Composition Operators and Multiplication Operators on Weighted Spaces of Analytic Functions

Let V be an arbitrary system of weights on an open connected subset G of CN (N ≥ 1) and let B(E) be the Banach algebra of all bounded linear operators on a Banach space E. LetHVb (G,E) andHV0 (G,E) be the weighted locally convex spaces of vector-valued analytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical spaces of analytic functions H(G) to the weighted spaces of analytic functions HVb (G,E) and HV0 (G,E).


Introduction
Multiplication operators (also known as multipliers) and composition operators, on different spaces of analytic functions, have been actively appearing in different areas of mathematical sciences like dynamical systems, theory of semigroups, isometries, and, in turn, the theory of weighted composition operators besides their role in the theory of operator algebras and operator spaces. Evard and Jafari [1] and Siskakis [2,3] have employed these operators to make a study of weighted composition semigroups and dynamical systems on Hardy Spaces. De Leeuw et al. [4] and Nagasawa [5] have described isometries of Hardy spaces H 1 (D) and H ∞ (D) as a product of multiplication operators and composition operators. Isometries on H p -spaces and Bergman spaces are very much related with multiplication operators and composition operators, and for details on these isometries, we refer to Forelli [6], Cambern and Jarosz [7], Kolaski [8], Mazur [9], and Lin [10]. In [11], Arveson has recently obtained Toeplitz C * -algebras and operator spaces associated with these multiplication operators on Hardy Spaces.
In [38], Contreras and Hernández-Díaz 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 operators 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 operators on the disc algebra and the Hardy space H ∞ (D). Also, recently, Montes-Rodríguez [44] and Contreras and Hernández-Díaz [35] have studied the behaviour of these operators on weighted Banach spaces of analytic functions. The applications of these operators 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 multiplication operators on the weighted spaces of analytic functions.

Weighted spaces of analytic functions
Let G be an open connected subset of C N (N ≥ 1) and let H(G,E) be the space of all vector-valued analytic functions from G into the Banach space E. Let V be a set of nonnegative upper semicontinuous functions on G. Then V is said to be directed upward, if for every pair u 1 ,u 2 ∈ V and λ > 0, there exists v ∈ V such that λu i ≤ v (pointwise on G), for i = 1,2. If V is directed upward and for each z ∈ G, there exists v ∈ V such that v(z) > 0, then we call V as an arbitrary system of weights on G. If U and V are two arbitrary systems of weights on G such that for each u ∈ U, there exists v ∈ V for which u ≤ v, then we write U ≤ V . If U ≤ V and V ≤ U, then we write U ∼ = V . Let V be an arbitrary system of weights on G. Then we define (2.1) For v ∈ V and f ∈ H(G,E), we define Clearly, the family { · v,E : v ∈ V } of seminorms defines a Hausdorff locally convex topology on each of theses spaces: HV b (G,E) and HV 0 (G,E). With this topology, the spaces HV b (G,E) and HV 0 (G,E) are called the weighted locally convex spaces of vectorvalued analytic functions. These spaces have a basis of closed absolutely convex neighbourhoods of the form If E = C, then we write HV b (G,E) = HV b (G), HV 0 (G,E) = HV 0 (G) and Throughout the paper, we assume for each z ∈ G, there exists f z ∈ HV 0 (G) such that f z (z) = 0.
If v : D → R + is a continuous weight and E = C, then the corresponding weighted Banach spaces of analytic functions are defined as follows: Now, using the definitions of weights given in [32,[49][50][51], we give definitions of some systems of weights which are required for characterizing some results in the remaining sections.
Let V be an arbitrary system of weights on G and let v ∈ V . Then define w : G → R + as , for every z ∈ G. (2.6) In case w(z) = 0, v is an upper semicontinuous, and we call it an associated weight of v. Let V denote the system of all associated weights of V . Then an arbitrary system of weights V is called a reasonable system as it satisfies the following properties: if v ∈ V , then for every z ∈ G, there exists Then v is called essential if there exists a constant λ > 0 such that v(z) ≤ v(z) ≤ λv(z), for each z ∈ G. A reasonable system of weights V is called an essential system if each v ∈ V is an essential weight. If V is an essential system of weights, then we have 4 International Journal of Mathematics and Mathematical Sciences V ∼ = V . For example, let G = D, the open unit disc, and let f ∈ H(D) be nonzero. Then define v f (z) = [sup{| f (z)| : |z| = r}] −1 , for every z ∈ D. Clearly, each v f is a weight satisfying v f = v f , and the family V = {v f : f ∈ H(D), f is nonzero} is an essential system of weights on D. For more details on these weights, we refer to [50]. Let G be any balanced (i.e., λz ∈ G, whenever z ∈ G and λ ∈ C with |λ| ≤ 1) open subset of C N (N ≥ 1). Then a weight v ∈ V is called radial and typical if v(z) = v(λz) for all z ∈ G and λ ∈ C with |λ| = 1, and vanishes at the boundary ∂G. In particular, a weight v on D is radial and typical if v(z) = v(|z|) and lim |z|→1 v(z) = 0. For instance, in [35], it is shown that v p (z) = (1 − |z| 2 ) p , (0 < p < ∞), for every z ∈ D, are essential typical weights. For more details on the weighted Banach spaces of analytic functions and the weighted locally convex spaces of analytic functions associated with these weights, we refer to [32,[49][50][51][52][53][54]. For basic definitions and facts in complex analysis and functional analysis, we refer to [55][56][57][58].
Let F(G,E) be a topological vector space of vector-valued analytic functions from G into E, and let L(G,E) be the vector space of all vector-valued functions from G 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-map φ : G → G, we define the linear map In case W Ψ,φ takes F(G,E) into itself and is continuous, we call W Ψ,φ , the weighted composition operator on F(G,E) induced by the symbols Ψ and φ. If Ψ(z) = I, the indentity operator on E for every z ∈ G, then W Ψ , φ is called the composition operator induced by φ and we denote it by C φ . In case φ(z) = z for every z ∈ G, W Ψ,φ is called the multiplication operator induced by Ψ, and we denote it by M Ψ .

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.
is bounded and both v and w are radial weights vanishing on the boundary, then Proof. It is shown by [51,59] . In [32], it is observed that the evaluation functional δ z : . Now, we present the generalizations of the above characterizations to the weighted spaces of vector-valued analytic functions for general systems of weights, which was obtained by Manhas in [60].
Remark 3.4. Proposition 3.3 makes it clear that every bounded analytic function Ψ : G → B(E) induces the multiplication operator M Ψ on HV b (G,E), for any system of weights V on G. Also, if V = {λχ K : λ ≥ 0, K ⊆ G, K compact set}, then every operator-valued analytic map Ψ : G → B(E) induces a multiplication operator M Ψ on HV b (G,E). This makes it clear that even unbounded analytic operator-valued mappings generate multiplication operators on some of weighted locally convex spaces HV b (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 functions H ∞ v (D) defined by a single continuous weights (see Proposition 3.1). Thus the behaviour of the multiplication operators on the weighted locally convex spaces of analytic functions is very much influenced by different systems of weights V on G.
Theorem 3.5. Let V be an arbitrary system of weights and U a reasonable system of weights on G. Let Ψ : G → B(E) be an analytic map. Then Proof. The sufficient part follows from Proposition 3.3. Conversely, suppose M Ψ :H U b (G, E) → HV b (G,E) is a multiplication operator. Let v ∈ V . Then by the continuity of M Ψ at the origin, there exists u ∈ U with u ∈ U such that u ≤ u and M Ψ (B u,E ) ⊆ B v,E . To establish the inequality v(z) Ψ(z) ≤ u(z) for every z ∈ G, it is enough to prove that v(z) Ψ z (y) ≤ u(z) y , for every z ∈ G and y ∈ E. Fix z 0 ∈ G and y 0 ∈ E. Then by (2.7c), there exists f z0 ∈ B u such that f z0 (z 0 ) = 1/ u(z 0 ). Let g 0 : G → E be defined as Clearly, g 0 ∈ B u,E and g 0 (z 0 ) = 1/ u(z 0 ). Also, according to (2.7b), f z0 ∈ B u and therefore g 0 ∈ B u,E . Thus it follows that M Ψ (g 0 ) ∈ B v,E . That is, v(z) Ψ z (g 0 (z)) ≤ 1, for every z ∈ G. In particular, for z = z 0 , we have v(z 0 ) Ψ z0 (y 0 ) ≤ u(z 0 ) y 0 . This completes the proof of the theorem.
Corollary 3.6. Let V be an arbitrary system of weights and let U be an essential system of weights on G. Let Ψ : G → B(E) be an operator-valued analytic map. Then M Ψ : Proof. It follows from Theorem 3.5 and from the relation that U ∼ = U.
Remark 3.7. All the results proved above are also hold for the spaces HV 0 (G,E) and HU 0 (G,E).

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 invertible multiplication operators on the weighted Banach spaces of scalar-valued analytic functions [31].
. This shows that the multiplication operator M Ψ is not compact.
In [13], Axler has characterized the Fredholm multiplication operators on Bergman spaces. Then on the spaces H ∞ v (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].
Further, Manhas has extended in [60] the characterizations of invertible multiplication operators to the weighted spaces of vector-valued analytic functions. We begin with stating an invertibility criterion on a Hausdorff topological vector space [62], which we have used for characterizing invertible multiplication operators on the spaces HV b (G,E).

Theorem 4.4. Let E be a complete Hausdorff topological vector space and let T : E → E be a continuous linear operator. Then T is invertible if and only if T is bounded below and has dense range. Or, let E be a Hausdorff topological vector space and let T : E → E be a continuous linear operator. Then T is invertible if and only if T is bounded below and onto.
In the above invertible criterion, a generalized definition of bounded below operators on Hausdorff topological 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 operator T on a Hausdorff topological vector space E is said to be bounded below if for every neighbourhood N of the origin in E, there exists a neighbourhood M of the origin in E such that T(N c ) ⊆ M c , where the symbol c stands for the complement of the neighborhood in E. We begin with the following proposition.
Proposition 4.5. Let V be an arbitrary system of weights on G and let Ψ : Corollary 4.6. Let V be an arbitrary system of weights on G and let Ψ : , for every z ∈ G and y ∈ E. Thus according to Proposition 4.5, it follows that M Ψ is invertible on The converse of the above Corollary 4.6 may not be true. That is, if an analytic map Ψ : G → B(E) is not bounded away from zero, even then M Ψ is invertible on some of the weighted spaces HV b (G,E). This can be easily seen from the following corollary.
is not bounded away from zero. Also, we note that invertible multiplication operators on Bergman spaces of analytic functions [13] and weighted Banach spaces of analytic functions (see Proposition 4.1) are generated only by the functions which are bounded away from zero. Thus in general, the invertible behaviour is very much controlled by different systems of weights V on G.
Theorem 4.9. Let V be a reasonable system of weights on G and let Ψ : G → B(E) be an operator-valued analytic map such that each Ψ(z) is one-to-one and M Ψ is a multiplication Proof. If conditions (i) and (ii) hold, then from Proposition 4.5, it clearly follows that M Ψ is invertible.
z0 (y 0 ) ≤ u(z 0 ) y 0 . This proves our claim. Since each Ψ(z) is invertible, we have v(z) y ≤ u(z) Ψ z (y) , for every z ∈ G and y ∈ E. This proves condition (ii). With this, the proof of the theorem is complete.
Proof. Follows from Theorem 4.9 since V ∼ = V . 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 operator S on a Banach space E is said to be quasicompact [63] if there exists an integer n and a compact operator K on E such that S n − K < 1. The essential norm of a continuous linear operator S on a Banach space E is defined by S e = inf{ S − K : K compact on E}. Clearly, S is compact if and only if S e = 0. The nth approximation number of S is defined as a n (S) = inf{ S − T n : T n is bounded on E, rank T n ≤ n}. Now, it readily follows that S e ≤ a n (S) ≤ S . For more details on the properties of the approximation numbers, we refer to [63].

Dynamical systems and multiplication operators
Let g ∈ H ∞ (G,B(E)) and let g ∞ = sup{ g(z) : z ∈ G}. Then for each t ∈ R, we define Ψ t : G → B(E) as Ψ t (z) = e tg(z) , for every z ∈ G. Clearly, Ψ t is an operator-valued bounded analytic map and hence by Proposition 3.3, M Ψt is a multiplication operator on HV b (G,E), for any arbitrary system of weights V on G. Proof. We have already observed that M Ψt is a multiplication operator on HV b (G,E), for every t ∈ R. Thus it follows that Π(t, f ) ∈ HV b (G,E), for every t ∈ R and f ∈ HV b (G,E). Clearly, Π is linear and Π(0, f ) = f , for every f ∈ HV b (G,E). Again, it is easy to see that Π(t + s, f ) = Π(t,Π(s, f )), for every t,s ∈ R and f ∈ HV b (G,E). To show that Π is a dynamical system, it is sufficient to prove that Π is jointly continuous.
10 International Journal of Mathematics and Mathematical Sciences This shows that Π is jointly continuous and hence Π is a (linear) dynamical system on HV b (G,E). Further, it implies that the family ᏹ is a C o -group of multiplication operators on the weighted spaces HV b (G,E). Now, we will show that the family ᏹ is locally equicontinuous in B(HV b (G,E)). For this, it is enough to see that for any fixed s ∈ R, the subfamily ᏹ s = {M Ψt : −s ≤ t ≤ s} is equicontinuous on HV b (G,E). Now, it is easy to see that the subfamily ᏹ s is a bounded set in B(HV b (G,E)) because the map t → M Ψt is continuous in the strong operator topology. Also, for each f ∈ HV b (G,E), the set E). Thus according to a corollary of the Banach-Steinhaus theorem [58], it follows that the family ᏹ is locally equicontinuous.

Characterizations of composition operators
Every self-analytic map φ : D → D induces a composition operator on the Hardy space H ∞ (D). But these maps do not necessarily induce composition operators on the weighted space H ∞ v (D), for general weights v (see [32]). For example, consider the weight v(z) = e −(1−|z|) −1 , for z ∈ D. Then v = v. Let φ : D → D be defined as φ(z) = (z + 1)/2, for every z ∈ D. Then for z = r ∈ R, we have v(z)/v(φ(z)) = v(r)/v(φ(r)) = re 1/(1−r) , for 0 < r < 1. Then as r → 1, v(r)/v(φ(r)) → ∞, so C φ is not bounded on H ∞ v (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. Let v and w be continuous bounded weights. Then the following are equivalent: If v and w are typical weights, then the above conditions are equivalent to is bounded. Further, Garcia et al. [53] have generalized the above characterization to the weighted Banach spaces of scalar-valued analytic functions defined on the open unit ball of a Banach space. For presenting this generalization, we need to fix some definitions and notations.
Let X be a complex Banach space and B X its open unit ball. Then clearly, the space H ∞ v (B X ) (defined in the same way as H ∞ v (D)) is a Banach space. By B v , we denote the closed unit ball of H ∞ v (B X ). It is well known that in H ∞ v (B X ), the τ v (norm) topology is finer than the τ 0 (compact-open) topology and that B v is τ 0 -compact [64]. A weight v satisfies Condition-I if inf x∈rBX v(x) > 0, for every 0 < r < 1 [65]. If v satisfies Condition-I, [65]. If X is finite dimensional, then all weights on B X satisfy Condition-I. Now, we can present the extended version of the above theorem [53].
, which is a contradiction to the fact that C φ is bounded.
Remark 6.3. In the above theorem, the first three conditions are equivalent even if Condition-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 generalized Theorem 6.1 and related results of [32] to the general weighted spaces of analytic functions, which are given below.
Remark 6.5. The condition V ≤ U • φ in the above theorem is not a sufficient condition for C φ to be a composition operator from HU 0 (G) → HV 0 (G). For instance, let G = {z ∈ C : z = x + iy, x > 0} be the right half plane. Let U = V be the system of constant weights on G. Let φ : G → G be defined as φ(z) = z 0 , for every z ∈ G, where z 0 ∈ G is fixed. Then, clearly, the inequality V ≤ U • φ is true. But C φ : HU 0 (G) → HV 0 (G) is not even an into 12 International Journal of Mathematics and Mathematical Sciences map. For instance, if we take f (z) = 1/z, for every z ∈ G, then f ∈ HU 0 (G) but C φ ( f ) / ∈ HV 0 (G). So, in order to show that C φ : HU 0 (G) → HV 0 (G) is a composition operator, we need an additional condition on φ. Let v ∈ V and ε > 0. Then consider the set F(v,ε) = {z ∈ G; v(z) ≥ ε}. Clearly F(v,ε) is a closed subset of G. In the next theorem, we have obtained a sufficient condition for C φ to be a composition operator from HU 0 (G) into HV 0 (G).
Theorem 6.6. Let U and V be arbitrary systems of weights on G.
Proof. In view of Theorem 6.4, condition (i) implies that C φ : HU b (G) → HV b (G) is a composition operator. To show that C φ : HU 0 (G) → HV 0 (G) is a composition operator, it is enough to prove that C φ is an into map. Let f ∈ HU 0 (G). Let v ∈ V and ε > 0. Then we consider the set . This completes the proof.
Corollary 6.7. Let U and V be arbitrary systems of weights on G.
The converse of the above corollary may not be true. That is, if C φ is a composition operator on HV b (G) and HV 0 (G), then φ ∈ H(G) may not be conformal mapping of G onto itself. For example, let V = {λχ K : λ ≥ 0, K ⊆ G, K is compact}, then it can be easily seen that C φ is a composition operator on HV 0 (G) if and only if φ : G → G is an analytic map.
In the next theorem, Manhas [66] has obtained a necessary and sufficient condition for C φ to be a composition operator on HV b (G) in terms of the inducing map φ and the system of weights V . Theorem 6.8. Let V be an arbitrary system of weights on G and let U be a reasonable system of weights on G.
Proof. Suppose that C φ : HU b (G) → HV b (G) is a composition operator. Let v ∈ V . Then by the continuity of C φ at the origin, there exists u ∈ U and a neighbourhood B u of the origin in HU b (G) such that C φ (B u ) ⊆ B v . Let u be the associated weight of u. Then u ∈ U. Now, we claim that v ≤ u • φ. Fix z 0 ∈ G. Then by (2.7c), there exists f 0 ∈ B u such that | f 0 (φ(z 0 ))| = 1/ u(φ(z 0 )). Further, it implies that C φ ( f 0 ) ∈ B v . That is, v(z)| f 0 (φ(z))| ≤ 1, for every z ∈ G. In particular, for z = z 0 , we have v(z 0 ) ≤ u(φ(z 0 )). This proves our claim and hence V ≤ U • φ.
Conversely, suppose that the condition is true. To show that C φ : HU b (G) → HV b (G) is a composition operator, it is sufficient to show that C φ is continuous at the origin. Let v ∈ V and B v be a neighbourhood of the origin in HV b (G). Then by the given condition, Then by (2.7b), f u ≤ 1 if and only if f u ≤ 1. Now, This proves that C φ f ∈ B v and hence C φ is a composition operator. This completes the proof of the theorem. Corollary 6.9. Let U and V be reasonable systems of weights on G. Let φ ∈ H(G) be such that φ(G) ⊆ G. Then the following statements are equivalent: Corollary 6.10. Let V be an arbitrary system of weights on G and let U be an essential system of weights on G. Let φ ∈ H(G) be such that φ(G) ⊆ G. Then C φ : is a composition operator if and only if V ≤ U • φ.
Theorem 6.11. Let V be an arbitrary system of weights on G and let U be an essential system of weights on G such that each weight of V and U vanishes at infinity. Let φ ∈ H(G) be such that φ(G) ⊆ G. Then C φ : HU 0 (G) → HV 0 (G) is a composition operator if and only if V ≤ U • φ.
Example 6.12. Let G = D, the open unit disc, and let v be a weight defined as v(z) = 1 − |z| 2 , for every z ∈ G. Let V = {λv : λ > 0}. Then clearly, V is an essential system of weights on G. Let φ : G → G be an analytic map defined by φ(z) = (z + 1)/2, for every z ∈ G. Now, by the Pick-Schwarz lemma, it follows that That is, υ(z) ≤ 2υ(φ(z)), for ever z ∈ G. Hence by Theorem 6.4, C φ is a composition operator on HV b (G).

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.
be a composition operator. Then the following statements are equivalent: Further, Galindo et al. [68] have obtained a characterization of weakly compact composition operators in terms of the inducing map φ : B X →B Y , which is stated below.
The converse holds, if moreover, Y has the approximation property. (By (Y ,σ(Y ,P(Y ))), we mean the space Y endowed with the weakest topology, making all p ∈ P(Y ) continuous, where P(Y ) denote the algebra of all continuous polynomials on Y .) 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.
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 when X = 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 X and Y . This is presented in the following theorem.
Theorem 7.4. Let v and w be weights on B Y and B X , respectively, with Condition-I. Let φ : B X →B Y be a holomorphic map. Then the following hold.
, then the improved result is as follows.
Proposition 7.5. Let v be a weight on Y and Let φ : B X →B Y be a holomorphic map. Then is relatively compact and φ ∞ < 1. Further, if w is taken as a norm-radial weight (i.e., w(x) = w(y), for every x, y, such that x = y ), then the compactness of C φ is better and given in the following corollary.
Corollary 7.6. Let v and w be weights on B Y and B X , respectively, such that w is normradial. Let φ : B X →B Y be a holomorphic map. Then we have the following: If Y is finite dimensional, then φ(B X ) is always relatively compact and in this case, the following corollary reduces to [32,Theorem 3.3], whenever X = Y = C.
Corollary 7.7. Let Y be a finite dimensional Banach space and X a complex Banach space.
Let v, w be weights and let φ : B X →B Y be a holomorphic map. Then we have the following.

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 HV b (G), which are composition operators and this parallels a standard result for functional Hilbert spaces. For each z ∈ G, the point evaluation δ z defines a continuous linear functional on HV b (G). If we put Δ(G) = {δ z : z ∈ G}, then Δ(G) is a subset of the continuous dual HV b (G) * .
This implies that Φ δ z = δ φ(z) . Conversely, let us suppose that Φ (Δ(G)) ⊂ Δ(G). For z ∈ G if we define φ(z) to be the unique element of G such that Φ δ z = δ φ(z) . Let f ∈ HV b (G). Then Thus Φ = C φ . Also, since the identity function f (z) = z belongs to HV b (G) and the range of C φ is contained in H(G), φ is necessarily an analytic map. Also, in view of Corollary 6.9, Proof. Let λ ∈ C and let K λ denote the constant function K λ (z) = λ, for every z ∈ G. Since each weight v ∈ V is bounded, it follows that each constant function K λ ∈ HV b (G). Let Φ : HV b (G) → C be a nonzero multiplicative linear functional. Then we have Φ(K 1 ) = Φ(K 1 · K 1 ) = Φ(K 1 )Φ(K 1 ). That is, Φ(K 1 ) is equal to zero or one. In case Φ(K 1 ) = 0, it follows that Φ( f ) = Φ( f · K 1 ) = Φ( f )Φ(K 1 ) = 0, for every f ∈ HV b (G). Thus Φ = 0, a contradiction. This shows that Φ(K 1 ) = 1. Further, it implies that Φ(K λ ) = Φ(K λ · K 1 ) = Φ(λ · K 1 ) = λΦ(K 1 ) = λ. Let f : G → C be defined as f (z) = z, for every z ∈ G. Then J. S. Manhas 17 clearly, f ∈ HV b (G). Now, we fix z 0 = Φ( f ). We will show that z 0 ∈ G. Suppose that z 0 / ∈ G. Then we define the function h z0 : G → C as h z0 (z) = 1/(z − z 0 ), for every z ∈ G. Again, since each weight v ∈ V is bounded and G is a bounded domain, it follows that h z0 ∈ HV b (G). Also, from the definition of h z0 , we have (z − z 0 )h z0 (z) = 1, for every z ∈ G. That is, which is a contradiction because Φ(K 1 ) = 1. This proves that z 0 ∈ G. Now, let g ∈ HV b (G). Then we define the function h : G → C as It can be easily seen that h ∈ HV b (G). Now, it readily follows that ( ). Thus it follows that Φ(g) = g(z 0 ) = δ z0 (g). This proves that Φ = δ z0 . With this, the proof of the theorem is complete.
Hence Φ = C φ . With this, the proof of the theorem is complete. Proof. If φ is a conformal mapping, then obviously, C φ is invertible on HV b (G). On the other hand, suppose A is the inverse of C φ . Then we have AC φ = C φ A = I. For f and g in HV b (G), we have C φ A( f g) = f g. Further, it implies that A( f g)oφ = f g = (C φ A f ) · (C φ Ag) = (A f )oφ · (Ag)oφ = (A f · Ag)oφ. That is, (A( f g) − A f · Ag)oφ = 0. Since C φ is invertible, φ is nonconstant and hence the range of φ is an open set. Thus it follows that A( f g) = A f · Ag. According to Theorem 8.3, there exists an analytic map ψ : G → G such that A = C ψ . Let f (z) = z, for every z ∈ G. Then f ∈ HV b (G) and we have (C ψ C φ f )(z) = ( f oφoψ)(z) = (φoψ)(z), for every z ∈ G. Also, (C φ C ψ )(z) = ( f oψoφ)(z) = (ψoφ)(z), for every z ∈ G. From this, we conclude that φ is invertible with an analytic inverse map as ψ. Hence φ is a conformal mapping of G onto itself.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable: Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009