Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 81–98

COMPOSITION OPERATORS BETWEEN WEIGHTED SPACES OF HOLOMORPHIC FUNCTIONS ON BANACH SPACES

Domingo García, Manuel Maestre, and Pablo Sevilla-Peris

Universidad de Valencia, Departamento de Análisis Matemático C/ Doctor Moliner 50, E-46100 Burjasot (Valencia), Spain Domingo.Garcia@uv.es; Manuel.Maestre@uv.es; Pablo.Sevilla@uv.es

Abstract. In this paper we study composition operators between weighted spaces of holomorphic functions defined on the open unit ball of a Banach space. Necessary and sufficient conditions are given for composition operators to be compact. We show that new phenomena appear in the infinite-dimensional setting different from the ones of the finite-dimensional one.

1. Weights. Weighted spaces

The starting idea of composition operators is a simple and very natural question. Consider D the open unit disc of C and a holomorphic map ϕ: D ⟶ D. If f: D ⟶ C is a holomorphic function, we can compose f ∘ ϕ and try to analyze what happens when we let the f vary; in other words we define an operator between spaces of holomorphic functions and we want to study what properties does this operator have (continuity, compactness, …). This obviously depends on which are the spaces considered. First candidates are the Hardy spaces and a full study of the situation in this case can be found in [18]. In the last few years a lot of research has been done studying the behavior of operators between weighted spaces of holomorphic functions H_{w}(B) whenever B is the unit disk of C or, more in general, an open subset of C^{n} (see below for definitions and notation). Among the operators between these spaces particular attention has been paid to composition operators. We refer to [4], [5], [7], [8], [9], [10], [15], [19] and particularly to the recent surveys [3], [6] for information about the subject. But also some interest has been given to the more general case where X is a Banach space and B_{X} is its open unit ball (see e.g. [1], [2], [13], [14], [17]). In this paper, strongly influenced by the work of Bonet, Domański, Lindström and Taskinen [8], we study composition operators between H_{w}(B_{X}) and H_{v}(B_{Y}) and we find that new phenomena appear in the infinite-dimensional setting different from the ones of the finite-dimensional one. In Section 2 we make an introductory study of composition operators. In Section 3 we study the compactness of a composition operator giving necessary and sufficient conditions for such an operator to be compact. Finally, in Section 4 we show that Hilbert spaces are a natural setting to extend [8, Theorem 2.3], a result that gives conditions on the weight v such that all composition operators from H_{v}(B_{X}) into itself are continuous.

2000 Mathematics Subject Classification: Primary 47B38; Secondary 46E15, 46G20.

The authors were supported by the MCYT and FEDER Project BFM2002-01423.

We fix the notation to be used in the rest of the article. Let X be a complex Banach space and B_{X} its open unit ball. Any continuous bounded mapping v: B_{X} →]0, ∞[ is called a weight.

We denote by H(B_{X}) the space of all holomorphic functions f: B_{X} ⟶ C. A set A ⊂ B_{X} is said to be B_{X} -bounded if there exists 0 < r < 1 such that A ⊂ rB_{X}. The subspace of H(B_{X}) of those functions that are bounded on the B_{X} -bounded sets is denoted by H_{b}(B_{X}).

Following [8] and [17] we consider

H_{v}(B_{X}) = {f ∈ H(B_{X}) : ∥f∥_{v} = sup_{x∈B_{X}} v(x)|f(x)| < ∞}.

With the norm ∥ · ∥_{v}, the space H_{v}(B_{X}) is a Banach space. We denote 𝔅_{v} 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 ([17, Section 3]) and that 𝔅_{v} is τ_{0} -compact ([17, p. 349]).

Following [4], [6], we say that a weight is radial if v(λx) = v(x) for every λ ∈ C with |λ| = 1 and every x ∈ B_{X}.

A weight v satisfies Condition I if inf_{x∈rB_{X}} v(x) > 0 for every 0 < r < 1 ([14]). If v satisfies Condition I, then H_{v}(B_{X}) ⊆ H_{b}(B_{X}) ([14, Proposition 2]). If X is finite-dimensional, then all weights on B_{X} satisfy Condition I. From now on, unless otherwise stated, every weight is assumed to satisfy Condition I.

Given any weight v, following [5], we consider an associated growth condition u: B_{X} ⟶]0, +∞[ defined by u(x) = 1/v(x). With this new function we can rewrite

𝔅_{v} = {f ∈ H_{v}(B_{X}) : |f| ≤ u}.

From this, ũ: B_{X} ⟶]0, +∞[ is defined by

ũ(x) = sup_{f∈𝔅_{v}} |f(x)|

and a new associated weight ṽ = 1/ũ. All these functions were defined by Bierstedt, Bonet and Taskinen for open subsets of C^{n} in [5]. In [5, Proposition 1.2], the following relations between weights for open sets on C^{n} are proved. The same arguments work for the unit ball of a Banach space.

Proposition 1.1 Let X be a Banach space and v a weight defined on B_{X}. The following hold:

(i) 0 < v ≤ ṽ and ṽ is bounded and continuous; i.e., ṽ is a weight.

(ii) ũ (respectively ṽ) is radial and decreasing or increasing whenever u (respectively v) is so.

(iii) ∥f∥_{v} ≤ 1 ⇔ ∥f∥ṽ ≤ 1.

(iv) For each x ∈ B_{X} there exists f_{x} ∈ 𝔅_{v} such that ũ(x) = |f_{x}(x)|.

As an immediate consequence of (iii) we have

Corollary 1.2 ([5, Observation 1.12]). If X is a Banach space and v is any weight defined on B_{X}, then H_{v}(B_{X}) = Hṽ(B_{X}) holds isometrically.

Since the constant function 1 belongs to H_{v}(B_{X}) we have

sup_{x∈B_{X}} v(x) = ∥1∥_{v} = ∥1∥ṽ = sup_{x∈B_{X}} ṽ(x).

Definition 1.3 ([19]). A weight v is said to be essential if there exists C > 0 such that v(x) ≤ ṽ(x) ≤ C v(x) for all x ∈ B_{X}.

We say that a weight v is norm-radial if v(x) = v(y) for every x, y such that ∥x∥ = ∥y∥. If v is norm-radial and non-increasing (with respect to the norm) then ṽ is also norm-radial. Indeed, if v is such a weight and T: X → X is a linear mapping, T ≠ 0, with ∥T ∥ ≤ 1, then for any f ∈ H_{v}(B_{X}) we can consider f_{T} = f ∘ T. Then

∥f_{T} ∥_{v} = sup_{z∈B_{X}} v(z)|f(T (z))| ≤ sup_{z∈B_{X}} v(T (z))|f(T (z))| ≤ ∥f∥_{v}.

Hence, for any x ∈ B_{X} we have sup_{∥f∥_{v}≤1} |f(x)| ≥ sup_{∥f∥_{v}≤1} |f(T (x))|. Now if y ∈ B_{X} with ∥x∥ = ∥y∥ we can take T such that T (x) = y to get that ṽ(x) ≤ ṽ(y). The converse inequality is proved in the same way.

Note that given a Banach space X such that for any two x, y ∈ B_{X} with ∥x∥ = ∥y∥ there exists a holomorphic isometry T: B_{X} ⟶ B_{X} with T (x) = y then any norm-radial weight v satisfies that ṽ is also norm-radial. This happens if X is a Hilbert space.

2. Composition operators

Let X, Y be Banach spaces. We denote by B_{X}, B_{Y} their open unit balls. Let ϕ: B_{X} → B_{Y} be a holomorphic mapping. The composition operator associated to ϕ is defined by

C_{ϕ}: H(B_{Y}) ⟶ H(B_{X}), f ⇝ C_{ϕ}(f) = f ∘ ϕ.

C_{ϕ} is clearly linear and (τ_{0}, τ_{0})-continuous. Given any two weights v, w we consider the restriction C_{ϕ}: H_{v}(B_{Y}) → H_{w}(B_{X}) whenever this is well defined. If h: B_{X} ⟶ Y is bounded we denote as usual ∥h∥_{∞} = sup{∥h(x)∥ : ∥x∥ < 1}.

Remark 2.1. Let H, E be two Banach spaces of holomorphic functions whose topologies are stronger than the pointwise convergence topology. If C_{ϕ}: H → E is well defined then, by the closed graph theorem, C_{ϕ} is continuous. As a consequence, to find out if the composition operator C_{ϕ} is continuous it is enough to find out if C_{ϕ} is well defined.

Proposition 2.2. If there is some 0 < r < 1 such that ϕ(B_{X}) ⊆ rB_{Y} , then C_{ϕ}: H_{v}(B_{Y}) → H_{w}(B_{X}) is well defined (and then continuous) for any two weights v with Condition I and w.

Proof. Since ϕ(B_{X}) ⊆ rB_{Y} , then for each f ∈ H_{v}(B_{Y}) there is K > 0 such that sup_{y∈ϕ(B_{X})} |f(y)| ≤ K. Hence

sup_{x∈B_{X}} w(x)|C_{ϕ}(f)(x)| = sup_{x∈B_{X}} w(x)|f(ϕ(x))| ≤ sup_{x∈B_{X}} w(x) sup_{x∈B_{X}} |f(ϕ(x))| < ∞.

Therefore C_{ϕ}(f) ∈ H_{w}(B_{X}) and C_{ϕ} is well defined.

The following proposition extends some of the results in [8, Proposition 2.1] (see also [7, Theorem 4]).

Proposition 2.3. Let v, w be two weights satisfying Condition I and ϕ: B_{X} ⟶ B_{Y} holomorphic. Then the following are equivalent:

(i) C_{ϕ}: H_{v}(B_{Y}) ⟶ H_{w}(B_{X}) is well defined and continuous.

(ii) sup_{x∈B_{X}} (w(x)/ṽ(ϕ(x))) < ∞. (iii) sup_{x∈B_{X}} (w̃(x)/ṽ(ϕ(x))) < ∞. (iv) sup_{∥ϕ(x)∥>r_{0}}(w(x)/ṽ(ϕ(x))) < ∞ for some 0 < r_{0} < 1.

Proof. The implication (iii) ⇒ (ii) is trivial, since w ≤ w̃. Let us assume now (ii). Let f ∈ H_{v}(B_{Y}); we have

w(x)|f(ϕ(x))| = \frac{w(x)}{ṽ(ϕ(x))}ṽ(ϕ(x))|f(ϕ(x))| ≤ M∥f∥ṽ = M∥f∥_{v}

for all x. Hence C_{ϕ} is continuous.

Suppose now that C_{ϕ} is continuous. If (iii) does not hold there exists (x_{n})_{n∈N} ⊆ B_{X} such that

lim_{n→∞} \frac{w̃(x_{n})}{ṽ(ϕ(x_{n}))} = ∞.

For each n ∈ N we can take f_{n} ∈ 𝔅_{v} so that |f_{n}(ϕ(x_{n}))| = ũ(ϕ(x_{n})) = 1/ṽ(ϕ(x_{n})). Hence

|f_{n}(ϕ(x_{n}))|w̃(x_{n}) = \frac{w̃(x_{n})}{ṽ(ϕ(x_{n}))}

which is a contradiction with the fact that C_{ϕ}(𝔅_{v}) is bounded.

Clearly (ii) implies (iv). Conversely, if (iv) holds, let

M = sup_{∥ϕ(x)∥>r_{0}} \frac{w(x)}{ṽ(ϕ(x))}.

We take x ∈ B_{X}; if ∥ϕ(x)∥ > r_{0} then

w(x)|f(ϕ(x))| = \frac{w(x)}{ṽ(ϕ(x))}ṽ(ϕ(x))|f(ϕ(x))| ≤ M∥f∥_{v}.

If ∥ϕ(x)∥ ≤ r_{0} , since f is bounded in \overline{r_{0}B_{Y}} , we have

w(x)|f(ϕ(x))| ≤ ( sup_{x∈B_{X}} w(x))( sup_{y∈\overline{r_{0}B_{Y}}} |f(y)|).

Joining both cases we have sup_{x∈B_{X}} w(x)|f(ϕ(x))| < ∞ and C_{ϕ}(f) ∈ H_{w}(B_{X}) for all f ∈ H_{v}(B_{Y}). By Remark 2.1, C_{ϕ} is continuous.

Note that (i), (ii) and (iii) above are equivalent even if Condition I does not hold. On the other hand, as Example 2.4 below shows, Condition I is necessary to prove that (iv) implies (i).

Example 2.4. Let X be any infinite-dimensional Banach space and let ϕ(x) = x for every x ∈ B_{X}. By the Josefson–Nissenzweig theorem [11, Chapter XII] we can choose (x_{n}^{∗})_{n} ⊆ X^{∗} with ∥x_{n}^{∗}∥ = 2 and weak-star converging to 0. We define a(x) = 1 + ∑_{n=1}^{∞} |x_{n}^{∗}(x)|^{n} and v(x) = 1/a(x).

For every b > \frac{1}{2} , sup_{∥x∥=b} a(x) = +∞; indeed, since ∥x_{n}^{∗}∥ = 2, for each n ≥ 2 there is x_{n} ∈ X such that ∥x_{n}∥ = 1 and |x_{n}^{∗}(x_{n})| > 2−1/n. Let y_{n} = bx_{n}. Then

a(y_{n}) > |x_{n}^{∗}(y_{n})|^{n} = (b|x_{n}^{∗}(x_{n})|)^{n} > (b(2 − \frac{1}{n}))^{n}

for all n ≥ 2. Therefore sup_{∥x∥=b} a(x) = ∞ for every b > \frac{1}{2} ; hence inf_{∥x∥=b} v(x) = 0 and v does not satisfy Condition I. We fix \frac{1}{2} < c < d < \frac{3}{4} and we consider continuous mappings φ, ψ: [0, 1] → [0, 1] such that

φ(t) = \{ \begin{matrix}1 & if |t| ≤ c, \\ > 0 & if c < |t| < \frac{3}{4}, \\ 0 & if \frac{3}{4} ≤ |t|, \end{matrix} ψ(t) = \{ \begin{matrix}0 & if |t| ≤ d, \\ > 0 & if d < |t| < \frac{3}{4}, \\ 1 & if \frac{3}{4} ≤ |t|. \end{matrix}

We define now

w(x) = ψ(∥x∥)\frac{1}{a(x)} + φ(∥x∥).

Clearly, if ∥x∥ ≤ c, then w(x) = 1 and if ∥x∥ ≥ \frac{3}{4} then w(x) = 1/a(x). Hence, for each b ≥ \frac{3}{4} we have

sup_{∥ϕ(x)∥>b} \frac{w(x)}{ṽ(ϕ(x))} ≤ sup_{∥ϕ(x)∥>b} \frac{w(x)}{v(ϕ(x))} = sup_{∥x∥>b} \frac{w(x)}{v(x)} = 1.

Let us see that C_{ϕ}: H_{v}(B_{X}) → H_{w}(B_{X}) is not well defined. We have that f(x) = ∑_{n=1}^{∞} x_{n}^{∗}(x)^{n} is an entire function on X (see ([12, p. 157]) and

∥f∥_{v} = sup_{x∈B_{X}} \frac{|∑_{n=1}^{∞} x_{n}^{∗}(x)^{n}|}{1 + ∑_{n=1}^{∞} |x_{n}^{∗}(x)|^{n}} ≤ sup_{x∈B_{X}} \frac{∑_{n=1}^{∞} |x_{n}^{∗}(x)|^{n}}{1 + ∑_{n=1}^{∞} |x_{n}^{∗}(x)|^{n}} < 1.

Hence f ∈ H_{v}(B_{X}). But, by the maximum modulus theorem and the Cauchy inequality, if \frac{1}{2} < b < c then

sup_{∥x∥=b} w(x)|f(x)| = sup_{∥x∥=b} |f(x)| = sup_{∥x∥≤b} |f(x)| ≥ sup_{∥x∥≤b}|∑_{n=1}^{∞} x_{n}^{∗}(x)^{n}| = ∞.

This obviously implies C_{ϕ}f = f ∉ H_{w}(B_{X}) and C_{ϕ} is not well defined.

3. Compactness

We now study conditions for the operators C_{ϕ} to be compact. The proof of the next result is easily adapted from those of [18, Section 2.4] and [8, Lemma 3.1] and will be used several times.

Lemma 3.1. Let C_{ϕ}: H_{v}(B_{Y}) ⟶ H_{w}(B_{X}) be continuous. Then the following are equivalent:

(i) C_{ϕ} is compact.

(ii) Each bounded sequence (f_{n})_{n} ⊆ H_{v}(B_{Y}) such that f_{n}⟶^{τ_{0}}0 satisfies that ∥C_{ϕ}f_{n}∥_{w} ⟶ 0.

Many authors, when working in the finite-dimensional setting, consider weights with the property that lim_{∥x∥→1^{−}} w(x) = 0 (e.g. w(x) = g(∥x∥) with g: [0, 1] → [0, +∞[ continuous and non-increasing such that g(0) = 1 and g(1) = 0, [15], [10]). This kind of weights are also considered in [8]. A characterization of compactness is given in [8, Theorem 3.3]. Strongly inspired by that we have the following.

Proposition 3.2. Let v, w be weights with lim_{∥x∥→1^{−}} w(x) = 0 and ϕ: B_{X} ⟶ B_{Y} . Then, C_{ϕ}: H_{v}(B_{Y}) → H_{w}(B_{X}) is compact if and only if

(1)  lim_{∥x∥→1^{−}} \frac{w(x)}{ṽ(ϕ(x))} = 0

and

(2)  ϕ(rB_{X}) is relatively compact for every 0 < r < 1.

Proof. Let us begin by assuming that C_{ϕ} is compact. Suppose that there is r_{0} such that ϕ(r_{0}B_{X}) is not relatively compact, then there is (x_{n})_{n} ⊆ r_{0}B_{X} and ε > 0 with ∥ϕ(x_{n}) − ϕ(x_{m})∥ > ε for every n ≠ m. For each pair (n, m), n ≠ m, we choose y_{nm}^{∗} ∈ Y^{∗} with ∥y_{nm}^{∗}∥ = 1 such that

|y_{nm}^{∗}(ϕ(x_{n})) − y_{nm}^{∗}(ϕ(x_{m}))| ≥ ε.

We have (y_{nm}^{∗}) ⊂ H_{v}(B_{Y}) and ∥y_{nm}^{∗}∥_{v} ≤ ∥v∥_{∞} for all n ≠ m.

The adjoint operator of C_{ϕ}, C_{ϕ}^{t}: H_{w}(B_{X})^{∗} → H_{v}(B_{Y})^{∗} is also compact. For each x ∈ B_{X} we denote by δ_{x} the evaluation functional. We denote by ∥ · ∥_{w}^{∗} the dual norm in H_{w}(B_{X})^{∗}; that is ∥γ∥_{w}^{∗} = sup_{∥f∥_{w}≤1} |γ(f)| for γ ∈ H_{w}(B_{X})^{∗}. Clearly ∥δ_{x}∥_{w}^{∗} = 1/w̃(x). Since w satisfies Condition I so does w̃ and {δ_{x} : x ∈ r_{0}B_{X}} is bounded in H_{w}(B_{X})^{∗}. Then {C_{ϕ}^{t}(δ_{x}) : x ∈ r_{0}B_{X}} = {δ_{ϕ(x)} : x ∈ r_{0}B_{X}} is relatively compact in H_{v}(B_{Y})^{∗}. On the other hand,

ε ≤ |δ_{ϕ(x_{n})}(y_{nm}^{∗}) − δ_{ϕ(x_{m})}(y_{nm}^{∗})| ≤ ∥δ_{ϕ(x_{n})} − δ_{ϕ(x_{m})}∥_{v}^{∗}∥y_{nm}^{∗}∥_{v}

for every n ≠ m. Hence, for all n ≠ m, ∥δ_{ϕ(x_{n})} − δ_{ϕ(x_{m})}∥_{v}^{∗} ≥ ε/∥v∥_{∞}. This is a contradiction.

Let us suppose that w(x)/ṽ(ϕ(x)) does not converge to 0 when ∥x∥ → 1^{−}. Then there is a sequence (x_{n})_{n} ⊆ B_{X} with lim_{n} ∥x_{n}∥ = 1 and c > 0 such that w(x_{n}) ≥ cṽ(ϕ(x_{n})) for all n ∈ N. Using Proposition 1.1, for each n ∈ N we can choose f_{n} ∈ 𝔅_{v} such that |f_{n}(ϕ(x_{n}))| = 1/ṽ(ϕ(x_{n})).

Suppose that there exists 0 < r_{0} < 1 such that ∥ϕ(x_{n})∥ ≤ r_{0} for every n. Since v satisfies Condition I, so does ṽ and M = inf_{y∈r_{0}B_{Y}} ṽ(y) > 0. Then

w(x_{n}) ≥ cṽ(ϕ(x_{n})) ≥ cM > 0.

But this contradicts the fact that lim_{n} w(x_{n}) = 0. Hence we can extract a subsequence of (x_{n})_{n} (that we denote the same) so that lim_{n} ∥ϕ(x_{n})∥ = 1.

We can assume that ∥ϕ(x_{n})∥ > ^{n} \sqrt{1 − 1/n} for every n. We choose y_{n}^{∗} ∈ Y^{∗} with ∥y_{n}^{∗}∥ = 1 such that |y_{n}^{∗}(ϕ(x_{n}))| > ^{n} \sqrt{1 − 1/n} and we define g_{n}(y) = y_{n}^{∗}(y)^{n}f_{n}(y), for all y ∈ B_{Y} . We have

sup_{y∈B_{Y}} v(y)|y_{n}^{∗}(y)|^{n}|f_{n}(y)| ≤ sup_{y∈B_{Y}} v(y)∥y∥^{n}|f_{n}(y)| ≤ ∥f_{n}∥_{v} ≤ 1.

Hence (g_{n})_{n} ⊆ H_{v}(B_{Y}) and it is bounded. Since 𝔅_{v} is τ_{0} -bounded ([17]), given any compact set K ⊆ B_{Y} there exists M > 0 such that sup_{y∈K} |f_{n}(y)| ≤ M for all n ∈ N. Since K is compact, K ⊆ rB_{Y} for some 0 < r < 1; hence

sup_{y∈K} |g_{n}(y)| = sup_{y∈K} |y_{n}^{∗}(y)|^{n}|f_{n}(y)| ≤ M sup_{y∈K} ∥y∥^{n} ≤ M r^{n}.

Thus, (g_{n})_{n} ⊆ H_{v}(B_{Y}) is bounded and τ_{0} convergent to 0. By Lemma 3.1, the sequence ∥C_{ϕ}(g_{n})∥_{w} must tend to 0. On the other hand we have, for every n ∈ N,

∥C_{ϕ}(g_{n})∥_{w} ≥ w(x_{n})|g_{n}(ϕ(x_{n}))| = \frac{w(x_{n})}{ṽ(ϕ(x_{n}))}|y_{n}^{∗}(ϕ(x_{n}))|^{n} > c\frac{n − 1}{n}.

This gives a contradiction and implies (1).

Assume now that (1) holds and ϕ(rB_{X}) is relatively compact for every r. We begin by showing that C_{ϕ} is continuous. By hypothesis there is 0 < r_{0} < 1 such that

sup_{∥x∥>r_{0}} \frac{w(x)}{ṽ(ϕ(x))} ≤ 1.

Since ϕ(r_{0}B_{X}) is relatively compact there is M > 0 such that 0 < M ≤ ṽ(ϕ(x)) for all ∥x∥ ≤ r_{0} . Therefore

sup_{∥x∥≤r_{0}} \frac{w(x)}{ṽ(ϕ(x))} ≤ \frac{1}{M} sup_{∥x∥≤r_{0}} w(x) < ∞.

This gives

sup_{x∈B_{X}} \frac{w(x)}{ṽ(ϕ(x))} < ∞

and C_{ϕ} is continuous.

Let us suppose that C_{ϕ} is not compact. From Lemma 3.1, there is a τ_{0} -null sequence (f_{n})_{n} ⊆ 𝔅_{v} such that (∥C_{ϕ}(f_{n})∥_{w})_{n} does not converge to 0. Going to a subsequence if necessary we can assume that there is λ > 0 such that

sup_{x∈B_{X}} w(x)|f_{n}(ϕ(x))| = ∥C_{ϕ}(f_{n})∥_{w} > λ > 0

for all n ∈ N. We choose (x_{n})_{n} ⊆ B_{X} with w(x_{n})|f_{n}(ϕ(x_{n}))| ≥ λ for all n and let us suppose that (x_{n})_{n} has a subsequence (x_{n_{k}})_{k} such that lim_{k} ∥x_{n_{k}}∥ = 1. Given any ε > 0 there is k_{1} such that

w(x_{n_{k}}) ≤ εṽ(ϕ(x_{n_{k}}))

for all k ≥ k_{1} . Hence

λ ≤ w(x_{n_{k}})|f_{n_{k}}(ϕ(x_{n_{k}}))| ≤ εṽ(ϕ(x_{n_{k}}))|f_{n_{k}}(ϕ(x_{n_{k}}))| ≤ ε∥f_{n_{k}}∥_{v} < ε.

This contradicts the fact that λ > 0. Therefore there exists 0 < s < 1 such that ∥x_{n}∥ ≤ s for every n. Since ϕ((x_{n})_{n}) ⊆ ϕ(sB_{X}) which is relatively compact, given ε > 0 and M = sup_{x∈B_{X}} w(x), there exists n_{2} such that, for n ≥ n_{2}

sup_{y∈ϕ(sB_{X})} |f_{n}(y)| < \frac{ε}{M} .

Therefore |f_{n}(ϕ(x_{n}))| < ε/M for all n ≥ n_{2} and

λ ≤ w(x_{n})|f_{n}(ϕ(x_{n}))| < ε.

This again gives a contradiction and finally shows that C_{ϕ} is compact.

Nevertheless many weights do not satisfy this condition on the limit (see [5], [8]). We are now interested in the study of the compactness of C_{ϕ} with general weights. So far, two different situations have been considered. First, the finite-dimensional case with general weights was studied in [8]. In this case the condition that \overline{ϕ(rB_{X})} is compact and contained in B_{Y} is trivial. For the infinitedimensional case, only composition operators between H^{∞}(B_{Y}) and H^{∞}(B_{X}) (i.e. v(x) = w(x) = 1) have been studied in [1] and [13]. There, it is proved that C_{ϕ}: H^{∞}(B_{Y}) → H^{∞}(B_{X}) is compact if and only if ϕ(B_{X}) ⊆ sB_{Y} for some 0 < s < 1 and ϕ(B_{X}) is relatively compact.

In [8, Theorem 3.3] a characterization of the compactness of a composition operator is obtained for general weights when X = Y = C. This characterization is given in terms of an analytical condition (see (3) below). Proposition 3.2 shows that some topological condition is also needed if we want to have a characterization whenever X and Y are general Banach spaces.

Theorem 3.3. Let v, w be weights with Condition I and ϕ: B_{X} → B_{Y} a holomorphic mapping. Then the following hold.

(a) If C_{ϕ}: H_{v}(B_{Y}) → H_{w}(B_{X}) is compact then ϕ(rB_{X}) is relatively compact for every 0 < r < 1.

(b) Suppose that ∥ϕ∥_{∞} < 1. If ϕ(B_{X}) is relatively compact, then C_{ϕ}: H_{v}(B_{Y}) → H_{w}(B_{X}) is compact.

(c) Suppose that ∥ϕ∥_{∞} = 1. (i) If C_{ϕ}: H_{v}(B_{Y}) → H_{w}(B_{X}) is compact, then

(3)  lim_{r→1^{−}} sup_{∥ϕ(x)∥>r} \frac{w(x)}{ṽ(ϕ(x))} = 0.

(ii) If ϕ(B_{X}) ∩ rB_{Y} is relatively compact for every 0 < r < 1 and

lim_{r→1^{−}} sup_{∥ϕ(x)∥>r} \frac{w(x)}{ṽ(ϕ(x))} = 0

then C_{ϕ}: H_{v}(B_{Y}) → H_{w}(B_{X}) is compact.

Proof. (a) Note that in Proposition 3.2 when we proved that if C_{ϕ} is compact then ϕ(rB_{X}) is relatively compact for every 0 < r < 1 we did not use the fact that lim_{∥x∥→1^{−}} w(x) = 0. Therefore, this implication remains true for any weight w.

(b) By assumption there is 0 < s < 1 such that ϕ(B_{X}) ⊆ sB_{Y} . If ϕ(B_{X}) is relatively compact then

sup_{x∈B_{X}} \frac{1}{ṽ(ϕ(x))} ≤ sup_{y∈\overline{ϕ(B_{X})}} \frac{1}{ṽ(y)} < ∞.

Hence

sup_{x∈B_{X}} \frac{w(x)}{ṽ(ϕ(x))} < ∞.

By Proposition 2.3, C_{ϕ} is continuous.

Let (f_{n})_{n} ⊆ H_{v}(B_{Y}) be bounded and τ_{0} -convergent to 0. We take ε > 0. Let us write M = sup_{x∈B_{X}} w(x) < ∞. We choose n_{0} ∈ N such that for all n ≥ n_{0} ,

sup_{y∈ϕ(B_{X})} |f_{n}(y)| < \frac{ε}{M} .

Hence, for n ≥ n_{0} , ∥C_{ϕ}f_{n}∥_{w} = sup_{x∈B_{X}} w(x)|f_{n}(ϕ(x))| ≤ M sup_{y∈ϕ(B_{X})} |f_{n}(y)| < ε. By Lemma 3.1, C_{ϕ} is compact.

(c) Let us suppose now that ∥ϕ∥_{∞} = 1. Let C_{ϕ} be compact and assume that

lim_{r→1^{−}} sup_{∥ϕ(x)∥>r} \frac{w(x)}{ṽ(ϕ(x))} ≠ 0.

So we can find (r_{n})_{n} ⊆]0, 1[ with lim_{n} r_{n} = 1 and c > 0 so that, for all n ∈ N,

sup_{∥ϕ(x)∥>r_{n}} \frac{w(x)}{ṽ(ϕ(x))} > c.

From this we get a sequence (x_{n})_{n} ⊆ B_{X} with ∥ϕ(x_{n})∥ > r_{n} and w(x_{n}) ≥ cṽ(ϕ(x_{n})) for all n ∈ N. Without loss of generality we can assume that r_{n} > n \sqrt{1 − 1/n}. Applying Proposition 1.1, for each n ∈ N, we can choose f_{n} ∈ 𝔅_{v} satisfying |f_{n}(ϕ(x_{n}))| = 1/ṽ(ϕ(x_{n})). We take y_{n}^{∗} ∈ Y^{∗} such that ∥y_{n}^{∗}∥ = 1 and |y_{n}^{∗}(ϕ(x_{n}))| > r_{n} and we define g_{n}(y) = y_{n}^{∗}(y)^{n}f_{n}(y). Proceeding now as in Proposition 3.2 we obtain the contradiction we are looking for. Hence lim_{r→1^{−}} sup_{∥ϕ(x)∥>r} w(x)/ṽ(ϕ(x)) = 0.

Now we assume that (3) holds and ϕ(B_{X}) ∩ rB_{Y} is relatively compact for every 0 < r < 1. By (3), given ε > 0, there is r_{0} such that, for every r_{0} < r < 1,

(4)  sup_{∥ϕ(x)∥>r} \frac{w(x)}{ṽ(ϕ(x))} < ε.

By Proposition 2.3(iv), C_{ϕ} is continuous. From (4), w(x) < εṽ(ϕ(x)) for all ∥ϕ(x)∥ > r_{0} . Suppose that C_{ϕ} is not compact. By Lemma 3.1 there exists (f_{n})_{n} ⊆ 𝔅_{v} τ_{0} -convergent to 0 such that (∥C_{ϕ}f_{n}∥_{w})_{n} does not converge to 0. Going to a subsequence if necessary, we can find λ > 0 such that ∥C_{ϕ}f_{n}∥_{w} > λ for all n. Let (x_{n})_{n} ⊆ B_{X} with w(x_{n})|f_{n}(ϕ(x_{n}))| ≥ λ for all n. If (x_{n})_{n} has a subsequence (x_{n_{k}})_{k} such that lim_{k} ∥ϕ(x_{n_{k}})∥ = 1, then there exists k_{1} ∈ N with ∥ϕ(x_{n_{k}})∥ > r_{0} for all k ≥ k_{1} . So, for k ≥ k_{1} , w(x_{n_{k}}) < εṽ(ϕ(x_{n_{k}})). Therefore

λ ≤ w(x_{n_{k}})|f_{n_{k}}(ϕ(x_{n_{k}}))| < εṽ(ϕ(x_{n_{k}}))|f_{n_{k}}(ϕ(x_{n_{k}}))| ≤ ε∥f_{n_{k}}∥ṽ = ε∥f_{n_{k}}∥_{v} ≤ ε.

Hence λ ≤ ε for every ε > 0. This leads to a contradiction.

Thus there exists 0 < s < 1 satisfying ∥ϕ(x_{n})∥ < s for all n. So, (ϕ(x_{n}))_{n} ⊆ ϕ(B_{X}) ∩ sB_{Y} which is relatively compact. Let M = sup_{x∈B_{X}} w(x), given any ε > 0 there is n_{2} ∈ N such that for all n ≥ n_{2}

sup_{y∈ϕ(B_{X})∩sB_{Y}} |f_{n}(y)| < \frac{ε}{M} .

Hence, if n ≥ n_{2} , then |f_{n}(ϕ(x_{n}))| < ε/M. As a consequence, if n ≥ n_{2} we have λ ≤ w(x_{n})|f_{n}(ϕ(x_{n}))| < ε. Thus λ ≤ ε for all ε > 0. This leads to a contradiction that shows that C_{ϕ} is compact.

If we want to get better results for part (b) of above theorem we need to add conditions on the weight w.

Proposition 3.4. Let v be a weight on B_{Y} and ϕ: B_{X} → B_{Y} a holomorphic mapping. Then C_{ϕ}: H_{v}(B_{Y}) → H^{∞}(B_{X}) is compact if and only if ϕ(B_{X}) is relatively compact and ∥ϕ∥_{∞} < 1.

Proof. If C_{ϕ} is compact, we take the canonical injection i: H^{∞}(B_{Y}) → H_{v}(B_{Y}). Composing i∘C_{ϕ} we get a compact composition operator from H^{∞}(B_{Y}) into H^{∞}(B_{X}). By [1, Proposition 3], ϕ(B_{X}) is relatively compact and ∥ϕ∥_{∞} < 1. The other implication is a particular case of Theorem 3.3(b).

Corollary 3.5 Let v, w be weights such that w is norm-radial and ϕ: B_{X} → B_{Y} a holomorphic mapping.

(a) If w(x) converges to 0 as ∥x∥ → 1− then C_{ϕ}: H_{v}(B_{Y}) → H_{w}(B_{X}) is compact if and only if ϕ(rB_{X}) is relatively compact for every 0 < r < 1 and lim_{∥x∥→1^{−}} w(x)/ṽ(ϕ(x)) = 0.

(b) If w(x) does not converge to 0 as ∥x∥ → 1− then C_{ϕ}: H_{v}(B_{Y}) → H_{w}(B_{X}) is compact if and only if ϕ(B_{X}) is relatively compact and ∥ϕ∥_{∞} < 1.

Proof. Part (a) is a particular case of Proposition 3.2.

If w(x) does not converge to 0 as ∥x∥ → 1− then there exist ε > 0 and a sequence (r_{n}) ⊂ (0, 1) convergent to 1 such that w(x) > ε for all x ∈ X with ∥x∥ = r_{n} and all n ∈ N. Given x ∈ B_{X} we consider n such that ∥x∥ < r_{n}, by the maximum modulus theorem, we have

|f(x)| ≤ max_{|λ|=r_{n}^{−1}} |f(λx)| ≤ \frac{1}{ε} max_{|λ|=r_{n}^{−1}} w(λx)|f(λx)| ≤ \frac{1}{ε}∥f∥_{w}.

Then ∥f∥_{∞} ≤ (1/ε)∥f∥_{w} ≤ (1/ε)∥w∥_{∞}∥f∥_{∞} for all f ∈ H_{w}(B_{X}). Thus H_{w}(B_{X}) and H^{∞}(B_{X}) coincide algebraically and topologically. Now Proposition 3.4 gives the conclusion.

Let us point out that if Y is finite-dimensional then, trivially, ϕ(B_{X}) is always relatively compact and hence we have the following corollary, which is exactly [8, Theorem 3.3] whenever X = Y = C.

Corollary 3.6. Let Y be a finite-dimensional Banach space and X a complex Banach space. Let v, w be weights and ϕ: B_{X} → B_{Y} a holomorphic mapping.

(a) If ∥ϕ∥_{∞} < 1, then C_{ϕ}: H_{v}(B_{Y}) → H_{w}(B_{X}) is compact.

(b) If ∥ϕ∥_{∞} = 1, then C_{ϕ}: H_{v}(B_{Y}) → H_{w}(B_{X}) is compact if and only if

lim_{r→1^{−}} sup_{∥ϕ(x)∥>r} \frac{w(x)}{ṽ(ϕ(x))} = 0.

After these corollaries it is natural to ask if the converse of (a), (b) and (c)(i) or (c)(ii) in Theorem 3.3 hold in general. The following two examples show that the answer is in the negative in all cases.

Example 3.7. There is a holomorphic mapping ϕ: B_{l_{p}} → B_{l_{p}} and weights v, w on B_{l_{p}} so that ∥ϕ∥_{∞} = 1, ϕ satisfies condition (3) and ϕ(rB_{l_{p}}) is relatively compact for 0 < r < 1, but C_{ϕ} is not compact H_{v}(B_{l_{p}}) → H_{w}(B_{l_{p}}). In addition, here ϕ(B_{l_{p}}) ∩ rB_{l_{p}} is not relatively compact for any 0 < r < 1. This shows that the converse of (c)(i) in Theorem 3.3 does hold in general. Take X = Y = l_{p} with 1 < p < ∞ and define ϕ: B_{l_{p}} → B_{l_{p}} by ϕ((x_{n})_{n}) = (x_{n}^{n})_{n}. This is a holomorphic mapping such that ϕ(B_{l_{p}}) is not relatively compact but ϕ(rB_{l_{p}}) is so for every 0 < r < 1. Take (x_{n})_{n} ⊆ l_{p} such that ∥x_{n}∥_{p} ≤ r for every n ∈ N. A standard diagonal method allows us to obtain a subsequence (x_{n_{m}})_{m} of (x_{n})_{n} such that (x_{n_{m}}(k))_{m} converges for every k.

The sequence (ϕ(x_{n_{m}}))_{m} converges in l_{p}. Indeed, as ϕ(x_{n_{m}}) = (x_{n_{m}}(k)^{k})_{k} and |x_{n_{m}}(k)^{k}| ≤ r^{k} for every k and m, given ε > 0, we can choose k_{0} such that

( ∑_{k=k_{0}+1}^{∞} r^{kp})^{1/p} < \frac{ε}{4}.

We denote, for each m, y_{m} = (x_{n_{m}}(k)^{k})_{k≤k_{0}} and z_{m} = (x_{n_{m}}(k)^{k})_{k>k_{0}} . We have a pointwise convergent sequence (y_{m})_{m} in C^{k_{0}} , thus it converges in the ∥ · ∥_{p} - norm of C^{k_{0}} . Let m_{0} be such that ∥y_{m_{1}} − y_{m_{2}}∥_{l_{p}^{k_{0}}} < \frac{1}{2}ε for every m_{1}, m_{2} ≥ m_{0} . Thus

\begin{aligned}∥ϕ(x_{n_{m_{1}}} ) − ϕ(x_{n_{m_{2}}} )∥_{l_{p}}^{p} & = ∥y_{m_{1}} − y_{m_{2}}∥_{l_{p}^{k_{0}}}^{p} + ∥z_{m_{1}} − z_{m_{2}}∥_{l_{p}}^{p} \\ & < (\frac{ε}{2})^{p} + ∑_{k=k_{0}}^{∞}(r^{k} + r^{k})^{p} < ε^{p}. \end{aligned}

Hence (ϕ(x_{n_{m}}))_{m} is convergent and ϕ(rB_{l_{p}}) is relatively compact.

We define v(x) = 1 − ∥x∥ and w(x) = (1 − ∥ϕ(x)∥)^{2} . We have ∥ϕ∥_{∞} = 1 and

\frac{w(x)}{ṽ(ϕ(x))} ≤ \frac{w(x)}{v(ϕ(x))} = 1 − ∥ϕ(x)∥,

hence lim_{r→1^{−}} sup_{∥ϕ(x)∥>r} w(x)/ṽ(ϕ(x)) = 0. We denote by (e_{n}^{∗})_{n} the canonical basis of l_{q}. The sequence (e_{n}^{∗})_{n} is clearly bounded in H_{v}(B_{l_{p}}). On the other hand, for every n ≠ m,

∥C_{ϕ}e_{n}^{∗} − C_{ϕ}e_{m}^{∗}∥_{w} = sup_{x∈B_{lp}} w(x)|e_{n}^{∗}(ϕ(x)) − e_{m}^{∗}(ϕ(x))|

≥ w( \frac{1}{\sqrt[n]{2}}e_{n})|e_{n}^{∗}(ϕ( \frac{1}{\sqrt[n]{2}}e_{n})) − e_{m}^{∗}(ϕ( \frac{1}{\sqrt[n]{2}}e_{n}))|

= (1 − ∥\frac{1}{2}e_{n}∥)^{2}|e_{n}^{∗}(\frac{1}{2}e_{n}) − e_{m}^{∗}(\frac{1}{2}e_{n})| = \frac{1}{8}.

Hence (C_{ϕ}e_{n}^{∗})_{n} does not have any convergent subsequence and the operator is not compact. Now, as a consequence of Theorem 3.3(c)(ii), there exists 0 < r < 1 such that ϕ(B_{l_{p}})∩rB_{l_{p}} is not relatively compact. Actually ϕ(B_{l_{p}})∩rB_{l_{p}} is not relatively compact for any 0 < r < 1. Indeed, fix 0 < r < 1 and let x_{n} = ^{n} \sqrt{r/2} e_{n}. Then ∥ϕ(x_{n})∥ = ∥\frac{1}{2}re_{n}∥ = \frac{1}{2}r and (ϕ(x_{n}))_{n} ⊆ ϕ(B_{l_{p}}) ∩ rB_{l_{p}}. On the other hand, for every n ≠ m,

∥ϕ(x_{n}) − ϕ(x_{m})∥ = ∥\frac{1}{2}re_{n} − \frac{1}{2}re_{m}∥ = 2^{(1−p)/p}r.

Example 3.8. Let 1 < p < ∞. We give now a holomorphic mapping ϕ: B_{l_{p}} → B_{l_{p}} with ∥ϕ∥_{∞} = 1 and weights v, w on B_{l_{p}} satisfying condition (3) such that C_{ϕ}: H_{v}(B_{l_{p}}) → H_{w}(B_{l_{p}}) is compact but ϕ(B_{X}) ∩ rB_{X} is not relatively compact for any 0 < r < 1. Let ϕ: B_{l_{p}} → B_{l_{p}} be defined by ϕ((x_{n})_{n}) = (x_{n}^{n})_{n}. By Example 3.7 we have that ϕ is a holomorphic mapping such that ϕ(rB_{X}) is relatively compact for every 0 < r < 1 but ϕ(B_{X}) ∩ rB_{X} is not. We take the weights v(x) = 1 − ∥x∥ and w(x) = (1 − ∥x∥)(1 − ∥ϕ(x)∥). Clearly both v and w satisfy Condition I and

\frac{w(x)}{v(ϕ(x))} = 1 − ∥x∥.

This tends to 0 as ∥x∥ → 1^{−}. Applying Proposition 3.2, C_{ϕ}: H_{v}(B_{X}) → H_{w}(B_{X}) is compact.

There is a holomorphic mapping ϕ: B_{l_{p}} → B_{l_{p}} with ∥ϕ∥_{∞} < 1 and weights v, w on B_{l_{p}} so that C_{ϕ}: H_{v}(B_{l_{p}}) → H_{w}(B_{l_{p}}) is compact even though ϕ(B_{l_{p}}) is not relatively compact. Indeed, we define ϕ: B_{l_{p}} → B_{l_{p}} by ϕ((x_{n})_{n}) = 2^{−1}(x_{n}^{n})_{n}, we have that ∥ϕ∥_{∞} = 2^{−1} , ϕ(B_{X}) is not relatively compact and, for the weights v(x) = 1−∥x∥ and w(x) = (1−∥x∥)(1−∥ϕ(x)∥), C_{ϕ} is compact. Hence, whenever ∥ϕ∥_{∞} < 1, the hypothesis of being ϕ(B_{X}) relatively compact is a sufficient but not necessary condition for C_{ϕ} to be compact.

Remark 3.9. If ϕ: B_{X} → B_{X} is holomorphic and satisfies that ϕ(B_{X})∩rB_{X} is relatively compact for every 0 < r < 1 then ϕ(rB_{X}) is relatively compact for every 0 < r < 1. Indeed, if ∥ϕ∥_{∞} < 1 then our assumption implies that ϕ(B_{X}) is relatively compact. If ∥ϕ∥_{∞} = 1 then we define the weights v(x) = 1 − ∥x∥ and w(x) = (1 − ∥ϕ(x)∥)^{2} . Clearly, lim_{r→1^{−}} sup_{∥ϕ(x)∥>r} w(x)/v(ϕ(x)) = 0. Applying Theorem 3.3(c)(ii) we have that C_{ϕ}: H_{v}(B_{X}) → H_{w}(B_{X}) is compact. By Theorem 3.3(a) we obtain the claim.

An open problem is the following. If we assume the analytical condition (3), does there exist an intermediate topological condition between the ones in Theorem 3.3(c)(i) and (ii) that give a characterization of the compactness of C_{ϕ}?

Finally, the next example shows that to assume ϕ(B_{X})∩rB_{X} to be relatively compact for every 0 < r < 1 is a strictly weaker condition than to assume ϕ(B_{X}) to be relatively compact.

Example 3.10. We give now an example of a holomorphic mapping ψ: B_{l_{p}} → B_{l_{p}} such that ψ(B_{l_{p}}) ∩ rB_{l_{p}} is relatively compact for every 0 < r < 1 but ψ(B_{l_{p}}) is not. Take l_{p} with 1 ≤ p < ∞ and define ψ by ψ((x_{n})_{n}) = (x_{1}, (x_{1}^{n}x_{n})_{n≥2}). Clearly ψ(B_{l_{p}}) is not relatively compact. On the other hand, ψ(B_{l_{p}}) ∩ rB_{l_{p}} is relatively compact for every 0 < r < 1. Its proof is analogous to the one given in Example 3.7 since (x_{1}^{n}x_{n}) ∈ ψ(B_{l_{p}}) ∩ rB_{l_{p}} implies that |x_{1}| < r and, from this, |x_{1}^{n}x_{n}| < r^{n} for every n ∈ N.

In Remark 3.9 we have obtained a purely topological result by using weights and composition operators in the case ∥ϕ∥_{∞} = 1. A strengthening of this topological result can nevertheless be obtained directly simply by adapting to our setting some known results for entire mappings due to Aron and Schottenlocher ([2]). We present here those adapted results.

Definition 3.11. A holomorphic mapping f: B_{X} → Y is called compact in x ∈ B_{X} if there is a neighborhood of x, V_{x}, such that f(V_{x}) is relatively compact in Y . A mapping f is said to be compact if it is compact in x for every x ∈ B_{X}.

If f is a holomorphic mapping in an open set U and x ∈ U, we denote by ∑_{n=0}^{∞} P^{n}f(x) the Taylor series expansion of f at x. The next lemma was obtained for entire functions by Aron and Schottenlocher in [2]. Their proof, except for trivial natural changes, works also for holomorphic functions on any balanced and convex open set U.

Lemma 3.12 ([2, Proposition 3.4]). Let f: U → Y be a holomorphic function. The following conditions are equivalent.

(i) f is compact.

(ii) For all x ∈ U and all n ∈ N, P^{n}f(x) is compact.

(iii) For all n ∈ N, P^{n}f(0) is compact.

(iv) There is a 0-neighborhood V_{0} in U such that f(V_{0}) is relatively compact.

Proposition 3.13. Let X be a Banach space and f: B_{X} → B_{Y} a compact holomorphic mapping. Then f(rB_{X}) is relatively compact for every 0 < r < 1.

Proof. Given r, we choose 1/r > s > 1. By the maximum modulus principle we have, for each y ∈ rB_{X},

∥P^{n}f(0)(y)∥ ≤ \frac{1}{s^{n}} ∥f∥_{rsB_{X}} .

This implies that ∑_{n=0}^{∞} P^{n}f(0) converges uniformly and absolutely on rB_{X}. Given ε > 0 let k_{0} be such that

∑_{n=k_{0}+1}^{∞} \frac{1}{s^{n}} < \frac{ε}{3∥f∥_{rsB_{X}}} .

Let g_{k} := ∑_{n=0}^{k} P^{n}f(0); we have

∥f(y) − g_{k_{0}}(y)∥ ≤ ∑_{n=k_{0}+1}^{∞} ∥P^{n}f(0)(y)∥ ≤ ∑_{n=k_{0}+1}^{∞} \frac{1}{s^{n}} ∥f∥_{rsB_{X}} < \frac{ε}{3}

for every y ∈ rB_{X}. Now, as g_{k_{0}} is a compact polynomial, g_{k_{0}}(rB_{X}) is relatively compact. Thus, there are {y_{1}, …, y_{m}} such that for each y ∈ rB_{X}, there exists y_{j_{0}} satisfying ∥g_{k_{0}}(y) − g_{k_{0}}(y_{j_{0}})∥ < \frac{1}{3}ε; hence

∥f(y) − g_{k_{0}}(y_{j_{0}})∥ ≤ ∥f(y) − g_{k_{0}}(y)∥ + ∥g_{k_{0}}(y) − g_{k_{0}}(y_{j_{0}})∥ < ε,

i.e. f(rB_{X}) is a precompact set.

4. A result for Hilbert spaces

The result in [8, Theorem 2.3] gives conditions on a weight v such that all composition operators from H_{v}(D) into itself are continuous. The proof of this is based on the Schwarz lemma and on the decomposition of every holomorphic mapping ϕ from D into D as ϕ = ψ ∘ α_{p}, where ψ ∈ H(D, D) with ψ(0) = 0 and α_{p} is a Möbius transform. This cannot be repeated for functions defined on the unit ball of an arbitrary Banach space. However, Renaud showed in [16] that a very close situation holds for Hilbert spaces.

Let B_{H} be the open unit ball of a Hilbert space H with a scalar product ( · | ·). For each a ∈ B_{H} a linear mapping Γ(a): B_{H} ⟶ B_{H} is defined by

Γ(a)(x) = \frac{1}{1 + ν(a)}(x | a)a + ν(a)x,

where ν(a) = \sqrt{1 − ∥a∥^{2}} . Using this mapping an automorphism of B_{H}, α_{a}: B_{H} → B_{H} is defined as

α_{a}(x) = Γ(a)(\frac{x − a}{1 − (x | a)}).

These are the Möbius transforms for Hilbert spaces defined by Renaud in [16], where a deep study can be found. Each one of them is holomorphic, and satisfies α_{a}(a) = 0, α_{a}(0) = −a, α_{a}^{−1} = α_{−a}.

The following version of the Schwarz lemma for Banach spaces is well known. It is proved by applying the classical Schwarz lemma to the family of functions {[λ ↦ x^{∗} ∘ f(λx/∥x∥)] : x^{∗} ∈ X^{∗}, ∥x^{∗}∥ ≤ 1, 0 < ∥x∥ < 1}.

Let X, Y be two Banach spaces and B_{X}, B_{Y} their open unit balls. Let f: B_{X} ⟶ B_{Y} be holomorphic such that f(0) = 0. Then for all x ∈ B_{X}

∥f(x)∥_{Y} ≤ ∥x∥_{X}.

Let h: [0, 1] →]0, ∞[ be continuous and non-increasing. Given any Banach space X a weight v can be defined on B_{X} by putting v(x) = h(∥x∥). Note that a weight defined in this way is clearly norm-radial. With such weights we have the following result.

Theorem 4.1. Let X be any Banach space and H a Hilbert space. Let h: [0, 1] →]0, ∞[ be continuous and non-increasing and consider the weights defined by h on B_{X} and B_{H}, both denoted by v. Then the following are equivalent:

(i) C_{ϕ}: H_{v}(B_{H}) ⟶ H_{v}(B_{X}) is continuous for all holomorphic ϕ: B_{X} → B_{H}. (ii) Each (x_{n})_{n∈N} ⊆ B_{H} such that ∥x_{n}∥ = 1 − 2^{−n} satisfies

(5)  inf_{n∈N} \frac{ṽ(x_{n+1})}{ṽ(x_{n})} > 0.

Proof. We will follow the pattern of the proof given in [8, Theorem 2.3] with the natural modifications using now Renaud’s Möbius mappings and a suitable version of Schwarz’ lemma. We present the details for the sake of completeness.

First of all if ϕ(0) = 0, by the general version of the Schwarz lemma, we have ∥ϕ(x)∥_{H} ≤ ∥x∥_{X} and C_{ϕ} is continuous. Now for each a ∈ B_{H} we have α_{a}: B_{H} → B_{H}. If every C_{α_{a}} is continuous then all C_{ϕ} are continuous. Indeed, given ϕ, let a = ϕ(0) and define ψ = α_{a} ∘ ϕ. Then ψ(0) = 0 and C_{ϕ} = C_{ψ} ∘ C_{α_{−a}} is continuous. Therefore it is enough to prove that C_{α_{a}}: H_{v}(B_{H}) → H_{v}(B_{H}) is continuous for all a ∈ B_{H} if and only if v satisfies (5).

Let us begin by assuming that all C_{α_{a}} are continuous. By Proposition 2.3, for each a ∈ B_{H} we can find M_{a} > 0 such that ṽ(x) ≤ M_{a}ṽ(α_{a}(x)) for all x ∈ B_{H}. We also know that

sup_{∥x∥=r} ∥α_{a}(x)∥ = \frac{∥a∥ + r}{1 + r∥a∥}

and it is attained at

x_{0} = \frac{−r}{∥a∥}a

(see [16, (9′)]). Since v is norm-radial and non-increasing so is also ṽ and

ṽ(x) ≤ M_{a}ṽ(α_{a}(\frac{−r}{∥a∥}a))

for every x ∈ B_{H} with ∥x∥ = r. We define a new function by l(r) = ṽ(x) with ∥x∥ = 1 − r. Let now s = 1 − r, then for s < \frac{1}{2} we have

(6)  l(s\frac{1 − ∥a∥}{1 + ∥a∥}) ≤ l(1 − \frac{∥a∥ + (1 − s)}{1 + (1 − s)∥a∥}) ≤ l(s\frac{1 − ∥a∥}{1 + \frac{1}{2}∥a∥}).

Taking ∥a∥ = \frac{2}{5} and using the second inequality in (6) we can find M_{a} and s_{0} > 0 such that l(s) ≤ M_{a}l(\frac{1}{2}s) for 0 < s < s_{0} . This implies (5).

Let us assume now that (5) holds. We define a function l exactly in the same way as we did before. Then there are M > 0 and 0 < t_{0} ≤ \frac{1}{2} such that l(t) ≤ Ml(\frac{1}{2}t) for all t < t_{0} . Given any c > 0 we can choose n ∈ N with c < 2^{n}. If t < t_{0} , then l(t) ≤ M^{n}l(t/c). Take c = (1 + ∥a∥)/(1 − ∥a∥) and use the first inequality in (6) to get that for each a ∈ B_{H} there exists K_{a} > 0 such that

l(t) ≤ K_{a}l(1 − \frac{∥a∥ + (1 − t)}{1 + (1 − t)∥a∥})

for t < t_{0} . With this, for any fixed a ∈ B_{H}, we can find a constant M_{a} > 0 such that for 0 < r < 1 and ∥x∥ = r,

ṽ(x) ≤ M_{a}l(1 − \frac{∥a∥ + r}{1 + r∥a∥}) ≤ M_{a}ṽ(α_{a}(x)).

Applying Proposition 2.3, C_{α_{a}} is continuous.

The authors wish to thank Richard Aron, José Bonet and Seán Dineen for fruitful conversations and helpful comments. We also thank the referee for her/his many corrections.

References

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Received 17 February 2003