# Toeplitz-Superposition Operators on Analytic Bloch Spaces

Authors: M. A. Bakhit; A. El-Sayed Ahmed
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The important purpose of this current work is to study a new class of operators, the so-called Toeplitz-superposition operators as an expansion of the weighted known composition operators, induced by such continuous entire functions mapping on bounded specific sets. Minutely, we have deeply discussed the conditions for boundedness of this new type of operators between certain types of some holomorphic Bloch classes with some specific values of the weighted functions.


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## Body

## 1. Introduction

Fundamentals of the needed analytic function spaces as well as the types of concerned operators are briefly introduced. The paper focuses first on the concerned setting of certain classes of function spaces and the new defined operator, which in turn is motivated essentially by some certain classical concepts of known operators such as superposition operators as well as Toeplitz operator. There is an emphasis in the concerned paper on intensive tying together the needed type of analytic function spaces and the concerned operators, to illustrate the roles of the obtained results.All of the needed information to justify the target of this research is collected in this concerned section. Moreover, here, basic concerned concepts, the Bloch space of analytic-type, certain needed concerned lemmas, and superposition and Toeplitz operators are presented.LetD=z∈ℂ:∣z∣<1 be the open unit disk in ℂ, and let HD denote the class of all analytic functions in D. Let dAz=dxdy denote the concerned Lebesgue measures on D.Numerous intensive studies on analytic Bloch-type spaces are researched in literature (see [1–5] and others).Leth∈HD and 0<b<∞, the b-Bloch space Bb is defined by
(1)Bb=f∈HD:∥h∥Bb=supζ∈D1−ζ2b∣f′ζ∣<∞.The spaceB1 is called the Bloch space and denoted by B (see [3]).The following interesting needed lemma has been proved in [6].Lemma 1.
For a given0<a<∞, let the functionh∈Ba. Then, we have(2)∣hζ∣≲∥h∥Ba,if0<a<1;∥h∥Balne1−ζ2,ifa=1;∥h∥Ba1−ζ2a−1,ifa>1.The following useful integral estimate is well known and can be found in [7].Lemma 2.
Lets>0andt>−1. Then(3)∫D1−w2tdAw1−z¯w2+t+s≲11−z2s,forallz∈D.Fora>−1 and p∈0,∞, the weighted Bergman spaces AapD is the space of all functions h∈HD, for which
(4)∥h∥Aapp=∫DhζpdAaζ<∞,wheredAaζ=1−ζ2adAζ.Whena=0, we simply write ApD for A0pD, and when p=2, Aa2D is a Hilbert space. It is well known that the Bergman kernel Kzw of the Hilbert space Aa2D is given by Kzw=1−w¯z−a−2, where z,w∈D. The Bergman projection Pa is the orthogonal projection from L2D,dAa onto Hilbert space Aa2D, which given as:
(5)Pahz=∫DKzwhwdAaw.Fora>−1 and h∈HD, the Toeplitz-type operator Tua with symbol u∈H∞D is defined by
(6)Tuahz=∫Duwhw1−w¯za+2dAaw.This paper is organized as follows: during Section2, we have defined the Toeplitz-superposition operators on the normed (metric) subspaces. Throughout Section 3, we establish the conditions for the Toeplitz-superposition operators to be bounded from a-Bloch space Ba into b-Bloch space Bb, in the case a∈0,1 and b>a or b<a. Section 4 is devoted to a study the boundedness of Toeplitz-superposition operators between weighted Bloch spaces in the case 0<a≤b or a=0,b>0.Remark 3.
It is concerned remarkable to say that two concerned quantitiesNhandNh∗, where both depending on the concerned functionh∈HD, the expressionNh≲Nh∗,can be satisfied when we have a concerned positive constantC1,for whichNh≤C1Nh∗. WhenNh∗≲Nh≲Nh∗, the expressionNh≈Nh∗can be written to say that there is an equivalence relation between the concerned quantitiesNhandNh∗.Furthermore, whenNh≈Nh∗,we deduce thatNh<∞⇔Nh∗<∞.


## 2. Toeplitz-Superposition Operators

LetEℂ denote the set of all entire functions on the complex plane ℂ. For a function ϕ∈Eℂ, the superposition operator Sϕ:HD→HD is defined by Sϕh=ϕ∘h. Moreover, if u∈HD and ϕ∈Eℂ, the weighted superposition operator Sϕ,u:HD→HD is defined by Sϕ,uhz=uzϕhz, for all h∈HD and z∈D. Note that, if uz=1, then Sϕ,u=Sϕ, for any z∈D.For any normed subspaceX⊂HD, we will consider the set KX, defined by
(7)KX=h∈X:ϕ∘h∈X,whereϕ∈Eℂ.Now, we define the Toeplitz-superposition operators acting onHD.Definition 4.
Let two functionsϕ∈Eℂandu∈HD. Then, the Toeplitz-superposition operatorsTuSϕon the normed (metric) subspaceXare given by(8)TuSϕh=Tuϕ∘h=Pu·ϕ∘h,forallh∈KX.Letα,β be the scalers if ϕ is a fixed entire function and u,v∈HD. Then, from the definition of Toeplitz-superposition operators, we have
(9)Tαu+βvSϕh=Tαu+βvϕ∘h=Pαu·ϕ∘h+βv·ϕ∘h=αPu·ϕ∘h+βPv·ϕ∘h=αTuSϕh+βTvSϕh,which holds for all h∈KX, and hence, the Toeplitz-Superposition operators are linear on the normed subspace X.It can be seen that wheneveru∈HD, then, the operator TuSϕ becomes the operator Sϕ,u. So, Toeplitz-superposition operators can be taken as an extension of weighted superposition operators. The present paper is interested in answering the following interesting questions.
(i)
Can we transform one holomorphic function space into another by what kinds of entire functions?(ii)
What are the holomorphic spaces that can be transformed one into another by certain weighted classes of entire functions such as specific analytic polynomials of a certain degree and certain entire-type functions of given type and order?(iii)
When does the holomorphic functionφ induces a Toeplitz-superposition operators to form one holomorphic function space into another?As a concerned result, the obtained results will introduce answers of the above mentioned questions by using the class of Toeplitz-superposition operators that are acting between different classes of Bloch functions.Also, the answers for some of these concerned questions have been introduced by several authors; the following citations can be stated for interesting and intensive studies [8–20].


## 3. Boundedness in the casea∈0,1 and b>a or b<a

Several important discussions on boundedness property of the new operator acting on the analytic Bloch spaces are presented in this concerned section. Furthermore, some essential equivalent characterizations for its boundedness are established too.Now, we will introduce the main results of boundedness.Theorem 5.
Fora∈0,1andb>a. Suppose thatu∈H∞Dand letϕ∈Eℂ, withϕ≠0. Then, the Toeplitz-superposition operatorTuSϕ:Ba→Bbis bounded.Proof.
First, assume thatu∈H∞D. Let h∈Bb, since
(10)∥h∥Bb≈supz∈D1−z2b−1hz, we have
(11)∥TuSϕh∥Bb≈supz∈D1−z2b−1∣TuSϕh∣=supz∈D1−z2b−1∫Duwϕ∘hw1−w¯z2dAw≲supz∈D1−z2b−1∫Duwϕhw1−w¯z2dAw≲∥u∥H∞supz∈D1−z2b−1∫D∣ϕhw∣1−w¯z2dAw.Now, let the constantR>0 where h∈Ba such that ∥h∥Ba≤R, by Lemma 1, we have ∣hz∣≲R. Set R1=max∣z∣=R∣ϕz∣, then ∣ϕhz∣≤R1. Since b>a, we have the fact that Ba⊂Bb, and since a∈0,1, we have that Ba⊂H∞D. Thus,
(12)∥TuSϕh∥Bb≲R1∥u∥H∞supz∈D1−z2b−1∫DdAw1−w¯z2≲∥u∥H∞<∥u∥Ba<∞,where R,R1 depended only on a,b, and ϕ. This shows that TuSϕ:Ba→Bb is bounded.Theorem 6.
For0<b<a<1, letu∈L1Dbe harmonic and letϕ∈Eℂ. Then, the Toeplitz-superposition operatorTuSϕ:Ba→Bbis bounded if and only ifu∈H∞Dandϕis a constant entire function.Proof.
It is trivial that ifu∈H∞D and ϕ is constant, then TuSϕ:Ba→Bb is bounded. If ϕ is constant, not identically 0, and TuSϕ maps Ba into Bb then it is clear that u∈H∞D. Assume now that u≠0 and ϕ is not constant, and set TuSϕ maps Ba into Bb. Let h be the constant function defined by hζ=λ, for all ζ∈D, such that ϕλ≠0. Since h∈Ba, it follows that TuSϕhζ=Tuϕλ∈Bb. This implies that u∈Bb⊂H∞D, since 0<b<1. Finally, since ϕ is not constant, then there is a disk ∣w−w0∣<ε and δ>0, on which ∣ϕw∣>δ∣w∣. Set the test function h0w=w0+r1−w1−a∈Ba. Then, for all w∈D, we have
(13)∥TuSϕh0∥Bb≈supz∈D1−z2b−1∣TuSϕh0∣=supz∈D1−z2b−1∫Duwϕ∘h0w1−w¯z2dAw≥supz∈D1−z2b−1∫D∣uw∣1−wa1−w¯z2dAw.But, along with the positive radius, we getuw/1−wa1−w¯z2→∞, as w→1. This shows that TuSϕ:Ba→Bb is not bounded.


## 4. Boundedness in the case0<a≤b or a=0,b>0

Theorem 7.
Fora>0, letu∈L1Dbe harmonic and letϕ∈Eℂ. Then,TuSϕis bounded onBaif and only ifu∈H∞Dandϕis an affine function (linear function plus a translation).Proof.
First, suppose thatu∈H∞D and ϕ is an affine function. It is easy to explain TuSϕ is bounded from Ba into itself.On the other hand, assume thatu∈H∞D and ϕ∈Eℂ does not linear function. Then, by using the Cauchy estimates for ϕ∈Eℂ, we can find a sequence wn⊂ℂ, for each n∈ℕ such that ∣wn∣→∞ as n→∞ and ϕwn=maxw=nϕw≳wn2. Also, since the weight 1−ζ2a is typical, we can find a sequence of points zn⊂D such that∣zn∣→1−, with 0.5<∣zn∣<1 and such that 1−∣zn∣∣wn∣=1, for all n∈ℕ. Now consider the sequence of functions hn contained in Ba satisfies ∥hn∥Ba≤1 and∣hnzn∣=∣wn∣. Furthermore, we can suppose that hnzn=wn. Hence,
(14)∥TuSϕhn∥Bb≈supzn∈D1−zn2b−1∣Tuϕhnzn∣≳∥u∥H∞supz∈D1−zn2b−1∫Dwn21−w¯zn2dAw≳∥u∥H∞supz∈D1−zn2b−1∫Dwn1−w¯z2dAw→∞,asn→∞.Because∣wn∣→∞ as n→∞. This shows that TuSϕ:Ba→Ba cannot be bounded if ϕ∈Eℂ is not a linear function.Theorem 8.
For0<a≤b, letu∈L1Dbe harmonic and letϕ∈Eℂbe an increasing and continuous function. Then,TuSϕ:Ba→Bbis bounded if and only ifu∈H∞D, and for eachλ∈0,1, there is a positive constantηwhenever∣w∣>η, such that(15)∣ϕw∣≲ϕλ∣w∣.Proof.
First, suppose thatu∈H∞D and (15) is true. Now, consider R1>0 and let h∈Ba satisfy ∥h∥Ba≤R1 and select λ∈0,1 such that λR1<1. Then, there is η>0, such that ∣ϕw∣≲ϕλ∣w∣, whenever ∣w∣>η. Thus, since D¯R=w∈ℂ:∣w∣≤R is a compact set and ϕ∈Eℂ is a continuous function, we can assume that ∣ϕw∣≤1, for all w∈D¯R. Hence,
(16)∥TuSϕh∥Bb≈supzn∈D1−zn2b−1∣Tuϕhz∣=supz∈D,∣hw∣≤R1−z2b−1∣TuSϕh∣+supz∈D,∣hw∣>R1−z2b−1∣Tuϕhz∣≲∥u∥H∞supz∈D1−z2b−1∫DdAw1−w¯z2+∥u∥H∞supz∈D,∣hw∣>R1−z2b−1∫Dϕλ∣∥h∥Ba1−w2a∣dAw1−w¯z2≲∥u∥H∞+∥u∥H∞supz∈D1−z2b−1∫Dϕλ∣11−w2a∣dAw1−w¯z2.Using that the functionϕ is increasing and the fact that λ<1, we have
(17)∥TuSϕh∥Bb≲∥u∥H∞+∥u∥H∞supz∈D1−z2b−1∫DdAw1−w¯z2≲∥u∥H∞.This shows thatTuSϕ:Ba→Bb is bounded.On the other hand, assume thatu∈H∞D and ϕ∈Eℂ does not satisfy (15). Then, we can find λ1∈0,1 and a sequence wn⊂ℂ such that ∣wn∣→∞ as n→∞ and ∣ϕwn∣≥ϕλ1∣wn∣, for all n∈ℕ. Since the weight 1−z2a is typical, we can find a sequence of points zn⊂D such that ∣zn∣→1− as n→∞. Thus, we can consider a sequence of functions hn contained in Ba satisfies ∥hn∥Ba≤1 and ∣hnzn∣≲∣wn∣. Now, let z∈D and set the function fnz=wnhnz/hnzn for all n∈ℕ. Then, we have fnzn=wn and ∥fn∥Ba≲1. For large enough n∈ℕ, we obtain
(18)∥TuSϕfn∥Bb≈supzn∈D1−zn2b−1∣TuSϕfnzn∣≳supzn∈D1−zn2b−1∫Duwϕwn1−w¯zn2dAw≳∥u∥H∞supzn∈D1−zn2b−1∫Dϕλ1wn1−w¯zn2dAw≳∥u∥H∞supzn∈D1−zn2b−1∫Dϕ11−wn2adAw1−w¯zn2→∞,asn→∞.Then, we conclude thatTuSϕ:Ba→Bb cannot be bounded.Theorem 9.
For1<a≤b, letu∈L1Dbe harmonic and letϕ∈Eℂ. Then, the following are equivalent:(i)
TuSϕ maps Ba into Bb(ii)
u∈H∞D and ϕ is a polynomial of degree at most b−1/a−1(iii)
TuSϕ:Ba→Bb is boundedProof.
First, suppose that (i) holds, letϕ=zn be a polynomial with n≤b−1/a−1, then, we have that u∈H∞D.Now, suppose that the entire functionϕ is a polynomial of degree m>b−1/a−1. Then, for an integer n and a positive constant δ, there is a sequence zn→∞ such that ∣ϕzn∣≥δznm. We may assume without losing generality that ∣zn∣>1 and ∣argzn∣<minaπ/4,π/2, for an integer n. Now, we let haw=1−w1−a∈Ba, then, we show that TuSϕha∉Bb. The point wn=1−zn−1/a such that ∣1−wn∣<1 and ∣arg1−wn∣<π/4, and satisfies that ∣1−wn∣≲1−wn, for an integer n. Thus, we have
(19)1−zn2b−1∣TuSϕhazn∣=1−zn2b−1∣∫Duwϕhaw1−w¯zn2dAw∣≳1−zn2b−1∫Dδ∣uw∣1−znma1−w¯zn2dAw≳∫Dδ∣uw∣1−znma−m−b+11−wa1−w¯zn2dAw.Sincema−m−b+1>0, then, we obtain
(20)1−zn2b−1∣TuSϕhazn∣→∞,asn→∞.This implies thatTuSϕha∉Bb. Based on the above it is clear that (i) ⇒ (ii).Second, assume thatu∈H∞D and ϕ=zn are a polynomial of degree n≤b−1/a−1. For all h∈Ba and b−1−na−n≥0, by Lemma 1 and 2, where n is bounded, we have
(21)∥TuSϕh∥Bb≈supz∈D1−z2b−1∣TuSϕhz∣=supz∈D1−z2b−1∫Duwϕ∘hw1−w¯z2dAw≤∥u∥H∞supz∈D1−z2b−1∫Dhwn1−w¯z2dAw≤∥u∥H∞∥h∥Bansupz∈D1−z2b−1−na−n∫DdAw1−w¯z2≤∥u∥H∞∥h∥Ban.This shows that (ii)⇒ (iii). Thus, the proof has been completed.Theorem 10.
Forb>−1, letu∈L1Dbe harmonic and letϕ∈Eℂ, with orderρand typeτ. Then, the following are equivalent:(i)
TuSϕ maps B into Bb(ii)
u∈H∞D and ϕ∈Eℂ with ρ<1 or (with ρ=1 and τ=0);(iii)
TuSϕ:B→Bb is boundedProof.
First, assume thatTuSϕ maps B into Bb. Now, we assume on the antithesis that (ii) does not hold. Then, the function ϕ∈Eℂ with ρ>1 or (with ρ=1 and τ>0). Thus, there is a positive constant λ and a sequence wn of complex numbers such that ∣wn∣→∞ and
(22)∣ϕwn∣≥expλ∣wn∣,foranyn∈ℕ.Hence, as in the proof of Theorem8, we can consider the sequence wn⊂D and hn⊂B satisfies ∥hn∥B≤1 and ∣hnzn∣≲∣wn∣. Now, let z∈D and set the function fnz=wnhnz/hnzn for all n∈ℕ. Then, we have fnzn=wn and ∥fn∥B≲1. For large enough, n∈ℕ, since ∣wn∣→∞, we obtain
(23)∥TuSϕfn∥Bb≈supz∈D1−zn2b−1∣TuSϕfnzn∣≳supz∈D1−zn2b−1∫Duwϕwn1−w¯zn2dAw≳∥u∥H∞supz∈D1−zn2b−1∫Dexpλ1∣wn∣1−w¯zn2dAw→∞,asn→∞.Then, we conclude thatTuSϕ:B→Bb cannot be bounded. Based on the above results, it is clear that (i) ⇒ (ii).Second, setMt,ϕ=max∣w∣=t∣ϕw∣, where t≥0, the order ρ of ϕ∈Eℂ is
(24)ρ=limsupt→∞loglogMt,ϕlogt.If0<ρ<∞, then the type τ of ϕ∈Eℂ is
(25)τ=limsupt→∞logMt,ϕtρ.For givenλ=b/R>0, the condition (ii) implies that (see for example [18])
(26)ϕw≲expλw,foranyw∈ℂ.Moreover, for a functionh∈B, with ∥h∥B≲1, we know that
(27)∣hw∣≲1+log11−∣w∣,forw∈D.Then,(28)∣ϕhw∣≲expλ∣hw∣≤e1−∣w∣b≤2eb.Thus, we have(29)∥TuSϕh∥Bb≈supz∈D1−z2b−1∣TuSϕhz∣≲2eb∥u∥H∞supz∈D1−z2b−1∫DdAw1−w¯z2≲2eb∥u∥H∞<∞.This shows thatTuSϕ:B→Bb is bounded. So, (ii) ⇒ (iii).


## 5. Conclusion and Future Study

This manuscript deals with a radical study of a concerned class of Toeplitz superposition operators acting between some certain classes of analytic function spaces of Bloch-type. Global discussions of the boundedness property of the new class of operators are presented class of the univalent Bloch functions. All concerned entire functions which transform a class of holomorphic Bloch-type spaces into another using the so-called Toeplitz superposition operators in terms of their order and type or the degree of polynomials are characterized in this paper. Moreover, all the defined Toeplitz-superposition operators induced by concerned entire functions are cleared to be bounded actually. We have cleared that for two spaces of normed-type which belonging toHD, where X=Ba and Y=Bb, we can find certain concerned functions ϕ and u, with ϕ∈Eℂ and u∈HD, for which the newly Toeplitz-superposition operators TuSϕ can map Ba into Bb for some specific values of a and b. Furthermore, the operator TuSϕ:X→Y is shown to be actually bounded.



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