Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 30, 2005, 337–352

THE v-BOUNDARY OF WEIGHTED SPACES OF HOLOMORPHIC FUNCTIONS

Christopher Boyd and Pilar Rueda

University College Dublin, Department of Mathematics Belfield, Dublin 4, Ireland; Christopher.Boyd@ucd.ie Universidad de Valencia, Departamento de Análisis Matemático Facultad de Matemáticas, ES-46100 Burjasot, Valencia, Spain; Pilar.Rueda@uv.es

Abstract. We commence a geometric theory of the weighted spaces of holomorphic functions on bounded open subsets of C^{n}, ℋ_{v}(U) and ℋ_{v_{o}}(U) by finding an upper bound for the set of extreme points of the unit ball of ℋ_{v_{o}}(U). When U is balanced and v is radial we show that ℋ_{v_{o}}(U) is not isometrically isomorphic to a subspace of c_{o}. We give a Choquet type theorem for ℋ_{v_{o}}(U) and use it to study the centraliser of ℋ_{v_{o}}(U).

1. Introduction

From a functional analysis point of view spaces of weighted holomorphic functions have been studied, to date, under at least four broad headings. The duality problem (see [37], [41], [42], [43], [19], [6], [38], [39]) which seeks to represent the dual and bidual of ℋ_{v_{o}}(U) (defined below) as spaces of analytic functions. The theory of M-ideals (see [45], [46], [24]) which studies ℋ_{v_{o}}(Δ) as an M-ideal in its bidual. The isomorphic theory (see [28], [29], [30], [31], [32], [33], [34], [9]) which seeks to classify ℋ_{v_{o}}(U) as a Banach space up to isomorphism. The theory of composition and multiplication operators on ℋ_{v_{o}}(U) and its bidual ℋ_{v}(U) (see [7], [8], [16], [17], [20], [40], [44], [15]). In this paper we examine a new aspect of weighted spaces of holomorphic functions—the isometric theory. As we shall see this new aspect leads to a deeper understanding of all the previous aspects mentioned above. The isometric theory of weighted spaces of holomorphic functions is further investigated in [11], [12] and [13].

Let U be a bounded open subset of C^{n}. A continuous weight v on U is a bounded, strictly positive real valued function on U. We shall mainly concentrate on weights which converge to 0 on the boundary of U. On occasions we will restrict our attention to balanced domains. Here radial weights (weights v with the property that v(z) = v(λz) whenever |λ| = 1) will play an important role in both our theory and examples. We will use ℋ_{v}(U) to denote the space of all holomorphic functions f on U which have the property that ∥f∥_{v} := sup_{z∈U} v(z)|f(z)| < ∞.

2000 Mathematics Subject Classification: Primary 46B04, 46E15.

The second author was supported by MCYT and FEDER Project BFM2002-01423.

Endowed with ∥ · ∥_{v}, ℋ_{v}(U) becomes a non-separable Banach space. A separable subspace of ℋ_{v}(U) is got by considering all f in ℋ_{v}(U) with the property that |f(z)|v(z) converges to 0 as z converges to the boundary of U i.e. given ε > 0 there is a compact subset K of U such that v(z)|f(z)| < ε for z in U \ K. This subspace is denoted by ℋ_{v_{o}}(U). Thus ℋ_{v}(U) may be regarded as all holomorphic functions on U which satisfy a growth condition of order O(1/v(z)) while ℋ_{v_{o}}(U) are those functions with a growth rate of order o(1/v(z)). Under moderate conditions on U and v (see [6]), ℋ_{v}(U) is the bidual of ℋ_{v_{o}}(U). When this happens the dual of ℋ_{v_{o}}(U) is denoted by G_{v}(U). The geometry of the unit ball of ℋ_{v_{o}}(U)′ is the primary object of study of this paper. An upper bound for the set of extreme points of the unit ball of ℋ_{v_{o}}(U)′ is given by {λv(z)δ_{z} : z ∈ U, |λ| = 1}. We shall use ℬ_{v}(U) to denote the set of z ∈ U for which v(z)δ_{z} is an extreme point of the unit ball of ℋ_{v_{o}}(U)′. We study the topological properties of this set showing that the mapping which takes z to v(z)δ_{z} is a homeomorphism onto its range.

The set ℬ_{v}(U) enables us to study the geometry of ℋ_{v_{o}}(U). An example of the type of result we obtain is: if v is radial on a balanced domain then a bounded sequence (f_{k})_{k} in ℋ_{v_{o}}(U) converges weakly to f in ℋ_{v_{o}}(U) if and only if (f_{k})_{k} converges pointwise to f. Bonet and Wolf, [9], and Lusky, [28], [29], [30], [31], [32], [33] have shown that ℋ_{v_{o}}(U) is isomorphic to a subspace of c_{o}. We will show that when v is either complete or radial this isomorphism is never an isometry. A Choquet type theorem allows us to recover the values of functions in ℋ_{v_{o}}(U) from the values it obtains on \overline{ℬ_{v}(U)} . This allows us to examine the centraliser of weighted spaces of holomorphic functions. In [11] we shall examine the weak^{∗} -exposed and weak^{∗} -strongly exposed points of the unit ball of ℋ_{v_{o}}(U) and in [12] and [13] we make use of the v -boundary to classify isometries between weighted spaces of holomorphic functions.

During the summers of 2001, 2002 and 2003 the first author made short visits to the University of Valencia. He wishes to thank the Department of Mathematical Analysis there for its hospitality during those visits.

2. Elementary theory of the v -boundary

Let U be a bounded open subset of C^{n} and v: U → R^{+} be a continuous strictly positive weight. We use G_{v}(U) to denote the space of linear functionals on ℋ_{v}(U) whose restriction to the unit ball of ℋ_{v}(U), B_{ℋ_{v}(U)} , are continuous for the compact open topology, τ_{o}. If we endow G_{v}(U) with the topology induced from ℋ_{v}(U)′ then it follows from [6, Theorem 1.1(a)] that G_{v}(U)′ is isometrically isomorphic to (ℋ_{v}(U), ∥ · ∥_{v}). We shall say that v converges to 0 as z converges to the boundary of U if given ε > 0 there is a compact subset K of U such that v(z) < ε for z in U \ K.

The v -boundary of weighted spaces of holomorphic functions 339

Given a bounded open subset U of C^{n} and v: U → R^{+} a continuous strictly positive weight we use ℋ_{v_{o}}(U) to denote the subspace of ℋ_{v}(U) defined by

ℋ_{v_{o}}(U) := {f ∈ ℋ_{v}(U) : lim_{z→∂U} v(z)|f(z)| = 0}.

We endow ℋ_{v_{o}}(U) with the norm induced from ℋ_{v}(U). If we assume that v(z) converges to 0 as z converges to the boundary of U, that U is balanced and v is radial, then ℋ_{v_{o}}(U) is equal to the closure of the polynomials with respect to the norm ∥ · ∥_{v}. Bierstedt and Summers, [6, Theorem 1.1(b)] show that the condition that B_{ℋ_{vo} (U)} is τ_{o} -dense in B_{ℋ_{v}(U)} is a necessary and sufficient condition to ensure that G_{v}(U) is isometrically isomorphic to the dual of ℋ_{v_{o}}(U). In particular, if B_{ℋ_{vo} (U)} is τ_{o} -dense in B_{ℋ_{v}(U)} then as G_{v}(U) is a dual Banach space its unit ball will have extreme points.

The purpose of this paper is to examine the geometric structure of the space ℋ_{v_{o}}(U)′. In particular, we will investigate how the geometric theory of ℋ_{v_{o}}(U)′ depends on the weight v. In the special case when U = Δ is the unit disc in C, and v(x) ≡ 1, (ℋ_{v}(Δ), ∥ · ∥_{v}) = (ℋ ^{∞}(Δ), ∥ · ∥_{∞}), the Banach space of all bounded holomorphic functions on the unit disc. We let L^{1}(δΔ) denote the space of all integrable functions on the unit circle and H_{o}^{1} denote the space of all holomorphic functions f on the unit disc with sup_{0<r<1}(∫_{0}^{2π} |f(re^{iθ})| dθ) < ∞ and f(0) = 0. The Banach space H_{o}^{1} can be identified with a subspace of L^{1}(δΔ). A classical result of Ando, [1], shows that L^{1}(δΔ)/H_{o}^{1}(Δ) is the unique isometric predual of (ℋ ^{∞}(Δ), ∥ · ∥_{∞}). Furthermore, Ando, [1], shows that the set of extreme points of the unit ball of L^{1}(δΔ)/H_{o}^{1}(Δ) is empty.

Let us begin our description of the geometry of the unit ball of ℋ_{v_{o}}(U)′ with an upper bound on the possible extreme points this set may have. We shall use Γ to denote the set {λ ∈ C : |λ| = 1}.

Given a Banach space E we use Ext B_{E} to denote the set of extreme points of the closed unit ball of E.

Proposition 1. Let U be a bounded open subset of C^{n} and v be a continuous strictly positive weight on U which converges to 0 on the boundary of U. Then the extreme points of the unit ball of ℋ_{v_{o}}(U)′ are contained in the set {λv(z)δ_{z} : z ∈ U, λ ∈ Γ}.

Proof. The mapping f → fv is an isometric isomorphism of ℋ_{v_{o}}(U) onto a subspace of C(\overline{U}) (fv tends to 0 on the boundary of U). Applying [18, Lemma V.8.6] we see that the set of extreme points of the unit ball of ℋ_{v_{o}}(U)′ is contained in the set of extreme points of the unit ball of C(\overline{U})′. However the set of extreme points of the unit ball of C(\overline{U})′ is \{λδ_{z} : z ∈ U, λ ∈ Γ\}. Restricting these to the image of the unit ball of ℋ_{v_{o}}(U)′ we get that the extreme points of the unit ball of ℋ_{v_{o}}(U)′ are contained in the set {λv(z)δ_{z} : z ∈ U, λ ∈ Γ}.

In particular, if U is a bounded balanced open subset of C^{n} and v is a continuous strictly positive radial weight on U which tends to 0 on the boundary of U then the extreme points of the unit ball of G_{v}(U) are contained in the set {λv(z)δ_{z} : z ∈ U, λ ∈ Γ}.

From the remark before [5, Proposition 1.2], [5, Theorem 1.5(c)] and [6, Theorem 1.1] (see also [38, Proposition 2.2.1]) we have the following result:

Proposition 2. Let U be a bounded open subset of C^{n} and v be a continuous strictly positive weight on U. Then ℋ_{v_{o}}(U) contains all polynomials on C^{n} if and only if v extends continuously to the boundary of U with v|_{∂U} ≡ 0. Furthermore, if U is balanced and v is radial, either of these equivalent conditions will imply that G_{v}(U) is isometrically isomorphic to the dual of (ℋ_{v_{o}}(U), ∥ · ∥_{v}).

Under the conditions of Proposition 2 we have:

Theorem 3. Let U be a bounded balanced open subset of C^{n} and v be a continuous strictly positive radial weight which converges to 0 on the boundary of U. Then G_{v}(U) is the unique isometric predual of ℋ_{v}(U).

Proof. Since U is separable and the mapping z → v(z)δ_{z} is continuous the set {v(z)δ_{z} : z ∈ U} is separable. Hence, its closed linear span, G_{v}(U), is also separable. By Proposition 2 G_{v}(U) is a separable dual space and so has the Radon–Nikodým Property. [21, Theorem 10] (see also [23, (b), p. 144]) implies that G_{v}(U) is the unique isometric predual of ℋ_{v}(U).

Suppose that z ∈ U is such that v(z)δ_{z} is not an extreme point of the unit ball of ℋ_{v_{o}}(U)′. Then v(z)δ_{z} = \frac{1}{2}(ϕ_{1} + ϕ_{2}) for some ϕ_{1} , ϕ_{2} in the unit ball of ℋ_{v_{o}}(U)′. As λv(z)δ_{z} = \frac{1}{2}(λϕ_{1} + λϕ_{2}) for every λ in C with |λ| = 1 we see that λv(z)δ_{z} will not be an extreme point of the unit ball of ℋ_{v_{o}}(U)′ for any λ in Γ. With this observation, we give the following definition:

Definition 4. Let U be a bounded open subset of C^{n} and v be a continuous strictly positive weight on U which converges to 0 on the boundary of U. The v -boundary of U, ℬ_{v}(U), is {z ∈ U : v(z)δ_{z} is an extreme point of the unit ball of ℋ_{v_{o}}(U)′}.

Radial weights have radial v -boundaries.

Lemma 5. Let U be a balanced bounded open subset of C^{n} and v be a continuous strictly positive weight on U which converges to 0 on the boundary of U. If v is radial then ℬ_{v}(U) is radial in the sense that z ∈ ℬ_{v}(U) implies λz ∈ ℬ_{v}(U) for all λ ∈ Γ.

Proof. Given f ∈ ℋ_{v_{o}}(U) and λ ∈ Γ we define f_{λ} by f_{λ}(z) = f(λz). We note that f ∈ B_{ℋ_{vo} (U)} if and only if f_{λ} ∈ B_{ℋ_{vo} (U)} . Given ϕ ∈ ℋ_{v_{o}}(U)′ we define ϕ_{λ} by ϕ_{λ}(f) = ϕ(f_{λ}). It follows from the definition of the norm on ℋ_{v_{o}}(U)′, that ∥ϕ_{λ}∥ = ∥ϕ∥.

Suppose that z ∈ U but z ∉ ℬ_{v}(U). Then we can find ϕ_{1} , ϕ_{2} in the unit ball of ℋ_{v_{o}}(U)′, ϕ_{1} ≠ ϕ_{2} , so that v(z)δ_{z} = \frac{1}{2}(ϕ_{1} + ϕ_{2}). Then for λ ∈ Γ and each f ∈ ℋ_{v_{o}}(U) we have

v(λz)δ_{λz}(f) = v(z)f(λz) = v(z)f_{λ}(z) = \frac{1}{2}(ϕ_{1}(f_{λ})+ϕ_{2}(f_{λ})) = \frac{1}{2}((ϕ_{1})_{λ}+(ϕ_{2})_{λ})(f).

Therefore v(λz)δ_{λz} = \frac{1}{2}((ϕ_{1})_{λ} + (ϕ_{2})_{λ}). Hence λz ∉ ℬ_{v}(U) and the result is proven.

Definition 6. Let U be a bounded open subset of C^{n} and v be a continuous strictly positive weight on U which converges to 0 on the boundary of U. We shall say that v is a complete weight if ℬ_{v}(U) = U.

We recall the following definition:

Definition 7. Let E be a complex Banach space. A point x in E is said to be an exposed point of the unit ball of E if there is ϕ ∈ E′ of norm 1 such that Re(ϕ(x)) = 1 and Re(ϕ(y)) < 1 for all y ∈ E, ∥y∥ ≤ 1, y ≠ x. When E = F′ is a dual space and the vector ϕ which exposes x in B_{E} is in F, we say that x is weak^{∗} -exposed and that ϕ weak^{∗} exposes the unit ball of E at x.

A continuous strictly positive weight v on B_{C^{n}} is said to be unitary if v(z) = v(Az) for every n × n unitary matrix A. Hence v is unitary if and only if v(z) = v(w) whenever ∥z∥ = ∥w∥. If n = 1 the concept of a unitary weight coincides with the concept of a radial weight. For n = 2 the weight v(z) = (1 − ∥z∥^{1+2/π tan^{−1}(|z_{2}|/|z_{1}|)}) is a radial weight which is not unitary. It is readily shown that if v is unitary then ℬ_{v}(B_{C^{n}}) is unitary in the sense that z ∈ ℬ_{v}(B_{C^{n}}) if and only if Az ∈ ℬ_{v}(B_{C^{n}}) for all unitary matrices A.

In [11] we obtain the following sufficient condition for completeness of a unitary weight on B_{C^{n}}.

Proposition 8. Let v: B_{C^{n}} → R be a continuous strictly positive strictly decreasing unitary weight on the unit ball of C^{n} which converges to 0 on the boundary of B_{C^{n}} such that v(x) is twice differentiable and (∂v(x)/∂x_{1})^{2} − v(x)∂^{2}v(x)/∂x_{1}^{2} > 0 for x of the form (x_{1}, 0, …, 0) with x_{1} in (0, 1). Then the weak^{∗} -exposed points (and hence the extreme points) of the unit ball of ℋ_{v_{o}}(B_{C^{n}})′ is the set {v(z)λδ_{z} : λ ∈ Γ, z ∈ B_{C^{n}}}.

This condition allows us to show that when α > 0, β ≥ 1 each of the following weights on the unit ball of C^{n} is complete.

(a) v_{α,β}(z) = (1 − ∥z∥^{β})^{α}.

(b) w_{α,β}(z) = e^{−α/(1−∥z∥^{β})} .

(c) v(z) = (log(2 − ∥z∥))^{α}.

(d) v(z) = (1 − log(1 − ∥z∥))^{−α}.

(e) v(z) = cos(\frac{1}{2}π∥z∥).

(f) v(z) = cos^{−1} ∥z∥.

(g) Finite products of the examples in (a) to (f).

3. Structure of the v -boundary

Lemma 9. Let U be a bounded open subset of C^{n} and v be a continuous strictly positive weight which converges to 0 on the boundary of U.

(a) If λ, μ ∈ Γ and z, w ∈ U then λv(z)δ_{z} = μv(w)δ_{w} on ℋ_{v_{o}}(U) implies z = w and λ = μ.

(b) Let z ∈ \overline{U} . If v(z)δ_{z} = 0 on ℋ_{v_{o}}(U) then z ∈ ∂U.

(c) Let λ, μ ∈ Γ and z, w ∈ \overline{U} . If λv(z)δ_{z} = μv(w)δ_{w} in ℋ_{v_{o}}(U)′ then z = w or z, w ∈ ∂U.

Proof. (a) If z ≠ w we may suppose without loss of generality that z_{1} ≠ w_{1} and take

p(t) = \frac{2}{v(w)μ} \frac{t_{1} − z_{1}}{w_{1} − z_{1}} + \frac{1}{v(z)λ} \frac{t_{1} − w_{1}}{z_{1} − w_{1}} .

Then p ∈ ℋ_{v_{o}}(U) and

λv(z)p(z) = 1 ≠ 2 = μv(w)p(w).

(b) If z ∈ U then v(z) > 0 and the constant map p(w) ≡ 1/v(z) ∈ ℋ_{v_{o}}(U). But then v(z)δ_{z}(p) = 1.

Part (c) follows from (a) and (b).

Lemma 10. Let U be a bounded open subset of C^{n} and v be a continuous strictly positive weight which converges to 0 on the boundary of U. Then the map

μ: U → (ℋ_{v_{o}}(U)′, σ(ℋ_{v_{o}}(U)′, ℋ_{v_{o}}(U)))

given by μ(z) := v(z)δ_{z} is a homeomorphism onto its image.

Proof. Consider μ: \overline{U} → (ℋ_{v_{o}}(U)′, σ(ℋ_{v_{o}}(U)′, ℋ_{v_{o}}(U))) given by μ(z) := v(z)δ_{z}. Define the relation ∼ on \overline{U} by z ∼ w if μ(z) = μ(w) and consider the quotient space \overline{U} /μ. Consider the map μ̄: \overline{U} /μ → ℋ_{v_{o}}(U)′ given by μ̄ ∘ q = μ, where q: \overline{U} → \overline{U} /μ is the natural quotient map. (The set \overline{U} /μ may also be regarded as the one-point compactification of U.) Let us show that μ̄ is continuous when ℋ_{v_{o}}(U)′ is endowed with the weak^{∗} -topology. Let z_{0} ∈ \overline{U} and consider the subbasic neighbourhood of v(z_{0})δ_{z_{0}}

N(z_{0}; f, ε) := {ϕ ∈ ℋ_{v_{o}}(U)′ : |ϕ(f) − v(z_{0})δ_{z_{0}}(f)| < ε},

where f ∈ ℋ_{v_{o}}(U) and ε > 0. Since vf is continuous on \overline{U} , the set

N(z_{0}; ε) := \{z ∈ \overline{U} : |v(z)f(z) − v(z_{0})f(z_{0})| < ε\}

is an open neighbourhood of z_{0} in \overline{U} and μ(N(z_{0}; ε)) ⊂ N(z_{0}; f, ε).

As \overline{U} is compact \overline{U} /μ is compact. Hence μ̄ is a uniform homeomorphism onto its image. On the other hand, by Lemma 9 μ|_{U} is injective. Let us show that μ|_{U} is an open map: given A ⊂ U open there exists an open subset B ⊂ \overline{U} such that A = B ∩ U. Then μ(B) is open in μ(\overline{U}). By Lemma 9 μ(A) = μ(B) ∩ μ(U) and therefore μ(A) is open in μ(U). Thus μ|_{U} is a homeomorphism onto its image.

Let U be a bounded open subset of C^{n}, and let v be a continuous strictly positive weight which converges to 0 on the boundary of U. We know that the extreme points of the unit ball of ℋ_{v_{o}}(U)′ are contained in the set {λv(z)δ_{z} : λ ∈ Γ, z ∈ U}. Let us investigate the topological structure of this set.

Lemma 11. Let U be an open subset of C^{n} and v be a continuous strictly positive weight which converges to 0 on the boundary of U. Then \overline{Ext(B_{ℋ_{vo} (U)′} )}^{σ^{∗}} = \{λv(z)δ_{z} : z ∈ \overline{ℬ_{v}(U)} , λ ∈ Γ\}.

Proof. As B_{ℋ_{vo} (U)} is separable it follows from [14, Proposition 2.5.12] that

(ℋ_{v_{o}}(U)′, σ(ℋ_{v_{o}}(U)′, ℋ_{v_{o}}(U))) is metrizable. Let ϕ ∈ \overline{Ext(B_{ℋ_{vo} (U)′} )}^{σ^{∗}}. Then there exist (λ_{n})_{n} ⊂ Γ and (z_{n})_{n} ⊂ ℬ_{v}(U) such that

ϕ = w^{∗} lim_{n} λ_{n}v(z_{n})δ_{z_{n}} = w^{∗} lim_{n} λ_{n}μ(z_{n}).

Since λ_{n} ∈ Γ, there exists a subsequence (λ_{n_{k}})_{k} of (λ_{n})_{n} converging to some λ_{o} of modulus 1. Then (μ(z_{n_{k}}))_{k} w^{∗} -converges to 1/λ_{o}ϕ. By Lemma 10 and using that μ̄ is a uniform homeomorphism onto its image, z_{n_{k}} = μ^{−1}μ(z_{n_{k}}) converges to some z_{o} ∈ \overline{ℬ_{v}(U)} . Hence ϕ = w^{∗} lim_{k→∞} λ_{n_{k}}μ(z_{n_{k}}) = λ_{o}v(z_{o})δ_{z_{o}}.

Conversely, if z ∈ \overline{ℬ_{v}(U)} we have z = lim_{n} z_{n} where z_{n} belongs to ℬ_{v}(U).

Then v(z)δ_{z} = w^{∗} lim_{n} v(z_{n})δ_{z_{n}} ∈ \overline{Ext(B_{ℋ_{v}(U)′} )}^{σ^{∗}}.

Note that if we start in the proof with ϕ ∈ \overline{Ext(B_{ℋ_{vo} (U)′} )}^{σ^{∗}} \ Ext(B_{ℋ_{vo} (U)′} ) then z_{0} = lim_{k} z_{n_{k}} ∈ \overline{ℬ_{v}(U)} \ ℬ_{v}(U).

Proposition 12. Let U be an open subset of C^{n} and v be a continuous strictly positive weight which converges to 0 on the boundary of U. Then ℬ_{v}(U) is a G_{δ} subset of U.

Proof. We use the fact that (ℋ_{v_{o}}(U)′, σ(ℋ_{v_{o}}(U)′, ℋ_{v_{o}}(U))) is metrizable (see Lemma 11). Applying [35, Proposition 1.3] it follows that ExtB_{ℋ_{vo} (U)′} is a G_{δ} set in ℋ_{v_{o}}(U)′ \ {0} endowed with the weak^{∗} -topology. Let

ExtB_{ℋ_{vo} (U)′} = ⋂_{n=1}^{∞} G_{n}

with each G_{n} open in ℋ_{v_{o}}(U)′ \ {0} endowed with the weak^{∗} -topology. We can assume by the proof of [35, Proposition 1.3] that each G_{n} is radial. Define an equivalence relation ≡ on ℋ_{v_{o}}(U)′ \ {0} by ϕ_{1} ≡ ϕ_{2} if ϕ_{1} = λϕ_{2} for some λ ∈ Γ and let π be the quotient mapping from ℋ_{v_{o}}(U)′\{0} onto ℋ_{v_{o}}(U)′\{0}/ ≡. We shall consider ℋ_{v_{o}}(U)′ \ {0}/ ≡ endowed with the quotient topology of the weak^{∗} topology on ℋ_{v_{o}}(U)′ \ {0}. Since π^{−1}(π(A)) = ⋃_{λ∈Γ} λA, π is an open mapping. As each G_{n} is radial it is easily checked that π(⋂_{n=1}^{∞} G_{n}) = ⋂_{n=1}^{∞} π(G_{n}) and then ExtB_{ℋ_{vo} (U)′} / ≡ is a G_{δ} set in ℋ_{v_{o}}(U)′ \ {0}/ ≡. Further, as the image of each z in U under μ lies in at most one ≡-equivalence class of ℋ_{v_{o}}(U)′ \ {0}, it is readily checked that π ∘ μ is a homeomorphism onto its image. Under this mapping ℬ_{v}(U) is mapped onto ExtB_{ℋ_{vo} (U)′} / ≡ and result follows.

We shall see in [11] that U is equal to the intersection of U with the ℋ ^{∞} - convex hull of ℬ_{v}(U).

4. Convergence of weighted holomorphic functions

The study of the v -boundary allows us to show in [11, Theorem 24 and 25] that ℋ_{v_{o}}(U) and ℋ_{v}(U) are never smooth and (under moderate conditions on the weight) that both spaces are not rotund. In [12] the v -boundary is central in the isometric classification of ℋ_{v_{o}}(U) and ℋ_{v}(U).

We present some applications of the v -boundary to the geometry of ℋ_{v_{o}}(U) and ℋ_{v}(U). The following theorem characterises weak convergence and weak compactness in ℋ_{v_{o}}(U).

Theorem 13. Let U be a bounded open subset of C^{n} and v be a continuous strictly positive weight on U which converges to 0 on the boundary of U. Then (a) A bounded sequence (f_{k})_{k} in ℋ_{v_{o}}(U) converges weakly to f in ℋ_{v_{o}}(U) if

and only if (f_{k})_{k} converges pointwise to f.

(b) A bounded subset of ℋ_{v_{o}}(U) is weakly relatively compact if and only if it is relatively countably compact for the topology of pointwise convergence on ℋ_{v_{o}}(U).

Proof. We showed in Proposition 1 that the extreme points of the unit ball of ℋ_{v_{o}}(U)′ are contained in the set {λv(z)δ_{z} : z ∈ U, λ ∈ Γ}. Part (a) now follows from Rainwater’s theorem, [36], while part (b) is a consequence of a theorem of Bourgain and Talagrand, [10].

Proposition 14. Let U be a balanced bounded open subset of C^{n} and let v be a continuous strictly positive radial weight which converges to 0 on the boundary of U. Let (z_{α})_{α} be a net in U and z be a point of U. If v(z_{α})f(z_{α}) → v(z)f(z) for all f ∈ ℋ_{v_{o}}(U) then f(z_{α})v(z_{α}) → f(z)v(z) for all f ∈ ℋ_{v}(U).

Proof. By [24, Examples III.1.4] ℋ_{v_{o}}(U) is an M-ideal in ℋ_{v}(U). (The proof in [24] is for the open unit disc Δ but is easily extended to arbitrary balanced domains in C^{n}.) The result now follows by [24, Corollary III.2.15] (see also [22]).

Proposition 15. Let U be a balanced bounded open subset of C^{n} and let v be a continuous strictly positive radial weight which converges to 0 on the boundary of U. Then given (z_{n})_{n} in U and (f_{m})_{m} in the unit ball of ℋ_{v}(U) we have that

lim_{m→∞} lim_{n→∞} v(z_{n})f_{m}(z_{n}) = lim_{n→∞} lim_{m→∞} v(z_{n})f_{m}(z_{n}).

Proof. It follows from Lemma 10 that the set {v(z)δ_{z} : z ∈ U} is relatively compact for the σ(ℋ_{v_{o}}(U)′, ℋ_{v_{o}}(U))-topology. Applying [24, Corollary III.2.15] we now see that it is therefore weakly relatively compact. It now follows from [25, Lemma 19.A.1] that for all (z_{n})_{n} in U and (f_{m})_{m} in the unit ball of ℋ_{v}(U) we have that

lim_{m→∞} lim_{n→∞} v(z_{n})f_{m}(z_{n}) = lim_{n→∞} lim_{m→∞} v(z_{n})f_{m}(z_{n}).

5. C(K)-spaces

Let X be a locally compact Hausdorff space and A be a closed subspace of C_{o}(X). According to Araujo and Font [2] A is strongly separating if for each pair of points x_{1} and x_{2} in X there is f ∈ A such that |f(x_{1})| ≠ |f(x_{2})|. They define the Choquet boundary of A as {x ∈ X : e_{x} is an extreme point of B_{A′}}, e_{x} is evaluation at x. It follows from [2, Corollary 4.2] that if ℋ_{v_{o}}(U) is isometrically isomorphic to a strongly separating subspace A of C_{o}(X) then ℬ_{v}(U) is homeomorphic to the Choquet boundary of A.

A compact set K is said to be perfect if it has no isolated points. A compact set K is said to be scattered (dispersed) if it contains no perfect subsets.

Theorem 16. Let U be a bounded open subset of C^{n} and v be a continuous strictly positive weight on U which converges to 0 on the boundary of U. If ℋ_{v_{o}}(U) is isometrically isomorphic to a subspace of (complex) C(K) with K scattered then ℬ_{v}(U) is a countable (and therefore discrete) subset of U.

Proof. Define an equivalence relation ≡ on ExtB_{ℋ_{vo} (U)′} by λ_{1}v(z)δ_{z} ≡ λ_{2}v(w)δ_{w} if z = w. Suppose that T is an isometry from ℋ_{v_{o}}(U) onto a subspace M of C(K). Then T^{∗} maps ExtB_{C(K)′} with the weak^{∗} -topology, which is homeomorphic to Γ × K, onto a set which contains {λv(z)δ_{z} : λ ∈ Γ, z ∈ ℬ_{v}(U)} endowed with the σ(ℋ_{v_{o}}(U)′, ℋ_{v_{o}}(U))-topology. As T^{∗} is linear it induces a map T_{1}^{∗} from K onto a set containing {λv(z)δ_{z} : λ ∈ Γ, z ∈ ℬ_{v}(U)}/ ≡ or equivalently a set containing ℬ_{v}(U). As T^{∗} is continuous and surjective, it is σ(M′, M) − σ(ℋ_{v_{o}}(U)′, ℋ_{v_{o}}(U)) open, see [26, Proposition 3.17.17]. Hence μ^{−1} ∘T_{1}^{∗} is a continuous, open mapping and so μ^{−1} ∘T_{1}^{∗}(K) is scattered. It follows that ℬ_{v}(U) is also scattered. As U is metrizable the proof of [25, Lemma 25.D] gives us that ℬ_{v}(U) is countable and hence must be discrete.

Corollary 17. Let U be a bounded open subset of C^{n} and v be a continuous strictly positive weight on U which converges to 0 on the boundary of U. If v is complete or if U is balanced and v is radial then ℋ_{v_{o}}(U) cannot be isometrically isomorphic to a subspace of C(K) with K scattered. In particular, ℋ_{v_{o}}(U) cannot be isometrically isomorphic to a subspace of c_{o}.

Proof. In either of the above cases ℬ_{v}(U) contains a non-trivial connected component and so cannot be scattered.

In [28, Corollary 2.4(i)] Lusky proved that if v is a continuous strictly positive radial weight on Δ which converges to 0 on the boundary of Δ then ℋ_{v_{o}}(Δ) is isomorphic to a subspace of c_{o}. The weight v(z) = (1 − log(1 − |z|))^{β}, β < 0, is an example of a weight on Δ which gives a Banach space ℋ_{v_{o}}(Δ) which is isomorphic to a subspace of c_{o} yet not isomorphic to c_{o}. See [29]. Bonet and Wolf [9] have recently extended this result by showing that if U is an open subset of C^{n} and v is a continuous strictly positive weight on U then ℋ_{v_{o}}(U) is almost isometrically isomorphic to a subspace of c_{o}. The previous corollary shows that this isomorphism can never be an isometric embedding whenever U is bounded and v is either a complete or radial weight which converges to 0 on the boundary of U.

6. A Choquet theorem

In this section we present an ‘analytic’ representation of the v -boundary obtained from an application of Choquet’s theorem.

Theorem 18 (A Choquet type theorem). Let U be a bounded open subset of C^{n} and v be a continuous strictly positive weight on U which converges to 0 on the boundary of U. Then for each z ∈ U there is a C-valued measure, ν_{z}, of bounded variation with support contained in \overline{ℬ_{v}(U)} ^{U} so that

(∗)  f(z) = ∫_{\overline{ℬ_{v}(U)} _{U}} f(w) dν_{z}(w)

for all f in ℋ_{v_{o}}(U). Moreover we have that ν_{z}(\overline{ℬ_{v}(U)} ^{U}) = 1.

Proof. We consider B_{ℋ_{vo} (U)′} with the weak^{∗} -topology. By definition the set of extreme points of the unit ball of ℋ_{v_{o}}(U)′ is equal to {λv(z)δ_{z} : λ ∈ Γ, z ∈ ℬ_{v}(U)}. Hence by the Choquet–Bishop–de Leeuw theorem, [35, Chapter 4, Theorem] and Lemma 11, for each z in U there is a probability measure, μ_{z}, with support contained in Γ × \overline{ℬ_{v}(U)} ^{U} so that

(∗∗)  f(z)v(z) = ∫_{Γ×\overline{ℬ_{v}(U)} ^{U}} λf(w)v(w) dμ_{z}(λ, w)

for all f in ℋ_{v_{o}}(U). Define a measure ν_{z} on \overline{ℬ_{v}(U)} ^{U} by

ν_{z}(E) = \frac{1}{v(z)} ∫_{Γ×E} λv(w) dμ_{z}(λ, w).

By the Radon–Nikodým theorem we have that

f(z) = ∫_{\overline{ℬ_{v}(U)} _{U}} f(w) dν_{z}(w)

and the first part of the result is proven.

Let M = sup_{w∈ℬ_{v}(U)} v(w) and E be a measurable subset of ℬ_{v}(U). Then |ν_{z}(E)| ≤ M/v(z) and thus ν_{z} has total variation at most 4M/v(z). Taking f ≡ 1 we get that ν_{z}(\overline{ℬ_{v}(U)} ^{U}) = 1.

If z belongs to ℬ_{v}(U) then a result of Bauer, [3], (see [35, Proposition 1.4]) shows that the point mass ε_{v(z)δ_{z}} is the unique probability measure with support contained in B_{ℋ_{vo} (U)′} which represents z. However, it is not possible to ‘lift’ this result and conclude δ_{z} is the unique probability measure on U which satisfies (∗). To see this consider any strictly positive weight v on Δ which converges to 0 on the boundary of Δ. Let z ∈ ℬ_{v}(Δ) and let r be such that |z| < r < 1. By Cauchy’s integral formula

f(z) = \frac{1}{2πi} ∫_{|λ|=r} \frac{f(λ)}{λ − z} dλ

for all f ∈ ℋ_{v_{o}}(Δ) and thus

ρ_{z}(E) = \frac{1}{2πi} ∫_{E∩\{λ:|λ|=r\}} \frac{dλ}{λ − z}

is another probability measure representing z.

Theorem 18 shows that the v -boundary is a determining set for ℋ_{v_{o}}(U) in the sense that if f, g ∈ ℋ_{v_{o}}(U) and f = g on ℬ_{v}(U) then f = g on U.

The above result should be compared with [6, Proposition 3.2]. Theorem 18 and [27, Chapter 9] explain our use of the term “v -boundary”.

7. The centralizer of weighted spaces of holomorphic functions

We introduce some notation of Behrends [4].

Definition 19. Let E be a Banach space and T: E → E be a continuous linear operator. Then T is a multiplier if every extreme point of the unit ball of E′ is an eigenvalue of T^{∗}. That is

T^{∗}(e) = a_{T} (e)e

for some real or complex number a_{T} (e) and every e ∈ ExtB_{E′}. We let Mult(E) denote the set of all multipliers on E.

Definition 20. Let E be a Banach space. The centralizer of E, Z(E), is the set of all T ∈ Mult(E) for which there is \overline{T} in Mult(E) with ( \overline{T} )^{∗}(e) = \overline{a_{T} (e)}e for all e ∈ ExtB_{E′}.

We say that Z(E) is trivial if Z(E) = K.Id, (K = R or C depending on whether E is a real or complex Banach space).

Let U be a bounded open subset of C^{n}, v be a continuous strictly positive weight on U which converges to 0 on the boundary of U. Given g ∈ ℋ ^{∞}(U) we let M_{g}: ℋ_{v_{o}}(U) → ℋ_{v_{o}}(U) be the pointwise multiplication operator

(M_{g}(f))(z) = g(z)f(z).

Proposition 21. Let U be a bounded connected open subset of C^{n}, v be a continuous strictly positive weight on U which converges to 0 on the boundary

of U. Suppose that the \overline{ℬ_{v}(U)}^{∘} is non-empty. Then Mult(ℋ_{v_{o}}(U)) = \{M_{g} : g ∈ ℋ ^{∞}(U)} and Z(ℋ_{v_{o}}(U)) is trivial.

Proof. Suppose that T ∈ Mult(ℋ_{v_{o}}(U)). Then for z ∈ ℬ_{v}(U) v(z)δ_{z} is an extreme point of the unit ball of ℋ_{v_{o}}(U)′. Therefore we have

T^{∗}v(z)δ_{z} = a(z)v(z)δ_{z}

for some a(z) in C. Hence for each f in ℋ_{v_{o}}(U) and z in ℬ_{v}(U) we have that

(T f)(z) = a(z)f(z).

Taking f ≡ 1 we see that a extends to a holomorphic function in ℋ_{v_{o}}(U) which we also denote by a. By continuity we get that (T f)(z) = a(z)f(z) for all z in

\overline{ℬ_{v}(U)} and all f ∈ ℋ_{v_{o}}(U) for some a(z) ∈ C. Since \overline{ℬ_{v}(U)}^{∘} is non-empty the principle of analytic continuation implies that

(T f)(z) = a(z)f(z)

for all z ∈ U. As

|a(z)| = \frac{∥T^{∗}(δ_{z})∥}{∥δ_{z}∥} ≤ ∥T^{∗}∥

a is bounded on U and this proves the first part of the proposition.

Suppose that T = M_{g} is in Z(ℋ_{v_{o}}(U)). Then \overline{M}_{g} is also a multiplier and so

(\overline{M}_{g}(f))(z) = ḡ(z)f(z) = M_{h}(f)(z) = h(z)f(z)

for all z in \overline{ℬ_{v}(U)}^{∘} ℬ(U) all f ∈ ℋ_{v_{o}}(U) and some h ∈ ℋ ^{∞}(U). Thus g is both

analytic and conjugate analytic on \overline{ℬ_{v}(U)}^{∘} ℬ(U) and therefore must be constant.

Corollary 22. Let U be a bounded open subset of C^{n}, v be a continuous strictly positive complete weight on U which converges to 0 on the boundary of U. Then Mult(ℋ_{v_{o}}(U)) = {M_{g} : g ∈ ℋ ^{∞}(U)} and Z(ℋ_{v_{o}}(U)) is trivial.

A different assumption also gives us trivial centralisers.

Proposition 23. Let U be a bounded open subset of C^{n} and let v be a continuous strictly positive weight on U which converges to 0 on the boundary of U. Suppose that ℋ ^{∞}(U) separates ℋ_{v_{o}}(U)′. Then Mult(ℋ_{v_{o}}(U)) = {M_{g} : g ∈ ℋ ^{∞}(U)}.

Proof. Arguing as in Proposition 21 we get a holomorphic function a in ℋ_{v_{o}}(U) so that

(T f)(z) = a(z)f(z)

for all z in \overline{ℬ_{v}(U)} and all f in ℋ_{v_{o}}(U). For each f in ℋ ^{∞}(U) we have that af belongs to ℋ_{v_{o}}(U). As

(T f)(z) = a(z)f(z)

for all z in \overline{ℬ_{v}(U)} applying Theorem 18 we see that

T (f)(z) = ∫_{\overline{ℬ_{v}(U)} _{U}} T (f)(w) dν_{z}(w) = ∫_{\overline{ℬ_{v}(U)} _{U}} a(w)f(w) dν_{z}(w) = a(z)f(z)

for all z in U. Hence, we have that

⟨T^{∗}δ_{z}, f⟩ = ⟨a(z)δ_{z}, f⟩

for all z in U and all f in ℋ ^{∞}(U). Since ℋ ^{∞}(U) separates ℋ_{v_{o}}(U)′ we have that T^{∗}δ_{z} = a(z)δ_{z} for all z in U.

The remainder of the proposition follows as in Proposition 21.

In particular we get:

Proposition 24. Let U be a balanced bounded open subset of C^{n} and v be a continuous strictly positive radial weight which converges to 0 on the boundary of U. Then Mult(ℋ_{v_{o}}(U)) = {M_{g} : g ∈ ℋ ^{∞}(U)}. Furthermore, when n = 1 Z(ℋ_{v_{o}}(U)) is trivial.

Proof. It is shown in [5, Proposition 1.2] that polynomials are dense in ℋ_{v_{o}}(U). They therefore will separate ℋ_{v_{o}}(U)′. The first part of the result now follows from Proposition 23. Let n = 1 and suppose that T = M_{g} is in the centraliser of ℋ_{v_{o}}(U). As v is radial, ℬ_{v}(U) contains a circle and so we have h in ℋ ^{∞}(U) so that

(\overline{M}_{g}(f))(z) = \overline{g}(z)f(z) = M_{h}(f)(z) = h(z)f(z)

for all z in \overline{ℬ_{v}(U)}^{∘} and all f ∈ ℋ_{v_{o}}(U). Consider z in \overline{ℬ_{v}(U)}^{∘} with |z| = r. As v is radial we have \overline{g}(z) = h(z) for |z| = r. Taking a Poisson integral we get \overline{g}(z) = h(z) for |z| ≤ r which proves that all g for which M_{g} is in the centraliser of ℋ_{v_{o}}(U) are constant.

From [24, Theorem III.2.3] we get:

Proposition 25. Let U be a balanced bounded open subset of C, v be a continuous strictly positive radial weight on U which converges to 0 on the boundary of U. Then Z(ℋ_{v}(U)) is trivial.

Definition 26. A Banach space E is a C_{σ} -space if there is a compact Hausdorff set K and an involutory homeomorphism σ: K → K (σ^{2} = Id) such that E is isometrically isomorphic to

{f ∈ C(K) : f(x) = −f(σ(x)) for all x ∈ K}.

From Proposition 21 and [24, Theorem II.5.9] we get:

Proposition 27. Let U be a bounded open subset of C^{n}, v be a continuous strictly positive complete weight on U which converges to 0 on the boundary of U. Then ℋ_{v_{o}}(U) is not a C_{σ} -space.

The above proposition is also valid for radial weights on balanced bounded open subsets of C.

Given a Banach space E we shall use Z_{E} to denote \overline{ExtB_{E′}}^{σ^{∗}} \ \{0\}. For a Banach space E there is a canonical isometric embedding of E into C_{o}(Z_{E}) (see [24, Examples I.3.4]). We have seen that \overline{ExtB_{ℋ_{v}(U)′}} ^{σ^{∗}}\\{0\} may be identified with \overline{ℬ_{v}(U)} . Thus in this case we are identifying ℋ_{v_{o}}(U) with a subspace of C_{o}(U).

Definition 28. A Banach space E has the strong Banach–Stone property if given locally compact Hausdorff spaces X and Y and an isometric isomorphism T: C_{o}(X; E) → C_{o}(Y; E) there is a homeomorphism ϕ: Y → X and a continuous function h from Y into the isometries of E endowed with the strong operator topology such that

(T (f))(y) = h(x)f(ϕ(y))

for all f ∈ C_{o}(X, E) and all y ∈ Y .

We have:

Proposition 29. Let U be a bounded open subset of C^{n}, v be a continuous strictly positive weight on U which converges to 0 on the boundary of U. Suppose

that \overline{ℬ_{v}(U)}^{∘} is non-empty. Then ℋ_{v_{o}}(U) has the strong Banach-Stone property. Alternatively, if U is a balanced bounded open subset of C and v is a continuous strictly positive radial weight then ℋ_{v_{o}}(U) and ℋ_{v}(U) have the strong Banach– Stone property.

Proof. Apply Proposition 21, Proposition 24, [4, Theorem 8.11] and Proposition 25.

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Received 28 July 2004