# Weighted Composition Groups on the Little Bloch Space

Authors: S. B. Mose; J. O. Bonyo
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We determine both the semigroup and spectral properties of a group of weighted composition operators on the little Bloch space. It turns out that these are strongly continuous groups of invertible isometries on the Bloch space. We then obtain the norm and spectra of the infinitesimal generator as well as the resulting resolvents which are given as integral operators. As a consequence, we complete the analysis of the adjoint composition group on the predual of the nonreflexive Bergman space and a group of isometries associated with a specific automorphism of the upper half-plane.
Dedicated to Prof. Len Miller (PhD advisor to J. O. Bonyo) and Prof. Vivien Miller of Mississippi State University on their retirement


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## Body

## 1. Introduction

The (open) unit discD of the complex plane ℂ is defined as D=z∈ℂ:z<1, while the upper half-plane of ℂ, denoted by U, is given by U=ω∈ℂ:ℑω>0, where ℑω stands for the imaginary part of ω. The Cayley transform ψz≔i1+z/1−z maps the unit disc D conformally onto the upper half-plane U with inverse ψ−1ω=ω−i/ω+i. For every α > −1, we define a positive Borel measure dmα on D by dmαz=1−z2αdAz, where dA denotes the area measure on D.For an open subset Ω ofℂ, let ℋΩ denote the Fréchet space of analytic functions f:Ω⟶ℂ endowed with the topology of uniform convergence on compact subsets of Ω. Let AutΩ⊂ℋΩ denote the group of biholomorphic maps f : Ω ⟶ Ω. For 1 ≤ p < ∞, α > −1, the weighted Bergman spaces of the unit disc D, LapD,mα, are defined by(1)LapD,mα≔f∈ℋD:fLapD,mα=∫Dfzpdmαz1/p<∞.Clearly,LapD,mα=LpD,mα∩ℋD, where LpD,mα denotes the classical Lebesgue spaces. For every f∈LapD,mα, the growth condition is given by(2)fz≤Kf1−z2γ,where K is a constant and γ=α+2/p, see, for example, [1], Theorem 4.14.The Bloch space of the unit disc, denoted byℬ∞D, is defined as the space of analytic functions f∈ℋD such that the seminorm(3)fℬ∞,1D≔supz∈D1−z2f′z<∞.Following [1, 2], ℬ∞D is a Banach space with respect to the norm fℬ∞D≔f0+fℬ∞,1D. On the contrary, the little Bloch space of the disc, denoted by ℬ∞,0D, is defined to be the closed subspace of ℬ∞D such that(4)ℬ∞,0D≔clℬ∞ℂz,where clℬ∞ℂz denotes ℬ∞ closure of the set of analytic polynomials in z. Equivalently,(5)ℬ∞,0D≔f∈ℋD:limz⟶1−,z∈D1−z2f′z=0,and possesses the same norm as ℬ∞D. Since ℬ∞,0D is a closed subspace of the Banach space ℬ∞D, it follows that ℬ∞,0D is a Banach space as well with respect to the norm ⋅ℬ∞D. Note that every f∈ℬ∞D (or f∈ℬ∞,0D) satisfies the growth condition:(6)fz≤1+12log1+z1−zfℬ∞D.See, for instance, [3] for details. Let 1 < p < ∞ and q be conjugate to p in the sense that 1/p+1/q=1. If LapD,mα∗ is the dual space of LapD,mα, then(7)LapD,mα∗≈LaqD,mα,α>−1,under the integral pairing(8)f,g=∫Dfzgz¯dmα,f∈LapD,mα,g∈LaqD,mα.It is well known that for 1 <p < ∞, LapD,mα is reflexive. The case p = 1 is the nonreflexive case and the duality relations have been determined as follows:(9)La1D,mα∗≈ℬ∞D,ℬ∞,0D∗≈La1D,mα,under the duality pairings given by, respectively:(10)f,g=∫Dfzgz¯dmα,f∈La1D,mα,g∈ℬ∞D,f,g=∫Dfzgz¯dmα,f∈ℬ∞,0D,g∈La1D,mα.In other words, the dual and predual spaces of the nonreflexive Bergman spaceLa1D,mα are the Bloch and little Bloch spaces, respectively. For a comprehensive account of the theory of Bloch and Bergman spaces, we refer to [1, 2, 4–6].In [7], all the self analytic maps φtt≥0⊆AutU of the upper half-plane U were identified and classified according to the location of their fixed points into three distinct classes, namely, scaling, translation, and rotation groups. For each self-analytic map φt, we define a corresponding group of weighted composition operator on ℋU by(11)Sφtfz=φt′zγfφtz,for some appropriate weight γ.It is noted in [7] Section 5 that for the rotation group, we consider the corresponding group of weighted composition operators defined on the analytic spaces of the disc ℋD given by(12)Ttfz=eictfeiktz,withc,k∈ℝ,k≠0.The study of composition operators on spaces of analytic functions still remains an active area of research. For Bloch spaces, most studies have only focussed on the boundedness and compactness of these operators. See, for instance, [3, 8–11]. In [7, 12], both the semigroup and spectral properties of the group Ttt∈ℝ were studied in detail on the Hardy and Bergman spaces. The aim of this paper is to extend the analysis of the group Ttt∈ℝ from the Hardy and Bergman spaces to the setting of the little Bloch space. Specifically, we apply the theory of semigroups as well as spectral theory of linear operators on Banach spaces to study the properties of the group of weighted composition operators given by equation (12) on the little Bloch space of the disk. As a consequence, we shall complete the analysis of the adjoint group on the dual of the nonreflexive Bergman space La1D,mα. The analysis of the adjoint group on the reflexive Bergman space, that is, LapD,mα for 1 < p < ∞, was considered exhaustively in [12]. We shall also consider a specific automorphism of U and carry out an analysis of the corresponding composition operator.IfX is an arbitrary Banach space, let ℒX denote the algebra of bounded linear operators on X. For a linear operator T with domain DT⊂X, denote the spectrum and point spectrum of T by σ(T) and σp(T), respectively. The resolvent set of T is ρT=ℂ\σT, while r(T) denotes its spectral radius. For a good account of the theory of spectra, see [13–15]. If X and Y are arbitrary Banach spaces and U∈ℒX,Y is an invertible operator, then clearly Att∈ℝ⊂ℒX is a strongly continuous group if and only if Bt ≔ UAtU−1, t∈ℝ, is a strongly continuous group in ℒY. In this case, if Att∈ℝ has generator Γ, then Btt∈ℝ has generator Δ = UΓU−1 with domain DΔ=UDΓ≔y∈Y:Uy∈DΓ. Moreover, σp(Δ) = σp(Γ) and σ(Δ) = σ(Γ), since if λ is in the resolvent set ρΓ≔ℂ\σΓ, we have that R(λ, Δ) = UR(λ, Γ)U−1. See, for example, [16], Chapter II and [15], Chapter 3.


## 2. Groups of Composition Operators on the Little Bloch Space

We consider the group of weighted composition operatorsTtt∈ℝ given by equation (12) and defined on the little Bloch space ℬ∞,0D as Ttf(z) = eictf(eiktz), where c,k∈ℝ, k ≠ 0 and ∀f∈ℬ∞,0D. We denote the infinitesimal generator of the group Ttt∈ℝ by Γ{c, k} and give some of its properties in the following proposition.Proposition 1.
(1)
T
t
t
∈
ℝ is a strongly continuous group of isometries on ℬ∞,0D
(2)
The infinitesimal generator Γc,k of Ttt∈ℝ on ℬ∞,0D is given by Γc,kfz=icfz+kzf′z. with domain DΓc,k=f∈ℬ∞,0D:zf′∈ℬ∞,0D.Proof.
To prove isometry, we have(13)Ttfℬ∞D=Ttf0+supz∈D1−z2Ttf′z=eictf0+supz∈D1−z2eicteiktf′eiktz=f0+supz∈D1−z2f′eiktz.
By change of variables, letω = eiktz. Then,(14)Ttfℬ∞D=f0+supω∈D1−ω2f′ω=fℬ∞D,as desired.
To prove strong continuity, we shall use the density of polynomials inℬ∞,0D. Therefore, it suffices to show that, for znn≥0,(15)limt⟶0+Ttzn−znℬ∞,0D=0.
Now,Ttzn−zn=eicteiktzn−zn=eic+knt−1zn. Therefore,(16)limt⟶0+Ttzn−znℬ∞,0D=limt⟶0+supz∈D1−z2Ttzn−zn′=limt⟶0+supz∈D1−z2neik+knt−1zn−1=0,as claimed.
Now, for the infinitesimal generator Γc,k, let f∈DΓc,k in ℬ∞,0D, then the growth condition (6) implies that(17)Γc,kfz=limt⟶0+eictfeiktz−fzt=∂∂teictfeiktzt=0=icfz−izf′z.
Therefore,DΓc,k⊆f∈ℬ∞,0D:zf′∈ℬ∞,0D. Conversely, if f∈ℬ∞,0D is such that zf′∈ℬ∞,0D, then Fz=icfz+kzf′z∈ℬ∞,0D and for all t > 0,(18)Ttfz−fzt=1t∫0t∂∂sTsfzds=1t∫0teicsicfeiksz+keikszf′eikszds=1t∫0tTsFzds.
Strong continuity ofTss≥0 implies that(19)1t∫0tTsFds−F≤1t∫0tTsF−Fds⟶0,ast⟶0+.
Thus,DΓc,k⊇f∈ℬ∞,0D:zf′∈ℬ∞,0D.DefineMz, Q on ℋD by Mzf(z) = zf(z) and Qfz=fz−f0/z, Qf0=f′0. More generally, Qmfz=∑k=m∞fk0/k!zk−m, Qmf0=fm0/m!. Then, MzmQmf=∑m∞fk0/k!zk and QmMzmf=f. We now give the following proposition.Proposition 2.
(1)
M
z
:
ℬ
∞
D
⟶
ℬ
∞
D is bounded.
(2)
M
z
ℬ
∞
,
0
D
⊆
ℬ
∞
,
0
D.
(3)
Q
:
ℬ
∞
,
0
D
⟶
ℬ
∞
,
0
D is bounded.
(4)
Form ≥ 1, Mzmℬ∞,0D=f∈ℬ∞,0D:fk0=0∀k<m. In particular, Mzℬ∞,0D is closed in ℬ∞,0D.Proof.
Iff∈ℬ∞D, then for all z∈D,(20)1−z2zf′=1−z2zf′z+fz≤1−z2f′z+1−z2fz≤1−z2f′z+1−z21+12log1+z/1−zfℬ∞D.
Therefore, assertions (1) and (2) follow. For (3), iff∈ℬ∞,0D, then for |z| < 1,(21)1−z2Qf′z=1−z2zf′z−fz+f0z2≤1−z2f′zz+1−z21+1/2log1+z/1−zfℬ∞Dz2+1−z2fℬ∞Dz2⟶0asz⟶1.
Thus,Qf∈ℬ∞,0D. To prove (4), let f∈ℬ∞,0D and f(0) = 0. Then, f=MzQf∈Mzℬ∞,0D. The reverse inclusion is obvious. Therefore, the one-to-one and onto mapping Mz:ℬ∞,0D⟶f∈ℬ∞,0D:f0=0 is bounded. So, the open mapping theorem implies that the inverse is bounded. It therefore follows that Q:span1⊕Mzℬ∞,0D⟶ℬ∞,0D is bounded.Proposition 3.
LetΓc,k be the infinitesimal generator of the group Tt∈ℝ given by (12) on ℬ∞,0D, then(1)
Γc,k = ic + kΓ0,1 with domain DΓc,k=DΓ0,1=f∈ℬ∞,0D:zf′∈ℬ∞,0D.(2)
σΓc,k=ic+kσΓ0,1 and σpΓc,k=ic+kσpΓ0,1.In fact,λ ∈ ρ(Γ0,1) if and only if ic + kλ ∈ ρ(Γc,k), and(22)Ric+kλ,Γc,k=1kRλ,Γ0,1.Proof.
See [12], Lemma 4.3.As a result of Proposition3 above and without loss of generality, we restrict our attention to the generator Γ0,1instead of Γc,k as the cases c ≠ 0 and k ≠ 1 where k ≠ 0 can be easily obtained from Γ0,1. Indeed, Γ0,1f(z) = izf′(z) with domain DΓ0,1=f∈ℬ∞,0D:zf′∈ℬ∞,0D is the infinitesimal generator of the group Tt = f(eitz) which is exactly the case when c = 0 and k = 1 in equation (12). We now give the spectral properties of the generator Γ0,1 as well as the resulting resolvents in the following theorem.Theorem 1.
(1)
σ
Γ
0,1
=
σ
p
Γ
0,1
=
in
:
n
∈
ℤ
+, and for each n ≥ 0, ker(in − Γ0,1) = span(zn).
(2)
Ifλ ∈ ρ(Γ0,1), then Mzℬ∞,0D is R(λ, Γ0,1), invariant ∀m∈ℤ+, m>ℑλ. Moreover, if h∈Mzmℬ∞,0D, then
(23)
R
λ
,
Γ
0,1
h
z
=
i
z
−
λ
t
∫
0
z
ω
i
λ
−
1
h
ω
d
ω
=
i
z
m
∫
0
1
t
m
+
i
λ
−
1
Q
m
h
t
z
d
t
.
(3)
Forλ ∈ ρ(Γ0,1), the resolvent operator R(λ, Γ0,1) is compact.
(4)
σ
R
λ
,
Γ
0,1
=
σ
p
R
λ
,
Γ
0,1
=
w
∈
ℂ
:
w
−
1
/
2
ℜ
λ
=
1
/
2
ℜ
λ. Moreover,
(24)
r
R
λ
,
Γ
0,1
=
R
λ
,
Γ
0,1
=
1
ℜ
λ
.Proof.
Since eachTt is an invertible isometry, its spectrum satisfies σTt⊆∂D, and the spectral mapping theorem for strongly continuous groups (see, for example, [16], Theorem V.2.5 or [17]) implies that etσΓ0,1⊆σTt. Thus, etσΓ0,1⊆∂D⟹etσΓ0,1=1⟹etℜω=1⟹ℜω=0 for ω ∈ σ(Γ0,1). It immediately follows that σΓ0,1⊆iℝ.
We now solve the resolvent equation: Ifλ∈ℂ and h∈ℋD, (λ − Γ)f = h. This is equivalent to(25)f′z+iλzfz=izhz,z≠0,ziλfz=iziλ−1hz,z∈D\−1,0.
In particular, (λ − Γ)f = 0 if and only f(z) = Kz−iλ, where K is a constant. Since z−iλ∈ℋD if and only if −iλ∈ℤ+, it follows that(26)σpΓ0,1=in:n∈ℤ+,with ker(in − Γ0,1) = span(zn). Moreover, if n∈ℤ+ and λ ∈ σp(Γ0,1), then(27)λ−Γf=zn,has a unique solution(28)fz=1λ−inzn.
Notice that, forλ ∉ σp(Γ0,1) and f∈DΓ0,1, (λ − Γ)f(0) = λf(0). More generally, if fz=zngz with g0≠0, then(29)λ−Γf=λf−zzmg′=zmλg−mzmg−zm+1g′.
Note that the functions (λ − Γ)f and f have the same order of zero at 0. Thus, Mzmℬ∞,0D is invariant under λ − Γ0,1.
Fixλ∈ℂ\σpΓ0,1 and let m>ℑλ. If h=zmg with g∈ℬ∞,0D, then(30)i∫0zωiλ−1hωdω=izm+iλ∫01tm+iλ−1gtzdt.
Thus, (λ − Γ)h has a unique solution:(31)fz=izm∫01tm+iλ−1Qmhtzdt.
Ifu∈ℬ∞D and 0 ≤ t < 1, then(32)utzℬ∞D=supz<11−z2tu′tz≤supz<11−t2z2u′tz≤uℬ∞D.
Thus,f≤1/m−ℑλMzmQmh. Now, ∀m ≥ 1,(33)ℬ∞,0D=spanzn⊕Mzmℬ∞,0D,Rλ,Γ0,1spanzn0≤n<m=1λ1λ−i0⋱01λ−m−1i.
Thus,λ ∉ σp(Γ0,1) implying that R(λ, Γ0,1) is bounded on ℬ∞,0D. Therefore, σ(Γ0,1) = σp(Γ0,1). This proves (1) and (2).
To prove the compactness of the resolvent operator, we argue as in [7], Theorem 5.2. Fix λ ∈ ρ(Γ0,1) and let m∈ℤ+ be such that ℑλ<m. Then, by equation (33), it suffices to show that Rmλ,Γ0,1=Rλ,Γ0,1Mzmℬ∞,0D is compact.
LetArD, r > 0, be the disc algebra ArD=CrD¯∩ℋrD, equipped with the supremum norm, and for each t, 0 ≤ t < 1, and f∈ℋD, let Htf(z) = ft(z) = f(tz). Then, by equation (32), for every t ∈ [0, 1), Ht is a contraction on ℬ∞,0D.
Now, by equation (23), Rmλ,Γ0,1=iMzm∫01tm+iλ−1HtQmdt with convergence in norm. Define Cr=iMzm∫0rtm+iλ−1HtQmdt on Mzmℬ∞,0D, for 0 < r < 1. Then,(34)Rm−Cr≤∫r1tm−ℑλ−1Qmdt=Qmm−ℑλ1−rm−ℑλ⟶0,as r ⟶ 1−. Choosing s so that 1 < s < r−1, we have that Cr:Mzmℬ∞,0D⟶Mzmℬ∞,0D factors through AsD. If B denotes the closed unit ball of Mzmℬ∞,0D, let h=Qmff∈Mzmℬ∞,0D. Then, ∀t, 0 ≤ t ≤ r, the growth condition (6) implies that, for |z| ≤ s,(35)htz≤1+12log1+rs1−rshℬ∞,0D,ddthtz≤hℬ∞,0D1−rs.
LetK=1+1/2log1+rs/1−rshℬ∞,0D. Thus, for |z| ≤ s,(36)Crfz≤Ksmrm−ℑλm−ℑλ,ddzCrfz≤Kmsm−1rm−ℑλm−ℑλ+smrm−ℑλm−ℑλhℬ∞,0D1−rs.
Thus, by Arzela-Ascoli,CrB is precompact in AsD which further implies that CrB is precompact in ℬ∞,0D by the continuous embeddedness of AsD in ℬ∞,0D. Therefore, each Cr is compact in ℒMzmℬ∞,0D and as a result, Rmλ,Γ0,1=normlimr⟶1−Cr is compact as well.
The spectral mapping theorem for resolvents as well as assertion (1) above implies that(37)σRλ,Γ0,1=σpRλ,Γ0,1=1λ−im:m∈ℤ+∪0=ω∈ℂ:ω−12ℜλ=12ℜλ.
Clearly, the spectral radiusrRλ,Γ0,1=1/ℜλ and therefore by the Hille–Yosida theorem, it follows that 1/ℜλ=rRλ,Γ0,1≤Rλ,Γ0,1≤1/ℜλ, as desired.As a consequence, the properties of the general groupTt given by equation (12) is as follows.Corollary 1.
(1)
σ
Γ
c
,
k
=
σ
p
Γ
c
,
k
=
i
c
+
k
n
:
n
∈
ℤ
+, and for each n ≥ 0, ker(i(c + kn) − Γc,k) = span(zn).
(2)
Ifμ ∈ ρ(Γc,k), then Mzℬ∞,0D is R(μ, Γc,k) -invariant ∀m∈ℤ+, m>ℑμ−ic/k. Moreover, if h∈Mzmℬ∞,0D, then
(38)
R
μ
,
Γ
c
,
k
h
z
=
i
k
z
−
μ
−
i
c
/
k
t
∫
0
z
ω
i
μ
−
i
c
/
k
−
1
h
ω
d
ω
=
i
k
z
m
∫
0
1
t
m
+
i
μ
−
i
c
/
k
−
1
Q
m
h
t
z
d
t
.
(3)
Forμ ∈ ρ(Γc,k), the resolvent R(μ, Γc,k) is compact.
(4)
σ
R
μ
,
Γ
c
,
k
=
σ
p
R
μ
,
Γ
c
,
k
=
w
∈
ℂ
:
w
−
1
/
2
ℜ
μ
=
1
/
2
ℜ
μ.
(5)
r
R
μ
,
Γ
c
,
k
=
R
μ
,
Γ
c
,
k
=
1
/
2
ℜ
μ
.Proof.
Following Proposition3, μ ∈ ρ(Γc,k) if and only if μ−ic/k∈ρΓ0,1. The proof now follows at once from Theorem 1. We omit the details.


## 3. Adjoint of the Composition Group on the Predual of Nonreflexive Bergman SpaceLa1D,mα

In studying the adjoint properties of the rotation group isometries given by equation (12) on Bergman spaces LapD,mα, 1 ≤ p < ∞; the second author in [12] considered the reflexive case, that is, when 1 < p < ∞. This was an extension of the investigation of adjoint properties of the Cesáro operator in [18] on Hardy spaces, and later generalized to Bergman spaces in [7]. For the nonreflexive Bergman space La1D,mα (that is, p = 1), the analysis of the adjoint of the rotation group isometries remains open and forms the basis of this section. Specifically, we complete the analysis of the adjoint group of the group of isometries Ttf(z) = eictf(eiktz), where c,k∈ℝ with k ≠ 0 and ∀f∈La1D,mα.Recall fromSection 1, the duality relation ℬ∞,0D∗≈La1D,mα under the integral pairing g,f=∫Dgzfz¯dmαg∈ℬ∞,0D,g∈La1D,mα. In particular, the predual of La1D,mα is the little Bloch space ℬ∞,0D. Thus, using this duality pairing, for every g∈ℬ∞,0D, we have(39)g,Ttf=∫Dgzeictfeiktz¯dmαz=∫De−ictgzfeiktz¯1−z2αdAz.By a change of variables argument: Letω = eiktz so that z = e−iktω and(40)g,Ttf=∫De−ictge−iktωfω¯1−e−iktω2αdAω=∫De−ictge−iktωfω¯dmαω=∫DT−tgωfω¯dmαω=T−tg,f,where T−tgω=e−ictge−iktω for all g∈ℬ∞,0D. Thus, the adjoint group Tt∗ of Tt for t∈ℝ is therefore given by(41)Tt∗gω≔T−tgω=e−ictge−iktω,for allg∈ℬ∞,0D.Let Γ denote the infinitesimal generator of the adjoint groupTt∗. Using the results of Section 2, we easily obtain the properties of the group Tt∗t∈ℝ as we give in the following theorem.Theorem 2.
LetTt∗t∈ℝ⊆ℒℬ∞,0D be the adjoint group of the group of weighted composition operators Ttt∈ℝ⊆ℒLa1D,mα given by (41). Then, the following hold:(1)
Tt∗t∈ℝ is strongly continuous group of isometries on ℬ∞,0D.(2)
The infinitesimal generator Γ ofTt∗t≥0 is given by Γgω=−icgω+kωg′ω with domain DΓ=g∈ℬ∞,0D:ωg′∈ℬ∞,0D.(3)
σΓ=σpΓ=−ic+kn:n∈ℤ+, and for each n ≥ 0, ker−ic+kn−Γ=spanωn(4)
Ifμ ∈ ρ(Γ), then Mωℬ∞,0D is R(μ, Γ)-invariant ∀m∈ℤ+, m>ℑ−μ−ic/k. Moreover, if h∈Mωmℬ∞,0D, then(42)Rμ,Γhw=−ikωμ+ic/kt∫0ωz−iμ+ic/k−1hzdz=−ikωm∫01tm−iμ+ic/k−1Qmhtωdt.(5)
σRμ,Γ=σpRμ,Γ=w∈ℂ:w−1/2ℜμ=1/2ℜμ.(6)
rRμ,Γ=Rμ,Γ=1/ℜμ.Proof.
The proof follows immediately by replacingc and k with −c and −k, respectively, in Proposition 3 and Corollary 1. We omit the details.


## 4. Specific Automorphism of the Half-Plane

In this section, we consider a specific automorphism groupφtt∈ℝ⊂AutU corresponding to the rotation group given by(43)φtz=zcost−sintzsint+cost.It can be easily verified thatφt(z) = ψ ∘ ut ∘ ψ−1(z), where ut(z) = e−2itz. The associated group of weighted composition operators on ℋU is given by Sφt, and by the chain rule, it follows that Sφt=Sψ−1SutSψ, where Sψ−1=Sψ−1.Now, forf∈ℬ∞,0D,(44)Sutfz=ut′zγfutz=e−2iγtfe−2itz.Apparently,Sut can be obtained as a special case of the group Ttt≥0 given by equation (12) when c = −2γ and k = −2. Let Γ = Γ−2γ,−2 be the infinitesimal generator of the group Sut, then the properties of Γ can be summarized by the following proposition.Proposition 4.
Let Γ be the infinitesimal generator of the group of isometriesSut on ℬ∞,0D. Then,(1)
Γfz=i−2γfz−2zf′z for every f∈ℬ∞,0D, with domain(45)DΓ=f∈ℬ∞,0D:f′∈ℬ∞,0D.(2)
σΓ=σpΓ=−2γ+ni:n∈ℤ+, and for each n ≥ 0,(46)ker−2γ+ni−Γ=spanzn.(3)
Ifμ ∈ ρ(Γ), then ℛMzm is R(μ, Γ)-invariant for every m∈ℤ+, m>ℑ−μ+2γi/2. Moreover, if h∈ℛMzm, then(47)Rμ,Γhz=−i2zμ+2iγ/2i∫0zω−μ+2iγ/2i−1hωdω≔Rμhz.Proof.
Takec = −2γ and k = −2 in Proposition 1 and Corollary 1. The proof follows immediately.
Now, using the similarity theory of semigroups, we detail the properties of the group of weighted composition operators associated with the automorphism groupφtt≥0 given by (43) in the following theorem.Theorem 3.
Letφt∈AutU be given by φtz=zcost−sint/zsint+cost, for all t∈ℝ,z∈U, and let Sφtfz≔φt′γfφtz be the corresponding group of isometries on ℬ∞,0U. Then,(1)
The infinitesimal generator Δ of the groupSφt on ℬ∞,0U is given by(48)Δhz=−2γzhz−1+z2h′z,with domainDΔ=h∈ℬ∞,0U:2γω+ih+ω+i2h′∈ℬ∞,0D.(2)
σpΔ=σΔ=−2γ+ni:n∈ℤ+, and for each n ≥ 0, ker−2γ+ni−Δ=spanSψ−1zn.(3)
Ifμ ∈ ρ(Δ) and if m∈ℤ+ is such that m>ℑ−μ/2−iγ. Then, if h∈ℛMzm, we have(49)Rμ,Δhz=z−iμ+2iγ/2iz+i−μ+2iγ/2i+2γ∫0zω−i−μ+2iγ/2i−1ω+iμ+2iγ/2i+2γ−1hωdω.(4)
R(μ, Δ) is compact on ℬ∞,0D.(5)
σRμ,Δ=σpRμ,Δ=w∈ℂ:w−1/2ℜμ=1/ℜμ. Moreover(50)rRμ,Δ=Rμ,Δ=1ℜμ.Proof.
Letgz=ψ−1z=z−i/z+i and g−1z=ψz=i1+z/1−z. Since φtz=g−1∘ut∘gz, it follows that Sφt=SgSutSg−1=SgSutSg−1, where Sg is invertible. Let Δ be the generator of Sφt and Γ ≔ Γ−2γ,−2 be the generator of Sut, then Δ=SgΓSg−1 with domain DΔ=SgDΓ.
Letf′∈B∞,0D, then f ∈ D(Γ) and define h≔Sgf belongs to D(Δ) with f=Sg−1h. Then,(51)Δhz=SgΓSg−1hz=SgΓfz=Sg−2iγfz−2izf′z=g′zγ−2iγfgz−i2gzf′gz.
As stated earlier,gz=z−i/z+i, implying that g′z=2i/z+i2, and thus(52)Δhz=2iγz+i2γ−2iγfgz−i2gzf′gz.
Sinceg−1z=i1+z/1−z and g−1z′=2i/1−z2, then we have fz=Sg−1hz=Sg−1hz=2iγ/1−z2γhg−1z implying that fgz=z+i2γ/2iγhz. Moreover, f′z=2iγ/1−z2γ+22γ1−zhg−1z+2ih′g−1z implying that(53)f′gz=z+i2γ+12iγ+12γhz+z+ih′z.
Therefore,(54)Δhz=2iγz+i2γ−iγz+i2γ2iγhz−iz−iz+iz+i2γ+12iγ+12γhz+z+ih′z=−2iγhz−2γz−ihz−z−iz+ih′z=−2γzhz−1+z2h′z.
As given earlier, the domain of Δ,D(Δ) is given by D(Δ) =SgDΓ=Sgf:∈DΔ. Now h ∈ D(Δ) implies that Sg−1h∈DΓ which implies that Sg−1h′∈B∞,0D. But(55)Sg−1h′=2iγ1−z2γhg−1z′=2γ2iγ1−z−2γ−1hg−1z+2iγ1−z2γ2i1−z2h′g−1z=2iγ1−z2γ2γ1−z−1hg−1z+2i1−z2h′g−1z.
Then, we have(56)Sg−1h′=2iγ1−z2γ2γ1−g∘g−1zhg−1z+2i1−g∘g−1z2h′g−1z.
By change of variables, letω=g−1z which implies gω=z=g∘g−1z and Sg−1h′=Sg−12γ/1−gωhω+2i/1−gω2h′ω. Therefore,(57)h∈DΔ⇔Sg−12γ1−gωhω+2i1−gω2h′ω∈B∞,0D⇔2γ1−gωhω+2i1−gω2h′ω∈B∞,0D⇔ω+i2i2γhω+ω+ih′ω∈B∞,0D,which implies that UΔ=h∈B∞,0D:2γhω+ω+ih′ω∈B∞,0D.
FromSection 1, the spectrum and point spectrum of Δ are given as σpΔ=σpΓ=σΓ=σΔ=−iγ+n:n∈ℤ+.
For the resolvents, ifμ ∈ ρ(Δ) = ρ(Γ), then for m∈ℤ+, m>ℑ−μ+iγ and if h∈RMzm, we have Rμ,Δ=SfRμ,ΓSf−1 and so(58)Rμ,Δhz=Sg−i2zμ+2iγ/2i∫0zω−iμ+2iγ/2−1Sg−1hωdω=Sg−i2zμ+2iγ/2i∫0zω−μ+2iγ/2i−12iγ1−ω2γhg−1ωdω=−i22iγz+i2γgzμ+2iγ/2i∫0zgωμ+2iγ/2i−12iγ1−gω2γhωdgdωdω=z−iz+i2γμ+2iγ/2i∫0zω−iμ+2iγ/2i−1ω+iμ+2iγ/2i+2γ−1hωdω.
Finally, from spectral mapping theorems it follows that, for allμ ∈ ρ(Δ), the spectrum of R(μ, Δ) is given by(59)σRμ,Δ=1μ−z:z∈σΔ∪0=1μ+iγ+n:n∈ℤ+∪0=ω∈ℂ:ω−12ℜμ=12ℜμ.
Similarly, the point spectrum is given by(60)σpRμ,Δ=1μ−z:z∈σpΔ∪0=ω∈ℂ:ω−12ℜμ=12ℜμ.
Therefore,σRμ,Δ=σpRμ,Δ=ω∈ℂ:ω−1/2ℜμ=1/2ℜμ. Finally, we conclude this section by proving the spectral radius rRμ,Δ=1/ℜμ and Rμ,Δ=1/ℜμ.
It is clear that the spectrum of the resolvent isrRμ,Δ=1/ℜμ. Hille–Yosida theorem yields rRμ,Δ=1/ℜμ≤Rμ,Δ≤1/ℜμ.



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