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J. Korean Math. Soc. 45 (2008), No. 5, pp. 1203–1220

WEIGHTED COMPOSITION OPERATORS ON WEIGHTED SPACES OF VECTOR-VALUED ANALYTIC FUNCTIONS Jasbir Singh Manhas

Reprinted from the Journal of the Korean Mathematical Society Vol. 45, No. 5, September 2008 c °2008 The Korean Mathematical Society

J. Korean Math. Soc. 45 (2008), No. 5, pp. 1203–1220

WEIGHTED COMPOSITION OPERATORS ON WEIGHTED SPACES OF VECTOR-VALUED ANALYTIC FUNCTIONS Jasbir Singh Manhas Abstract. 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. Let HVb (G, E) and HV0 (G, E) be the weighted locally convex spaces of vector-valued analytic functions. In this paper, we characterize self-analytic mappings φ : G → G and operator-valued analytic mappings Ψ : G → B (E) which generate weighted composition operators and invertible weighted composition operators on the spaces HVb (G, E) and HV0 (G, E) for different systems of weights V on G. Also, we obtained compact weighted composition operators on these spaces for some nice classes of weights.

1. Introduction Weighted composition operators have been appearing in a natural way on different spaces of analytic functions. For example, De Leeuw, Rudin and Wermer [12] and Nagasawa [27] have shown that the isometries of Hardy spaces H 1 (D) and H ∞ (D) are weighted composition operators whereas Forelli [13] has shown the same behaviour on the Hardy spaces H p for 1 < p < ∞, p 6= 2. Also, the same type of results were obtained by Kolaski [17] and Mazur [24] on Bergman spaces. These results have further been extended to the spaces of vector-valued analytic functions by Cambern and Jarosz [6] and Lin [18]. In recent years, the theory of weighted composition operators on spaces of analytic functions is gaining importance as it includes two nice classes of operators such as composition operators and multiplication operators and a very less information is known about weighted composition operators as compared to the theory of composition operators which has been presented in three monographs (see Cowen and MacCluer [11], Shapiro [33] and Singh and Manhas [34]). Received January 11, 2006. 2000 Mathematics Subject Classification. Primary 47B37, 47B38, 47B07, 46E40, 46E10; Secondary 47D03, 37B05, 32A10, 30H05. Key words and phrases. system of weights, Banach algebra, weighted locally convex spaces of vector-valued analytic functions, weighted composition operators, invertible and compact operators. The research of the author was supported by SQU grant No. IG/SCI/DOMS/07/02. c °2008 The Korean Mathematical Society

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Contreras and Hernandez-Diaz [9] have made a study of weighted composition operators on Hardy spaces whereas Mirzakarimi and Siddighi [25] have studied these operators on Bergman and Dirichlet spaces. On Bloch-type spaces, these operators are explored by MacCluer and Zhao [21], Ohno [28], Ohno and Zhao [31] and Ohno, Stroethoff and Zhao [29]. Also Ohno and Takagi [30] have obtained a study of these operators on the disc algebra and the Hardy space H ∞ (D). Recently, Montes-Rodriguez [26] and Contreras and Hernandez-Diaz [8] have obtained some nice properties 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 [16] and [36]). We have organized this paper into six sections. The preliminaries required for proving the results in the remaining sections are presented in Section 2. In Section 3, we characterize the weighted composition operators on the weighted locally convex spaces HVb (G, E) and HV0 (G, E) of vector-valued analytic functions generalizing the boundedness results of Contreras and Hernandez-Diaz [8], Monte-Rodriguez [26] and Manhas [23] obtained on the spaces of scalar-valued analytic functions. We have investigated the invertibility of these operators on the spaces HVb (G, E) and HV0 (G, E) in Section 4 which generalize the results of Ohno and Takagi [30] obtained on the Hardy space H ∞ (D). In Section 5, we have presented some results on compact weighted composition operators on weighted Banach spaces of vector-valued analytic functions generalizing some of the results obtained in [7] and [8]. Finally, in Section 6, we have given some examples to illustrate the theory. 2. Preliminaries Let G be an open connected subset of CN (N ≥ 1) and 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 non-negative upper semicontinuous functions on G. Then V is said to be directed upward if for every pair u1 , u2 ∈ V and λ > 0, there exists v ∈ V such that λui ≤ 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 HVb (G, E) = {f ∈ H(G, E) : vf (G) is bounded in E for each v ∈ V } and HV0 (G, E) = {f ∈ H(G, E) : vf vanishes at infinity on G for each v ∈ V } . For v ∈ V and f ∈ H(G, E), we define kf kv,E = sup{v(z) kf (z)k : z ∈ G}.

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Clearly the family {k·kv,E : v ∈ V } of seminorms defines a Hausdorff locally convex topology on each of theses spaces HVb (G, E) and HV0 (G, E). With this topology the spaces HVb (G, E) and HV0 (G, E) are called the weighted locally convex spaces of vector-valued analytic functions. These spaces have a basis of closed absolutely convex neighborhoods of the form Bv,E = {f ∈ HVb (G, E) (resp. HV0 (G, E)) : kf kv,E ≤ 1}. If E = C, then we write HVb (G, E) = HVb (G), HV0 (G, E) = HV0 (G) and Bv = {f ∈ HVb (G) (resp. HV0 (G)) : kf kv ≤ 1}. Throughout the paper, we assume for each z ∈ G, there exists fz ∈ HV0 (G) such that fz (z) 6= 0. Now using the definitions of weights given in ([2], [4], [5]), 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 1 w(z) ˜ = sup {|f (z)| : kf kv ≤ 1} ≤ v(z) and 1 v˜(z) = for every z ∈ G. w(z) ˜ In case w(z) ˜ 6= 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 if it satisfies the following properties: (2.a) for each v ∈ V, there exists v˜ ∈ V˜ such that v ≤ v˜; (2.b)

for each v ∈ V, kf kv ≤ 1 if and only if kf kv˜ ≤ 1, for every f ∈ HVb (G) ;

(2.c)

if v ∈ V, then for every z ∈ G, there exists 1 fz ∈ Bv such that |fz (z)| = . v˜ (z) Let v ∈ V. 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 V ∼ = V˜ . For example, let G = D, the open unit disc and let f ∈ H(D) be non-zero. Then define vf (z) = [sup{|f (z)| : |z| = r}]−1 for every z ∈ D. Clearly each vf is a weight satisfying v˜f = vf and the family V = {vf : f ∈ H(D), f is non-zero} is an essential system of weights on D. For more details on these weights, we refer to [2]. Let G be any balanced (i.e., λz ∈ G, whenever z ∈ G and λ ∈ C with |λ| ≤ 1) open subset of CN (N ≥ 1). Then a weight v ∈ V is called radial and typical if v(z) = v(λz) for all z ∈ G

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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 2 instance in [8], it is shown that vp (z) = (1−|z| )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 [1], [2], [3], [4], [5], [14], [19], [20]. For basic definitions and facts in complex analysis and functional analysis, we refer to [10], [15], [32]. 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 WΨ,φ : F (G, E) → L(G, E) as WΨ,φ (f ) = Ψ · f ◦ φ, for every f ∈ F (G, E) , where the product Ψ · f ◦ φ is defined pointwise on G as (Ψ · f ◦ φ) (z) = Ψz (f (φ(z))) for every z ∈ G. In case WΨ,φ takes F (G, E) into itself and is continuous, we call WΨ,φ , the weighted composition operator on F (G, E) induced by the symbols Ψ and φ. If Ψ (z) = I, the identity operator on E for every z ∈ G, then WΨ ,φ is called the composition operator and in case φ (z) = z for every z ∈ G, WΨ,φ is called the multiplication operator. 3. Analytic mappings inducing weighted composition operators In this section we characterize the operator-valued analytic mappings Ψ : G → B (E) and the self-analytic mappings φ : G → G which induce weighted composition operators on the spaces HVb (G, E) and HV0 (G, E) for different systems of weights V on G. We begin with the following proposition. Proposition 3.1. Let U and V be arbitrary systems of weights on G. Let Ψ : G → B (E) and φ : G → G be analytic mappings. Then WΨ,φ : HUb (G, E) → HVb (G, E) is a weighted composition operator if for every v ∈ V, there exists u ∈ U such that v (z) kΨz k ≤ u (φ (z)) for every z ∈ G. Proof. We shall show that WΨ,φ is continuous at the origin. Let v ∈ V. Then by the given condition there exists u ∈ U such that v (z) kΨz k ≤ u (φ (z)) for every z ∈ G. We claim that WΨ,φ (Bu,E ) ⊆ Bv,E. For f ∈ Bu,E , we have kWΨ,φ f kv,E = sup {v (z) kΨz (f (φ (z)))k : z ∈ G} ≤ sup {v (z) kΨz k kf (φ (z))k : z ∈ G} ≤ sup {u (φ (z)) kf (φ (z))k : z ∈ G} ≤ sup {u (z) kf (z)k : z ∈ G} = kf ku,E ≤ 1. This proves that WΨ,φ is continuous at the origin and hence WΨ,φ is a weighted composition operator. ¤


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Remark 3.1. The condition in Proposition 3.1 is not sufficient for WΨ,φ to induce a weighted composition operator from HU0 (G, E) → HV0 (G, E). For instance, let G = {z ∈ G : z = x + iy, y > 0} be the upper half-plane. Let U = V be the system of constant weights on G. Let E = H ∞ (D) be the Banach algebra of bonded analytic functions on the unit disc D. Let z0 ∈ G be fixed and let Ψ : G → B (E) be defined as Ψ (z) = Mz0 for every z ∈ G, where Mz0 : E → E is given by Mz0 f = z0 f for every f ∈ E. Define φ : G → G as φ (z) = z0 for every z ∈ G. Now, it is easy to see that for every v ∈ V , there exists u ∈ V such that v (z) kΨz k ≤ u (φ (z)) for every z ∈ G. But WΨ,φ is not a weighted composition operator on HV0 (G, E). For instance, let f ∈ E be such 1 that kf k∞ = 1 and define F : G → E as F (z) = f for every z ∈ G. Clearly z F ∈ HV0 (G, E). But WΨ,φ (F ) ∈ / HV0 (G, E). So, we need an additional condition for WΨ,φ to be a weighted composition operator from HU0 (G, E) to HV0 (G, E). Now, we define a set which we need for our additional condition. Let v ∈ V and ε > 0. Then define the set N (v, ε) = {z ∈ G : v (z) ≥ ε} . Proposition 3.2. Let U and V be arbitrary systems of weights on G. Let Ψ : G → B (E) and φ : G → G be analytic mappings. Then WΨ,φ : HU0 (G, E) → HV0 (G, E) is a weighted composition operator if (i) for every v ∈ V, there exists u ∈ U such that v (z) kΨz k ≤ u (φ (z)) for every z ∈ G; (ii) for every v ∈ V, ε > 0 and compact set K ⊆ G, the set φ−1 (K) ∩ N (v kΨk , ε) is compact, where N (v kΨk , ε) = {z ∈ G : v (z) kΨz k ≥ ε}. Proof. According to Proposition 3.1, Condition (i) implies that WΨ,φ : HUb (G, E) → HVb (G, E) is a weighted composition operator. Now, it is enough to show that WΨ,φ : HU0 (G, E) → HV0 (G, E) is an into map. For, let g ∈ HU0 (G, E). Fix v ∈ V and ε > 0. We shall show that the set K = {z ∈ G : v (z) kΨz (f (φ (z)))k ≥ ε} is a compact subset of G. Now, condition (i) implies that there exists u ∈ U such that v (z) kΨz k ≤ u (φ (z)) for every z ∈ G. Clearly the set S = {z ∈ G : u (z) kf (z)k ≥ ε} is compact in G and ¡ φ ε(K) ¢ ⊆ S. If we put m = sup {kf (z)k : z ∈ S}, then m > 0 and S ⊆ N u, m . From condition (ii), it ε follows that the set H = φ−1 (S) ∩ N (v kΨk , m ) is compact and hence K being a closed subset of H is compact. This completes the proof. ¤ The following corollaries are immediate consequences of Proposition 3.1 and Proposition 3.2. Corollary 3.1. Let U and V be arbitrary systems of weights on G. Let Ψ ∈ H ∞ (G, B (E)) and φ ∈ H (G) be such that φ (G) ⊆ G. Then (i) WΨ,φ : HUb (G, E) → HVb (G, E) is a weighted composition operator if V ≤ U ◦ φ.

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(ii) WΨ,φ : HU0 (G, E) → HV0 (G, E) is a weighted composition operator if V ≤ U ◦ φ and φ is a conformal mapping of G onto itself. Corollary 3.2. Let V = {λχK : λ ≥ 0, K ⊆ G, K is compact}. Then every operator-valued analytic map Ψ : G → B (E) and self-analytic map φ : G → G induces a weighted composition operator WΨ,φ on HV0 (G, E). Remark 3.2. Corollary 3.2 makes it clear that the converse of Corollary 3.1 may not be true. That is, if WΨ,φ is a weighted composition operator on HVb (G, E), then Ψ : G → B (E) may not be bounded and φ : G → G may not be conformal mapping. Thus the behaviour of the weighted composition operator is very much influenced by different systems of weights on G. Now in the following theorems, we obtained a necessary and sufficient conditions for WΨ,φ to be a weighted composition operator on the spaces HVb (G, E) and HV0 (G, E). Theorem 3.1. Let V be an arbitrary system of weights on G and let U be a reasonable system of weights on G. Let Ψ : G → B (E) and φ : G → G ˜b (G, E) → HVb (G, E) is a weighted be analytic mappings. Then WΨ,φ : H U composition operator if and only if for every v ∈ V, there exists u ∈ U such that v (z) kΨz k ≤ u ˜ (φ (z)) for every z ∈ G. Proof. We first show that WΨ,φ is a weighted composition operator. For, let v ∈ V and let Bv,E be a neighborhood of the origin in HVb (G, E). Then by the given condition, there exists u ∈ U such that v (z) kΨz k ≤ u ˜ (φ (z)) for every z ∈ G. We claim that WΨ,φ (Bu˜,E ) ⊆ Bv,E . Let f ∈ Bu˜,E . Then kf ku˜,E ≤ 1. Now kWΨ,φ f kv,E = sup {v (z) kΨz (f (φ (z)))k : z ∈ G} ≤ sup {v (z) kΨz k kf (φ (z))k : z ∈ G} ≤ sup {˜ u (φ (z)) kf (φ (z))k : z ∈ G} ≤ sup {˜ u (z) kf (z)k : z ∈ G} = kf ku˜,E ≤ 1. This shows that WΨ,φ is a weighted composition operator. Conversely, suppose that WΨ,φ is a weighted composition operator. To establish the given condition, let v ∈ V. Then by the continuity of WΨ,φ at the ˜ with u ∈ U such that u ≤ u origin, there exists u ˜∈U ˜ and WΨ,φ (Bu˜,E ) ⊆ Bv,E . Now, we claim that v (z) kΨz k ≤ u ˜ (φ (z)) for every z ∈ G. Fix z0 ∈ G. Then 1 by (2.c), there exists fz0 ∈ Bu such that |fz0 (φ (z0 ))| = . Let y ∈ E u ˜ (φ (z0 )) be such that kyk = 1 and let gz0 : G → E be defined as gz0 (z) = fz0 (z) y for 1 . Also, acevery z ∈ G. Then clearly gz0 ∈ Bu,E and kgz0 (φ (z0 ))k = u ˜ (φ (z0 )) cording to (2.b), fz0 ∈ Bu˜ and therefore gz0 ∈ Bu˜,E . Thus WΨ,φ (gz0 ) ∈ Bv,E . That is, v (z) kΨz (gz0 (φ (z)))k ≤ 1 for every z ∈ G. For z = z0 , we have


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v (z0 ) kΨz0 (y)k ≤ u ˜ (φ (z0 )). Further, it implies that v (z0 ) kΨz0 k ≤ u ˜ (φ (z0 )). This completes the proof of the theorem. ¤ Corollary 3.3. Let V be an arbitrary system of weights on G and let U be an essential system of weights on G. Let Ψ : G → B (E) and φ : G → G be analytic mappings. Then WΨ,φ : HUb (G, E) → HVb (G, E) is a weighted composition operator if and only if for every v ∈ V, there exists u ∈ U such that v (z) kΨz k ≤ u (φ (z)) for every z ∈ G. ˜. ¤ Proof. Follows from Theorem 3.1 since U ∼ =U Corollary 3.4. Let U and V be reasonable systems of weights on G. Let Ψ : G → B (E) and φ : G → G be analytic mappings. Then the following statements are equivalent: ˜b (G, E) → H V˜b (G, E) is a weighted composition operator; (i) WΨ,φ : H U (ii) for every v ∈ V , there exists u ∈ U such that v (z) kΨz k ≤ u ˜ (φ (z)) for every z ∈ G; ˜ (φ (z)) for (iii) for every v ∈ V , there exists u ∈ U such that v˜ (z) kΨz k ≤ u every z ∈ G. Proof. The proof can be established by using the same technique as in Theorem 3.1. ¤ Theorem 3.2. 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 U vanishes at infinity. Let Ψ : G → B (E) and φ : G → G be analytic mappings. Then WΨ,φ : HU0 (G, E) → HV0 (G, E) is a weighted composition operator if and only if (i) for every v ∈ V , there exists u ∈ U such that v (z) kΨz (y)k ≤ u (φ (z)) kyk for every z ∈ G and y ∈ E; (ii) for each v ∈ V and ε > 0, the set {z ∈ G : v (z) kΨz (y)k ≥ ε} is compact for every 0 6= y ∈ E. Proof. Suppose that conditions (i) and (ii) hold. From condition (i), it follows that v (z) kΨz k ≤ u (φ (z)) for every z ∈ G. According to Proposition 3.1, it readily follows that WΨ,φ is a weighted composition operator from HUb (G, E) to HVb (G, E). To show that WΨ,φ : HU0 (G, E) → HV0 (G, E) is a weighted composition operator, it is enough to prove that WΨ,φ (HU0 (G, E)) ⊆ HV0 (G, E). Let f ∈ HU0 (G, E) and let v ∈ V , ε > 0. We shall show that the set S = {z ∈ G : v (z) kΨz (f (φ (z)))k ≥ ε} is compact in G. Further, condition (i) implies that there exists u ∈ U such that v (z) kΨz (y)k ≤ u (φ (z)) kyk for every z ∈ G and y ∈ E. Let H = {z ∈ G : u (z) kf (z)k ≥ ε}. Then H © is compact and ªφ (S) ⊆ H. Let m = sup {u (z) : z ∈ H} and let Nε = ε y ∈ E : kyk < 2m . Clearly Nε is an open neighborhood of zero in E. Now, n since f (φ (S)) is totally bounded in E, there exists a finite set {f (φ (zi ))}i=1 n S in f (φ (S)) such that f (φ (S)) ⊆ [f (φ (zi )) + Nε ] . Further, condition (ii) i=1

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© ª implies that the set Ki = z ∈ G : v (z) kΨz (f (φ (zi )))k ≥ 2ε is compact for each i ∈ {1, 2, . . . , n}. Let z0 ∈ S. Then we may choose j ∈ {1, 2, . . . , n} such that f (φ (z0 )) − f (φ (zj )) ∈ Nε . Also, we have ε ≤ v (z0 ) kΨz0 (f (φ (z0 )))k ≤ v (z0 ) kΨz0 (f (φ (z0 ))) − Ψz0 (f (φ (zj )))k + v (z0 ) kΨz0 (f (φ (zj )))k ≤ u (φ (z0 )) kf (φ (z0 )) − f (φ (zj ))k + v (z0 ) kΨz0 (f (φ (zj )))k ε ≤ + v (z0 ) kΨz0 (f (φ (zj )))k . 2 Thus it follows that z0 ∈ Kj . Hence S is compact being a closed subset of the n S union Ki . This shows that WΨ,φ (f ) ∈ HV0 (G, E). i=1

Conversely, suppose that WΨ,φ : HU0 (G, E) → HV0 (G, E) is a weighted ˜ , therefore, by using the same argument composition operator. Since U ∼ = U as in Theorem 3.1, we can prove that for every v ∈ V, there exists u ∈ U such that v (z) kΨz k ≤ u (φ (z)) for every z ∈ G. Further, it is immediate that v (z) kΨz (y)k ≤ u (φ (z)) kyk for every z ∈ G and y ∈ E. This proves condition (i). To prove condition (ii), we fix v ∈ V and ε > 0. Let 0 6= y ∈ E. Consider the constant function 1y : G → E as 1y (z) = y for every z ∈ G. Then clearly 1y ∈ HU0 (G, E) and hence, WΨ,φ (1y ) ∈ HV0 (G, E). Clearly the set C = {z ∈ G : v (z) kΨz (y)k ≥ ε} is compact in G. This proves condition (ii). With this the proof of the theorem is complete. ¤ Remark 3.3. Theorem 3.1, Theorem 3.2, Corollary 3.3, and Corollary 3.4 are generalizations of boundedness results of weighted composition operators given in ([8], [23], [26]). If G = D, E = C, Ψ (z) = 1 for every z ∈ D and U and V consists of single continuous weights only, then Corollary 3.3, Corollary 3.4, and Theorem 3.2 reduces to the boundedness results of composition operators given in [5]. If φ : G → G is the identity map, then Proposition 3.1, Corollary 3.1, Corollary 3.2, Corollary 3.3, and Theorem 3.1 reduces to the boundedness results of multiplication operators given in [22]. 4. Invertible weighted composition operators In [30], Ohno and Takagi have characterized the invertible weighted composi¡ ¢ ¯ and the Hardy space H ∞ (D). Further, tion operators on the disc algebra A D in [23], the author has generalized these invertible results to the weighted spaces HV0 (G) and HVb (G) of scalar-valued analytic functions. In this section, we characterize the operator-valued analytic mappings Ψ : G → B (E) and the self-analytic mappings φ : G → G which generate invertible weighted composition operators on the spaces HV0 (G, E) and HVb (G, E) of vector-valued analytic functions. We begin with stating a generalized definition of bounded below operator and an invertibility criterion on a Hausdorff topological vector space [35] which we shall use for characterizing invertible weighted composition operators.


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A continuous linear operator T on a Hausdorff topological vector space E is said to be bounded below if for every neighborhood N of the origin in E, there exists a neighborhood 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. Theorem 4.1. (a) 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. (b) 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. Theorem 4.2. Let V be an arbitrary system of weights on G. Let Ψ : G → B (E) and φ : G → G be analytic mappings such that WΨ,φ is a weighted composition operator on HVb (G, E). Then WΨ,φ is invertible if (i) for each v ∈ V , there exists u ∈ V such that v (φ (z)) kyk ≤ u (z) kΨz (y)k for every z ∈ G and y ∈ E; (ii) for each z ∈ G, Ψ (z) : E → E is onto and φ is a conformal mapping of G onto itself. Proof. According to Theorem 4.1, it is enough to show that WΨ,φ is bounded below and onto. For, let v ∈ V and Bv,E be a neighborhood of the origin in HVb (G, E). Then by condition (i), there exists u ∈ V such that v (φ (z)) kyk ≤ u (z) kΨz (y)k for every ¡z ∈ G¢and y ∈ E. c c c We claim that WΨ,φ Bv,E ⊆ Bu,E . Let f ∈ Bv,E . Then 1 < kf kv,E = sup {v (z) kf (z)k : z ∈ G} = sup {v (φ (z)) kf (φ (z))k : z ∈ G} ≤ sup {u (z) kΨz (f (φ (z)))k : z ∈ G} = kWΨ,φ f ku,E . c This shows that WΨ,φ (f ) ∈ Bu,E and hence WΨ,φ is bounded below. Now, we shall show that WΨ,φ is onto. For, let g ∈ HVb (G, E). Fix z0 ∈ G and let v ∈ V be such that v (φ (z0 )) > 0. Then by condition (i), there exists u ∈ V such that v (φ (z0 )) kyk ≤ u (z0 ) kΨz0 (y)k for every y ∈ E. That is, kΨz0 (y)k ≥ λ kyk v (φ (z0 )) for every y ∈ E, where λ = > 0. This proves that Ψz0 is bounded u (z0 ) below on E and hence, by condition (ii), Ψz0 is invertible in B (E). We denote −1 the inverse of Ψz0 by Ψ−1 : G → B (E) z0 . We define an analytic map Ψ −1 −1 as Ψ (z) = Ψz for every z ∈ G. Also, by condition (ii), we have inverse analytic map φ−1 : G → G. Now, we define an analytic map h : G → E as h = Ψ−1 ◦φ−1 ·g◦φ−1 . Then we show that h ∈ HVb (G, E). For, let v ∈ V. Then by condition (i), there exists u ∈ V such that v (φ (z)) kyk ≤ u (z) kΨz (y)k for every z ∈ G and y ∈ E.

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Now, we have khkv,E = sup {v (z) kh (z)k : z ∈ G} = sup {v (φ (z)) kh (φ (z))k : z ∈ G} ≤ sup {u (z) kΨz (h (φ (z)))k : z ∈ G} ° ¡ © ¢° ª °:z∈G = sup u (z) °Ψz Ψ−1 z (g (z)) = sup {u (z) kg (z)k : z ∈ G} = kgku,E < ∞. This proves that h ∈ HVb (G, E) and clearly WΨ,φ (h) = g. Hence WΨ,φ is invertible. ¤ Corollary 4.1. Let V be an arbitrary system of weights on G. Let Ψ : G → B (E) and φ : G → G be analytic mappings. Then WΨ,φ : HV0 (G, E) → HV0 (G, E) is an invertible weighted composition operator if (i) Ψ is bounded, bounded away from zero and for each z ∈ G, Ψ (z) : E → E is onto; (ii) φ is a conformal mapping of G onto itself such that V ◦ φ ∼ =V. Proof. Since Ψ : G → B (E) is bounded analytic function and φ : G → G is a conformal mapping such that V ≤ V ◦ φ, therefore Corollary 3.1 implies that WΨ,φ is a weighted composition operator on HV0 (G, E). Also, since Ψ : G → B (E) is bounded away from zero, there exists m > 0 such that kΨz (y)k ≥ m kyk for every z ∈ G and y ∈ E. Let v ∈ V . Then, since V ◦ φ ≤ V , there exists u ∈ V such that v (φ (z)) ≤ u (z) for every z ∈ G. Further, there exists 1 w ∈ V such that v ≤ w and v (φ (z)) kyk ≤ w (z) kΨz (y)k for every z ∈ G m and y ∈ E. Thus, according to the given conditions and Theorem 4.2, it follows that WΨ,φ is an invertible weighted composition operator on HVb (G, E) . ¤ Corollary 4.2. Let G be a simply connected open subset of C and let V = {λχK : λ ≥ 0, K ⊆ G, K is compact}. Let Ψ : G → B (E) and φ : G → G be analytic mappings. Then WΨ,φ is an invertible weighted composition operator on HVb (G, E) if and only if Ψ (z) is invertible in B (E) for every z ∈ G and φ is a conformal mapping of G onto itself. Proof. In view of Corollary 3.2, we can conclude that WΨ,φ is a weighted composition operator on HV0 (G, E). Since Ψ (z) is invertible in B (E) for every −1 z ∈ G, we define an analytic map Ψ−1 : G → B (E) as Ψ−1 (z) = (Ψ (z)) for every z ∈ G. Also, since φ : G → G is a conformal mapping, it follows that φ−1 : G → G is analytic. Again, according to Corollary 3.2, it follows that WΨ−1 ◦φ−1 ,φ−1 is a weighted composition operator on HV0 (G, E) and it is the inverse of WΨ,φ . This proves that WΨ,φ is invertible. Conversely, suppose that WΨ,φ is an invertible weighted composition operator on HV0 (G, E). To show that Ψ (z) is invertible in B (E) for every z ∈ G, it is enough to prove that Ψ (z) is bounded below and onto. For, let z0 ∈ G and let K = {z0 }. Let v = χK and let y ∈ E. Define the constant function


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1y : G → E as 1y (z) = y for every z ∈ G. Clearly 1y ∈ HV0 (G, E). Since WΨ,φ is invertible on HV0 (G, E) and hence by Theorem 4.1, WΨ,φ is bounded 1 kf kv for below and onto. Now, there exists m > 0 such that kWΨ,φ f kv ≥ m every f ∈ HV0 (G, E). In particular, for f = 1y , we have 1 kWΨ,φ 1y kv ≥ k1y kv . m 1 That is, kΨz0 (y)k ≥ kyk for every y ∈ E. Thus, Ψz0 is bounded below. To m show that Ψz0 : E → E is onto, we fix y ∈ E. Then for 1y ∈ HV0 (G, E), there exists g ∈ HV0 (G, E) such that WΨ,φ g = 1y . That is, Ψz0 (g (φ (z0 ))) = y. This proves that Ψz0 is invertible. In order to show that φ is a conformal mapping, it is enough to prove that φ is bijective. Let z1 , z2 ∈ G be such that z1 6= z2 . Then by the Riemann Mapping Theorem, there exists one-one analytic function g ∈ H (G) such that g (z1 ) 6= g (z2 ). Now, if we define an analytic function h : G → C as h (z) = g (z) − g (z1 ) , then clearly h ∈ HV0 (G) such that and h (z1 ) = 0 and h (z2 ) 6= 0. Let y ∈ E be such that kyk = 1. Then define hy : G → E as hy (z) = h (z) y for every z ∈ G. Then clearly hy ∈ HV0 (G, E) such that hy (z1 ) = 0 and hy (z2 ) 6= 0. Since Ψz1 and Ψz2 are bounded below operators on E, there exist δ1 > 0 and δ2 > 0 such that 1 1 kΨz1 (y)k ≥ kyk and kΨz2 (y)k ≥ kyk δ1 δ2 for every y ∈ E. Let 1 F = hy . δ2 khy (z2 )k Then F ∈ HV0 (G, E). Let δ = δ1 + δ2 and let v = δχK , where K = {z1 , z2 }. Since WΨ,φ (HV0 (G, E)) is dense in HV0 (G, E), there exists f ∈ HV0 (G, E) 1 such that kWΨ,φ f − F kv < . That is, we have 2 ¯ ¯ ¯ 1 1 ¯¯ 1 ¯ δ kΨz1 (f (φ (z1 )))k < and δ ¯kΨz2 (f (φ (z2 )))k − ¯ < . 2 δ2 2 δ 1 δ 1 Further, it implies that kf (φ (z1 ))k < and |kf (φ (z2 ))k − 1| < . δ1 2 δ2 2 Thus, 1 1 kf (φ (z1 ))k < and |kf (φ (z2 ))k − 1| < . 2 2 Now, if φ (z1 ) = φ (z2 ), then from the last two inequalities we get a contradiction. Hence, φ (z1 ) 6= φ (z2 ). Also, using the injectivity argument of WΨ,φ , we can prove that φ is onto. With this the proof is complete. ¤ Remark 4.1. Corollary 4.2 makes it clear that the converse of Corollary 4.1 may not be true. For instance, let G = {z = x + iy : y > 0} be the upper half plane and let V = {λχK : λ ≥ 0, K ⊆ G, K is compact}. Let E = H ∞ (G) and let Ψ : G → B (E) be defined as Ψ (z) = Mz for every z ∈ G, where

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Mz : H ∞ (G) → H ∞ (G) is defined as Mz f = zf for every f ∈ H ∞ (G). Then clearly kΨ (z)k = kMz k = |z| for every z ∈ G. Let φ : G → G be defined as φ (z) = z + 1 for every z ∈ G. Then clearly φ is a conformal mapping of G onto itself and Ψ (z) is invertible in B (E) for every z ∈ G. Thus according to Corollary 4.2, WΨ,φ is an invertible weighted composition operator. But Ψ is neither bounded nor bounded away from zero. This shows that the theory of weighted composition operators is very much influenced by different systems of weights. Theorem 4.3. Let G be a simply connected open subset of C and let V be a reasonable system of weights on G. Let φ : G → G and Ψ : G → B (E) be analytic maps such that each Ψ (z) is one-one and WΨ,φ is a weighted composition operator on H V˜b (G, E). Then WΨ,φ is invertible if and only if (i) φ is a conformal mapping of G onto itself and Ψ (z) : E → E is onto for every z ∈ G; (ii) for each v ∈ V , there exists u ∈ V such that v˜ (φ (z)) kyk ≤ u˜ (z) kΨz (y)k for every z ∈ G and y ∈ E. Proof. Suppose conditions (i) and (ii) hold. Let z0 ∈ G and let v ∈ V be such that v (φ (z0 )) > 0. Then v˜ (φ (z0 )) > 0 and by condition (ii), there exists u ∈ V such that v˜ (φ (z0 )) kyk ≤ u ˜ (z0 ) kΨz0 (y)k for every y ∈ E. Thus 0 )) kΨz0 (y)k ≥ λ kyk for every y ∈ E, where λ = v˜(φ(z > 0. This implies u ˜(z0 ) that each Ψz0 is bounded below and hence, by condition (i), it follows that each Ψz0 is invertible and we denote its inverse by Ψ−1 z0 . Define an analytic map Ψ−1 : G → B (E) as Ψ°−1 (z) =°Ψ−1 for every z ∈ G. From condition z −1 ° (ii). It follows that° v˜ (φ (z)) °Ψ (y) ≤ u ˜ (z) kyk . Further by condition (i), °z ¡ −1 ¢ ° −1 ° we have that v˜ (z) °Ψφ−1 (z) (y)° ≤ u ˜ φ (z) kyk for every z ∈ G and y ∈ E. °¡ −1 ¢ ° ¡ ¢ −1 ° ° That is, v˜ (z) Ψ ◦ φ (z) ≤ u ˜ φ−1 (z) for every z ∈ G. According to Theorem 3.1, WΨ−1 ◦φ−1 ,φ−1 is a weighted composition operator on H V˜b (G, E) such that WΨ,φ WΨ−1 ◦φ−1 ,φ−1 = WΨ−1 ◦φ−1 ,φ−1 WΨ,φ = I, the identity operator. Hence, WΨ,φ is invertible. Conversely, suppose that WΨ,φ is invertible on H V˜b (G, E). We first show ˜ b (G) be such that that for each z0 ∈ G, Ψz0 : E → E is onto. Let fz0 ∈ HV fz0 (z0 ) = 1. Choose y ∈ E such that kyk = 1. If we define gy : G → E as gy (z) = fz0 (z) y for every z ∈ G, then clearly gy ∈ H V˜b (G, E). Since ˜ b (G, E) such that WΨ,φ (f ) = gy . That is, WΨ,φ is onto, there exists f ∈ HV Ψz0 (f (φ (z0 ))) = y. This proves that Ψz0 is onto. Now, to show that φ is a conformal mapping, we need to prove that φ is bijective. Let z1 , z2 ∈ G be such that z1 6= z2 . Then by the Riemann Mapping Theorem, there exists one-one analytic function g ∈ H ∞ (G) such that g (z1 ) 6= g (z2 ). Define h : G → C as h (z) = g (z) − g (z1 ) for every z ∈ G. Let gz2 ∈ H V˜b (G) be such that gz2 (z2 ) 6= 0 and let h0 = hgz2 . Since H V˜b (G) is a module over H ∞ (G) , it follows that h0 ∈ H V˜b (G) such that h0 (z1 ) = 0 and h0 (z2 ) 6= 0. Let y ∈ E


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be such that kyk = 1. Then define hy : G → E as hy (z) = h0 (z) y for every z ∈ G. Then clearly hy ∈ H V˜b (G, E) such that by hy (z1 ) = 0 and hy (z2 ) 6= 0. Since Ψz1 and Ψz2 are invertible operators and hence bounded below, there exist ε1 > 0 and ε2 > 0 such that 1 1 kyk and kΨz2 (y)k ≥ kyk ε1 ε2

kΨz1 (y)k ≥ for every y ∈ E. Let

H=

1 ε2 khy (z2 )k

hy .

Then H ∈ H V˜b (G, E). Let ε = ε1 + ε2 and let v ∈ V be such that v (z1 ) ≥ 1 ˜ and v (z2 ) ≥³ 1. Then there ´ exists v˜ ∈ V such that v˜ (z1 ) ≥ 1 and v˜ (z2 ) ≥ 1. ˜ Since WΨ,φ H Vb (G, E) is dense in H V˜b (G, E) , there exists F ∈ H V˜b (G, E) such that kWΨ,φ F − Hkv˜
0 such that kWΨ,φ 1y k∞ ≤ M k1y k∞ for every y ∈ E. That is, kΨz (y)k ≤ M kyk for every z ∈ D and y ∈ E. Thus kΨz k ≤ M for every z ∈ D and hence Ψ is bounded. Condition (ii) follows from Theorem 5.1. Conversely, suppose that conditions (i) and (ii) are true. Condition (i) clearly implies that WΨ,φ is a weighted composition operator on HVb (D, E). Now, using condition (ii), we shall prove that WΨ,φ is compact. Let K ⊆ HVb (D, E) be a bounded subset. Then we shall show that WΨ,φ (B) is relatively compact in HVb (D, E). For, let {fn } be a sequence in K. Since K is bounded, it is relatively compact with respect to the topology of the uniform convergence on compact subsets of D. By Montel’s Theorem, there exists a subsequence {fnk } which converges uniformly on compact subsets of D to some holomorphic function f . Clearly f ∈ HVb (D, E) We shall show that kWΨ,φ fnk − WΨ,φ f k∞ → 0, as k → ∞. Suppose this is not true. Then there exists ε > 0 and a subsequence of {fnk } which we still denote by {fnk } such that kWΨ,φ fnk − WΨ,φ f k∞ > ε for all k. That is, sup {kΨz (fnk (φ (z))) − Ψz (f (φ (z)))k : z ∈ D} > ε for a sequence {znk } in D such that ° all k. Further, it implies that there exists ° ° ° °Ψznk (fnk (φ (znk ))) − Ψznk (f (φ (znk )))° > ε. That is, ° ° ° ° ε < °Ψznk ° kfnk (φ (znk )) − f (φ (znk ))k .

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We can assume that φ (znk ) → z0 . Since fnk → f uniformly on every compact subset of D, it follows that z0 °∈ T. Thus |φ (znk )| → 1, as k → ∞. Further, by ° ° ° the given condition, we have °Ψznk ° → 0 and hence, we get a contradiction. This proves that WΨ,φ is a compact weighted composition operator. ¤ 6. Examples Example 6.1. Let G = D, the open unit disc and let E = H ∞ (D). Let V = {λχK : λ ≥ 0, K ⊆ D, K a compact set}. Then HV0 (D, E) = (H (D, E) , k), where k denotes the compact open topology. Now, we define Ψ : D → B(H ∞ (D)) as Ψ (z) = Mz for every z ∈ D, where Mz : H ∞ (D) → H ∞ (D) is defined as Mz f = ez f for every f ∈ H ∞ (D) . Let φ : D → D be defined as φ (z) = 1−2z 2−z for every z ∈ D. Since Ψ : D → B (E) is an operator-valued analytic map such that each Ψ (z) is invertible in B (E) and φ is a conformal mapping of D onto itself, according to Corollary 4.2, it follows that WΨ,φ is an invertible weighted composition operator on HV0 (D, E). 2

Example 6.2. Let G = D and let v be a weight defined as v (z) = 1 − |z| for every z ∈ D. Let V = {λv : λ ≥ 0} and let E = H ∞ (D). Then clearly V is an essential system of weights. Let φ : D → D be defined as φ (z) = z 2 for every z ∈ D and let Ψ : D → B (H ∞ (D)) be defined as Ψ (z) = Mz for every z ∈ D, where Mz : H ∞ (D) → H ∞ (D) is defined as Mz f = 2zf for³every f ´ ∈ H ∞ (D). 2 Now, by the Pick-Schwarz Lemma, we have v (z) kΨ (z)k = 1 − |z| |φ0 (z)| ≤ 2

1 − |φ (z)| = v (φ (z)) for every z ∈ D. Hence, by Corollary 3.3, WΨ,φ is a weighted composition operator on HVb (D, E). But, according to Corollary 4.3, WΨ,φ is not an invertible weighted composition operator. References [1] K. D. Bierstedt, J. Bonet, and A. Galbis, Weighted spaces of holomorphic functions on balanced domains, Michigan Math. J. 40 (1993), no. 2, 271–297. [2] K. D. Bierstedt, J. Bonet, and J. Taskinen, Associated weights and spaces of holomorphic functions, Studia Math. 127 (1998), no. 2, 137–168. [3] K. D. Bierstedt, R. Meise, and W. H. Summers, A projective description of weighted inductive limits, Trans. Amer. Math. Soc. 272 (1982), no. 1, 107–160. [4] K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 54 (1993), no. 1, 70–79. [5] J. Bonet, P. Domanski, M. Lindstr¨ om, and J. Taskinen, Composition operators between weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 64 (1998), no. 1, 101–118. 1 , Proc. Amer. Math. Soc. 107 (1989), [6] M. Cambern and K. Jarosz, The isometries of HH no. 1, 205–214. [7] M. D. Contreras and S. Diaz-Madrigal, Compact-type operators defined on H ∞ , Function spaces (Edwardsville, IL, 1998), 111–118, Contemp. Math., 232, Amer. Math. Soc., Providence, RI, 1999. [8] M. D. Contreras and A. G. Hernandez-Diaz, Weighted composition operators in weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 69 (2000), no. 1, 41– 60.


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Department of Mathematics and Statistics College of Science, Sultan Qaboos University P.O. Box 36, P. C. 123 Al-Khod, Sultanate of Oman E-mail address: [email protected]