# Composition Semigroups on Weighted Bergman Spaces Induced by Doubling Weights

Authors: Fanglei Wu
Journal: Journal of Mathematics
 (2021)

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## Abstract

We prove that composition semigroups are strongly continuous on weighted Bergman spaces with doubling weights. Point spectra and compact resolvent operators of infinitesimal generators of composition semigroups are characterized.


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## Body

## 1. Introduction

LetℋD denote the space of analytic functions in the unit disc D=z∈ℂ:z<1. For a nonnegative function ω∈L10,1, the extension to D, defined by ωz=ωz for all z∈D, is called a radial weight. For 0<p<∞ and a radial weight ω, the weighted Bergman space Aωp consists of f∈ℋD such that(1)fAωpp=∫DfzpωzdAz<∞,where dAz=dxdy/π is the normalized Lebesgue area measure on D. As usual, Aαp stands for the classical weighted Bergman space induced by the standard radial weight ωz=1−z2α, where −1<α<∞.For a radial weightω, write ω^z=∫z1ωsds for all z∈D. In this paper, we always assume ω^z>0, for otherwise Aωp=ℋD for each 0<p<∞. A weight ω belongs to the class D^ if there exists a constant C=Cω≥1 such that ω^r≤Cω^1+r/2 for all 0≤r<1. For more knowledge about those Bergman spaces, see [1–4] and the reference therein.A familyφtt≥0 of analytic self-maps of the unit disk D in the complex plane ℂ is said to be a semigroup if the following conditions hold:(i)
φ0 is the identity map of D(i)
φt°φs=φt+s, for t,s≥0(ii)
For eachz∈D, φtz⟶z, as t⟶0+A semigroupφtt≥0 is said to be trivial if each φt is the identity of D. The infinitesimal generator of φtt≥0 is defined as the function(2)Gz=limt↦0+φtz−zt,z∈D.For any nontrivial semigroupφtt≥0, there exist a point b∈D¯ and an analytic function P:D↦ℂ with ReP≥0 such that(3)Gz=b¯z−1z−bPz.Refer [5] for the details. Representation (3) is unique, and the point b is said to be the Denjoy–Wolff point of φtt≥0.Notice that each semigroupφtt≥0 gives rise to a semigroup Ctt≥0 consisting of composition operators on HD, the set of analytic functions on D, where(4)Ctf≔Cφtf=f°φt,f∈HD.Given a semigroupφtt≥0 and a Banach space X of analytic functions on D, we say that φtt≥0 generates a strongly continuous composition operator on X if Ct is bounded on X and(5)limt⟶0+Ctx−xX=0for allx∈X.Ctt≥0 is called uniformly continuous on X if limt⟶0+Ct−I=0, where I is the identity map on X. The infinitesimal generator of a strongly continuous semigroup Ctt≥0 on a Banach space X is the operator(6)Γx≔limt⟶0+Ctx−xt,which is densely defined for every x in the domain(7)DΓ≔x∈X:limt⟶0+Ctx−xtexists.Refer [6] for more information about operator semigroup.In 1978, Berkson and Porta [5] initially studied the strong continuity of semigroups of composition operators acting on the classical Hardy space HpD. They proved that for 1≤p<∞,Ctt≥0 is strongly continuous on Hp. Later, Siskakis demonstrated that Ctt≥0 is strongly continuous on the Bergman space Aαp (1≤p<∞; −1<α<∞) and the Dirichlet space D in [7, 8]. Moreover, he showed that the infinitesimal generator Γ of a semigroup of composition operators on these spaces is of the form Γf=Gf′ with a certain domain. Refer [9–16] for more results of composition semigroups on other various spaces.In this paper, we consider composition semigroups on weighted Bergman spacesAωp with ω∈D^. We will show that every Ctt≥0 is strongly continuous on Aωp if 1≤p<∞ and ω∈D^. The corresponding infinitesimal generator Γ of Ctt≥0 and its point spectrum can also be identified. In addition, if the Denjoy–Wolff point of φtt≥0 belongs to D, then we can also characterize the compactness of resolvent operator Rλ,Γ of Γ, provided λ belongs to the resolvent set of Γ.Throughout the paper, the symbolA≈B means that A≲B≲A. We say that A≲B if there exists a constant C such that A≤CB.


## 2. Strongly Continuous Composition Semigroup

We need more information aboutφtt≥0 before presenting our results.Assume a semigroupφtt≥0 consisting of analytic self-maps of D with infinitesimal generator G and Denjoy–Wolff point b. In general, φtt≥0 can be classified into two classes: b∈D and b∈∂D, the boundary of D. In particular, there exists a unique univalent function h:D⟶ℂ, called Koenigs function, such that(1)
Ifb∈D, then hb=0, h′b=1 and(8)h°γbφtz=eG′bth°γbzforz∈Dand≥0,whereγbz=z−b/1−b¯z. Moreover,(9)Gzh°γbz=G′bh°γbz,z∈D.(2)
Ifb∈∂D, then h0=0 and(10)hφtz=hz+G0tforz∈Dandt≥0.Moreover,(11)h′zGz=G0,z∈D.Lemma1 shows that every composition operator is bounded on Aωp if 0<p<∞ and ω∈D^. The proof can be easily obtained by a simple combination of Theorem 15 and (4.7) in [17].Lemma 1.
(see [17]). Let 0<p<∞ and ω∈D^, and let φ be an analytic self-map of D. Then, the composition Cφ is bounded on Aωp. Moreover, there exist constants η=ηω>1 and C=Cη,ω,p such that(12)Cφ≤C1+φ01−φ0η.Now, we are ready to show our results. Forf∈ℋD and 0<r<1, set(13)Mpr,f=12π∫02πfreiθp1/p,0<p<∞,M∞r,f=supz=rfz.Theorem 1.
Let1≤p<∞ and ω∈D^. Suppose φtt≥0 is a semigroup of analytic self-maps of D with infinitesimal generator G. Then, the induced composition semigroup Ctt≥0 defined in (4) is strongly continuous on Aωp with infinitesimal generator Γ:(14)Γf=Gf′on its domain(15)DΓ=f∈Aωp:Gf′∈Aωp.Moreover, Ctt≥0 is uniformly continuous on Aωp if and only if φtt≥0 is trivial.Proof.
Sinceω∈D^, the polynomials are dense in Aωp. Therefore, for any f∈Aωp, there exists a sequence of polynomials Pn such that limn⟶∞f−PnAωp=0. It follows triangle inequality that(16)Ctf−fAωp≤Ctf−CtPnAωp+CtPn−PnAωp+f−PnAωp≤Ct+1f−PnAωp+CtPn−PnAωp.
According to Lemma1, we know that supt∈0,1Ct<∞. Therefore, to prove limt⟶0+Ctf−fAωp=0, it suffices to prove limt⟶0+CtP−PAωp=0 for each polynomial P. Equivalently, we only need to show that for each n≥0,​limt⟶0+φtn−enAωp=0, where enz=zn, while it can be easily obtained by Lebesgue-dominated convergence theorem. Thus, Ctt≥0 is strongly continuous on Aωp.
By definition, the domain ofΓ is(17)DΓ=f∈Aωp:limt⟶0+Ctf−ftexists inAωp.
Let, now,D=f∈Aωp:Gf′∈Aωp. We are going to show that if f∈DΓ, then Gf′∈Aωp. Indeed, if f∈DΓ, then Γf∈Aωp and(18)limt⟶0+Ctf−ft−ΓfAωp=0.
Since for a fixed0<r<1, the well-known inequality M∞r,f≲Mp1+r/2,f1−r−1/p yields(19)M∞pr,f≲fAωpp1−rω^r.Convergence in the norm of Aωp implies the pointwise convergence. Therefore, for every z∈D,(20)Γfz=limt⟶0+fφtz−fzt=limt⟶0+fφtz−fφ0zt∂fφtz∂t|t=0=Gzf′z.
That is,Gzf′z=Γfz∈Aωp and DΓ⊆D. On the other hand, for λ∈ρΓ, the resolvent set of Γ, we have(21)D=f∈Aωp:Gf′∈Aωp=f∈Aωp:Gf′−λf∈Aωp=Rλ,Γ,where Rλ,Γ=λI−Γ−1 is the resolvent operator of Γ. Since Rλ,ΓAωp⊆DΓ, D⊆DΓ. Hence, D=DΓ.
IfCtt≥0 is uniformly continuous on Aωp, then the infinitesimal generator Γ is bounded on Aωp. To show φtt≥0 is trivial, it is equivalent to show G≡0. To the end, we may consider polynomials enz=zn. Then, Γen=nGen−1, and taking n=1, we see that G∈Aωp. Since Γ is bounded on Aωp, for n≥1, we have Γen≲en, that is,(22)np∫01Mppr,Grn−1p+1ωrdr≲∫01rnp+1ωrdr.
It follows thatnpMpp1/2,G≲1. Since G∈Aωp, we have Mp1/2,G=0. Thus, G≡0. The proof is complete.Theorem 2.
Let1≤p<∞ and ω∈D^. Suppose φtt≥0 is a semigroup of analytic self-maps of D with Denjoy–Wolff point b, infinitesimal generator G, and associated Koenigs function h.(i)
Ifb∈D, then PσΓ⊆kG′b:k=0,1,2,…. And, kG′b∈PσΓ if and only if h°γbk∈Aωp.(ii)
Ifb∈∂D, then PσΓ=λG0:eλhz∈Aωp.Proof.
(i)
By (7),(23)Gz=G′bh°γbh°γb.Now, supposef∈ℋD and λ≠0 such that Γf=Gf′=λf. Then,(24)f′f=λh°γb′G′bh°γb.Pickr such that b<r<1 and f has no zeros on z=r. We have(25)12πi∫z=rf′zfzdz=λG′b12πi∫z=rh°γb′zh°γbzdz.From this and the argument principle, it follows thatλ/G′b=k, a nonnegative integer, which shows the first part of (i). Also, notice that the differential equation(26)f′z=kh°γb′zh°γbzfzhas solutionfz=ch°γbk, where c≠0. This shows the second part of (i).(ii)
By (10), we have Gz=G0/h′z. If fz=eλhz∈Aωp, then Γf=λG0f, so λG0∈PσΓ. Conversely, if Γf=G′f=λG0f, then f′z=λh′zfz, which follows that fz=ceλhz, where c≠0. The proof is complete.


## 3. Resolvent Operator

In [18], Siskakis characterized the compactness of Rλ,Γ on the Hardy space Hp and the weighted Bergman space Aαp if the Denjoy–Wolff point of φtt≥0 is in the interior of D. In this section, we will consider the compactness of Rλ,Γ on Aωp with ω∈D^.Forα≥1 and ω∈D^, Cαω∗ consists of all f∈ℋD such that(27)fCαω∗2=g02+supI∈∂D∫SIfz2ω∗zdAzωSIα<∞.C0αω∗ consists of all f∈ℋD such that(28)limI⟶0∫SIf′z2ω∗zdAzωSIα=0,where SI=reit∈D:eit∈I,1−I≤r<1 is Carleson square associated with I and ω∗z=∫z1logs/zsdωs for all z∈D/0. Refer [1, 2] for more information about these spaces.To prove the main result in this section, we need the following lemma, which characterizes the boundedness and compactness of integral operatorVg on Aωp. Here, for a g∈ℋD, Vg is defined as(29)Vgfz=1z∫0zfζg′ζfζ,z∈D,f∈ℋD.Lemma 2.
(see [1, 2]). Let 0<p<∞, ω∈D^, and g∈ℋD. Then,(i)
Vg is bounded on Aωp if and only if g∈C1ω∗(ii)
Vg is compact on Aωp if and only if g∈C01ω∗Lemma 3 is critical to our result.Lemma 3.
Let1≤p<∞ and ω∈D^, and let φtt≥0 be a nontrivial semigroup of self-maps on D with Denjoy–Wolff point 0, infinitesimal generator G, and Koenigs function h. Suppose Ctt≥0 is the corresponding composition semigroup on Aωp, with the generator Γ. Then, for all λ∈ρΓ, the resolvent operator of Γ has the following representation:(30)Rλ,Γfz=−1G′01hzλ/−G′0∫0zfζhζλ/−G′0−1h′ζdζ.
In particular,−G′0 belongs to ρΓ, and hence,(31)R−G0,Γfz=−1G′0hz∫0zfζh′ζdζ.Proof.
Let(32)R≔−1G′01hzλ/−G′0∫0zfζhζλ/−G′0−1h′ζdζ.
It is elementary to compute thatλI−ΓR=RλI−Γ=I, which shows that R is the resolvent operator of Γ. Since the Denjoy–Wolff point of φtt≥0 is 0, it is easy to see that Re−G′0≥0. If Re−G′0>0, by Lemma 1, we have(33)ω0≔limt⟶∞logCtt=0.
So,−G′0∈ρΓ by Theorem 1.10 in ([6], p42). If −G′0 is a pure imaginary number, write Gz=−iαz, where α∈ℝ\,0 and Γf=−iαzf′z. In this case, iα−Γf=g has the unique analytic solution(34)fz=1iαz∫0zgζdζ.
It is not difficult to see that the operator(35)g⟶1iαz∫0zgζdζis bounded on Aωp. Hence, −G′0 belongs to ρΓ. The proof is complete.Now, we are ready to prove the main result in this section.Theorem 3.
Let1≤p<∞ and ω∈D^. Suppose φtt≥0 is a semigroup of analytic self-maps of D with Denjoy–Wolff point 0, infinitesimal generator G, and associated Koenigs function h. Denote by Γ the infinitesimal generator if the corresponding composition semigroup Ctt≥0 on Aωp and denote by Rλ,Γ the resolvent operator for λ∈ρΓ. Then, the following statements are equivalent:(i)
Rλ,Γ is compact on Aωp(ii)
loghz/z∈C01ω∗Proof.
The well-known resolvent equation(36)Rλ,Γ−Rμ,Γ=μ−λRλ,ΓRμ,Γλ,μ∈ρΓ,shows that Rλ,Γ is compact on Aωp for all λ∈ρΓ if and only if it is compact for a certain λ0∈ρΓ. So, by Lemma 3, to prove the compactness of Rλ,Γ, it is suffices to identify the compactness of R−G′0,Γ, or equivalently, the compactness of Rh:(37)Rhfz=1hz∫0zfζh′ζdζ.
To this end, we will use the technology mentioned in [18], which points out that Rh can be decomposed as follows:(38)MzPh=RhMz,Qh=Ph+QhPh,where(39)Mzfz=zfz,Phfz=1zhz∫0zfζζh′ζdζ,Qhfz=1z∫0zfζζh′ζhζdζ.
Moreover, it is elementary to see that(40)Qhfz=Jfz+LhMzfz,where Jfz=1/z∫0zfζdζ and(41)Lhfz=1z∫0zfζloghζζ′dζ.
SinceRh is bounded on Aωp, Ph is bounded on Aωp by (38). Furthermore, since h is univalent and h0=0, it is well known that loghz/z∈BMOA. Hence, loghz/z∈C1ω∗. So by Lemma 2, Lh is bounded on Aωp. Therefore, the compactness of J and Lh and (40) yield the boundedness of Qh on Aωp. Consequently, (38) and (40) imply that Rh is compact if and only if Lh is compact if and only if loghz/z∈C01ω∗ by Lemma 2. The proof is finished.Remark 1.
By [1], we immediately know that BMOA⊊C1ω∗⊊ℬ and VMOA⊊C01ω∗⊊ℬ0. Therefore, in the above theorem, loghz/z∈C01ω∗ if and only if loghz/z∈VMOA if and only if loghz/z∈ℬ0, which indicates that Rλ,Γ is compact on Aωp if and only if it is compact on the Hardy space Hp if and only if it is compact on the classical Bergman space Ap by Theorem 6.1 in [17]. This makes sense since Aωp induced by ω∈D^ lies between Hp and Ap.It is well known that ifRλ,Γ is compact on Aωp, then the spectrum σΓ of Γ is only the point spectrum PσΓ. By a simple combination of Theorems 2 and 3, we are in a position to depict the spectra of infinitesimal generators of some special composition semigroups. We end up the paper by showing some examples.Example 1.
For aRec≥0,consider the semigroup(42)φtz=e−ctz,t≥0,z∈D,with infinitesimal generator Gz=−cz and Koenigs function hz=z. Apparently, its Denjoy–Wolff point is 0 and h∈C01ω∗. Therefore, by Theorems 2 and 3, the spectrum σΓ of infinitesimal generator Γ of the corresponding composition semigroup Ctt≥0 is(43)σΓ=−ck:k=k=0,1,2,⋯.Example 2.
Consider the semigroup(44)φtz=1−1−ze−t,t≥0,z∈D,with Denjoy–Wolff point 0, infinitesimal generator Gz=−1−zlog1/1−z, and Koenigs function hz=log1/1−z. Similarly, the spectrum σΓ of infinitesimal generator Γ of the corresponding composition semigroup Ctt≥0 is(45)σΓ=−k:k=k=0,1,2,⋯.



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