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Volume 2007, Article ID 92070,21pages doi:10.1155/2007/92070

Research Article

Composition Operators and Multiplication Operators on

Weighted Spaces of Analytic Functions

J. S. Manhas

Received 17 June 2006; Accepted 26 April 2007

Recommended by Marianna A. Shubov

LetV be an arbitrary system of weights on an open connected subsetGofCN(N1) and letB(E) be the Banach algebra of all bounded linear operators on a Banach spaceE. LetHVb(G,E) andHV0(G,E) be the weighted locally convex spaces of vector-valued an-alytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical spaces of analytic functionsH(G) to the weighted spaces of analytic functionsHVb(G,E) andHV0(G,E).

Copyright © 2007 J. S. Manhas. This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

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In recent years, many authors like Attele [12], Axler [13–16], Bercovici [17], Eschmeier [18], Luecking [19], Vukoti´c [20], and Zhu [21] have made a study of multiplication op-erators on Bergman spaces, whereas Campbell and Leach [22], Feldman [23], Lin [10], and Ohno and Takagi [24] have obtained a study of these operators on Hardy spaces. On Bloch spaces, these operators are studied by Arazy [25], Axler [15] and Brown and Shields [26]. Also, Axler and Shields [16] and Stegenga [27] have explored multiplication oper-ators on Dirichlet spaces. On BMOA, these operoper-ators are studied by Ortega and Fabreg´a [28]. Further, on Nevanlinna classes of analytic functions, these operators are studied by Jarchow et al. [29] and Yanagihara [30]. Besides these well-known analytic function spaces, a study of these operators on some other Banach spaces of analytic functions has also been pursued by Bonet et al. [31–34], Contreras and Hern´andez-D´ıaz [35], Ohno and Takagi [24], and Shields and Williams [36,37].

In [38], Contreras and Hern´andez-D´ıaz have made a study of weighted composition operators on Hardy spaces, whereas Mirzakarimi and Siddighi [39] have considered these operators on Bergman and Dirichlet spaces. On Bloch and Block-type spaces, these op-erators are studied by MacCluer and Zhao [40], Ohno [41], Ohno and Zhao [42], and Ohno et al. [43]. In [24], Ohno and Takagi have obtained some properties of these oper-ators on the disc algebra and the Hardy spaceH∞(D). Also, recently, Montes-Rodr´ıguez [44] and Contreras and Hern´andez-D´ıaz [35] have studied the behaviour of these opera-tors on weighted Banach spaces of analytic functions. The applications of these operaopera-tors can be found in the theory of semigroups and dynamical systems (see [2,3,45]). For more information on composition operators on spaces of analytic functions, we refer to three monographs (see Cowen and MacCluer [46], Shapiro [47], and Singh and Manhas [48]).

In the present survey, we report on a recent study of composition operators and mul-tiplication operators on the weighted spaces of analytic functions.

2. Weighted spaces of analytic functions

LetG be an open connected subset ofCN (N1) and letH(G,E) be the space of all vector-valued analytic functions fromGinto the Banach spaceE. LetV be a set of non-negative upper semicontinuous functions onG. ThenV is said to be directed upward, if for every pairu1,u2∈V andλ >0, there existsv∈V such thatλui≤v(pointwise on

G), fori=1, 2. IfV is directed upward and for eachz∈G, there existsv∈V such that

v(z)>0, then we callV as an arbitrary system of weights onG. IfUandV are two arbi-trary systems of weights onGsuch that for eachu∈U, there existsv∈Vfor whichu≤v, then we writeU≤V. IfU≤V andV≤U, then we writeU∼=V. LetV be an arbitrary system of weights onG. Then we define

HVb(G,E)=f ∈H(G,E) :v f(G) is bounded inE, for eachv∈V,

HV0(G,E)=f ∈H(G,E) :v f vanishes at infinity onG, for eachv∈V. (2.1)

Forv∈Vand f ∈H(G,E), we define

fv,E=sup

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Clearly, the family { · v,E:v∈V} of seminorms defines a Hausdorff locally convex topology on each of theses spaces: HVb(G,E) andHV0(G,E). With this topology, the spacesHVb(G,E) andHV0(G,E) are called the weighted locally convex spaces of vector-valued analytic functions. These spaces have a basis of closed absolutely convex neigh-bourhoods of the form

Bv,E=f ∈HVb(G,E)resp.,HV0(G,E):fv,E≤1. (2.3)

IfE=C, then we writeHVb(G,E)=HVb(G),HV0(G,E)=HV0(G) and

Bv=f ∈HVb(G)resp.,HV0(G):fv≤1. (2.4)

Throughout the paper, we assume for each z∈G, there exists fz∈HV0(G) such that

fz(z) =0.

Ifv:DR+ is a continuous weight andE=C, then the corresponding weighted

Banach spaces of analytic functions are defined as follows:

H∞

v (D)=f ∈H(D) :v f(D) is bounded,

H0

v(D)=

f ∈H(D) : lim |z|→1−v(z)

f(z)=0 . (2.5)

Now, using the definitions of weights given in [32,49–51], we give definitions of some systems of weights which are required for characterizing some results in the remaining sections.

LetV be an arbitrary system of weights onGand letv∈V. Then definew:G→R+as

w(z)=supf(z):fv≤1v1 (z),

v(z)=w1

(z), for everyz∈G.

(2.6)

In casew(z) =0,vis an upper semicontinuous, and we call it an associated weight of

v. LetV denote the system of all associated weights ofV. Then an arbitrary system of weightsVis called a reasonable system as it satisfies the following properties:

for eachv∈V, there existsv∈V such thatv≤v; (2.7a)

for eachv∈V,fv≤1 ifffv≤1,

for every f ∈HVb(G); (2.7b)

ifv∈V, then for everyz∈G, there exists

fz∈Bvsuch thatfz(z)= 1

v(z).

(2.7c)

Letv∈V. Thenvis called essential if there exists a constantλ >0 such thatv(z)

v(z)≤λv(z), for eachz∈G. A reasonable system of weightsV is called an essential

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V∼=V. For example, letG=D, the open unit disc, and let f ∈H(D) be nonzero. Then definevf(z)=[sup{|f(z)|:|z| =r}]1, for everyz∈D. Clearly, eachvf is a weight sat-isfyingvf =vf, and the familyV= {vf :f ∈H(D), f is nonzero}is an essential system of weights onD. For more details on these weights, we refer to [50]. LetGbe any bal-anced (i.e.,λz∈G, wheneverz∈Gandλ∈Cwith|λ| ≤1) open subset ofCN(N≥1). Then a weightv∈V is called radial and typical ifv(z)=v(λz) for allz∈Gandλ∈C with|λ| =1, and vanishes at the boundary∂G. In particular, a weightvonDis radial and typical ifv(z)=v(|z|) and lim|z|→1v(z)=0. For instance, in [35], it is shown that vp(z)=(1− |z|2)p, (0< p <), for everyzD, are essential typical weights. For more

details on the weighted Banach spaces of analytic functions and the weighted locally con-vex spaces of analytic functions associated with these weights, we refer to [32,49–54]. For basic definitions and facts in complex analysis and functional analysis, we refer to [55–58].

LetF(G,E) be a topological vector space of vector-valued analytic functions fromG intoE, and let L(G,E) be the vector space of all vector-valued functions fromG into

E. Let B(E) be the Banach algebra of all bounded linear operators on E. Then for an operator-valued map Ψ:G→B(E) and self-mapφ:G→G, we define the linear map

WΨ,φ:F(G,E)→L(G,E) asWΨ,φ(f)=Ψ·f ◦φfor every f ∈F(G,E), where the

prod-uctΨ· f◦φis defined pointwise onGas (Ψ·f ◦φ)(z)=Ψz(f(φ(z))) for everyz∈G. In caseWΨ,φtakesF(G,E) into itself and is continuous, we callWΨ,φ, the weighted

com-position operator onF(G,E) induced by the symbolsΨandφ. IfΨ(z)=I, the indentity operator onEfor everyz∈G, thenWΨ,φis called the composition operator induced byφ and we denote it by. In caseφ(z)=zfor everyz∈G,WΨ,φis called the multiplication

operator induced byΨ, and we denote it byMΨ.

3. Characterizations of multiplication operators

In this section, we give characterizations of multiplication operators on the weighted spaces of analytic functions. We begin with the following straightforward observations obtained by [31] on the weighted Banach spaces of scalar-valued analytic functions.

Proposition 3.1. Let Ψ:DC be analytic function. ThenMΨ:Hv∞(D)→Hv∞(D) is

bounded if and only ifΨ∈H∞(D). IfH0

v(D) = {0}, then the previous statement is

equiva-lent to thatMΨ:H0

v(D)→Hv0(D) is bounded. Also,MΨ = Ψ∞.

Proposition 3.2. IfMΨ:Hv0(D)→Hw0(D) is bounded and bothvandware radial weights

vanishing on the boundary, thenMΨ=MΨ:Hv∞(D)→Hw∞(D).

Proof. It is shown by [51,59] that (H0

v(D))=Hv∞(D) and (Hw0(D))=Hw∞(D). In [32], it is observed that the evaluation functionalδz:Hv0(D)C defined asδz(f)= f(z) is also acting as the evaluation functional onHv∞(D). It is obvious thatMΨ(δz)=Ψ(z)δz. Now, forf ∈H∞

v (D), we haveMΨf,δz = f,Ψ(z)δz =f(z)Ψ(z).

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Proposition 3.3. LetUandVbe arbitrary systems of weights onG, and letΨ:G→B(E)

be an analytic map. ThenMΨ:HUb(G,E)→HVb(G,E) is a multiplication operator, if for

everyv∈V, there existsu∈Usuch thatv(z)Ψ(z) ≤u(z), for everyz∈G.

Remark 3.4. Proposition 3.3makes it clear that every bounded analytic functionΨ:G→

B(E) induces the multiplication operatorMΨonHVb(G,E), for any system of weights

V onG. Also, ifV = {λχK :λ≥0, K⊆G, Kcompact set}, then every operator-valued analytic mapΨ:G→B(E) induces a multiplication operatorMΨ on HVb(G,E). This makes it clear that even unbounded analytic operator-valued mappings generate multi-plication operators on some of weighted locally convex spacesHVb(G,E), whereas it is not true for other spaces of analytic functions. For instance, Arveson [11] and Axler [13] have shown that only bounded analytic functions give rise to multiplication operators on Hardy spaces and Bergman spaces, respectively. Also, the same behaviour has been observed on the weighted Banach spaces of analytic functionsHv∞(D) defined by a single continuous weights (seeProposition 3.1). Thus the behaviour of the multiplication oper-ators on the weighted locally convex spaces of analytic functions is very much influenced by different systems of weightsVonG.

Theorem 3.5. LetVbe an arbitrary system of weights andUa reasonable system of weights onG. LetΨ:G→B(E) be an analytic map. ThenMΨ:HUb(G,E)→HVb(G,E) is a

multi-plication operator if and only if for everyv∈V, there existsu∈Usuch thatv(z)Ψ(z)

u(z), for everyz∈G.

Proof. The sufficient part follows fromProposition 3.3. Conversely, supposeMΨ:HUb(G,

E)→HVb(G,E) is a multiplication operator. Letv∈V. Then by the continuity ofMΨ at the origin, there existsu∈U withu∈U such that u≤u andMΨ(Bu,E)⊆Bv,E. To establish the inequality v(z)Ψ(z) ≤u(z) for every z∈G, it is enough to prove that

v(z)Ψz(y) ≤u(z)y, for everyz∈Gandy∈E. Fixz0∈Gandy0∈E. Then by (2.7c), there exists fz0∈Busuch thatfz0(z0) =1/u(z0). Letg0:G→Ebe defined as

g0(z)=y01fz0(z)y0, for everyz∈G. (3.1)

Clearly,g0∈Bu,Eandg0(z0) =1/u(z0). Also, according to (2.7b), fz0∈Buand therefore

g0∈Bu,E. Thus it follows thatMΨ(g0)∈Bv,E. That is,v(z)Ψz(g0(z))1, for everyz∈

G. In particular, forz=z0, we havev(z0)Ψz0(y0) ≤u(z0)y0. This completes the proof

of the theorem.

Corollary 3.6. LetV be an arbitrary system of weights and letU be an essential sys-tem of weights on G. LetΨ:G→B(E) be an operator-valued analytic map. Then MΨ:

HUb(G,E)→HVb(G,E) is a multiplication operator if and only if for everyv∈V, there existsu∈Usuch thatv(z)Ψ(z) ≤u(z), for everyz∈G.

Proof. It follows fromTheorem 3.5and from the relation thatU∼=U.

Remark 3.7. All the results proved above are also hold for the spaces HV0(G,E) and

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4. Invertible multiplication operators

In this section, we present characterizations of invertible multiplication operators on the weighted spaces of analytic functions. We begin with the following characterization of in-vertible multiplication operators on the weighted Banach spaces of scalar-valued analytic functions [31].

Proposition 4.1. LetΨ∈H∞. ThenMΨ:H

v (D)→Hv∞(D) is invertible if and only if 1/Ψ∈H∞(or equivalently, there exists>0 such that|Ψ(z)| ≥, for allzD). The same

is true forH0

v(D) ifHv0(D) = {0}.

Remark 4.2. Sinceλ−MΨ=Mλ−Ψ, the above result shows that the spectrum ofMΨ sat-isfiesσ(MΨ)=Ψ(D). This shows that the multiplication operatorMΨis not compact.

In [13], Axler has characterized the Fredholm multiplication operators on Bergman spaces. Then on the spacesHv∞(D), the Fredholm multiplication operators and closed range multiplication operators are characterized by [31]. Further, in [61], Cichon and Seip have proved the following theorem related to closed range multiplication operators, which was conjectured by [31].

Theorem 4.3. Letv∈C2(D) be a radial weight such that(1− |z2|)2Δlogv(z)+as

z →1−, whereΔdenotes the Laplacian. ThenMΨ:Hv∞(D)→Hv∞(D) has closed range if

and only ifΨ=hb, wherehis invertible inH∞andbis finite Blaschke product.

Further, Manhas has extended in [60] the characterizations of invertible multiplica-tion operators to the weighted spaces of vector-valued analytic funcmultiplica-tions. We begin with stating an invertibility criterion on a Hausdorfftopological vector space [62], which we have used for characterizing invertible multiplication operators on the spacesHVb(G,E).

Theorem 4.4. LetEbe a complete Hausdorfftopological vector space and letT:E→Ebe a continuous linear operator. ThenT is invertible if and only ifT is bounded below and has dense range. Or, letEbe a Hausdorfftopological vector space and letT:E→Ebe a continuous linear operator. ThenTis invertible if and only ifTis bounded below and onto.

In the above invertible criterion, a generalized definition of bounded below opera-tors on Hausdorfftopological vector spaces is used. Now, we give this definition as it is needed for proving some of the results of this section. A continuous linear operatorTon a Hausdorfftopological vector spaceEis said to be bounded below if for every neighbour-hoodNof the origin inE, there exists a neighbourhoodMof the origin inEsuch that

T(Nc)⊆Mc, where the symbolcstands for the complement of the neighborhood inE. We begin with the following proposition.

Proposition 4.5. LetVbe an arbitrary system of weights onGand letΨ:G→B(E) be an

analytic map such thatMΨis a multiplication operator onHVb(G,E). ThenMΨis invertible

if

(i) for eachz∈G,Ψ(z) :E→Eis onto;

(ii) for each v∈V, there exists u∈V such that v(z)y ≤u(z)Ψz(y), for every

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Proof. Fixz0∈G. Letv∈Vsuch thatv(z0)>0. Then by condition (ii), there existsu∈V

such thatv(z0)y ≤u(z0)Ψz0(y), for everyy∈E. That is,Ψz0(y) ≥λ0y, for every

y∈E, whereλ0=v(z0)/u(z0)>0. This proves thatΨ(z0) is bounded below on Eand hence by condition (i),Ψ(z0) is invertible inB(E). We denote the inverse of Ψ(z0) by

Ψ1

z0. Now, we defineΨ

1:GB(E) asΨ1(z)=Ψ1

z , for everyz∈G. Clearly,Ψ1is an analytic map. Again, by condition (ii), it follows that

v(z1

z (y)≤u(z)y, for everyz∈G,y∈E. (4.1)

That is,v(z)Ψ1(z)u(z), for everyzG. Thus according to Proposition 3.3,Ψ1

induces the multiplication operatorMΨ1onHVb(G,E) such thatMΨMΨ1=MΨ1MΨ=

I, the identity operator. HenceMΨis invertible.

Corollary 4.6. LetV be an arbitrary system of weights onGand letΨ:G→B(E) be a

bounded analytic map. ThenMΨis an invertible multiplication operator onHVb(G,E) if (i) for eachz∈G,Ψ(z) :E→Eis onto;

(ii)Ψis bounded away from zero.

Proof. SinceΨ:G→B(E) is a bounded analytic function,Proposition 3.3implies that

MΨ is a multiplication operator on HVb(G,E), for any arbitrary system of weightsV on G. By condition (ii), there existsλ >0 such thatΨz(y) ≥λy, for everyz∈G and y∈E. Let v∈V. Then v(z)y ≤(1)v(z)Ψz(y), for every z∈G and y∈E. Further, it implies that there exists u∈V such thatv(z)y ≤u(z)Ψz(y), for every

z∈Gandy∈E. Thus according toProposition 4.5, it follows thatMΨis invertible on

HVb(G,E).

The converse of the aboveCorollary 4.6may not be true. That is, if an analytic map

Ψ:G→B(E) is not bounded away from zero, even thenMΨis invertible on some of the weighted spacesHVb(G,E). This can be easily seen from the following corollary.

Corollary 4.7. LetV= {λχK:λ≥0 andK⊆G,Kcompact set}. Then every analytic map

Ψ:G→B(E) induces an invertible multiplication operator MΨ onHVb(G,E) ifΨ(z) is

invertible for everyz∈G.

Remark 4.8. Let G= {z∈C:z=x+iyandx >0} and let V = {λχK:λ≥0, K⊆G,

Kcompact set}. LetE=H∞(G) be the Banach space of bounded analytic functions on

G. We define an analytic mapΨ:G→B(E) asΨ(z)=Mz, forz∈G, where the bounded operatorMz:E→Eis defined asMzf =z f, for every f ∈E. Clearly, eachΨ(z) is invert-ible inB(E) and hence byCorollary 4.7,MΨis an invertible multiplication operator on

HVb(G,E). ButΨ:G→B(E) is not bounded away from zero. Also, we note that invert-ible multiplication operators on Bergman spaces of analytic functions [13] and weighted Banach spaces of analytic functions (seeProposition 4.1) are generated only by the func-tions which are bounded away from zero. Thus in general, the invertible behaviour is very much controlled by different systems of weightsVonG.

Theorem 4.9. LetV be a reasonable system of weights onGand letΨ:G→B(E) be an

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operator onHVb(G,E). ThenMΨis invertible if and only if (i) for eachz∈G,Ψ(z) :E→Eis onto;

(ii) for each v∈V, there exists u∈V such that v(z)y ≤u(z)Ψz(y), for every

z∈Gandy∈E.

Proof. If conditions (i) and (ii) hold, then fromProposition 4.5, it clearly follows thatMΨ is invertible.

Conversely, suppose thatMΨis invertible on HVb(G,E). To establish condition (i), letz0∈Gand let fz0∈HVb(G) such that fz0(z0)=1. Fix 0 =y0∈E. Define an analytic

mapgz0:G→Easgz0(z)=fz0(z)y0, for everyz∈G. Clearly,gz0∈HVb(G,E). SinceMΨis

onto, there exists f0∈HVb(G,E) such thatMΨ(f0)=gz0. That is,Ψz0(f0(z0))=y0. This

shows that eachΨ(z) is onto. Now, to prove condition (ii), we fixv∈V. Then there exists

v∈V such thatv≤v. In view ofTheorem 4.4, we conclude thatMΨis bounded below and onto. Further, it implies that there existsu∈V withu∈V such that MΨ(Bvc,E)

Bc

u,E. Now, we claim thatv(z)Ψ−z1(y) ≤u(z)y, for everyz∈G and y∈E. We fix

z0∈Gandy0∈E. According to (2.7c), there exists fz0∈Busuch that|fz0(z0)| =1/u(z0).

Letg0:G→Ebe defined asg0(z)=(1/y0)fz0(z)y0, for everyz∈G. Clearly,g0∈Bu,E

and g0(z0) =1/u(z0). Also, (2.7b) implies that fz0 ∈Bu and hence g0∈Bu,E. Since

MΨ is onto, there existsh0∈HVb(G,E) such that MΨ(h0)=g0. That is,Ψz0(h0(z0))=

g0(z0). Since eachΨz0is invertible, we haveh0(z0)=Ψz−01(y0)fz0(z0)1/y0. Again, since g0u,E≤1, we conclude thatMΨ(h0)∈/ Bcu,E. Further, it implies thath0∈/ Bvc,E. That is,

v(z)h0(z)1, for everyz∈G. In particular, for z=z0, we havev(z0)h0(z0)1. That is,v(z0)Ψ1

z0(y0) ≤u(z0)y0. This proves our claim. Since eachΨ(z) is

invert-ible, we havev(z)y ≤u(z)Ψz(y), for everyz∈Gandy∈E. This proves condition (ii). With this, the proof of the theorem is complete.

Corollary 4.10. LetV be an essential system of weights onG and letΨ:G→B(E) be

an analytic map such that eachΨ(z) is one-to-one andMΨis a multiplication operator on

HVb(G,E). ThenMΨis invertible if and only if (i) for eachz∈G,Ψ(z) :E→Eis onto;

(ii) for each v∈V, there exists u∈V such that v(z)y ≤u(z)Ψz(y), for every

z∈Gandy∈E.

Proof. Follows fromTheorem 4.9sinceV∼=V.

Theorem 4.11. LetV= {λχK:λ≥0, K⊆G, Kcompact set}and letΨ∈H(G) be

non-zero. ThenMΨis not a compact multiplication operator onHVb(G).

Proof. Suppose thatMΨis a compact multiplication operator onHVb(G). SinceCorollary 4.7 implies thatMΨ is invertible onHVb(G) if and only if Ψ(z) =0, for every z∈G, we conclude that the spectrum ofMΨ satisfiesσ(MΨ)=Ψ(G). Now, from [57, Theo-rem 4], it follows that each point in the spectrum of a compact operator on a locally convex Hausdorffspace is an isolated point which is a contradiction to the fact that

σ(MΨ)=Ψ(G) is a connected set. Thus there is no nonzero compact multiplication

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Now, we will characterize quasicompact multiplication operators on weighted Banach spaces of analytic functions. For this, we need to give some definitions: a continuous linear operatorSon a Banach spaceEis said to be quasicompact [63] if there exists an integernand a compact operatorK onEsuch thatSnK<1. The essential norm of a continuous linear operatorSon a Banach spaceEis defined bySe=inf{S−K:

Kcompact onE}. Clearly,Sis compact if and only ifSe=0. Thenth approximation number ofSis defined asan(S)=inf{S−Tn:Tnis bounded onE, rankTn≤n}. Now, it readily follows thatSe≤an(S)≤ S. For more details on the properties of the ap-proximation numbers, we refer to [63].

Theorem 4.12. Letvbe a continuous weight onDand letΨ∈H∞(D). Then the

multipli-cation operatorMΨonHvb(D) is quasicompact if and only ifΨ∞<1.

Proof. IfΨ∞<1, then by choosingKas the zero operator onHvb(D), it follows that Mn

Ψ0 = MΨn = Ψn<1. ThusMΨis quasicompact. Conversely, suppose thatMΨ is a quasicompact multiplication operator onHvb(D). Then there exists an integernand a compact operatorK onHvb(D) such thatMΨn−K<1. Now, from [31, Corollary 2.5], it follows thatMΨe= MΨ = Ψ∞=an(MΨ), for alln. Further, it implies that 1>Mn

Ψ−K ≥ MΨne= Ψn. ThusΨ<1.

5. Dynamical systems and multiplication operators

Letg∈H∞(G,B(E)) and letg=sup{g(z):zG}. Then for eachtR, we de-fine Ψt:G→B(E) asΨt(z)=etg(z), for everyz∈G. Clearly,Ψt is an operator-valued bounded analytic map and hence byProposition 3.3,MΨt is a multiplication operator on

HVb(G,E), for any arbitrary system of weightsV onG.

Theorem 5.1. LetV be an arbitrary system of weights onGand letΠ:R×HVb(G,E)

H(G,E) be defined asΠ(t,f)=MΨtf, for everyt∈Rand f ∈HVb(G,E). ThenΠ is a

linear dynamical system onHVb(G,E). Moreover, ifVis a system of weights onGsuch that

HVb(G,E) is completely metrizable, then the family= {MΨt :t∈R}is locally

equicon-tinuousC0-group of multiplication operators inB(HVb(G,E)).

Proof. We have already observed thatMΨtis a multiplication operator onHVb(G,E), for everyt∈R. Thus it follows thatΠ(t,f)∈HVb(G,E), for everyt∈Randf ∈HVb(G,E). Clearly, Πis linear and Π(0,f)= f, for every f ∈HVb(G,E). Again, it is easy to see thatΠ(t+s,f)=Π(t,Π(s,f)), for everyt,s∈Rand f ∈HVb(G,E). To show thatΠis a dynamical system, it is sufficient to prove thatΠis jointly continuous. Let{(,)}be a net inR×HVb(G,E) such that (,)(t,f) inR×HVb(G,E). Letv∈V.

Then

Π

,fα−Π(t,f)v,E

=Ψtαfα−Ψtfv,E=sup

v(z(z)(z)Ψt(z)f(z):z∈G

supv(z(z)Ψt(z)

f

α(z):z∈G

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supΨ(z)Ψt(z)v(z)(z):z∈G

+ supΨt(z)v(z)(z)−f(z):z∈G

supe|t|g∞e|tα−t|g∞1v(z)fα(z):zG

+ supe|t|g∞v(z)fα(z)f(z):zG

=e|t|g∞e|tα−t|g∞1fα

v,E

+e|t|g∞fαf

v,E−→0 as|tα−t| −→0,fα−fv,E−→0.

(5.1)

This shows thatΠis jointly continuous and henceΠis a (linear) dynamical system on

HVb(G,E). Further, it implies that the familyᏹis aCo-group of multiplication opera-tors on the weighted spacesHVb(G,E). Now, we will show that the familyᏹis locally equicontinuous inB(HVb(G,E)). For this, it is enough to see that for any fixeds∈R, the subfamilyᏹs= {MΨt:−s≤t≤s}is equicontinuous onHVb(G,E). Now, it is easy to see that the subfamilyᏹsis a bounded set inB(HVb(G,E)) because the mapt→MΨt is continuous in the strong operator topology. Also, for each f ∈HVb(G,E), the set

s(f)= {MΨtf :−s≤t≤s}is bounded in HVb(G,E). Thus according to a corollary of the Banach-Steinhaus theorem [58], it follows that the familyᏹis locally

equicontin-uous.

6. Characterizations of composition operators

Every self-analytic mapφ:DDinduces a composition operator on the Hardy space

H∞(D). But these maps do not necessarily induce composition operators on the weighted spaceHv∞(D), for general weightsv(see [32]). For example, consider the weightv(z)=

e−(1−|z|)1

, forz∈D. Thenv=v. Letφ:DDbe defined asφ(z)=(z+ 1)/2, for every

z∈D. Then forz=r∈R, we havev(z)/v(φ(z))=v(r)/v(φ(r))=re1/(1−r), for 0< r <1.

Then asr→1,v(r)/v(φ(r))→ ∞, sois not bounded onHv∞(D).

In this section, we give characterizations of composition operators on the weighted spaces of analytic functions. We begin with a characterization of composition operators obtained in [32] on the weighted Banach spaces of scalar-valued analytic functions.

Theorem 6.1. Letvandwbe continuous bounded weights. Then the following are equiva-lent:

(i):Hv∞(D)→Hw∞(D) is bounded; (ii) supzD(w(z)/v(φ(z)))<∞;

(iii) supzD(w(z)/v(φ(z)))<∞.

Ifvandware typical weights, then the above conditions are equivalent to

(iv):Hv0(D)→Hw0(D) is bounded.

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LetX be a complex Banach space andBX its open unit ball. Then clearly, the space

H∞

v (BX) (defined in the same way asHv∞(D)) is a Banach space. ByBv, we denote the closed unit ball ofHv∞(BX). It is well known that inHv∞(BX), theτv(norm) topology is finer than theτ0(compact-open) topology and thatBvisτ0-compact [64]. A weightv satisfies Condition-I if infx∈rBXv(x)>0, for every 0< r <1 [65]. Ifvsatisfies Condition-I, thenHv∞(BX)⊆H∞(BX) [65]. IfX is finite dimensional, then all weights onBX satisfy

Condition-I.

Now, we can present the extended version of the above theorem [53].

Theorem 6.2. LetBXandBYbe open unit balls of the Banach spacesXandY, respectively. Letwandvbe weights onBXandBY, respectively, satisfying Condition-I. Letφ:BX→BY

be a holomorphic map. Then the following are equivalent:

(i):Hv∞(BY)→Hw∞(BX) is bounded; (ii) supxBX(w(x)/v(φ(x)))<∞;

(iii) supxBX(w(x)/v(φ(x)))<∞;

(iv) supφ(x)>r0(w(x)/v(φ(x)))<∞, for some 0< r0<1.

Proof. (iii)⇒(ii) is obvious becausew≤w.

(ii)(i) let f ∈Hv(BY). Then we have w(x)|f(φ(x))| =(w(x)/v(φ(x)))v(φ(x))× |f(φ(x))| ≤Mfv=Mfv, for allx. Henceis bounded.

(i)(iii) if (iii) does not hold, then there exists a sequence (xn)n∈N⊆BX such that limn→∞w(xn)/v(φ(xn))=∞. Fixn∈N. Letfn∈Bvbe such that|fn(φ(xn))| =u(φ(xn))= 1/v(φ(xn)). Hence |fn(φ(xn))|w(xn)=w(xn)/v(φ(xn)), which is a contradiction to the fact thatis bounded.

(ii)(iv) is straightforward.

(iv)(i) letM=supφ(x)>r0(w(x)/v(φ(x))). Letx∈X. Ifφ(x)> r0, then we have

w(x)|f(φ(x))|=(w(x)/v(φ(x)))v(φ(x))|f(φ(x))| ≤Mfv. Ifφ(x) ≤r0, then we have

w(x)|f(φ(x))| ≤(supx∈BXw(x))(supx∈r0BY|f(y)|) becausef is bounded inr0BY. Thus we

have supx∈BXw(x)|f(φ(x))|<∞and(f)∈Hw∞(BX), for all f ∈Hv∞(BY). Henceis

bounded.

Remark 6.3. In the above theorem, the first three conditions are equivalent even if Condi-tion-I does not holds. On the other hand, in [53], Garcia et al. have given an example, which shows that Condition-I is necessary to prove that (iv) implies (i).

Also, Manhas [66] has further generalizedTheorem 6.1and related results of [32] to the general weighted spaces of analytic functions, which are given below.

Theorem 6.4. LetUandVbe arbitrary systems of weights onG. Letφ∈H(G) be such that

φ(G)⊆G. ThenCφ:HUb(G)→HVb(G) is a composition operator ifV≤U◦φ.

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map. For instance, if we take f(z)=1/z, for everyz∈G, then f ∈HU0(G) but(f)∈/

HV0(G). So, in order to show that:HU0(G)→HV0(G) is a composition operator, we need an additional condition onφ. Letv∈V andε >0. Then consider the setF(v,ε)= {z∈G; v(z)≥ε}. ClearlyF(v,ε) is a closed subset ofG. In the next theorem, we have obtained a sufficient condition forto be a composition operator fromHU0(G) into

HV0(G).

Theorem 6.6. LetUandVbe arbitrary systems of weights onG. Letφ∈H(G) be such that

φ(G)⊆G. ThenCφ:HU0(G)→HV0(G) is a composition operator if (i)V≤U◦φ;

(ii) for everyv∈V,ε >0, and compact setK⊆G, the setφ−1(K)F(v,ε) is compact.

Proof. In view ofTheorem 6.4, condition (i) implies that :HUb(G)→HVb(G) is a composition operator. To show that:HU0(G)→HV0(G) is a composition operator, it is enough to prove thatis an into map. Let f ∈HU0(G). Letv∈V andε >0. Then we consider the setK= {z∈G:v(z)|f(φ(z))| ≥ε}. We will show that K is a compact subset ofG. By condition (i), there existsu∈Usuch thatv(z)≤u(φ(z)), for everyz∈G. LetS= {z∈G:u(z)|f(z)| ≥ε}. Then clearly,Sis a compact subset ofGandφ(K)⊆S. LetM=sup{|f(z)|:z∈S}. ThenM >0 andS⊆F(u,ε/M). By condition (ii), the set

φ−1(S)F(v,ε/M) is compact. SinceK is a closed subset of the setφ1(S)F(v,ε/M), it

follows thatKis compact. Thus(f)∈HV0(G). This completes the proof.

Corollary 6.7. LetU andV be arbitrary systems of weights onG. Letφ∈H(G) be such

thatφ(G)⊆G. Then

(i):HUb(G)→HVb(G) is a composition operator ifV≤U◦φ;

(ii):HU0(G)→HV0(G) is a composition operator ifV≤U◦φandφis a

confor-mal mapping ofGonto itself.

The converse of the above corollary may not be true. That is, ifis a composition operator onHVb(G) andHV0(G), thenφ∈H(G) may not be conformal mapping ofG onto itself. For example, letV= {λχK:λ≥0,K⊆G,Kis compact}, then it can be easily seen thatis a composition operator onHV0(G) if and only ifφ:G→Gis an analytic map.

In the next theorem, Manhas [66] has obtained a necessary and sufficient condition forto be a composition operator onHVb(G) in terms of the inducing mapφand the system of weightsV.

Theorem 6.8. LetVbe an arbitrary system of weights onGand letUbe a reasonable system of weights onG. Letφ∈H(G) be such thatφ(G)⊆G. ThenCφ:HUb(G)→HVb(G) is a

composition operator if and only ifV≤U◦φ.

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for everyz∈G. In particular, forz=z0, we havev(z0)≤u(φ(z0)). This proves our claim and henceV≤U◦φ.

Conversely, suppose that the condition is true. To show that:HUb(G)→HVb(G) is a composition operator, it is sufficient to show thatis continuous at the origin. Let

v∈V andBvbe a neighbourhood of the origin inHVb(G). Then by the given condition, there existsu∈Uwithu∈Usuch thatv≤u◦φ. That is,v(z)≤u(φ(z)), for everyz∈G. Now, we claim that (Bu)⊆Bv. Let f ∈Bu. Then by (2.7b), fu≤1 if and only if fu≤1. Now,

Cφfv=sup

v(z)(z):z∈G

sup(z)(z):z∈G

supu(z)f(z):z∈G= fu≤1.

(6.1)

This proves thatCφf ∈Bvand henceis a composition operator. This completes the

proof of the theorem.

Corollary 6.9. LetUandVbe reasonable systems of weights onG. Letφ∈H(G) be such

thatφ(G)⊆G. Then the following statements are equivalent:

(i):HUb(G)→HVb(G) is a composition operator; (ii)V≤U◦φ;

(iii)V≤U◦φ.

Corollary 6.10. LetV be an arbitrary system of weights onGand letUbe an essential system of weights onG. Letφ∈H(G) be such thatφ(G)⊆G. ThenCφ:HUb(G)→HVb(G)

is a composition operator if and only ifV≤U◦φ.

Theorem 6.11. LetV be an arbitrary system of weights onG and letU be an essential system of weights onGsuch that each weight ofV andUvanishes at infinity. Letφ∈H(G)

be such thatφ(G)⊆G. ThenCφ:HU0(G)→HV0(G) is a composition operator if and only

ifV≤U◦φ.

Example 6.12. LetG=D, the open unit disc, and letv be a weight defined asv(z)= 1− |z|2, for everyzG. LetV = {λv:λ >0}. Then clearly,V is an essential system of

weights onG. Letφ:G→G be an analytic map defined byφ(z)=(z+ 1)/2, for every

z∈G. Now, by the Pick-Schwarz lemma, it follows that

1− |z|2φ(z)1φ(z)2

, for everyz∈G. (6.2)

That is,υ(z)2υ(φ(z)), for everz∈G. Hence byTheorem 6.4,is a composition op-erator onHVb(G).

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7. Compact and weakly compact composition operators

In [67], Aron et al. have characterized the compact composition operators on the Banach algebra of bounded analytic functions. This is recorded in the following theorem.

Theorem 7.1. LetCφ:H∞(BY)→H∞(BX) be a composition operator. Then the following

statements are equivalent:

(i)Cφis compact;

(ii)Cφis weakly compact andφ(BX) is relatively compact inY; (iii)φ(BX) lies strictly insideBYandφ(BX) is relatively compact inY.

Further, Galindo et al. [68] have obtained a characterization of weakly compact com-position operators in terms of the inducing mapφ:BX→BY, which is stated below.

Theorem 7.2. The Composition operatorCφ:H∞(BY)→H∞(BX) is weakly compact if (i)

φ(BX)⊂rBY, for some 0< r <1 and (ii)φ(BX) is relatively compact in (Y,σ(Y,P(Y))).

The converse holds, if moreover,Y has the approximation property. (By (Y,σ(Y,P(Y))),

we mean the spaceYendowed with the weakest topology, making allp∈P(Y) continuous,

whereP(Y) denote the algebra of all continuous polynomials onY.)

Recently, Garcia et al. [53] has obtained characterizations of compact composition operators on weighted Banach spaces of analytic functions, which generalizes the above Theorem 7.1 and [32, Theorem 3.3]. This generalization is presented in the following theorem.

Theorem 7.3. Letvandwbe weights onBYandBX, respectively, with limx→1−w(x)=0.

Letφ:BX→BYbe a holomorphic map. ThenCφ:Hv∞(BY)→Hw∞(BX) is compact if and only

if

(i) limx→1(w(x)/v(φ(x)))=0;

(ii)φ(rBX) is relatively compact, for every 0< r <1.

It has been observed in [32,50] that many weights do not satisfy this condition on the limit. In [32], Bonet et al. have characterized compact composition operators for general weights whenX=Y=C. This characterization is given in terms of an analytic condition (see (A) below in part (c)). Then by using a topological condition, Garcia et al. [53] have obtained a characterization of compact composition operators for general Banach spaces

XandY. This is presented in the following theorem.

Theorem 7.4. Letvandwbe weights onBY andBX, respectively, with Condition-I. Let

φ:BX→BYbe a holomorphic map. Then the following hold.

(a) IfCφ:Hv∞(BY)→Hw∞(BX) is compact, thenφ(rBX) is relatively compact, for every 0< r <1.

(b) Suppose thatφ∞<1. Ifφ(BX) is relatively compact, thenCφ:Hv∞(BY)→Hw∞(BX)

is compact.

(c) Suppose thatφ∞=1.

(i) IfCφ:Hv∞(BY)→Hw∞(BX) is compact, then

lim r→1 sup

φ(x)>r

w(x)

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(ii) Ifφ(BX)∩rBYis relatively compact, for every 0< r <1, and

lim r→1 sup

φ(x)>r

w(x)

(x)=0, (7.1)

thenCφ:Hv∞(BY)→H∞

w(BX) is compact.

In part (b) of the above theorem, if the spaceHw∞(BX) is replaced byH∞(BX), then the improved result is as follows.

Proposition 7.5. Letvbe a weight onY and Letφ:BX→BYbe a holomorphic map. Then

:Hv∞(BY)→H∞(BX) is compact if and only ifφ(BX) is relatively compact andφ∞<1.

Further, ifwis taken as a norm-radial weight (i.e.,w(x)=w(y), for everyx, y, such thatx = y), then the compactness ofis better and given in the following corollary.

Corollary 7.6. Letvandwbe weights onBY andBX, respectively, such thatwis

norm-radial. Letφ:BX→BYbe a holomorphic map. Then we have the following:

(a) Ifw(x)→0 asx →1−, thenCφ:Hv∞(BY)→Hw∞(BX) is compact if and only if

φ(rBX) is relatively compact, for every 0< r <1 and limx→1(w(x)/v(φ(x)))=0.

(b) Ifw(x)0 asx →1−, thenCφ:Hv∞(BY)→Hw∞(BX) is compact if and only if

φ(BX) is relatively compact andφ∞<1.

IfY is finite dimensional, thenφ(BX) is always relatively compact and in this case, the following corollary reduces to [32, Theorem 3.3], wheneverX=Y =C.

Corollary 7.7. LetYbe a finite dimensional Banach space andXa complex Banach space. Letv,wbe weights and letφ:BX→BYbe a holomorphic map. Then we have the following.

(a) Ifφ∞<1, thenCφ:Hv∞(BY)→H∞

w(BX) is compact.

(b) Ifφ∞=1, thenCφ:Hv∞(BY)→Hw∞(BX) is compact if and only if

lim r→1 sup

φ(x)>r

w(x)

(x)=0. (7.2)

Remark 7.8. Garcia et al. [53] have given examples to show that the converse of (a), (b), and (c)(i) or (c)(ii) inTheorem 7.4does not hold in general.

In [34], Bonet et al. have further generalizedTheorem 7.2to the Weighted Banach spaces of vector-valued analytic functions besides characterizing weakly compact com-position operators on vector-valued Hardy spaces, Bergman spaces, and Bloch spaces. Here, we present a vector-valued version ofTheorem 7.2.

Theorem 7.9. Let vbe an essential weight. Then :Hv∞(D,E)→Hw∞(D,E) is weakly

compact if and only if the Banach spaceEis reflexive and

lim r→1 sup

(z)|>r

v(z)

(z)=0 or φ∞<1. (7.3)

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HV(D,E) of vector-valued analytic functions, whereV= {vn}is an increasing sequence of strictly positive, radial, continuous, bounded weights and E is a complete, barralled locally convex space.

8. Composition operators and homomorphisms

In this section, we present a few results which relate homomorphisms with composition operators [66]. We will begin with a characterization of all continuous linear operators on

HVb(G), which are composition operators and this parallels a standard result for func-tional Hilbert spaces.

For eachz∈G, the point evaluation δz defines a continuous linear functional on

HVb(G). If we putΔ(G)= {δz :z∈G}, thenΔ(G) is a subset of the continuous dual

HVb(G).

Theorem 8.1. LetΦ:HVb(G)→HVb(G) be a linear transformation. Then there exists

φ:G→Gsuch thatΦ=Cφif and only if the transpose mappingΦfromHVb(G)∗into the

algebraic dualHVb(G)leavesΔ(G) invariant. In caseVis a reasonable system of bounded

weights onGandΦ(Δ(G))Δ(G),φis necessarily analytic andΦ=Cφis continuous if and only ifV≤Voφ .

Proof. Suppose thatΦ=Cφ, for someφ:G→G. Letz∈Gand f ∈HVb(G). Then

Φδz(f)=δz(f)=δzΦ(f)=δzCφf=fφ(z)=δφ

(z)(f). (8.1)

This implies thatΦδz=δφ(z). Conversely, let us suppose thatΦ(Δ(G))Δ(G). Forz∈ Gif we defineφ(z) to be the unique element ofGsuch thatΦδz=δφ(z). Letf ∈HVb(G).

Then

Φ(f)(z)=δzΦ(f)=δzoΦ(f)=Φδz(f)=δφ(z)(f)= f

φ(z)=Cφ(f)(z). (8.2)

ThusΦ=Cφ. Also, since the identity function f(z)=zbelongs toHVb(G) and the range ofis contained inH(G),φis necessarily an analytic map. Also, in view ofCorollary 6.9,

Φ=Cφis continuous whenV≤Voφ .

Theorem 8.2. LetGbe an open connected bounded subset ofCand letV be a system of bounded weights onGsuch thatV≤V2. LetΦ:HVb(G)Cbe a nonzero multiplicative

linear functional. Then there existsz0∈Gsuch thatΦ=δz0.

Proof. Letλ∈Cand letdenote the constant function(z), for everyz∈G. Since each weightv∈V is bounded, it follows that each constant functionKλ∈HVb(G). Let

Φ:HVb(G)Cbe a nonzero multiplicative linear functional. Then we haveΦ(K1)=

Φ(K1·K1)=Φ(K1)Φ(K1). That is,Φ(K1) is equal to zero or one. In caseΦ(K1)=0, it follows thatΦ(f)=Φ(f ·K1)=Φ(f)Φ(K1)=0, for every f ∈HVb(G). ThusΦ=0, a contradiction. This shows thatΦ(K1)=1. Further, it implies thatΦ()=Φ(Kλ·K1)=

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clearly, f ∈HVb(G). Now, we fixz0=Φ(f). We will show thatz0∈G. Suppose thatz0∈/

G. Then we define the functionhz0:G→Cashz0(z)=1/(z−z0), for everyz∈G. Again,

since each weightv∈V is bounded andGis a bounded domain, it follows thathz0

HVb(G). Also, from the definition ofhz0, we have (z−z0)hz0(z)=1, for everyz∈G. That

is, (f(z)−Kz0(z))hz0(z)=1, for everyz∈G. Thus (f −Kz0)hz0=K1. Further, it implies

thatΦ(K1)=Φ(f−Kz0)Φ(hz0)=(Φ(f)Φ(Kz0))Φ(hz0)=(z0−z0)Φ(hz0)=0, which is

a contradiction becauseΦ(K1)=1. This proves thatz0∈G. Now, letg∈HVb(G). Then we define the functionh:G→Cas

h(z)=hz0(z)

g(z)−gz0, forz =z0, h(z)=gz0, forz=z0. (8.3)

It can be easily seen thath∈HVb(G). Now, it readily follows that (f−Kz0)h=g−Kg(z0).

Further, we have Φ((f −Kz0)h)=Φ(g−Kg(z0)). That is, 0=Φ(g)Φ(Kg(z0)). Thus it

follows thatΦ(g)=g(z0)=δz0(g). This proves thatΦ=δz0. With this, the proof of the

theorem is complete.

Theorem 8.3. LetG1 andG2 be open connected bounded subsets ofC. LetV andU be sytems of bounded weights onG1 andG2, respectively, such thatV≤V2 andUU2. Let

Φ:HVb(G1)→HVb(G2) be a nonzero algebra homomorphism. Then there exists a

holo-morphic mapφ:G2→G1such thatΦ=Cφ.

Proof. SinceHVb(G1) and HVb(G2) contains constant functions, it follows thatK1∈

HVb(G1) andΦ(K1)=Φ(K1)·Φ(K1). Then using connectedness ofG2, we can conclude thatΦ(K1)=K1. Further, it implies thatΦ()=Kλ, for everyλ∈C. Now, letz0∈G2. Then defineδz0:HVb(G1)Casδz0(f)=(Φ f)(z0). Clearly,δz0is a multiplicative linear

functional onHVb(G1). Hence byTheorem 8.2, there existsα∈G1 such thatδz0(f)=

δα(f)= f(α), for every f ∈HVb(G1). Letg:G1→Cbe defined asg(z)=z, for every

z∈G1. Then clearly,g∈HVb(G1) andδz0(g)=g(α). Thus it follows that (Φg)(z0).

Let us defineφ=Φ(g). Thusφ:G2→G1is an analytic map such that (Φ f)(z0)=f(α)=

f(Φg)(z0)=(f oφ)(z0),z0∈G2. This shows thatΦ(f)=Cφ(f), for every f ∈HVb(G1). HenceΦ=Cφ. With this, the proof of the theorem is complete.

Theorem 8.4. LetGbe an open connected bounded subset ofCand letV be a system of bounded weights onGsuch thatV≤V2. Then a composition transformationCφonHVb(G)

is invertible if and only ifφ:G→Gis a conformal mapping.

Proof. Ifφis a conformal mapping, then obviously,is invertible onHVb(G). On the other hand, supposeAis the inverse of. Then we haveACφ=CφA=I. For f andg inHVb(G), we haveCφA(f g)= f g. Further, it implies thatA(f g)oφ= f g=(CφA f)· (CφAg)=(A f)oφ·(Ag)oφ=(A f·Ag). That is, (A(f g)−A f·Ag)oφ=0. Sinceis invertible,φis nonconstant and hence the range ofφis an open set. Thus it follows that

A(f g)=A f·Ag. According toTheorem 8.3, there exists an analytic mapψ:G→Gsuch thatA=Cψ. Let f(z)=z, for everyz∈G. Then f ∈HVb(G) and we have (CψCφf)(z)= (f oφoψ)(z)=(φoψ)(z), for everyz∈G. Also, (CφCψ)(z)=(f oψoφ)(z)=(ψoφ)(z), for everyz∈G. From this, we conclude thatφis invertible with an analytic inverse map as

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