# Certain Classes of Operators on Some Weighted Hyperbolic Function Spaces

Authors: A. El-Sayed Ahmed; Amnah E. Shammaky
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, some classes of concerned multiplication operators consisting of analytic and hyperbolic functions are defined and considered. Furthermore, some properties such as boundedness and compactness of the new operators are discussed. Finally, a general class of weighted hyperbolic Bloch functions is characterized by metric spaces.


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## Body

## 1. Introduction

Complex function spaces are one of the interesting core subjects in mathematical analysis. This subject has a lot of various generalizations with many joyful branches. The study of hyperbolic function spaces has been at the rigorous research activity. The presented paper deals with discussion of some hyperbolic function classes.The known open-unit disc that defined in the concerned complex planeℂ and its specific boundary are symbolized by Δ=z:z<1 and ∂Δ, respectively. The specific space of all holomorphic functions in Δ is denoted by ℋΔ. Also, assuming that BΔ defines the subset of ℋΔ, which contains those functions f∈ℋΔ with fz<1 for all concerned points z∈Δ.IfX,D is a concerned metric space, the specific open and closed concerned balls with center a and radius R>0 are denoted by(1)Ba,R≔u∈X:du,a<R,B¯a,R≔u∈X:Du,a=R,respectively.Assume thatf∗z=f′z/1−fz2 defines the specific hyperbolic derivative of f∈BΔ.Throughout this paper, we will suppose that0<α<1 and ω:0,1⟶0,∞ to be a nondecreasing and continuous function with ω≠0.A concerned functionf∈BΔ is said to belong to the specific hyperbolic α-Bloch class ℬω,α∗, when(2)fℬω,α∗=supz∈Δf∗z1−z2αω1−z<∞.The little specific hyperbolic Bloch-type classℬω,α,0∗ contains all f∈ℬωα∗, with(3)limz⟶1f∗z1−z2αω1−z=0.One of the major aims of this study is to give emerging treatments of some properties of a certain class of operators acting between hyperbolic Bloch functions using the concerned framework of hyperbolic spaces, which is based on the concerned technique of their specific functions. The multiplication operator between two different types of functions, one is analytic and the other is hyperbolic, is treated in this paper. Dealing with corresponding concerned spaces of weighted functions on the diskΔ, essential properties of the new type of operators are discussed. There are some certain attempts to study hyperbolic function spaces (see [1–14] and others). Most research studies were on composition operators. In this paper, by defining some new classes of operators, we will study hyperbolic logarithmic Bloch functions.


## 2. Boundedness and Compactness

Assumingfz and gz are two functions in ℋΔ, the following operators are(4)Ffz=zfz∗=zf′z+fz1−zfz2.Further, the operator(5)ℒgfz=fzg∗z,is one of the concerned aims in this study.Remark 1.
The operatorsFfz and ℒgfz are introduced for the first time to study hyperbolic function classes in this paper.
An interesting approach of unified criteria for boundedness and compactness properties of new type operators acting between hyperbolic logarithmic Bloch spaces is investigated in the present section.Lemma 1.
Suppose thatf∈ℋΔ. Thus, f∈ℬω,α∗⇔Ffz∈ℬω,α∗.Proof.
Clearly, we have that when the functiong∈ℬω,α∗, we can have(6)supz∈Δg∗zFω,z≤λ<∞,where(7)Fω,z=1−z2αω1−z.
Letf∈ℬω,α∗. Therefore,(8)fℬω,α∗=supz∈Δf∗zFω,z<∞.
Hence,(9)Ffzℬω,α∗=supz∈Δzf′z+fz1−zfz2Fω,z≤supz∈Δzf′z1−zfz2Fω,z+supz∈Δfz1−zfz2Fω,z≤fℬω,α∗+λ<∞.
Thus,Ffz∈ℬω,α∗.
For the other direction, whenFfz∈ℬω,α∗, we infer that(10)Ffzℬω,α∗=supz∈ΔFfz∗zLω,z<∞.
Since the concerned functionfz=Ffz/z is a hyperbolic function on Δ, then we deduce(11)supz∈U1/4f∗zFω,z≤λ1<∞,where U1/4=z:z<1/4 and λ1>0 (positive concerned constant). Then, regarding to the specific function Ffz∈ℬω,α∗, we clearly obtain(12)fℬω,α∗=supz∈Δf∗zFω,z<∞,≤λ1+supz∈Δ\U1/4f∗zFω,z,=λ1+supz∈Δ\U1/4zFfz′−Ffzz2Fω,zz2−Ffz2,≤λ1+4supz∈Δ\U1/4Fz∗zLω,z+16supz∈Δ\U1/4FfzFω,z,≤λ1+4Ffℬω,α∗+16λ<∞,where λ>0andλ1>0. Then, f∈ℬω,α∗.Lemma 2.
Suppose that the functionω maps from 0,1 to 0,∞. Thus, we have the following inclusion:(13)ℬω,α,ln∗⊂ℬω,α,ln,0∗.Proof.
Assume thatf∈ℬω,α,ln∗, so(14)fℬω,ln∗=supz∈Δf∗zFω,z<∞.
Thus, we infer that(15)limz⟶1−f∗zFω,z=limz⟶1−f∗z|Lω,zln1/1−z2ln1/1−z2≤fℬω,ln∗limz⟶1−log11−z2−1=0.
Thus,f∈ℬω,α,ln,0∗.Theorem 1.
Letg∈ℋΔ. Then, we have equivalence between the following:(a)
The boundedness of the operatorℒg:ℬω∗⟶ℬω,α∗ holds bounded(b)
The boundedness of the operatorℒg:ℬω,0∗⟶ℬω,α,0∗ holds(c)
The boundedness of the operatorℒg:ℬω,0∗⟶ℬω,α∗ holds(d)
The concerned functiong∈ℬω,α.ln∗Proof.
Because of(16)ℬω,0∗⊂ℬω∗,ℬω,α,0∗⊂ℬω,α∗.
Hence, clearly, we can deduce thata⇒c and b⇒c. Therefore, we now aim to deduce that d⇒a,d⇒b, and c⇒d.
Using Lemma1 and in view of the operator,(17)ℒg=FfzJg.
Assume the concerned functiong∈ℬω,α,ln∗. Hence,(18)gℬω,α,ln∗=supz∈Δg∗zFω,zln11−z2<∞.
Forf∈ℬω,α∗, we have(19)fz≤f0+fℬω∗ln11−z2.
From(20)ℒgfz=fzg∗z,we obtain that(21)ℒgfℬω,α∗=supz∈Δfzg∗zFω,z,≤f0supz∈Δg∗zFω,z+fℬω,α∗supz∈Δg∗zFω,zln11−z2,≤f0×gℬω,α∗+fℬω∗gℬω,α,ln∗.
Applying Lemma2 and considering that g∈ℬω,α,ln∗, we obtain that(22)gℬω,α∗<∞.
Hence,(23)ℒgfℬω,α∗<∞.
The closed-graph theorem yields the boundedness of the concerned operatorℒg:ℬω∗⟶ℬω,α∗, so d⇒a.
Now, assume thatf∈ℬω,0α. Therefore, for every ɛ>0, we can find R∈0,1, for which R<z<1 as well as(24)fz≤ɛln11−z2.
Then, we can infer that(25)ℒgf∗zFω,z=fzg∗zFω,z≤ɛg∗zFω,z≤ɛgℬω,α,ln∗,where ℒgf∈ℬω,α,0∗, and we deduce d⇒b.
Let us assume thatℒg:ℬω,0∗⟶ℬω,α∗ is bounded. Also, let us choose arbitrary w0∈Δ0 and suppose that fw=ln1/1−w. It is not hard to see that f∈ℬω,α,0∗. Then,(26)∞>ℒgfℬω∗=supz∈Δfzg∗zFω,z,=supz∈Δg∗zFω,zln11−z2,and put z=w0, we obtain that(27)g∗w01−w0αω1−w0ln11−w02≤ℒgfℬω,α∗<∞.
Becausew0 is arbitrary on Δ0, we have(28)gℬω,α,ln∗≤ℒgfℬω,α∗<∞.
Hence,g∈ℬω,logα. Thus, the assertion c⇒d follows, and therefore, the proof is completely established.
As in [15], the following lemma can be proved.Lemma 3.
Suppose thatM is a closed concerned set in ℬω,α,0∗. Then, M is compact, and ⇔M is bounded, and the following condition holds:(29)limz⟶1−supf∈Mf∗zLω,z=0.Theorem 2.
For the concerned functiong∈ℋΔ, we can find equivalence between the following:(i)
The compactness of the operatorℒg:ℬω∗⟶ℬω,α,0∗ holds(ii)
The compactness of the operatorℒg:ℬω,0∗⟶ℬω,α,0∗ holds(iii)
The concerned functiong∈ℬω,α,ln,0∗Proof.
Becauseℬω,α,0∗⊂ℬω,α∗, then I⇒II. Therefore, the specific proof of III⇒I and II⇒III are needed.
By using concerned Lemma1 and considering the operator ℒg for g∈ℬω,α,ln,0∗ using Lemma 2, we infer that g∈ℬω,α,0∗. Hence,(30)limz⟶1−g∗zFω,zln11−z2=0,limz⟶1−g∗zFω,z=0.
Therefore, by (19), for every f∈ℬω, we have that(31)limz⟶1−ℒgfzFω,z=limz⟶1−fzg∗zFω,z≤f0limz⟶1−g∗zFω,z+fℬω,α∗limz⟶1−g∗zFω,zln11−z2=0.
Thus,ℒg:ℬω∗⟶ℬω,α,0∗ is actually bounded. To prove that the concerned operator ℒg is actually compact, assume that fn⊂ℬω∗ is defined such that fnℬω∗≤1. We have to clear that ℒgfn has a concerned subsequence which is converging on ℬω,α,0∗. Using (19), we can find a concerned subsequence of the sequence fn which converges uniformly on concerned compact subsets on the disc Δ to the concerned function f. For this purpose, we suppose that the concerned sequence fn converges also to the concerned function f. For any fixed z∈Δ, we infer that(32)f∗zFω,z=limn⟶∞fn∗zFω,z≤1.
Then,f∈ℬω,α∗, where fℬω,α∗≤1. Thus, ℒgf∈ℬω,α,0∗, and it is enough to be clear that(33)limn⟶∞ℒgfn−ℒgfℬω,α∗=0.
Becauseg∈ℬω,,α,ln∗, we deduce for every ɛ>0 and we can find an R∈0,1, for which(34)supz∈Δ\ΔRfnz−fzg∗zFω,z≤fn0−f0supz∈Δ\ΔRg∗zFω,z+fnzℬω,α∗+fzℬω,α∗supz∈Δ\ΔRg∗zFω,zln11−z2<ɛ,n≥1.
Because the sequencefn converges to the function f uniformly on each compact subset on Δ, we deduce that there exists an N>0, for every n>N and every z∈Δ,fnz−fz<ɛ. Then, for n>N, we deduce that(35)supz∈ΔR¯|fnz−fzg∗zFω,z≤ɛsupz∈Δ¯g∗zFω,z≤ɛgℬω,α∗.
From (34) and (35), we infer that(36)limn⟶∞supz∈ΔRfnz−fzg∗zFω,z=0.
Then,(37)limn⟶∞ℒgfn−ℒgfℬω,α∗=0.
Hence,ℒg:ℬω∗⟶ℬω,α,0∗ is actually compact. Then, the assertion III⇒I is completely proved.
Next, we are clear thatII⇒III and assume that ℒg:ℬω,0∗⟶ℬω,α,0∗ is compact. Suppose that g∈ℬω,α,ln∗. Thus, we can find a specific sequence of points zn in Δ,zn⟶1, for which(38)g∗znF∗ω,zn⟶k>0,where k is a positive constant and(39)F∗ω,zn=1−znαω1−zn.
Now, assume thatfnz=ln1/1−zn. Then, fn∈ℬω,0∗ with fnℬω≤1. Because ℒg is actually compact, then ℒgfn is also a compact concerned subset of ℬω,α,0∗. In view of Lemma 3, we obtain that(40)limz⟶1−supnℒgfnzFω,z=0.
This is equivalent to(41)limz⟶1−supnfnzg∗zFω,z=0.
Puttingɛ=k/2, then we can find ρ>0 such that(42)supz>ρsupnfnzg∗zFω,z<ɛ=k2.
Assume thatN>0, so for n>N,zn>ρ, we deduce that(43)g∗znfnznF∗ω,zn≤supz>ρsupng∗znfnzF∗ω,zn<k2,and this contradicts (38). Hence, the proof is established.


## 3. Hyperbolic Bloch Classes and Usual Distance

Connections between hyperbolic Bloch spaces and the metric spaces are investigated in the following emerging result.The concept of the general hyperbolic derivative is defined in [2] by(44)hk∗w=hkw1−hwk+1;k∈ℕ.Remarking that, whenk=1, we get the known usual hyperbolic derivative as given above.Now, we introduce the following concepts:The logarithmic spaceℬk,ω,ln∗ is defined by(45)ℬk,ln∗=hk∗∈BΔ:hℬk,ω,ln∗=supz∈ΔLω,z,lnhk∗<∞,where(46)Lω,z,ln=1−z2αω1−z2ln11−z2.Theorem 3.
The considered spaceℬk,ω,ln∗ endowed by the usual metric (distance) D.,.;ℬk,ω,ln∗ is a complete metric space. Furthermore, ℬk,ω,ln,0∗ is a concerned specific closed subspace of the concerned space ℬk,ω,ln∗.Proof.
Leth,h1,andh2∈ℬk,ω,ln∗; hence, we infer that(47)Dh,h1;ℬk,ω,ln∗≥0,Dh,h,ℬk,ω,ln∗=0.
Therefore, we can deduce also that(48)Dh,h1;ℬk,ω,ln∗=Dh,h1;ℬk,ω,ln∗,Dh,h2;ℬk,ω,ln∗≤Dh,h1;ℬk,ω,ln∗+Dh1,h2;ℬk,ω,ln∗.
Since, the classes are defined by an usual distance on the hyperbolic classℬk,ω,ln∗, Dh,h1;ℬk,ω,ln∗=0, which results in h=h1. Therefore, the distance D forms a metric space on the class ℬk,ω,ln∗.
Lethnn=0∞ be a concerned Cauchy sequence in the defined metric space ℬk,ω,ln∗; therefore, for any ɛ>0, there exists an N=Nɛ, for which(49)Dhn,hm;ℬk,ω,ln∗<ɛ,for all m,n>N because fn⊂BΔ; thus, the concerned family hn is actually uniformly bounded in Δ. Therefore, we can obtain h∈BΔ with the concerned subsequence hnj∞=1 such that hnj converges uniformly on some compact concerned subsets of Δ to the specific function h, which implies that hn converges on compact subsets in the uniform type to the function h.
Now, letm>N, using the uniform convergence, we infer that(50)hnkz1−hnzk+1−hmkz1−hmzk+1Lω,z,ln=limn⟶∞hnkz1−hnzk+1−hmkz1−hmzk+1Lω,z,ln≤limn⟶∞Dhn,hm;ℬk,ω,ln∗≤ɛ,
for allz∈Δ, and hence, h∈ℬk,ω,ln∗ as we need.
Moreover, the above inequality and the compactness of the usualℬk,ω,ln∗ space give us that hnn=1∞ converges to h with respect to the distance D. Because(51)limn⟶∞Dhn,hm;ℬk,ω,ln∗≤ɛ, the second assertion follows, and hence, the proof is established.Corollary 1.
The spaceℬ∗ endowed by the usual metric (distance) D.,.;ℬ∗ is also a complete metric space. Further, ℬ0∗ is a concerned closed subspace of ℬ∗.Proof.
The proof can be obtained by lettingα=1,ω1−z≡1, and ln1/1−z2=1 in Theorem 3.Remark 2.
Complex function spaces have an interesting and joyful extension in quaternion analysis (see [16–20] and others).For this direction of study, can we use quaternion functions instead of hyperbolic functions to study the properties of the defined operators in this paper?


## 4. Conclusions

This paper presents a variety of techniques for defining and studying classes of operators on hyperbolic function spaces and features a lot of their properties, focusing on the new concepts and methods of clearing proofs. The paper begins with presenting basic concepts and notation to the studied hyperbolic spaces, followed by introducing the new definitions of the considered operators and detailed proofs of the new results. After giving the boundedness and compactness properties of the defined operators, it lastly presents a metric space characterization of the hyperbolic-type Bloch space. The results are excellently and carefully presented.With the help of a certain nondecreasing and continuous function, some weighted hyperbolic classes are defined and researched in this study. Essential characterizations for the weighted Bloch space by new multiplication operators are introduced in the sense of hyperbolic settings. Completeness of metric spaces is also established on certain hyperbolic function spaces.



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