Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 233–248

WEAKLY COMPACT COMPOSITION OPERATORS ON ANALYTIC VECTOR-VALUED FUNCTION SPACES

José Bonet, Paweł Domański, and Mikael Lindström

Universidad Politécnica de Valencia, E. T. S. Arquitectura, Dpto. de Matemática Aplicada ES-46071 Valencia, Spain; jbonet@pleiades.upv.es A. Mickiewicz University, Faculty of Mathematics and Computer Science ul. Matejki 48/49, PL-60-769 Poznań, Poland; domanski@amu.edu.pl Åbo Akademi University, Department of Mathematics FI-20500 Åbo, Finland; mlindstr@abo.fi

Abstract. Let X be a Banach space. It is proved that the composition operator on X - valued Hardy spaces, weighted Bergman spaces and Bloch spaces is weakly compact or Rosenthal if and only if both id: X → X and the corresponding composition operator on scalar valued spaces are weakly compact or Rosenthal, respectively.

1. Introduction

Let φ: D → D be an analytic self map of the complex unit disc D. It can be easily proved that if the composition operator C_{φ}: f ↦ f ∘φ on vector-valued (i.e. with values in a Banach space X) Hardy, Bergman or Bloch spaces belongs to some operator ideal, then both its scalar version and the identity operator on X belong to the same ideal. For the ideal of weakly compact operators Liu, Saksman and Tylli [LST] proved the converse for vector-valued Hardy spaces H_{1}(X), Bergman spaces B_{1}(X) and B_{∞}(X) = H^{∞}(X) as well as for Bloch spaces using analytic methods.

If a vector-valued space of analytic functions E[X] can be represented as the space L(^{∗}E, X) of all linear bounded operators from the predual of the scalar version of E[X] into X, then we give a very simple functional analytic argument which replaces the more analytic ones in [LST]. In this way we obtain the results for Bloch spaces and extend the results of [LST] to weighted Bergman spaces of infinite order B_{∞}^{v}(X). In that part of the paper our main idea is to use the following result due to Saksman and Tylli in [ST], see also [R], [LS]. Let E, F, E_{1} , F_{1} be Banach spaces and let R ∈ ℒ (E, F) and B ∈ ℒ (E_{1}, F_{1}) be two weakly compact operators. If B or R is compact, then the map T ↦ R ∘ T ∘ B from ℒ (F_{1}, E) into ℒ (E_{1}, F) is weakly compact.

1991 Mathematics Subject Classification: Primary 47B38, 47B10, 46E40, 46E15.

The research of the authors was partially supported by DGESIC project no. PB97-0333, the Committee of Scientific Research (KBN), Poland, grant 2P03A 051 15 and Academy of Finland, respectively.

Unfortunately the operator representation mentioned above does not hold in general, for instance, for Hardy spaces H_{1}(X) or Bergman spaces B_{1}(X). Thus the main part of the paper is devoted to that case. We are able to extend the methods and the results of [LST] to the classical weighted Bergman spaces B_{1}^{α}(X), α ≥ −1, a class which includes both H_{1}(X) and B_{1}(X). An essential improvement is done in a formula derived from the so-called Stanton formula (see Lemma 3).

Let us observe that for 1 < p < ∞ the weighted Bergman space B_{p}^{v}(X) and H_{p}(X) are reflexive whenever X is reflexive. Thus C_{φ} on these spaces is automatically weakly compact if and only if X is reflexive.

Further, we prove a characterization of compact composition operators on the Bloch space. The sets of interpolation for the Bloch space [Ro] play a crucial role in the proof.

2. Preliminaries

We denote by H(D, X) the space of holomorphic functions from the unit disc D into a Banach space X. As usual H_{p}(X) stands for the Hardy space of X -valued functions in H(D, X) such that

∥f∥_{H_{p}(X)}^{p} := sup_{0≤r<1} \frac{1}{2π} ∫_{0}^{2π} ∥f(re^{iθ})∥_{X}^{p} dθ < ∞  for p < ∞,
∥f∥_{H^{∞}(X)} := sup_{z∈D} ∥f(z)∥_{X} < ∞  for p = ∞.

Let v: D → R_{+} be an arbitrary weight, i.e., bounded continuous positive (which means strictly positive throughout the paper) function. We define the weighted Bergman space B_{p}^{v}(X) as the space of those functions f ∈ H(D, X) with

∥f∥_{B_{p}^{v}(X)}^{p} := \frac{1}{π} ∫_{D} ∥f(z)∥_{X}^{p}v(z) dA(z) < ∞  for p < ∞,
∥f∥_{B_{∞}^{v}(X)} := sup_{z∈D} ∥f(z)∥_{X}v(z) < ∞  for p = ∞,

where dA denotes the Lebesque area measure on the plane. If v(z) = (1 − |z|^{2})^{α}, α > −1, then we write B_{p}^{α}(X) and if α = 0 we just omit α. If X = C, then we omit X in the notation. For the definition of B_{p}^{α} cf. [CM]. The Bergman spaces B_{∞}^{v} appear naturally in the study of growth conditions on analytic functions and in the scalar-case have been considered in many papers, see for example, [BBG], [BBT], [BS], [BDL], [BDLT], [SW1], [SW2].

We denote by ℬ(X) the X -valued Bloch space of analytic functions f: D → X with the norm

∥f∥_{ℬ(X)} = ∥f(0)∥_{X} + sup_{z∈D}(1 − |z|^{2})∥f′(z)∥_{X} < ∞.

In [CH] the composition operators on the Bloch space are treated as weighted composition operators on B_{∞}^{v} spaces.

A map T ∈ ℒ (X) from the Banach space X into X is called compact, weakly compact, Rosenthal, if it maps the closed unit ball of X onto a relatively compact, a relatively weakly compact, a conditionally weakly compact set in X. A subset A in X is called conditionally weakly compact, if every sequence in A admits a weak Cauchy subsequence. Clearly every weakly compact operator is Rosenthal. Rosenthal’s l_{1} theorem implies that T: X → Y is Rosenthal if and only if T is not an isomorphism on any copy of l_{1} in X.

When we write f ∼ g for two functions f and g we mean there are strictly positive constants a, b such that af ≤ g ≤ bf for all the values of the variable.

For the sake of completeness we give a general argument why the considered conditions are necessary for C_{φ} to belong to the considered ideals.

Proposition 1. If 𝒥 is an operator ideal and C_{φ}: E(X) → E(X) belongs to 𝒥 whenever E(X) is one of the spaces of vector-valued analytic functions B_{p}^{v}(X), H_{p}(X), ℬ(X), then both id: X → X and C_{φ}: E → E, E the scalar version of the space, belongs to 𝒥 .

Proof. Let 0 ≠ x_{0} ∈ X, l_{0} ∈ X^{∗} with l_{0}(x_{0}) = 1 and z_{0} ∈ D. We define the operators

\begin{aligned}γ & : E → E(X),  γ(f)(z) = f(z)x_{0}; \\ η & : E(X) → E,  η(f) = l_{0} ∘ f; \\ p & : X → E(X),  p(x)(z) = x; \\ r & : E(X) → X,  r(f) = f(z_{0}). \end{aligned}

All these operators are continuous, id_{X} = r ∘ C_{φ} ∘ p and η ∘ C_{φ} ∘ γ is exactly the scalar composition operator on E.

3. Consequences of the Stanton formula

In [Sh1] Shapiro introduced the generalized Nevanlinna counting function N_{φ,α} for α > 0. It is defined by

N_{φ,α}(w) = ∑_{z∈φ^{−1}(w)}(log(\frac{1}{|z|}))^{α}, w ∈ D \ \{φ(0)\}.

For our purpose it is convenient to introduce the modified Nevanlinna counting function

Ñ_{φ,α}(w) = ∑_{z∈φ^{−1}(w)}(1 − |z|^{2})^{α−1} log(\frac{1}{|z|}), w ∈ D \ \{φ(0)\}.

The standard Nevanlinna counting function is N_{φ} = N_{φ,1} = Ñ_{φ,1} and the partial Nevanlinna counting function of φ is defined for 0 < r < 1 by

N_{φ}(r, w) = ∑_{z∈φ^{−1}(w), |z|≤r} log(\frac{r}{|z|}),  w ∈ D \ \{φ(0)\}.

The following formula for a continuous subharmonic function u is due to Stanton [St, Theorem 2]:

\frac{1}{2π} ∫_{0}^{2π} u(φ(re^{iθ})) dθ = u(0) + \frac{1}{2π} ∫_{D} N_{φ}(r, w) d[Δ(u)](w),

where r ∈ (0, 1) and φ: D → D is analytic, φ(0) = 0. When f ∈ H(D, X), d[Δ∥f∥_{X}](w) denotes integration with respect to the distributional Laplacian of ∥f∥_{X}, which is a positive measure on D since the map z ↦ ∥f(z)∥_{X} is subharmonic. This means that for every test function (infinitely differentiable function on C with compact support) τ we have

∫ τ(w) d[Δ∥f∥_{X}](w) = \frac{1}{2π} ∫ ∥f(w)∥_{X}Δτ(w) dA(w).

The Stanton formula was applied to composition operators first by Shapiro [Sh1], see also [SS]. We use it to characterize weakly compact operators with the help of the following lemmas.

Lemma 2 [LST, p. 300–301]. If f: D → X is analytic, φ(0) = 0 and 0 < r < 1, then

(1) \frac{1}{2π} ∫_{0}^{2π} ∥f(φ(re^{iθ})∥_{X} dθ = ∥f(0)∥_{X} + \frac{1}{2π} ∫_{D} N_{φ}(r, w) d[Δ(∥f∥_{X})](w),
(2)  ∥C_{φ}(f)∥_{H_{1}(X)} = ∥f(0)∥_{X} + \frac{1}{2π} ∫_{D} N_{φ}(w) d[Δ(∥f∥_{X})](w).

The next result was proved in [LST] only for α = 0:

Weakly compact composition operators 237

Lemma 3. If f: D → X is analytic, φ(0) = 0 and α > −1, then

(3)  ∥C_{φ}(f)∥_{B_{1}^{α}(X)} ∼ ∥f(0)∥_{X} + \frac{1}{2π} ∫_{D} Ñ_{φ,α+2}(w) d[Δ(∥f∥_{X})](w).

Proof. If 0 < r_{0} ≤ r < 1, then \frac{1}{2}(1 − r^{2}) ≤ log(1/r) ≤ C(1 − r^{2}) for some C. By partial integration, for z ∈ D away from the origin, we have

∫_{|z|}^{1} 2r(1 − r^{2})^{α} log(\frac{r}{|z|}) dr = ∫_{|z|}^{1} \frac{(1 − r^{2})^{α+1}}{r(α + 1)} dr

∼ ∫_{|z|}^{1}(log(\frac{1}{r}))^{α+1} \frac{dr}{r} ∼ (log \frac{1}{|z|})^{α+2}

∼ (1 − |z|^{2})^{α+1} log(\frac{1}{|z|}).

Further, we have that

lim_{|z|→0^{+}} \frac{∫_{|z|}^{1} 2r(1 − r^{2})^{α} log(r/|z|) dr}{(1 − |z|^{2})^{α+1} log(1/|z|)} = \frac{1}{α + 1}.

Indeed, by partial integration

I(|z|) := ∫_{|z|}^{1} 2r(1 − r^{2})^{α} log(\frac{r}{|z|}) dr = ∫_{|z|}^{1} \frac{(1 − r^{2})^{α+1}}{r(α + 1)} dr.

Further, let J(|z|) := (1 − |z|^{2})^{α+1} log(1/|z|). Then, by l’Hôpital’s rule,

lim_{|z|→0^{+}} \frac{I(|z|)}{J(|z|)} = lim_{|z|→0^{+}} \frac{I′(|z|)}{J′(|z|)}

= lim_{|z|→0^{+}} \frac{1}{2(α + 1)^{2}|z|^{2} log(1/|z|)(1 − |z|^{2})^{−1} + α + 1} = \frac{1}{α + 1}.

Hence ∫_{|z|}^{1} 2r(1 − r^{2})^{α} log(r/|z|) dr and (1 − |z|^{2})^{α+1} log(1/|z|) are comparable with uniform constant for all |z| > 0. Thus

(4)  \begin{aligned}∫_{0}^{1} 2r(1 − r^{2})^{α}N_{φ}(r, w) dr & = ∑_{z∈φ^{−1}(w)} ∫_{|z|}^{1} 2r(1 − r^{2})^{α} log(\frac{r}{|z|}) dr \\ & ∼ ∑_{z∈φ^{−1}(w)}(1 − |z|^{2})^{α+1} log(\frac{1}{|z|}). \end{aligned}

Now multiplying (1) by 2r(1 − r^{2})^{α}, integrating with respect to r from 0 to 1 and applying Fubini’s theorem, we get

\begin{aligned}_{∫D} & ∥f(φ(w))∥_{X}(1 − |w|^{2})^{α} dA(w) \\ & ∼ ∥f(0)∥_{X} + \frac{1}{2π} ∫_{D}(∫_{0}^{1} N_{φ}(r, w)2r(1 − r^{2})^{α} dr) d[Δ(∥f∥_{X})](w) \end{aligned}

and we conclude from (4).

For the special case that φ is the identity map we obtain the following formulas:

(5) \frac{1}{2π} ∫_{0}^{2π} ∥f(re^{iθ})∥_{X} dθ = ∥f(0)∥_{X} + \frac{1}{2π} ∫_{rD} log(\frac{r}{|w|}) d[Δ(∥f∥_{X})](w),
(6)  ∥f∥_{H_{1}(X)} = ∥f(0)∥_{X} + \frac{1}{2π} ∫_{D} log(\frac{1}{|w|}) d[Δ(∥f∥_{X})](w)

and

(7) ∥f∥_{B_{1}^{α}(X)} ∼ ∥f(0)∥_{X} + \frac{1}{2π} ∫_{D}(1 − |w|^{2})^{α+1} log(\frac{1}{|w|}) d[Δ(∥f∥_{X})](w).

The estimates (6) and (7) permit to define B_{1}^{−1}(X) as H_{1}(X), and therefore we can consider these Hardy spaces as weighted Bergman spaces.

4. Composition operators on weighted Bergman spaces

First we consider the continuity of C_{φ} on B_{1}^{α}(X). We start with the following result.

Lemma 4. Let α ≥ −1. If z ∈ D and f ∈ B_{1}^{α}(X), then
∥f(z)∥_{X} ≤ C∥f∥_{B_{1}^{α}(X)}(1 − |z|^{2})^{−(α+2)},

where C is independent of f.

Proof. By [Sm, Lemma 2.5],

|l(f(z))| ≤ C∥l ∘ f∥_{B_{1}^{α}}(1 − |z|^{2})^{−(α+2)}, l ∈ X^{∗},

and C does not depend on l and f. Since ∥l ∘ f∥_{B_{1}^{α}} ≤ ∥l∥ ∥f∥_{B_{1}^{α}(X)} we are done.

It follows immediately from Lemma 4 that evaluations are continuous on B_{1}^{α}(X), the compact open topology is weaker than the norm one and that B_{1}^{α}(X) is a Banach space.

Proposition 5. Let α ≥ −1. The composition operator C_{φ}: B_{1}^{α}(X) → B_{1}^{α}(X) is continuous. In fact, for each α ≥ −1 there exists a constant C(α) such that

∥C_{φ}∥ ≤ C(α)(\frac{1 + |φ(0)|}{1 − |φ(0)|})^{α+2}.

Proof. The proof is standard but for completeness we include it. The cases α = 0 and α = −1 are proved in [LST, Proposition 1]. For a = φ(0) let φ_{a}(z) := (a − z)/(1 − \overline{a}z). Then ψ := φ_{a} ∘φ: D → D is analytic, ψ(0) = 0 and φ = φ_{a} ∘ψ.

Since z ↦ ∥f ∘ φ_{a}(z)∥_{X} is subharmonic, Littlewood subordination theorem [CM, p. 30] yields

∫_{0}^{2π} ∥f ∘ φ(re^{iθ})∥_{X} dθ ≤ ∫_{0}^{2π} ∥f ∘ φ_{a}(re^{iθ})∥_{X} dθ

for all 0 < r < 1. Therefore,

∥C_{φ}f∥_{B_{1}^{α}(X)} ≤ \frac{1}{π} ∫_{D} ∥f ∘ φ_{a}(z)∥_{X}(1 − |z|^{2})^{α} dA(z).

By changing the variable in the last integral, we get

\begin{aligned}∥C_{φ}f∥_{B_{1}^{α}(X)} & ≤ \frac{1}{π} ∫_{D} ∥f(w)∥_{X}(1 − |φ_{a}(w)|^{2})^{α} \frac{(1 − |a|^{2})^{2}}{|1 − \overline{a}w|^{4}} dA(w) \\ & ≤ C(α)(\frac{1 + |a|}{1 − |a|})^{α+2}∥f∥_{B_{1}^{α}(X)}. \end{aligned}

By [L, Corollary 2.7], it follows for α > −1 that the space B_{1}^{α} is isomorphic to l_{1} . Therefore, by the well-known properties of l_{1} , we have:

Proposition 6. Let α > −1. The following statements are equivalent:

(a) C_{φ}: B_{1}^{α} → B_{1}^{α} is non-compact.

(b) C_{φ}: B_{1}^{α} → B_{1}^{α} is non-Rosenthal.

(c) There exist continuous linear operators S: l^{1} → B_{1}^{α} and T: B_{1}^{α} → l^{1} such that T ∘ C_{φ} ∘ S = id_{l_{1}} .

The proposition above can also be obtained using interpolating sequences in B_{1}^{α} (cf. [HRS, Theorem 3.1]) without referring to the isomorphic classification of B_{1}^{α} due to Lusky [L].

The following result was proved by Liu, Saksman and Tylli in [LST] for the spaces H_{1}(X) and B_{1}(X). We only have to check that the same argument is valid for all spaces B_{1}^{α}(X). Let us note that the Banach–Steinhaus theorem cannot be used to obtain (9) for any infinite dimensional Banach space X.

The operators V_{k} defined in the next proposition are related to de la Vallée– Poussin summability kernels.

Proposition 7. Let α ≥ −1, k ∈ N and X a Banach space. Define the operator V_{k} by setting

V_{k}f(z) = ∑_{n=0}^{k} f̂_{n}z^{n} + ∑_{n=k+1}^{2k−1} \frac{2k − n}{k}f̂_{n}z^{n}

for analytic f: D → X with the Taylor expansion f = ∑_{n=0}^{∞} f̂_{n}z^{n}. Then there is C > 0 such that

(8)  ∥V_{k}f∥_{B_{1}^{α}(X)} ≤ C∥f∥_{B_{1}^{α}(X)}

for all f ∈ B_{1}^{α}(X). Moreover, given ε > 0 and r ∈ (0, 1) there is k_{0} = k_{0}(ε, r) > 0 such that for k ≥ k_{0}

(9) ∥f(z) − V_{k}f(z)∥_{X} ≤ ε∥f∥_{B_{1}^{α}(X)}  for all |z| ≤ r and f ∈ B_{1}^{α}(X).

Further, if X is reflexive, respectively does not contain a copy of l_{1} , then the operator V_{k}: B_{1}^{α}(X) → B_{1}^{α}(X) is weakly compact, respectively Rosenthal.

Proof. By [LST, Proposition 2] we know that (8) and (9) are valid for H_{1}(X) with C = 2. Let f ∈ B_{1}^{α}(X) and α > −1. It is easily seen that

(10)  ∥f∥_{B_{1}^{α}(X)} = 2 ∫_{0}^{1} ∥f_{r}∥_{H_{1}(X)}r(1 − r^{2})^{α} dr,

where g_{s}(z) = g(sz) for 0 < s < 1. Thus (8) is a direct consequence of the corresponding result for H_{1}(X) and the relation V_{k}f_{r} = (V_{k}f)_{r}.

For completeness we give the argument from [LST] to obtain (9). Assume that r ∈ (\frac{1}{2}, 1) and ε > 0 are given. Let f ∈ B_{1}^{α}(X) with ∥f∥_{B_{1}^{α}(X)} ≤ 1. It follows from (10) that there exist a radius r′ ∈ (\sqrt{r} , 1) and a constant C with ∥f_{r′}∥_{H_{1}(X)} ≤ C(α + 1)(1 − \sqrt{r})^{−(α+1)} . Further we can choose k_{0} such that for k ≥ k_{0} we have ∥g(z) − V_{k}g(z)∥_{X} ≤ ε(α + 1)^{−1}(1 − \sqrt{r})^{α+1}C^{−1}∥g∥_{H_{1}(X)} for |z| ≤ \sqrt{r} and all g ∈ H_{1}(X). Thus, for |z| ≤ r we have that |z/r′| ≤ \sqrt{r}, so we get

∥f(z) − V_{k}f(z)∥_{X} = ∥f_{r′}(\frac{z}{r′} ) − V_{k}f_{r′}(\frac{z}{r′} )∥_{X} ≤ ε.

The final statement follows exactly as in [LST].

Let 𝒰 be a closed ideal of operators between Banach spaces. For T ∈ ℒ (X) define ∥T ∥_{𝒰} = inf{∥T − S∥ : S ∈ 𝒰 }. Let 𝒲 and ℛ be the closed ideal of weakly compact respectively Rosenthal operators on X.

Theorem 8. Let X be reflexive, respectively a Banach space not containing a copy of l_{1} . For each α ≥ −1 there exists a constant C(α) such that for C_{φ} acting on B_{1}^{α}(X) we have

∥C_{φ}∥_{𝒰} ≤ C(α) lim sup_{|w|→1} \frac{N_{φ,α+2}(w)}{(− log |w|)^{α+2}} ,

where 𝒰 is 𝒲 respectively ℛ.

Proof. Let f ∈ B_{1}^{α}(X) be arbitrary and fix an arbitrary r ∈ (0, 1). Without loss of generality, we may assume that φ(0) = 0. We have that ∥f(0)−V_{k}f(0)∥_{X} = 0. By (2) and (3) we get

∥C_{φ}(f − V_{k}f)∥_{B_{1}^{α}(X)} ∼ \frac{1}{2π} ∫_{rD} Ñ_{φ,α+2}(w) d[Δ(∥f − V_{k}f∥_{X})](w)

+ \frac{1}{2π} ∫_{D\rD} Ñ_{φ,α+2}(w) d[Δ(∥f − V_{k}f∥_{X})](w)

:= I_{r,k} + J_{r,k}.

To estimate the first term I_{r,k} observe that by [Sh2, Corollary 10.4(b)], N_{φ}(w) ≤ log(1/|w|) for each w ∈ D. Hence for all w ∈ D

Ñ_{φ,α+2}(w) ≤ N_{φ}(w) ≤ log(\frac{1}{|w|}).

Therefore we get,

I_{r,k} ≤ \frac{1}{2π} ∫_{rD} log(\frac{r}{|w|}) d[Δ(∥f − V_{k}f∥_{X})](w)

+ \frac{1}{2π} log(\frac{1}{r}) ∫_{rD} d[Δ(∥f − V_{k}f∥_{X})](w).

Hence by (6),

\frac{1}{2π} ∫_{rD} log(\frac{r}{|w|}) d[Δ(∥f − V_{k}f∥_{X})](w)

= \frac{1}{2π} ∫_{D} log(\frac{1}{|w′|}) d[Δ(∥(f − V_{k}f)_{r}∥_{X})](w′) = ∥(f − V_{k}f)_{r}∥_{H_{1}(X)}.

Further,

∥(f − V_{k}f)_{r}∥_{H_{1}(X)} = \frac{1}{2π} ∫_{0}^{2π} ∥(f − V_{k}f)_{r}(e^{iθ})∥_{X} dθ ≤ sup_{|w|=r} ∥f(w) − V_{k}f(w)∥_{X}.

Let now τ be a test function on the plane with 0 ≤ τ ≤ 1, the support of τ is contained in \frac{1}{2}(r + 1)D and τ ≡ 1 on rD. Then

∫_{rD} d[Δ(∥f − V_{k}f∥_{X})](w) ≤ ∫ τ(w) d[Δ(∥f − V_{k}f∥_{X})](w)

= \frac{1}{2π} ∫_{(r+1)D/2} ∥f(w) − V_{k}f(w)∥_{X}Δτ(w) dA(w)

≤ M ∫_{(r+1)D/2} ∥f(w) − V_{k}f(w)∥_{X} dA(w),

where M := (1/2π) max{|Δτ(w)| : w ∈ C} is finite. By Proposition 7, we get for every r ∈ (0, 1) that

lim_{k→∞} sup_{∥f∥_{B^{α}_{1}(X)}≤1} I_{r,k} = 0.

For the second term J_{r,k} we first notice that Ñ_{φ,α+2}(w) ≤ 2^{α+1}N_{φ,α+2}(w) for all w ∈ D. Therefore

J_{r,k} ≤ sup_{w∈D\rD}(\frac{N_{φ,α+2}(w)}{(− log |w|)^{α+2}} )\frac{2^{α+1}}{2π}∫_{D\rD}(log(\frac{1}{|w|}))^{α+2}d[Δ(∥f − V_{k}f∥_{X})](w).

Since log(1/|w|) and 1 − |w|^{2} are comparable for all w ∈ D \ rD, there is M(α, r) such that by (6), (7) and (8),

J_{r,k} ≤ M(α, r) sup_{w∈D\rD} \frac{N_{φ,α+2}(w)}{(− log |w|)^{α+2}} ∥f − V_{k}f∥_{B_{1}^{α}(X)}

≤ CM(α, r) sup_{w∈D\rD} \frac{N_{φ,α+2}(w)}{(− log |w|)^{α+2}} ∥f∥_{B_{1}^{α}(X)}.

Consequently,

∥C_{φ}∥_{𝒰} ≤ C(α){ lim_{k→∞} sup_{∥f∥_{B^{α}_{1}(X)}≤1} I_{k,r} + lim_{r→1} sup_{∥f∥_{B^{α}_{1}(X)}≤1} J_{r,k}}

≤ C(α) lim sup_{|w|→1} \frac{N_{φ,α+2}(w)}{(− log |w|)^{α+2}} .

Corollary 9. Let α ≥ −1. Then C_{φ}: B_{1}^{α}(X) → B_{1}^{α}(X) is weakly compact, respectively Rosenthal, if and only if X is reflexive, respectively does not contain a copy of l_{1} , and

(11)  lim sup_{|w|→1} \frac{N_{φ,α+2}(w)}{(− log |w|)^{α+2}} = 0.

Proof. One direction follows from Proposition 1. Indeed, by Proposition 6 and Sarason [S] (cf. also [J]) for α = −1, every Rosenthal operator C_{φ} on B_{1}^{α} is compact. By [CM, Example 3.2.6, Theorem 3.12] compactness of C_{φ} on B_{p}^{α} is independent of 0 < p < ∞ for α ≥ −1. Thus with p = 2 [Sh1, Theorems 6.8 and 2.3] give that C_{φ} on B_{1}^{α} is compact if and only if condition (11) is valid.

The converse statement follows directly from Theorem 8.

5. Composition operators on general vector-valued spaces

In this section E denotes a Banach space of analytic functions on the unit disc D which contains the constant functions and such that its closed unit ball U(E) is compact for the compact open topology co. These assumptions imply the following properties of the space E which will be frequently used later.

(a) For every z ∈ D the evaluation map δ_{z}: E → C, δ_{z}(f) = f(z), is continuous and non-zero.

(b) The map Δ: D → E^{∗}, Δ(z) = δ_{z}, z ∈ D, is a vector valued analytic function. Indeed, since E is a separating subset of the dual E^{∗∗} of E^{∗}, we can apply a result of Grosse-Erdmann [GE, Theorem 5.2] which ensures it is enough to check f ∘ Δ ∈ H(D) for every f ∈ E. This is trivially satisfied.

(c) By the Dixmier–Ng theorem [N], the space

∗E := {u ∈ E^{∗} : u | U(E) is co-continuous},

endowed with the norm induced by E^{∗}, is a Banach space and the evaluation map E → (^{∗}E)^{∗}, f ↦ [u ↦ u(f)] is an isometric isomorphism. In particular ^{∗}E is a predual of E.

(d) The linear span of the set {δ_{z} : z ∈ D} is contained and norm dense in ^{∗}E. This follows easily from the Hahn–Banach theorem: if f ∈ E = (^{∗}E)^{∗} vanishes on all the evaluation maps it must be zero.

Let X be a Banach space. The vector valued space E[X] associated with E is defined as

E[X] := {f ∈ H(D, X) : x^{∗} ∘ f ∈ E for every x^{∗} ∈ X^{∗} }.

Given f ∈ E[X], the map T_{f}: X^{∗} → E, T_{f}(x^{∗}) = x^{∗} ∘ f, is well defined, linear and weak^{∗} -pointwise continuous. By the closed graph theorem T_{f} is continuous and the supremum ∥f∥_{E[X]} := sup_{∥x^{∗}∥≤1} ∥x^{∗} ∘f∥_{E} is finite. We endow E[X] with this norm. Observe that the map Δ: D → ^{∗}E defined in (b) above (also see (d)) belongs to E[^{∗}E] and ∥Δ∥_{E[^{∗}E]} = 1.

A version of the following linearization result for E = H^{∞} can be found in [M] and for E = B_{∞}^{v} in [BBG].

Lemma 10. The space E[X] is isomorphic to the space of operators L(^{∗}E, X) in a canonical way. In particular, it is a Banach space.

Proof. First we define χ: L(^{∗}E, X) → E[X] by χ(T) := T ∘ Δ. The map χ is well defined, linear, continuous and its norm is less than or equal to 1.

Fix g ∈ E[X] and u ∈ ^{∗}E and define ψ(g)(u) : X^{∗} → C by (ψ(g)(u))(x^{∗}) := u(x^{∗} ∘ g) for x^{∗} ∈ X^{∗}. Clearly

|(ψ(g)(u))(x^{∗})| ≤ ∥u∥_{^{∗}E}∥x^{∗} ∘ g∥_{E} ≤ ∥u∥_{^{∗}E}∥x^{∗}∥_{X^{∗}}∥g∥_{E[X]},

for all x^{∗} ∈ X^{∗}, by the definition of the norm in E[X]. This yields ψ(g)(u) ∈ X^{∗∗} and ψ(g) ∈ L(^{∗}E, X^{∗∗}) with ∥ψ(g)∥ ≤ ∥g∥_{E[X]} . On the other hand ψ(g)(δ_{z}) = g(z) ∈ X for all z ∈ D. By the property (d) above we conclude ψ(g) ∈ L(^{∗}E, X), and the map ψ: E[X] → L(^{∗}E, X) is well defined, linear continuous and its norm is less than or equal to 1.

To complete the proof it is enough to observe that ψ ∘ χ and χ ∘ ψ coincide with the identities on L(^{∗}E, X) and E[X] respectively.

Let φ: D → D be holomorphic. The closed graph theorem and the argument in Proposition 1 imply that the composition operator C_{φ}: E[X] → E[X] is continuous if and only if C_{φ}: E → E is continuous. Moreover the result stated in Proposition 1 remains valid for the spaces of type E[X]. In order to obtain a converse we proceed as follows. Assume C_{φ} is continuous on E. The transpose map C′_{φ}: E^{∗} → E^{∗} maps ^{∗}E into itself; indeed, by the property (d) above it is enough to check that C′_{φ}(δ_{z}) = δ_{φ(z)} belongs to ^{∗}E for all z ∈ D which is trivial. Now the isomorphism proved in Lemma 10 transforms the operator C_{φ} on E[X] into the wedge operator W_{φ}: L(^{∗}E, X) → L(^{∗}E, X), W_{φ}(T) = id_{X} ∘ T ∘ (C′_{φ}|^{∗}E). More precisely, with the notations introduced in the proof of Lemma 10, (ψ ∘ C_{φ} ∘ χ)(S) = S ∘ (C′_{φ}|^{∗}E) for every S ∈ L(^{∗}E, X) which implies C_{φ} = χ ∘ W_{φ} ∘ ψ. We are ready to prove the main results in this section.

Proposition 11. Let C_{φ}: E → E be compact and let X be a Banach space.

(1) If X is reflexive, then C_{φ}: E[X] → E[X] is weakly compact.

(2) If X does not contain a copy of l_{1} , then C_{φ}: E[X] → E[X] is a Rosenthal operator.

Proof. Since C′_{φ}|^{∗}E is a compact operator on ^{∗}E, we can apply [ST, Theorem 2.9] for part (1) and [LS, Corollary 2.13] for part (2) to the wedge operator W_{φ} to reach the conclusion.

Corollary 12 [LST, Theorem 4]. Let φ: D → D be holomorphic and let X be a Banach space. The operator C_{φ} on the Bloch space ℬ(X) is weakly compact (respectively Rosenthal) if and only if C_{φ} is Rosenthal on ℬ and X is reflexive (respectively X does not contain a copy of l_{1} ).

Proof. First observe that the Bloch space ℬ satisfies the assumptions we impose on the general space E considered in this section. In fact, if f ∈ ℬ, it follows by integration that

max_{|z|≤r} |f(z)| ≤ \{1 + \frac{1}{2} log(\frac{1 + r}{1 − r} )\}∥f∥_{ℬ} (0 ≤ r < 1).

Therefore, every bounded set in ℬ is relatively compact with respect to the compact-open topology and point evaluations are bounded linear functionals on ℬ. To see that the closed unit ball U(ℬ) of ℬ is a compact subset of (ℬ, co) it is enough to observe that U(ℬ) is a normal family by Montel’s theorem. If f_{n} → f with respect to the co-topology and ∥f_{n}∥_{ℬ} ≤ 1 for all n, then also f′_{n} → f′ in the co-topology and consequently ∥f∥_{ℬ} ≤ 1.

It is now easy to see that the vector valued Bloch space ℬ(X) coincides with the space ℬ[X] defined in this section and that

∥f∥_{ℬ[X]} ≤ ∥f∥_{ℬ(X)} ≤ 2∥f∥_{ℬ[X]}

for every f ∈ ℬ[X].

By Proposition 11, it remains to show that every Rosenthal composition operator on ℬ is compact. This is proved below.

A sequence (z_{n}) ⊂ D is called δ -separated if inf_{n≠k} |(z_{n} − z_{k})/(1 − \overline{z}_{k}z_{n})| > δ > 0.

Proposition 13. There is a constant δ > 0 such that if (w_{n}) in D is δ - separated, then there exist a continuous linear operator R: l^{∞} → ℬ and functions h_{k} := R(e_{k}) ∈ ℬ such that

h′_{k}(w_{n}) = 0, if n ≠ k, (1 − |w_{n}|^{2})h′_{n}(w_{n}) = 1.

Proof. By the proof of Proposition 1 in [MM] (see [Ro]), there are two continuous linear operators

S: ℬ → l^{∞}, S(f) = ((1 − |w_{n}|^{2})f′(w_{n}))_{n}

and

T: l^{∞} → ℬ, T ((ξ_{n}))z = ∑_{n=1}^{∞} ξ_{n} \frac{1}{3\overline{w}_{n}} \frac{(1 − |w_{n}|^{2})^{3}}{(1 − \overline{w}_{n}z)^{3}}

such that ∥id − ST ∥ < 1. Thus ST has an inverse (ST)^{−1}: l^{∞} → l^{∞}, and therefore S has a right inverse R := T (ST)^{−1}: l^{∞} → ℬ. Since SR(e_{k}) = e_{k} for all k, we get that (1 − |w_{n}|^{2})h′_{k}(w_{n}) = δ_{nk} for all n and k.

Proposition 14. The following statements are equivalent:

(a) C_{φ}: ℬ → ℬ is non-compact.

(b) There exist continuous linear operators R: l^{∞} → ℬ and Q: ℬ → l^{∞} such that Q ∘ C_{φ} ∘ R = id_{l_{∞}}.

(c) C_{φ}: ℬ → ℬ is not a Rosenthal operator.

In [LST] the equivalence (a) ⇔ (c) is obtained by other methods.

Proof. (a) ⇒ (b): Since C_{φ} is non-compact, by [MM, Theorem 2], there is a sequence (z_{n}) ∈ D and a constant ε > 0 so that |φ(z_{n})| → 1 and

\frac{(1 − |z_{n}|^{2})|φ′(z_{n})|}{1 − |φ(z_{n})|^{2}} ≥ ε for all n ≥ 1.

Since |φ(z_{n})| → 1, passing to a subsequence, we can apply Proposition 13 and get a continuous linear operator R: l^{∞} → ℬ and functions h_{k} := R(e_{k}) ∈ ℬ such that

h′_{k}(φ(z_{n})) = 0, if n ≠ k, (1 − |φ(z_{n})|^{2})h′_{n}(φ(z_{n})) = 1.

Hence R(ξ) = ∑_{k=1}^{∞} ξ_{k}h_{k} for all ξ = (ξ_{k}) ∈ c_{0} . Now we define a map

Q: ℬ → l^{∞}, Q(f) = (\frac{1 − |φ(z_{n})|^{2}}{φ′(z_{n})} f′(z_{n}))_{n}.

Since

∥Q(f)∥ ≤ \frac{1}{ε} sup_{n} |f′(z_{n})|(1 − |z_{n}|^{2}) ≤ \frac{1}{ε}∥f∥_{ℬ} for all f ∈ ℬ ,

the map is well defined, linear and continuous. For every ξ = (ξ_{n}) ∈ c_{0} ,

Q ∘ C_{φ} ∘ R(ξ) = ((1 − |φ(z_{n})|^{2}) ∑_{k=1}^{∞} ξ_{k}h′_{k}(φ(z_{n})))_{n}.

Consequently, we get that Q ∘ C_{φ} ∘ R(ξ) = ξ for all ξ ∈ c_{0} . Using a result of Rosenthal [Rs, Proposition 1.2] we get the conclusion.

The implications (b) ⇒ (c) and (c) ⇒ (a) are obvious.

Corollary 15. Let v be a weight on D. Let C_{φ} be continuous on B_{∞}^{v}. The operator C_{φ} is weakly compact (respectively Rosenthal) on B_{∞}^{v}(X) if and only if C_{φ} is Rosenthal on B_{∞}^{v} and X is reflexive (respectively X does not contain a copy of l_{1} ).

Proof. It is well known (e.g. [BS], [BBT]) that the space B_{∞}^{v} satisfies the conditions imposed on the general space E considered in this section. Moreover it is easy to see that the vector valued space B_{∞}^{v}(X) coincides isometrically with the space B_{∞}^{v}[X] defined here.

The associated weight is defined by

ṽ(z) = (sup{|f(z)| : ∥f∥_{v} ≤ 1})^{−1}, z ∈ D.

It is better tied to the space B_{∞}^{v} than v itself [BBT], and B_{∞}^{v} = Bṽ ; ∞ holds isometrically. By [BDLT] the operator C_{φ} is continuous on B_{∞}^{v} if and only if

sup_{z∈D} \frac{v(z)}{ṽ(φ(z))} < ∞.

Moreover, by [BDL, Theorem 1], the operator C_{φ} is Rosenthal on B_{∞}^{v} if and only if it is compact. Hence the conclusion follows from Proposition 11.

Weakly compact composition operators 247

If we take v(z) = 1 for every z ∈ D in Corollary 15, we obtain as a particular case Theorem 6 and part of Theorem 7 in [LST].

To conclude we consider only radial weights v, that is, v(z) = v(|z|). A radial weight v is called essential, if there exists a C > 0 such that v(z) ≤ ṽ(z) ≤ Cv(z). We can now apply [BDLT, Theorem 3.3] to get the following corollary.

Corollary 16. Let v be an essential weight. Then C_{φ}: B_{∞}^{v}(X) → B_{∞}^{v}(X) is weakly compact (respectively Rosenthal) if and only if X is reflexive (respectively does not contain a copy of l_{1} ) and

lim_{r→1} sup_{\{z:|φ(z)|>r\}} \frac{v(z)}{v(φ(z))} = 0  or  ∥φ∥_{∞} < 1.

As a consequence of Lemma 4 and Fatou’s lemma, the weighted Bergman spaces B_{p}^{α}, 1 ≤ p < ∞, α ≥ −1, satisfy the conditions imposed on the scalar valued Banach space E. This permits to use Proposition 6 and Proposition 11 to get consequences on vector-valued composition operators on spaces of type B_{p}^{α}[X] as defined in this section. It is important to point out that the classical vectorvalued space B_{p}^{α}(X) is continuously included in but different from B_{p}^{α}[X]. This is the reason why we had to treat composition operators defined on them with another method.

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Received 13 September 1999