi↗∩ℓ∞ is not verified which explain a negative example of the converse statement. This finishes the proof.
We can reformulate Theorem24 as follows: if pv∈mi↗∩ℓ∞ with p0>0, then every compact operators can be approximated by finite rank operators and the converse is not always true.


### 4.2. Banach and Closed Prequasi-Ideal

In this part, enough settings onHp.ρ so that the prequasioperator ideal SHρ is Banach and closed are investigated.Theorem 25.
Assumepv∈mi↗∩ℓ∞ with p0>0, then the function gP=ρfs is a prequasinorm on SHp.ρ, where fsz=∑v=0∞svPzv converges for any z∈ℂ and
(31)ρfs=∑v=0∞1pvsvPpv,foreveryfs∈Hp⋅.Proof.
One hasg verifies the next setups:
(1)
LetP∈SHp⋅ρX,Y,gP=ρfs≥0 and gP=ρfs=0⇔svP=0, for all v∈N⇔P=0(2)
There isl≥1 with gλP=ρλfs≤lλρfs=lλgP, for every P∈SHp⋅ρX,Y and λ∈ℝ(3)
One hasKK0≥1, for P1,P2∈SHp⋅ρX,Y. Then, f1sz=∑v=0∞svP1zv and f2sz=∑v=0∞svP2zv converge for any z∈ℂ. Therefore, for hsz≕∑v=0∞svP1+P2zv, one has(32)gP1+P2=ρhs≤ρf1s.+f2s.≤Kρf1s.+ρf2s.≤KK0gP1+gP2(4)
We haveC≥1, let A∈LX0,X,B∈SHp.ρX,Y and D∈LY,Y0. Then, fsz=∑v=0∞svBzv converges for all z∈ℂ. Then, for hsz≕∑v=0∞svDBAzv, one has(33)gDBA=ρhs≤ρADfs≤CAgBDTheorem 26.
AssumeX and Y are Banach spaces, and pv∈mi↗∩ℓ∞ with p0>0, then SHp.ρ,g is a prequasi-Banach operator ideal, where gP=ρfs,fsz=∑v=0∞svPzv converges for any z∈ℂ and
(34)ρfs=∑v=0∞1pvsvPpv,foreveryfs∈Hp⋅.Proof.
Aspv∈mi↗∩ℓ∞ with p0>0, one has the function gP=ρfs is a prequasinorm on SHp⋅ρ. Let Pm be a Cauchy sequence in SHp⋅ρX,Y. Therefore, fsm∈Hp⋅ρ and fsmz=∑v=0∞svPmzv converges for any z∈ℂ. Suppose hsz≕∑v=0∞svPi−Pjzv, then from parts (iv) and (vii) of Definition 14 and since LX,Y⊇SHp.ρX,Y, we have
(35)gPi−Pj=ρhs≥ρs0Pi−Pje0=ρPi−Pje0≥ξPi−Pjρe0,then Pmm∈ℕ is a Cauchy sequence in LX,Y. Since the space LX,Y is a Banach space, there is P∈LX,Y with limm⟶∞Pm−P=0 and as fsm∈Hp.ρ, for every m∈ℕ. Hence, by using Theorem 25 and the continuity of ρ at θ, we have
(36)gP=gP−Pm+Pm≤KK0gPm−P+gPm=KK0ρPm−P∑m=−∞em+KK0ρfsm<ε,so fs∈Hp.ρ, which implies P∈SHp.ρX,Y.Theorem 27.
SupposeX and Y are Banach spaces, and pv∈mi↗∩ℓ∞ with p0>0, then SHp.ρ,g is a prequasiclosed operator ideal, where gP=ρfs,fsz=∑v=0∞svPzv converges for any z∈ℂ and
(37)ρfs=∑v=0∞1pvsvPpv,foreveryfs∈Hp⋅.Proof.
Aspv∈mi↗∩ℓ∞ with p0>0, so the function gP=ρfs is a prequasinorm on SHp.ρ. Let Pm∈SHp.ρX,Y, with m∈ℕ and limm⟶∞gPm−P=0. Then, fsm∈Hp.ρ and fsmz=∑v=0∞svPmzv converges for any z∈ℂ. Suppose hsz≕∑v=0∞svPi−Pjzv, then from parts (iv) and (vii) of Definition 14 and since LX,Y⊇SHp.ρX,Y, one obtains
(38)gP−Pj=ρhs≥ρs0P−Pje0=ρP−Pje0≥ξP−Pjρe0,then Pmm∈ℕ is a convergent sequence in LX,Y. Since the space LX,Y is a Banach space, then there is P∈LX,Y with limm⟶∞Pm−P=0 and as fsm∈Hp.ρ, for every m∈ℕ, by using Theorem 25 and the continuity of ρ at θ, one has
(39)gP=gP−Pm+Pm≤KK0gPm−P+gPm=KK0ρPm−P∑m=0∞em+KK0ρfsm<ε,hence, fs∈Hp.ρ, which gives P∈SHp.ρX,Y.According to Theorem9, we introduce the following properties of the s-type Hp.ρ.Theorem 28.
Fors-type Hp.ρ≔svT∈ℝℕ:T∈SHp.ρX,Y. The next settings are verified.
(1)
We haves-type Hp.ρ⊃F(2)
SupposesrT1r=0∞∈s‐typeHp⋅ρ and srT2r=0∞∈s‐typeHp⋅ρ, then srT1+T2r=0∞∈s‐typeHp.ρ(3)
One hasλ∈ℝ and srTr=0∞∈s‐typeHp⋅ρ, then λsrTr=0∞∈s−typeHp.ρ(4)
Thes-type Hp.ρ is solid


### 4.3. Small Prequasi-Banach Ideal

We introduce here some inclusion relations concerning the spaceSHp∙ρ for different pv.Theorem 29.
LetX and Y be Banach spaces with dimX=dimY=∞, and pv,qv∈mi↗∩ℓ∞ with p0>0 and pv<qv, for all v∈ℕ, we have
(40)SHp⋅ρX,Y⊂≠SHp⋅ρX,Y⊂≠LX,Y.Proof.
AssumeT∈SHp.ρX,Y. Therefore, fs∈Hp⋅ρ and fsz=∑v=0∞svTzv converges for any z∈ℂ. Then,
(41)∑v=0∞1qvsvTqv<∑v=0∞1pvsvTqv<∞,hence, T∈SHq.ρX,Y. Next, by taking T with svT=pv/v+11/pv, one has T∉SHp.ρX,Y and T∈SHq.ρX,Y. Clearly, SHq.ρX,Y⊂LX,Y. Again, by choosing svT=qv/v+11/qv, one has T∉SHq.ρX,Y and T∈LX,Y. This finishes the proof.
In this part, we examine the sufficient setting for whichSHp⋅ρapp is small.Theorem 30.
LetX and Y be Banach spaces with dimX=dimY=∞. Assume pv∈mi↗∩ℓ∞ with p0>0, then SHp⋅ρapp is small.Proof.
Obviously, the spaceSHp⋅ρapp,g generates a prequasi-Banach operator ideal, with gT=∑v=0∞1/pvαvTpv. Let SHp⋅ρappX,Y=LX,Y. Hence, there is C>0 with gT≤CT, for all T∈LX,Y. According to Dvoretzky’s theorem [15] with r∈ℕ, there are quotient spaces X/λr and subspaces ηr of Y that operated onto ℓ2r by isomorphisms Dr and Br with DrDr−1≤2 and BrBr−1≤2. Suppose 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 [16], we have
(42)1=haIr=haBrBr−1IrDrDr−1≤BrhaBr−1IrDrDr−1=BrhaJrBr−1IrDrDr−1≤BrdaJrBr−1IrDrDr−1=BrdaJrBr−1IrDrζrDr−1≤BrαaJrBr−1IrDrζrDr−1,for 0≤j≤r. Then for l≥1, one has
(43)1pj≤BrDr−1pj1pjαjJrBr−1IrDrζrpj⇒1pj≤lBr1pjαjJrBr−1IrDrζrpjDr−1⇒∑j=0r1pj≤lBrDr−1∑j=0r1pjαjJrBr−1IrDrζrpj⇒∑j=0r1pj≤lCBrDr−1gJrBr−1IrDrζr⇒∑j=0r1pj≤lCBrDr−1JrBr−1IrDrζr⇒∑j=0r1pj≤lCBrDr−1JrBr−1IrDrζr=lCBrDr−1Br−1IrDr⇒∑j=0r1pj≤4lC.
Asr⟶∞, we get ∑j=0∞1/pj<∞. Since ∑j=0∞a/pj≥1/suppj∑j=0∞1=∞. Hence, the space SHp⋅ρapp is small.
By the same manner, we can easily conclude the next theorem.Theorem 31.
AssumeX and Y be Banach spaces with dimX=dimY=∞. Suppose pv∈mi↗∩ℓ∞ with p0>0, then SHp⋅ρKol is small.


### 4.4. Simple Prequasi-Ideal

In this part, we offer enough settings onHp⋅ρ so that the space SHp⋅ρ is simple.Theorem 32.
Letpv, qv∈mi↗∩ℓ∞ with 1≤pv<qv, for every v∈ℕ, then
(44)LSHq⋅ρ,SHp⋅ρ=ΛSHq⋅ρ,SHp⋅ρ.Proof.
ConsiderT∈LSHq⋅ρ,SHp⋅ρ and T∉ΛSHq⋅ρ,SHp⋅ρ. According to Lemma 3, one has G∈LSHq⋅ρ and B∈LSHp⋅ρ with BTGIm=Im. For every m∈ℕ, one obtains
(45)ImSHp⋅ρ=∑v=0∞1pvsvImpv1/suppv=∑v=0m−11pv1/suppv≤BTGImSHq⋅ρ≤∑v=0∞1qvsvImqv1/supqv=∑v=0m−11qv1/supqv.
This defies Theorem29.Corollary 33.
Letpv,qv∈mi↗∩ℓ∞ with 1≤pv<qv, for each v∈ℕ, then
(46)LSHq⋅ρ,SHp⋅ρ=LCSHq⋅ρ,SHp⋅ρ.Proof.
Clearly, asΛ⊆Lc.Theorem 34.
Assumepv∈mi↗∩ℓ∞ with p0≥1, then SHp.ρ is simple.Proof.
SupposeT∈LCSHp⋅ρ and T∉ΛSHp⋅ρ. In view of Lemma 3, we have G,B∈LSHp⋅ρ so as to BTGIk=Ik. One gets ISHp⋅ρ∈LCSHp⋅ρ. Therefore, LSHp⋅ρ=LCSHp⋅ρ. This implies one and only one nontrivial closed ideal ΛSHp⋅ρ in LSHp⋅ρ.


### 4.5. Spectrum of Prequasi-Ideal

In this part, we introduce enough settings onHp⋅ρ so that the class L with sequence of eigenvalues in Hp⋅ρ equals SHp⋅ρ.Theorem 35.
IfX and Y are Banach spaces with dimX=dimY=∞. Suppose pv∈mi↗∩ℓ∞ with p0>0, we have
(47)SHp⋅ρλX,Y=SHp⋅ρX,Y.Proof.
LetT∈SHp⋅ρλX,Y, then fλ∈Hp⋅ρ, where fλz=∑v=0∞λvTzv converges for all z∈ℂ with ρfλ=∑v=0∞1/pvλvTpv<∞, and T−λvTI=0 for all v∈ℕ. We have T=λvTI, with v∈ℕ, hence svT=svλvTI=λvT, with v∈ℕ. As a result, fs∈Hp.ρ, then T∈SHp.ρX,Y. Secondly, assume T∈SHp.ρX,Y. Hence, fs∈Hp.ρ, where fsz=∑v=0∞svTzv converges for all z∈ℂ with ρfs=∑v=0∞1/pvsvTpv<∞. One has
(48)∑v=0∞1pvsvTpv≥1supvpv∑v=0∞svTpv.
Therefore,limv⟶∞svT=0. Let T−svTI−1 exists, for all v∈ℕ. Hence, T−svTI−1 exists and bounded, for all v∈ℕ. Therefore, limv⟶∞T−svTI−1=T−1 exists and bounded. By using the prequasioperator ideal of SHp⋅ρ,g, one has
(49)I=TT−1∈SHp⋅ρX,Y⇒svIv=0∞∈Hp⋅ρ⇒limv⟶∞svI=0.
Sincelimv⟶∞svI=1. Hence, T−svTI=0, for all v∈ℕ. This gives T∈SHp⋅ρλX,Y.
This shows the proof.


## 4.1. Ideal of Finite Rank Operators

In this part, enough settings (not necessary) onHp.ρ so that F¯=SHp.ρ are given. This explains the nonlinearity of the s-type Hp.ρ spaces (Rhoades open problem [10]).In view of Theorems13 and 20, we conclude the next theorem.Theorem 23.
Considerpv∈mi↗∩ℓ∞ with p0>0, then SHp. is an operator ideal.Theorem 24.
Ifpv∈mi↗∩ℓ∞ with p0>0, then F¯=SHp.ρ, where
(26)ρf=∑v=0∞1pvfv^,foreveryf∈Hp⋅.Proof.
Clearly,FX,Y¯⊂SHp.ρX,Y, since the space SHp.ρ is an operator ideal. Therefore, we have to show that SHp.ρX,Y⊆FX,Y¯. By letting T∈SHp.ρX,Y, then, fs∈Hp.ρ, with fsz=∑v=0∞svTzv converges for any z∈ℂ. So, ρfs<∞, fix ε∈0,1, we have m∈ℕ−0 with ρfs−∑v=0m−1ev<ε/4. As svTv∈ℕ is decreasing, we have
(27)∑v=m+12m1pvs2mTpv≤∑v=m+12m1pvsvTpv≤∑v=m∞1pvsvTpv<ε4.
Therefore, we haveA∈F2mX,Y, rank A≤2m and
(28)∑v=2m+13m1pvT−Apv≤∑v=m+12m1pvT−Apv<ε4.
Aspv∈ℓ∞, then
(29)∑v=0m1pvT−Apv<ε4.
SinceT−A∈SHp.pX,Y, then hs∈Hp.p, where hsz≔∑v=0∞svT−Azv converges for any z∈ℂ. Because pv is increasing and from the inequalities (27)–(29), we get
(30)dT,A=ρhs=∑v=03m−11pvsvT−Apv+∑v=3m∞1pvsvT−Apv≤∑v=03m1pvT−Apv+∑v=m∞1pv+2msv+2mT−Apv+2m≤3∑v=0m1pvT−Apv+∑v=m∞1pvsvTpv<ε.
SinceI2∈SH1/n+1pX,Y but the condition pv∈mi↗∩ℓ∞ is not verified which explain a negative example of the converse statement. This finishes the proof.
We can reformulate Theorem24 as follows: if pv∈mi↗∩ℓ∞ with p0>0, then every compact operators can be approximated by finite rank operators and the converse is not always true.


## 4.2. Banach and Closed Prequasi-Ideal

In this part, enough settings onHp.ρ so that the prequasioperator ideal SHρ is Banach and closed are investigated.Theorem 25.
Assumepv∈mi↗∩ℓ∞ with p0>0, then the function gP=ρfs is a prequasinorm on SHp.ρ, where fsz=∑v=0∞svPzv converges for any z∈ℂ and
(31)ρfs=∑v=0∞1pvsvPpv,foreveryfs∈Hp⋅.Proof.
One hasg verifies the next setups:
(1)
LetP∈SHp⋅ρX,Y,gP=ρfs≥0 and gP=ρfs=0⇔svP=0, for all v∈N⇔P=0(2)
There isl≥1 with gλP=ρλfs≤lλρfs=lλgP, for every P∈SHp⋅ρX,Y and λ∈ℝ(3)
One hasKK0≥1, for P1,P2∈SHp⋅ρX,Y. Then, f1sz=∑v=0∞svP1zv and f2sz=∑v=0∞svP2zv converge for any z∈ℂ. Therefore, for hsz≕∑v=0∞svP1+P2zv, one has(32)gP1+P2=ρhs≤ρf1s.+f2s.≤Kρf1s.+ρf2s.≤KK0gP1+gP2(4)
We haveC≥1, let A∈LX0,X,B∈SHp.ρX,Y and D∈LY,Y0. Then, fsz=∑v=0∞svBzv converges for all z∈ℂ. Then, for hsz≕∑v=0∞svDBAzv, one has(33)gDBA=ρhs≤ρADfs≤CAgBDTheorem 26.
AssumeX and Y are Banach spaces, and pv∈mi↗∩ℓ∞ with p0>0, then SHp.ρ,g is a prequasi-Banach operator ideal, where gP=ρfs,fsz=∑v=0∞svPzv converges for any z∈ℂ and
(34)ρfs=∑v=0∞1pvsvPpv,foreveryfs∈Hp⋅.Proof.
Aspv∈mi↗∩ℓ∞ with p0>0, one has the function gP=ρfs is a prequasinorm on SHp⋅ρ. Let Pm be a Cauchy sequence in SHp⋅ρX,Y. Therefore, fsm∈Hp⋅ρ and fsmz=∑v=0∞svPmzv converges for any z∈ℂ. Suppose hsz≕∑v=0∞svPi−Pjzv, then from parts (iv) and (vii) of Definition 14 and since LX,Y⊇SHp.ρX,Y, we have
(35)gPi−Pj=ρhs≥ρs0Pi−Pje0=ρPi−Pje0≥ξPi−Pjρe0,then Pmm∈ℕ is a Cauchy sequence in LX,Y. Since the space LX,Y is a Banach space, there is P∈LX,Y with limm⟶∞Pm−P=0 and as fsm∈Hp.ρ, for every m∈ℕ. Hence, by using Theorem 25 and the continuity of ρ at θ, we have
(36)gP=gP−Pm+Pm≤KK0gPm−P+gPm=KK0ρPm−P∑m=−∞em+KK0ρfsm<ε,so fs∈Hp.ρ, which implies P∈SHp.ρX,Y.Theorem 27.
SupposeX and Y are Banach spaces, and pv∈mi↗∩ℓ∞ with p0>0, then SHp.ρ,g is a prequasiclosed operator ideal, where gP=ρfs,fsz=∑v=0∞svPzv converges for any z∈ℂ and
(37)ρfs=∑v=0∞1pvsvPpv,foreveryfs∈Hp⋅.Proof.
Aspv∈mi↗∩ℓ∞ with p0>0, so the function gP=ρfs is a prequasinorm on SHp.ρ. Let Pm∈SHp.ρX,Y, with m∈ℕ and limm⟶∞gPm−P=0. Then, fsm∈Hp.ρ and fsmz=∑v=0∞svPmzv converges for any z∈ℂ. Suppose hsz≕∑v=0∞svPi−Pjzv, then from parts (iv) and (vii) of Definition 14 and since LX,Y⊇SHp.ρX,Y, one obtains
(38)gP−Pj=ρhs≥ρs0P−Pje0=ρP−Pje0≥ξP−Pjρe0,then Pmm∈ℕ is a convergent sequence in LX,Y. Since the space LX,Y is a Banach space, then there is P∈LX,Y with limm⟶∞Pm−P=0 and as fsm∈Hp.ρ, for every m∈ℕ, by using Theorem 25 and the continuity of ρ at θ, one has
(39)gP=gP−Pm+Pm≤KK0gPm−P+gPm=KK0ρPm−P∑m=0∞em+KK0ρfsm<ε,hence, fs∈Hp.ρ, which gives P∈SHp.ρX,Y.According to Theorem9, we introduce the following properties of the s-type Hp.ρ.Theorem 28.
Fors-type Hp.ρ≔svT∈ℝℕ:T∈SHp.ρX,Y. The next settings are verified.
(1)
We haves-type Hp.ρ⊃F(2)
SupposesrT1r=0∞∈s‐typeHp⋅ρ and srT2r=0∞∈s‐typeHp⋅ρ, then srT1+T2r=0∞∈s‐typeHp.ρ(3)
One hasλ∈ℝ and srTr=0∞∈s‐typeHp⋅ρ, then λsrTr=0∞∈s−typeHp.ρ(4)
Thes-type Hp.ρ is solid


## 4.3. Small Prequasi-Banach Ideal

We introduce here some inclusion relations concerning the spaceSHp∙ρ for different pv.Theorem 29.
LetX and Y be Banach spaces with dimX=dimY=∞, and pv,qv∈mi↗∩ℓ∞ with p0>0 and pv<qv, for all v∈ℕ, we have
(40)SHp⋅ρX,Y⊂≠SHp⋅ρX,Y⊂≠LX,Y.Proof.
AssumeT∈SHp.ρX,Y. Therefore, fs∈Hp⋅ρ and fsz=∑v=0∞svTzv converges for any z∈ℂ. Then,
(41)∑v=0∞1qvsvTqv<∑v=0∞1pvsvTqv<∞,hence, T∈SHq.ρX,Y. Next, by taking T with svT=pv/v+11/pv, one has T∉SHp.ρX,Y and T∈SHq.ρX,Y. Clearly, SHq.ρX,Y⊂LX,Y. Again, by choosing svT=qv/v+11/qv, one has T∉SHq.ρX,Y and T∈LX,Y. This finishes the proof.
In this part, we examine the sufficient setting for whichSHp⋅ρapp is small.Theorem 30.
LetX and Y be Banach spaces with dimX=dimY=∞. Assume pv∈mi↗∩ℓ∞ with p0>0, then SHp⋅ρapp is small.Proof.
Obviously, the spaceSHp⋅ρapp,g generates a prequasi-Banach operator ideal, with gT=∑v=0∞1/pvαvTpv. Let SHp⋅ρappX,Y=LX,Y. Hence, there is C>0 with gT≤CT, for all T∈LX,Y. According to Dvoretzky’s theorem [15] with r∈ℕ, there are quotient spaces X/λr and subspaces ηr of Y that operated onto ℓ2r by isomorphisms Dr and Br with DrDr−1≤2 and BrBr−1≤2. Suppose 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 [16], we have
(42)1=haIr=haBrBr−1IrDrDr−1≤BrhaBr−1IrDrDr−1=BrhaJrBr−1IrDrDr−1≤BrdaJrBr−1IrDrDr−1=BrdaJrBr−1IrDrζrDr−1≤BrαaJrBr−1IrDrζrDr−1,for 0≤j≤r. Then for l≥1, one has
(43)1pj≤BrDr−1pj1pjαjJrBr−1IrDrζrpj⇒1pj≤lBr1pjαjJrBr−1IrDrζrpjDr−1⇒∑j=0r1pj≤lBrDr−1∑j=0r1pjαjJrBr−1IrDrζrpj⇒∑j=0r1pj≤lCBrDr−1gJrBr−1IrDrζr⇒∑j=0r1pj≤lCBrDr−1JrBr−1IrDrζr⇒∑j=0r1pj≤lCBrDr−1JrBr−1IrDrζr=lCBrDr−1Br−1IrDr⇒∑j=0r1pj≤4lC.
Asr⟶∞, we get ∑j=0∞1/pj<∞. Since ∑j=0∞a/pj≥1/suppj∑j=0∞1=∞. Hence, the space SHp⋅ρapp is small.
By the same manner, we can easily conclude the next theorem.Theorem 31.
AssumeX and Y be Banach spaces with dimX=dimY=∞. Suppose pv∈mi↗∩ℓ∞ with p0>0, then SHp⋅ρKol is small.


## 4.4. Simple Prequasi-Ideal

In this part, we offer enough settings onHp⋅ρ so that the space SHp⋅ρ is simple.Theorem 32.
Letpv, qv∈mi↗∩ℓ∞ with 1≤pv<qv, for every v∈ℕ, then
(44)LSHq⋅ρ,SHp⋅ρ=ΛSHq⋅ρ,SHp⋅ρ.Proof.
ConsiderT∈LSHq⋅ρ,SHp⋅ρ and T∉ΛSHq⋅ρ,SHp⋅ρ. According to Lemma 3, one has G∈LSHq⋅ρ and B∈LSHp⋅ρ with BTGIm=Im. For every m∈ℕ, one obtains
(45)ImSHp⋅ρ=∑v=0∞1pvsvImpv1/suppv=∑v=0m−11pv1/suppv≤BTGImSHq⋅ρ≤∑v=0∞1qvsvImqv1/supqv=∑v=0m−11qv1/supqv.
This defies Theorem29.Corollary 33.
Letpv,qv∈mi↗∩ℓ∞ with 1≤pv<qv, for each v∈ℕ, then
(46)LSHq⋅ρ,SHp⋅ρ=LCSHq⋅ρ,SHp⋅ρ.Proof.
Clearly, asΛ⊆Lc.Theorem 34.
Assumepv∈mi↗∩ℓ∞ with p0≥1, then SHp.ρ is simple.Proof.
SupposeT∈LCSHp⋅ρ and T∉ΛSHp⋅ρ. In view of Lemma 3, we have G,B∈LSHp⋅ρ so as to BTGIk=Ik. One gets ISHp⋅ρ∈LCSHp⋅ρ. Therefore, LSHp⋅ρ=LCSHp⋅ρ. This implies one and only one nontrivial closed ideal ΛSHp⋅ρ in LSHp⋅ρ.


## 4.5. Spectrum of Prequasi-Ideal

In this part, we introduce enough settings onHp⋅ρ so that the class L with sequence of eigenvalues in Hp⋅ρ equals SHp⋅ρ.Theorem 35.
IfX and Y are Banach spaces with dimX=dimY=∞. Suppose pv∈mi↗∩ℓ∞ with p0>0, we have
(47)SHp⋅ρλX,Y=SHp⋅ρX,Y.Proof.
LetT∈SHp⋅ρλX,Y, then fλ∈Hp⋅ρ, where fλz=∑v=0∞λvTzv converges for all z∈ℂ with ρfλ=∑v=0∞1/pvλvTpv<∞, and T−λvTI=0 for all v∈ℕ. We have T=λvTI, with v∈ℕ, hence svT=svλvTI=λvT, with v∈ℕ. As a result, fs∈Hp.ρ, then T∈SHp.ρX,Y. Secondly, assume T∈SHp.ρX,Y. Hence, fs∈Hp.ρ, where fsz=∑v=0∞svTzv converges for all z∈ℂ with ρfs=∑v=0∞1/pvsvTpv<∞. One has
(48)∑v=0∞1pvsvTpv≥1supvpv∑v=0∞svTpv.
Therefore,limv⟶∞svT=0. Let T−svTI−1 exists, for all v∈ℕ. Hence, T−svTI−1 exists and bounded, for all v∈ℕ. Therefore, limv⟶∞T−svTI−1=T−1 exists and bounded. By using the prequasioperator ideal of SHp⋅ρ,g, one has
(49)I=TT−1∈SHp⋅ρX,Y⇒svIv=0∞∈Hp⋅ρ⇒limv⟶∞svI=0.
Sincelimv⟶∞svI=1. Hence, T−svTI=0, for all v∈ℕ. This gives T∈SHp⋅ρλX,Y.
This shows the proof.


## 5. Weighted Shift Operators onHp⋅ρ

In this section, we present the upper bounds ofs-numbers for infinite series of the weighted nth power forward and backward shift operator on Hp. with applications to some entire functions.Theorem 36.
Assumepv∈mi↗∩ℓ∞ with p0>0, then Vz∈LHp.ρ with
(50)Vz=suprprpr+11/ϖp,where ρf=∑r=0∞1/prfr∧pr1/ϖp, for all f∈Hp.ρ.Proof.
Suppose the setups are verified. Forf∈Hp.ρ. Since pv∈mi↗∩ℓ∞ with p0>0, then ρVzf=ρzf=∑r=0∞1/pr+1fr∧pr+11/ϖp≤∑r=0∞1/pr+1fr∧pr+11/ϖp≤suprpr/pr+11/ϖp∑r=0∞1/prfr∧pr+11/ϖp=suprpr/pr+11/ϖpρf.
Therefore,Vz∈LHp⋅p with Vz≤suprpr/pr+11/ϖp. Since Vz∈LHp.ρ. Then, there is A>0 with ρVzf≤Aρf, for all f∈Hp.ρ. Hence, ρVzer≤Aρer, one gets suprpr/pr+11/ϖp≤Vz.
This completes the proof.Theorem 37.
Considerpv∈mi↗∩ℓ∞ with p0>0, then Bz∈LHp.ρ with
(51)Bz=suprpr+1pr1/ϖp,where ρf=∑r=0∞1/prfr∧pr1/ϖp, for every f∈Hp.ρ.Proof.
Let the given settings hold for everyf∈Hp.ρ. Since pv∈mi↗∩ℓ∞ with p0>0, then
(52)ρBzf=∑r=0∞1prfr+1∧pr1/ϖp≤suprpr+1pr1/ϖp∑r=0∞1pr+1fr+1∧pr1/ϖp≤suprpr+1pr1/ϖp∑r=0∞1prfr∧pr1/ϖp=suprpr+1pr1/ϖpρf.
Therefore,Bz∈LHp⋅ρ with Bz≤suprpr+1/pr1/ϖp. Since Bz∈LHp.ρ. Then, there is A>0 with ρBzf≤Aρf, for all f∈Hp.ρ. Hence, ρBzer+1≤Aρer+1, then suprpr+1/pr1/ϖp≤Bz. This completes the proof.Theorem 38.
Letpv∈mi↗∩ℓ∞ with p0≥1. Suppose limsupv⟶∞1/pvv=1, then every function in Hp.ρ is analytic on the open unit disc D. Moreover, the convergence in Hp⋅ρ implies the uniform convergence on compact subsets of D, where ρf=∑r=0∞1/prfr∧pr1/ϖp, for any f∈Hp⋅ρ.Proof.
Supposelimsupv⟶∞1/pvv=1, and f∈Hp⋅ρ.Therefore, fz=∑v=0∞fv^zv converges for every z∈ℂ and ρf=∑v=0∞1/pvfv∧pv1/ϖp<∞. Hence, limsupv⟶∞1/pvfv∧pv<1v. We have
(53)limsupv⟶∞fv∧pvv<1limsup1/pvvv⟶∞=1.
Sincepv∈mi↗∩ℓ∞ with p0≥1, we obtain limsupv⟶∞fv^vz<z<1, for all z∈D. Hence, fz=∑v=0∞fv^zv converges for every complex value of z∈D. Assume A is a compact subset of D and fkz∈A, for all k∈ℕ. Let fk converges to f∈Hp∙ρ, we have
(54)fkz−fz=∑v=0∞fvk^−fv^zv≤∑v=0∞fvk^−fv^zv≤∑v=0∞1pvfvk∧−fv∧pv1/ϖp∑v=0∞pvqvzvqv1/ϖp=∑v=0∞pvqvzvqv1/ϖpρfk−f,where qv∈mi↗⋂ℓ∞ with q0≥1 and 1/pv+1/qv=1, for all v∈ℕ. Clearly, limsupv⟶∞pvqv/vzqv<1, then ∑v=0∞pvqvzvqv1/ϖp<∞. So limk⟶∞fkz=fz∈A.Theorem 39.
AssumeVz is the forward shift operator on Hp⋅ρ, with ρf=∑r=0∞1/prfr∧pr1/ϖp, for all f∈Hp.ρ. Then,
(55)supcardξ=r+1infk∈ξpkpk+n1/ϖp1An≤srVzn≤supcardξ=r+1infk∈ξpkpk+n1/ϖp,where An=∑k=0∞1/pkfk∧pk1/ϖp/∑k=ξ∞1/pkfk∧pk+n1/ϖp..Proof.
Let cardξ=r+1 and as Vznf∈Hp.ρ, for all f∈Hp.ρ, where fz=∑k=0∞fk^zk converges for every z∈ℂ and ρf=∑k=0∞1/pkfk¯pk1/ϖp<∞. Hence, Vznfz=∑k=0∞fk^zk+n and ρVznf=∑k=0∞a/pk+nfk∧pk+n1/ϖp<∞.Assume Pξ is and operator on Hp⋅ρ with rank Pξ=r+1 defined by
(56)pξgz=Pξ∑k=0∞fk^zk+n=∑k∈ξifk^zk+n.
SinceρPξg=∑k∈ξ1/pk+nfk∧pk+n1/ϖp≤∑k=0∞1/pk+nfk∧pk+n1/ϖp=ρg. This implies Pξ≤1.
Define an operatorSzn by Sznhz=Szn∑k∈ξfk^zk+n=∑k=0∞fk^zk, then
(57)ρSznh=∑k=0∞1pkfk∧pk1/ϖp≤Un∑k∈ξ1pk+nfk∧pk+n1/ϖp=Unρh.
Hence,Szn≤Un, where
(58)1≤Un=∑k=0∞1/pkfk∧pk1/ϖp∑k∈ξ1/pk+nfk∧pk+n1/ϖp<∞.
Therefore, the identity map will beIr+1=PξVznSzn, according to the definition of s-numbers, we have
(59)srIr+1=1≤PξsrVznszn≤srVznszn⇒srVzn≥1Szn≤1Un=∑k∈ξ1/pk+nfk∧pk+n1/ϖp∑k=0∞1/pkfk∧pk1/ϖp≥infk∈ξpkpk+n1/ϖp1An.
This inequality is satisfied for all card,ξ=r+1 and one has
(60)srVzn≥supcardξ=r+1infk∈ξpkpk+n1/ϖp1An.
On the other hand, letξ be a subset of ℕ with card ξ=r. Define the finite rank map Rzn by Rznvz=Rzn∑k=0∞f⌢kzk=∑k∈ξf⌢kzk+n. In view of the definition of approximation numbers, we have
(61)srVzn≤αrVzn≤Vzn−Rzn≤supfz≠0Vzn−Rznfzfz=supfz≠0∑k∉ξf⌢kzk+nfz≤supfz≠0∑k∉ξ1/pk+nf⌢kpk+n1/ϖpfz≤supk∉ξpkpk+n1/ϖp.
This inequality is verified for every cardξ=r and by using Lemma 10, one has
(62)supcardξ=r+1infk∈ξpkpk+n1/ϖp1An≤srVzn≤infcardξ=rsupk∉ξpkpk+n1/ϖp=supcardξ=r+1infk∈ξpkpk+n1/ϖp.
This completes the proof.Theorem 40.
IfBz is the backward shift operator on Hp.ρ, with ρf=∑r=0∞1/prf⌢rpr1/ϖp, for all f∈Hp.ρ Then,
(63)supcardξ=r+1infk∈ξpk+npk1/ϖp1Gn≤srBzn≤supcardξ=r+1infk∈ξpk+npk1/ϖp,where Gn=∑k=0∞1/pkfk∧pk1/ϖp/∑k∈ξ1/pk+nfk∧pk+n1/ϖp.Proof.
Assume cardξ=r+1 and since Bznf∈Hp.ρ, for every f∈Hp.ρ, where fz=∑k=0∞f⌢kzk converges for any z∈ℂandρf=∑k=0∞1/pkfk∧pk1/ϖp<∞. Therefore, Bznfz∑k=0∞f⌢k+nzkandρBznf=∑k=0∞1/pkfk∧pk1/ϖp<∞.Suppose Pξ is an operator on Hp.ρ with rank Pξ=r+1 evident by
(64)pξgz=Pξ∑k=0∞f⌢k+nzk=∑k∈ξ∞f⌢k+nzk.
AsρPξg=∑k∈ξ1/pkf⌢k+npk1/ϖp≤∑k=0∞1/pkfk+n∧pk1/ϖp=pg. This implies pξ≤1. Define an operator SznbySznhz=Szn∑k∈ξf⌢k+nzk=∑k=0∞f⌢kzk, one gets
(65)ρSznh=∑k=0∞1pkf⌢kpk1/ϖp≤Un∑k∈ξf⌢k+npk1/ϖp=Unρh.
Therefore,∥Szn∥≤Un, where 1≤Un=∑k=0∞1/pkf⌢kpk1/ϖp/∑k∈ξ1/pkf⌢k+npk1/ϖp<∞. Hence, the identity operator will be Ir+1=PξBznSzn, in view of the definition ofs-numbers, one has
(66)srIr+1=1≤PξsrBznSzn≤srBznSzn⇒srBzn≥1Szn≥1Un=∑k∈ξ1/pkf⌢k+npk1/ϖp∑k=0∞1/pkf⌢kpk1/ϖp≥infk∈ξpk+npk1/ϖp1Gn.
This inequality is confirmed for all cardξ=r+1, and we have
(67)srBzn≥supcardξ=r+1infk∈ξpk+npk1/ϖp1Gn.
On the other hand, supposeξ is a subset of ℕ with card ξ=r. Define the finite rank operator Rzn by Rznvz=Rzn∑k=0∞f⌢kzk=∑k∈ξf⌢k+nzk. From the definition of approximation numbers, one gets
(68)srBzn≤αrBzn≤Bzn−Rzn≤supfz≠0Bzn−Rznfzfz=supfz≠0∑k∉ξf⌢k+nzkfz≤supfz≠0∑k∉ξ1/pkf⌢k+npkfz1/ϖp≤supk∉ξpk+npk1/ϖp.
This inequality is satisfied for any cardξ=r and from Lemma 10, we have
(69)supcardξ=r+1infk∈ξpk+npk1/ϖp1Gn≤srBzn≤infcardξ=rsupk∈ξpk+npk1/ϖp=supcardξ=r+1infk∈ξpk+npk1/ϖp.
This finishes the proof.Next, the upper and lower bounds of norm∑m=0∞cmVzmonthespaceHp.ρ have been explained.Theorem 41.
The effect of∑m=0∞cmVzmonthespaceHp.p,whereρf=∑r=0∞1/prf⌢rpr1/ϖp,forallf∈Hp.ρ,cmm=0∞∈ℓpm/ϖp,andpv∈mi↗∩ℓ∞withp0≥1,wehave(70)supk∑m=0∞cmpm+kpkpk+m1/ϖp≤∑m0∞cmVzm≤supm,kpkpk+m1/ϖp∑m=0∞cmpm/ϖp.Proof.
Assumef∈Hp.ρ,wehave∑m=0∞cmVzmfz=∑k=0∞∑m=0∞cmf⌢kzk+m. Then,
(71)∑m=0∞cmVzm≥ρ∑m=0∞cmVzmekρek=∑m=0∞1/pm+kcmpm+k1/pk1/ϖp≥supk∑m=0∞cmpm+kpkpk+m1/ϖp.
Sinceρ satisfies the triangle inequality, we get
(72)∑m=0∞cmVzm=suppf≠0ρ∑m=0∞cmVzmfρf≤suppf≠0∑m=0∞∑k=0∞1/pm+kcmf⌢kpm+k1/ϖp∑k=0∞1/pkf⌢kpk1/ϖp≤supm,kpkpk+m1/ϖp∑m=0∞∑k=0∞1/pkcmf⌢kpm+k1/ϖp∑k=0∞1/pk1/pkf⌢kpk1/ϖp≤supm,kpkpk+m1/ϖp∑m=0∞cmpm/ϖp.
Next the upper and lower bounds of norm∑m=0∞cmBzmonthespaceHp.ρ have been investigated.Theorem 42.
The effect of∑m=0∞cmBzmonthespaceHp.ρ,whereρf=∑r=0∞1/prf⌢rpr1/ϖp,forallf∈Hp.ρ,cmm=0∞∈ℓpm/ϖp,andpv∈mi↗∩ℓ∞withp0≥1, we have
(73)supk∑m=0∞cmpkpk+mpk1/ϖp≤∑m=0∞cmBzm≤supm,kpk+mpk1/ϖp∑m=0∞cmpm/ϖp.Proof.
Supposef∈Hp.ρ,onehas∑m=0∞cmBzmfz=∑k=0∞∑m=0∞cmf⌢k+mzk. We have
(74)∑m=0∞cmBzm≥ρ∑m=0∞cmBzmekρek=∑m=0∞1/pk−mcmpk−m1/pk1/ϖp≥supk∑m=0∞cmpkpk+mpk1/ϖp.
Asρ verifies the triangle inequality, one can see
(75)∑m=0∞cmBzm=suppf≠0ρ∑m=0∞cmBzmfρf≤suppf≠0∑m=0∞∑k=0∞1/pkcmf⌢k+mpk1/ϖp∑k=0∞1/pkf⌢kpk1/ϖp≤supm,kpk+mpk1/ϖp∑m=0∞∑k=0∞1/pk+mcmf⌢k+mpk1/ϖp∑k=0∞1/pkf⌢kpk1/ϖp≤supm,kpk+mpk1/ϖp∑m=0∞cmpm/ϖp.
The following theorem indicates an upper estimation to thes-numbers of ∑m=0∞cmVzmonthespaceHp.ρ.Theorem 43.
The effect of∑m=0∞cmVzmonthespaceHp.ρ,wherepf=∑r=0∞1/prf⌢rpr1/ϖp,forallf∈Hp.ρ, the s-numbers of this operator are presented by
(76)sr∑m=0∞cmVzm≤supcardξ=r+1infk∈ξsupmpkpk+m1/ϖp∑m=0∞cmpm/ϖp,for all cmm=0∞∈ℓpm/ϖpandpv∈mi↗∩ℓ∞withp0≥1.Proof.
Letξ be a subset of ℕ and card ξ=r. By using the definition of s-numbers. Define the finite rank operator RbyRfz=R∑k=0∞f⌢kzk=∑k∈ξ∑m=0kcmf⌢k−mzk. In view of the definition of approximation numbers and since ρ satisfies the triangle inequality, we have
(77)sr∑m=0∞cmVzm≤αr∑m=0∞cmVzm≤∑m=0∞cmVzm−R≤supρf≠0ρ∑m=0∞cmVzmf−Rfρf≤supρf≠0∑m=0∞∑k∉ξ1/pm+kcmf⌢kpk+m1/ϖpρf≤supk∉ξ,mpkpk+m1/ϖp∑m=0∞cmpm/ϖp.
This inequality is verified for every cardξ=r, and one has
(78)sr∑m=0∞cmVzm≤infcardξ=rsupk∉ξ,mpkpk+m1/ϖp∑m=0∞cmpm/ϖp=supcardξ=r+1infk∉ξsupmpkpk+m1/ϖp∑m=0∞cmpm/ϖp.
This implies the proof.The next theorem investigates an upper estimation to thes-numbers of ∑m=0∞cmBzmonthespaceHp.ρ.Theorem 44.
Acting∑m=0∞cmBzm on the space Hp.ρ, where ρf=∑r=0∞1/prf⌢rpr1/ϖp,foreveryf∈Hp.ρ, the s-numbers of this operator satisfy
(79)sr∑m=0∞cmBzm≤supcardξ=r+1infk∈ξsupmpk+mpk1/ϖp∑m=0∞cmpm/ϖp,for all cmm=0∞∈ℓpm/ϖpandpv∈mi↗∩ℓ∞withp0≥1.Proof.
Assumeξ is a subset of ℕ and card ξ=r. From the definition of s-numbers. Define the finite rank operator RbyRfz=R∑k=0∞f⌢kzk=∑k∈ξ∑m=0kcmf⌢k−mzk. From the definition of approximation numbers and as ρ verifies the triangle inequality, one has
(80)sr∑m=0∞cmBzm≤αr∑m=0∞cmBzm≤∑m=0∞cmBzm−R≤supρf≠0ρ∑m=0∞cmBzmf−Rfρf≤supρf≠0∑m=0∞∑k∉ξ1/pkcmf⌢k+mpk1/ϖpρf≤supk∉ξ,mpk+mpk1/ϖp∑m=0∞cmpm/ϖp.
This inequality is satisfied for all cardξ=r, and we have
(81)sr∑m=0∞cmBzm≤infcardξ=rsupk∉ξ,mpk+mpk1/ϖp∑m=0∞cmpm/ϖp=supcardξ=r+1infk∉ξsupmpk+mpk1/ϖp∑m=0∞cmpm/ϖp.
This completes the proof.The following theorems are direct consequences of Theorem43 and Definition 18, for some entire functions, for example, the exponential and the sine functions.Theorem 45.
Letpv∈mi↗∩ℓ∞withp0≥1. Assume Bez is a shift operator on Hp.ρ, where for ρf=∑r=0∞1/prf⌢rpr1/ϖp,forallf∈Hp.ρandez=∑m=0∞zm/m!. The upper estimation of the s-number of Vez is given by
(82)srVez≤supcardξ=r+1infk∈ξsupmpk+mpk1/ϖp∑m=0∞1m!pm/ϖp.Theorem 46.
Letpv∈mi↗∩ℓ∞withp0≥1. Suppose Bsinz is a shift operator on Hp.ρ, where ρf=∑r=0∞1/prf⌢rpr1/ϖp,forallf∈Hp.ρandsinz=∑m=0∞−1mz2m+1/2m+1!. The upper estimation of the s-numbers of Vsinz is presented by
(83)srVsinz≤supcardξ=r+1infk∈ξsupmpk+mpk1/ϖp∑m=0∞12m+1!pm/ϖp.The following theorems are direct consequences of Theorem44 and Definition 19, for some entire functions, for example, the exponential and the sine functions.Theorem 47.
Assumepv∈mi↗∩ℓ∞withp0≥1. Suppose Bez is a shift operator on Hp.ρ, where ρf=∑r=0∞1/prf⌢rpr1/ϖp,foreveryf∈Hp.ρandez=∑m=0∞zm/m!. The upper estimation of the s-numbers of Bez is pretended by
(84)srBez≤supcardξ=r+1infk∈ξsupmpk+mpk1/ϖp∑m=0∞1m!pm/ϖp.Theorem 48.
Supposepv∈mi↗∩ℓ∞withp0≥1. Assume Bsinz is a shift operator on Hp.ρ, where ρf=∑r=0∞1/prf⌢rpr1/ϖp,foreveryf∈Hp.ρandsinz=∑m=0∞−1mz2m+1/2m+1!. The upper estimation of the Bsinz is presented by
(85)srBsinz≤supcardξ=r+1infk∈ξsupmpk+mpk1/ϖp∑m=0∞12m+1!pm/ϖp.



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*Source: 3526217-2021-10-29.xml*