R A Dana
email: dana@ceremade.dauphine.fr

C Le Van
email: levan@univ-paris1.fr

E cient allocations and Equilibria with short-selling and Incomplete Preferences

Keywords: Uncertainty, risk, risk adjusted prior, no arbitrage, equilibrium with short-selling, incomplete preferences, equilibrium with inertia. JEL Classification: C62, D50, D81, D84, G1

This article reconsiders the theory of existence of e cient allocations and equilibria when consumption sets are unbounded below under the assumption that agents have incomplete preferences. It is motivated by an example in the theory of assets with short-selling where there is risk and ambiguity. Agents have Bewley's incomplete preferences. As an inertia principle is assumed in markets, equilibria are individually rational. It is shown that a necessary and su cient condition for the existence of an individually rational e cient allocation or of an equilibrium is that the relative interiors of the risk adjusted sets of probabilities intersect. The more risk averse, the more ambiguity averse the agents, the more likely is an equilibrium to exist. The paper then turns to incomplete preferences represented by a family of concave utility functions. Several definitions of e ciency and of equilibrium with inertia are considered. Su cient conditions and necessary and su cient conditions are given for the existence of e cient allocations and equilibria with inertia.

Introduction

The issue of existence of an equilibrium for finite markets with short-selling is an old problem first considered in the early seventies by Grandmont [START_REF] Grandmont | On the temporary competitive equilibrium[END_REF], Hart [START_REF] Hart | On the existence of an equilibrium in a securities model[END_REF] and Green [START_REF] Green | Temporary general equilibrium in a sequential trading model with spot and futures transactions[END_REF] and reconsidered later by Hammond [START_REF] Hammond | Overlapping expectations and Hart's condition for equilibrium in a securities model[END_REF] and Page [START_REF] Jr | On equilibrium in Hart's securities exchange model[END_REF], [START_REF] Jr | Arbitrage and asset prices[END_REF]. In these early papers, investors were assumed to hold a single probabilistic belief (homogeneous or heterogeneous) and be risk averse von Neumann-Morgenstern (vNM) utility maximizers. Two su cient conditions for the existence of an equilibrium were given:

1. a condition which expresses that investors are su ciently similar in their beliefs and risk aversions so that there exists a non empty set of prices (the no-arbitrage prices) for which no agent can make costless unbounded utility nondecreasing purchases, 2. a collective absence of arbitrage condition, which requires that investors do not engage in mutually compatible, utility nondecreasing trades.

These conditions have been generalized to abstract economies and are known as the existence of no-arbitrage prices condition (see Werner [START_REF] Werner | Arbitrage and the Existence of Competitive Equilibrium[END_REF]) and the no unbounded utility arbitrage condition (NUBA) ( see Page [START_REF] Jr | Arbitrage and asset prices[END_REF]). They were shown to be equivalent under adequate hypotheses. Other su cient conditions were given. For a review of the subject in finite dimension, see Allouch et al [START_REF] Allouch | The geometry of arbitrage and the existence of competitive equilibrium[END_REF], Dana et al [START_REF] Dana | On the di↵erent notions of arbitrage and existence of equilibrium[END_REF], Page [START_REF] Jr | On equilibrium in Hart's securities exchange model[END_REF], [START_REF] Jr | Arbitrage and asset prices[END_REF]. All this trend of literature assumes complete preferences.

This paper extends the previous theory to incomplete preferences represented by families of concave utility functions. Such incomplete preferences include Bewley's [START_REF] Bewley | Knightian Decision Theory: Part I[END_REF] and Rigotti and Shannon [START_REF] Rigotti | Uncertainty and Risk in Financial Markets[END_REF] incomplete preferences. Under this representation assumption, it is easy to define and characterize the concepts of no-arbitrage prices and no unbounded utility arbitrage. Weak and strong concepts of e ciency and equilibria are defined. As in the case of complete preferences, existence of a no-arbitrage price is shown to be equivalent to NUBA and to be a su cient condition for the existence of weakly e cient allocations and weak equilibria. Under further adequate assumptions (for example strict concavity and monotonicity of utilities), it is shown to be necessary and su cient for the existence of e cient allocations and equilibria.

As Page [START_REF] Jr | Arbitrage and asset prices[END_REF] and Werner [START_REF] Werner | Arbitrage and the Existence of Competitive Equilibrium[END_REF], the paper is motivated by an example in the theory of assets with short-selling. Agents are assumed not to have enough information to quantify uncertainty by a single probability, hence each agent has a set of priors. Agents are further assumed to have risk averse Bewley's [START_REF] Bewley | Knightian Decision Theory: Part I[END_REF] (or Rigotti and Shannon [START_REF] Rigotti | Uncertainty and Risk in Financial Markets[END_REF] ) incomplete preferences. Under standard conditions on utility indices (strict concavity and increasingness) and sets of priors (convexity and compactness), it is shown that a necessary and su cient for the existence of an individually rational e cient allocation or equilibrium is that the relative interiors of the agents' sets of risk adjusted probabilities intersect or that agents do not engage in mutually compatible trades that have non negative expectations with respect to their risk adjusted probabilities. The first condition generalizes the overlapping expectations given by Grandmont [START_REF] Grandmont | On the temporary competitive equilibrium[END_REF], Green [START_REF] Green | Temporary general equilibrium in a sequential trading model with spot and futures transactions[END_REF] and Hammond [START_REF] Hammond | Overlapping expectations and Hart's condition for equilibrium in a securities model[END_REF] for the case of single belief. The second generalizes the NUBA type of condition given by Hart [START_REF] Hart | On the existence of an equilibrium in a securities model[END_REF]. In a vNM framework, Hart [START_REF] Hart | On the existence of an equilibrium in a securities model[END_REF] further shows that the more risk averse the agents, the more likely is an equilibrium to exist. In the Bewley's setting, it is shown that the more risk averse, the more ambiguity averse the agents, the more likely is an equilibrium to exist.

The example is also shown to have an interesting feature: the arbitrage concepts coincide with those of a Gilboa-Schmeidler's agent with same sets of priors and utility indices. Hence the condition that the relative interiors of the sets of risk adjusted probabilities intersect is also necessary and su cient for the existence of an equilibrium in a model with Gilboa-Schmeidler's preferences (with same sets of priors and utility indices). This may seem at odds with the literature on e ciency or equilibria with Bewley's incomplete preferences (see Bewley's [START_REF] Bewley | Knightian Decision Theory: Part I[END_REF], Rigotti and Shannon [START_REF] Rigotti | Uncertainty and Risk in Financial Markets[END_REF] or Dana and Riedel [START_REF] Dana | Intertemporal Equilibria with Knightian Uncertainty[END_REF]) which highlights the di↵erences with using maxmin complete preferences. However it is not as a di↵erent issue is addressed, that of existence of equilibrium with short selling.

The paper is organized as follows: section 2 deals with the example and provides an existence theorem while section 3 deals with its generalization.

An example 2.1 Bewley preferences

We consider a standard Arrow-Debreu model of complete contingent security markets with short selling. There are two dates, 0 and 1. At date 0, there is uncertainty about which state s from a state space ⌦= {1, ..., k} will occur at date 1. At date 0, agents who are uncertain about their future endowments trade contingent claims for date 1. The space of contingent claims is the set of random variables from⌦ ! R. The random variable X which equals x 1 in state 1, x 2 in state 2 and x k in state k, is identified with the vector in

X 2 R k , X = (x 1 , . . . , x k ). Let 4 = {⇡ 2 R k + : P k s=1 ⇡ s = 1} be the probability simplex in R k . Let int4 = {⇡ 2 4, ⇡ s > 0, 8 s}. For A ✓ 4, intA = {p 2 A | E ⇡ (X) := P k l=1 ⇡ l x l the expectation of X.
Finally, for a given price p 2 R k , p • X := P k l=1 p l x l , the price of X. There are m agents indexed by i = 1, . . . , m. Agent i has an endowment

E i 2 R k of contingent claims. Let (E i ) m
i=1 be the m-tuple of endowments and E = P m i=1 E i be the aggregate endowment. We assume that agent i has a convex compact set of priors P i ✓int4 and an incomplete Bewley preference relation ⌫ over R k defined by

X ⌫ i Y () E ⇡ (u i (X)) E ⇡ (u i (Y )), 8 ⇡ 2 P i (1)
where u i : R ! R is a strictly concave, increasing di↵erentiable utility index fulfilling u i (0) = 0. The associated strict preference is

X i Y if X ⌫ i Y and E ⇡ (u i (X)) > E ⇡ (u i (Y )) for some ⇡ 2 P i .

Individual and collective absence of arbitrage

In this subsection, we define and characterize the useful vectors of a Bewley preference relation of type [START_REF] Allouch | The geometry of arbitrage and the existence of competitive equilibrium[END_REF]. They are the directions such that trading at any positive scale makes the agent better o↵. Our main result is that they coincide with those of a Gilboa-Schmeidler utility defined by (u, P ) ( see (2) below) .

We then recall the concepts of no-arbitrage prices and of collective absence of arbitrage (NUBA). As these concepts only depend on useful vectors, we obtain that two economies with Bewley preferences or Gilboa-Schmeidler utilities with same indices and sets of priors have same sets of no-arbitrage prices and same NUBA condition.

Useful vectors

To simplify notations, in this subsection, the agent's index is omitted. We consider an agent described by a pair (u, P ) of a utility index and a set of priors. For ⇡ 2 P , let

b P ⇡ (X) = {Y 2 R k | E ⇡ (u(Y )) E ⇡ (u(X))} be the set of contingent claims preferred to X for the utility E ⇡ (u(.)) and R ⇡ (X) = {W 2 R k | E ⇡ (u(X + W )) u(X), 8 2 IR + } be its asymptotic cone. As E ⇡ (u(.)) is concave, R ⇡ (X) is independent of X and denoted R ⇡ .
Taking X = 0, we obtain:

R ⇡ = {W 2 R k | E ⇡ (u( W )) u(0) = 0, 8 2 IR + } Let b P (X) = {Y 2 R k | E ⇡ (u(Y )) E ⇡ (u(X)), 8 ⇡ 2 P }
be the set of contingent claims preferred to X for the Bewley's preference defined by (1) and R(X) be its asymptotic cone. From Rockafellar's [START_REF] Rockafellar | Convex Analysis[END_REF]

corollary 8.3.3, R(X) = \ ⇡2P R ⇡ (X) = \ ⇡2P R ⇡ . Hence it is independent of X and denoted R. R = {W 2 R k | E ⇡ (u( W )) 0, 8 2 IR + , ⇡ 2 P }
and is called the set of useful vectors for ⌫. For a given pair (u, P ), let

V (X) = min ⇡2P E ⇡ (u(X)) (2)
be the Gilboa-Schmeidler's utility.

{W 2 R k | V ( W ) 0, 8 2 IR + } = {W 2 R k | E ⇡ (u( W )) 0, 8 2 IR + , ⇡ 2 P }
Hence it coincides with R. Furthermore given C 2 IR k a reference point, the C-reference dependent ambiguity averse (RAA) utility, axiomatized by Mihm [START_REF] Mihm | Reference-Dependent Ambiguity Aversion[END_REF] is defined by

V C (X) = min ⇡2P [E ⇡ (u(X)) E ⇡ (u(C))] (3)
This is a concave variational utility, hence the set of useful vectors is independent of X.

As V C (C) = 0, R V C = {W 2 R k | V C (C + W ) 0, 8 2 IR + }. Equivalently R V C = {W 2 R k | E ⇡ (u(C + W )) E ⇡ (u(C)), 8 2 IR + , ⇡ 2 P } = R
The previous discussion is summarized in the following lemma:

Lemma 1 The set of useful vectors for a Bewley preference of type (1) coincides with the set of useful vectors for a Gilboa-Schmeidler preference (2) or of an RAA utility (3) for any reference point C 2 IR k .

Let us recall a characterization of useful vectors proven in Dana and Le Van [START_REF] Dana | Overlapping risk adjusted sets of priors and the existence of e cient allocations and equilibria with short-selling[END_REF]. To this end, let

e P = ⇢ p 2 4 | 9 ⇡ 2 P, Z 2 R k s. t. p s = ⇡ s u 0 (z s ) E ⇡ (u 0 (Z)) , 8 s = 1, . . . , k (4)
be the set of risk adjusted probabilities. We have (see Dana and Le Van [4]) :

Lemma 2 R = {W 2 R k | E p (W ) 0, for all p 2 P }.
In the next two subsections, we introduce two concepts of absence of arbitrage, a concept of individual no-arbitrage and a concept of collective absence arbitrage. These concepts only depend on agents' useful vectors.

We first define the concept of no-arbitrage price (see Werner [START_REF] Werner | Arbitrage and the Existence of Competitive Equilibrium[END_REF]).

Definition 1 A price vector p 2 R k is a " no-arbitrage price" for agent i if p • W > 0, for all W 2 R i \{0}. A price vector p 2 R k is a " no-arbitrage price" for the economy if it is a no-arbitrage price for each agent.

Let S i denote the set of no-arbitrage prices for i. Using Lemma 2, Dana and Le Van [START_REF] Dana | Overlapping risk adjusted sets of priors and the existence of e cient allocations and equilibria with short-selling[END_REF] characterize S i in terms of risk adjusted probabilities.

Lemma 3

1. The set of no-arbitrage prices for agent i is S i = cone int e P i where p 2 int e P i if and only if 9

⇡ 2 P i \ int 4, Z 2 R k , 8s, a < u 0 (z s ) < b and p s = ⇡su 0 (zs)
E⇡(u 0 (Z)) . The more risk averse, the more ambiguity averse the agent, the larger is S i .

The set of no-arbitrage prices for the economy is

\ i S i = cone \ i int e P i .
The more risk averse, the more ambiguity averse the agents, the larger is the set of no-arbitrage prices for the economy.

Collective absence of arbitrage

From now on, a feasible trade is an m tuple W 1 , . . . , W m with W i 2 R k for all i and P i W i = 0. We recall the no-unbounded-arbitrage condition (NUBA) introduced by Page [START_REF] Jr | On equilibrium in Hart's securities exchange model[END_REF] which requires inexistence of unbounded utility nondecreasing feasible trades.

Definition 2

The economy satisfies the NUBA condition if

P i W i = 0 and W i 2 R i for all i, implies W i = 0 for all i.
From Lemma 2, we may characterize the NUBA condition .

Corollary 1 NUBA is equivalent to: there exists no feasible trade W 1 , . . . , W m with W i 6 = 0 for some i that fulfills E ⇡ (W i ) 0 for all i and ⇡ 2 e P i .

Existence of e cient allocations and equilibria

Concepts in equilibrium theory

We next recall standard concepts in equilibrium theory.

Given the allocation of initial endowments

(E i ) m i=1 , an allocation (X i ) m i=1 2 (R k ) m is attainable (or feasible) if P m i=1 X i = E. The set of B-individually rational attainable allocations A((E i ) m i=1 ) is defined by A((E i ) m i=1 ) = ( (X i ) m i=1 2 (R k ) m | m X i=1 X i = E and X i ⌫ i E i , 8 i
) .

Definition 3

Given (E i ) m i=1 , an attainable allocation (X i ) m i=1 is B-e cient if there does not exist (X 0i ) m i=1 attainable such that X 0 i ⌫ i X i for all i with a strict inequality for some i. It is weakly B-e cient if there does not exist

(X 0i ) m i=1 attainable such that E ⇡ (u i (X 0i )) > E ⇡ (u i (X i )), 8 ⇡ 2 P i , 8 i.
It is individually rational (weakly) B-e cient if it is (weakly) B-e cient and X i ⌫ i E i for all i.

Since u i is strictly concave and strictly increasing and P i is compact for all i, (X i ) m i=1 is B-e cient if and only if (X i ) m i=1 is weakly B-e cient (see Lemma 4 and Remark 3 below).

Definition 4 A pair (X ⇤ , p ⇤ ) 2 A((E i ) m i=1 )⇥R k \{0} is a (weak) B-equilibrium with inertia if 1. for each agent i and X i 2 R k , X i i X i⇤ (E ⇡ (u i (X i )) > E ⇡ (u i (X i⇤ )), 8 ⇡ 2 P i , 8 i) implies p ⇤ • X i > p ⇤ • X i⇤ , 2. for each agent i, p ⇤ • X i⇤ = p ⇤ • E i .
Since u i is strictly concave for all i, it may easily be verified that (X ⇤ , p ⇤ ) is a B-equilibrium with inertia if and only if it is a weak B-equilibrium with inertia. Note that at an equilibrium with inertia, the allocation is individually rational.

Existence of e cient allocations and equilibria

The following theorem fully characterizes existence of individually rational Be cient allocations as well as B-equilibria with inertia.

Theorem 1

The following assertions are equivalent:

1. there exists a no-arbitrage price, equivalently \ i int e P i 6 = ;, 2. there exists no feasible trade W 1 , . . . , W m with W i 6 = 0 for some i and E ⇡ (W i ) 0 for all ⇡ 2 e P i and for all i, 3. the set of B-individually rational attainable allocations is compact, 4. there exists a B-individually rational e cient allocation, 5. there exists a B-equilibrium with inertia.

Furthermore any equilibrium price is a no-arbitrage price.

Proof : Since X i ⌫ i E i is equivalent to V E i (X) V E i (E i ) = 0
, the set of individually rational attainable allocations for the economy with Bewley's preferences coincides with the set of individually rational attainable allocations for the economy with RAA utilities with reference points (E i ) m i=1 . Applying Dana and Le Van [START_REF] Dana | Overlapping risk adjusted sets of priors and the existence of e cient allocations and equilibria with short-selling[END_REF] to these utilities, we obtain the equivalence between 1, 2 and 3.

Let us prove that 1 implies 4. Applying Dana and Le Van [START_REF] Dana | Overlapping risk adjusted sets of priors and the existence of e cient allocations and equilibria with short-selling[END_REF] to RAA utilities with reference points (E i ) m i=1 , we obtain the existence of an individually rational e cient allocation ( X0i ) m i=1 for RAA utilities. Let us show that it is Bweakly e cient. Suppose not. Then there exists (X 0i ) m i=1 attainable such that

E ⇡ (u i (X 0i )) > E ⇡ (u i ( Xi )
) for all i and ⇡ 2 P i but then V E i (X 0i ) > V E i ( Xi ) for all i contradicting the e ciency of ( X0i ) m i=1 for RAA utilities. Therefore ( X0i ) m i=1 is a B-individually rational weakly e cient allocation, hence a B-individually rational e cient allocation.

Let us next show that 4 implies 2. Let ( Xi ) m i=1 be a B-e cient allocation. Suppose that there exists a feasible trade W 1 , . . . , W m with W i 2 R i for all i and W i 6 = 0 for some i. We then have E ⇡ (u i ( Xi + tW i )) E ⇡ (u i ( Xi )), for all ⇡ 2 P i for all i and as u i is strictly concave, E ⇡ (u i ( Xi + tW i )) > E ⇡ (u i ( Xi )) for all i such that W i 6 = 0 and ⇡ 2 P i . The allocation ( Xi + tW i ) i2I being feasible, this contradicts the B-e ciency of ( Xi ) m i=1 . Hence 1-4 are equivalent.

Finally let us show that 1 is equivalent to 5. Let us first remark that if (X ⇤ , p ⇤ ) is an equilibrium with inertia, then p ⇤ is a no arbitrage price. Indeed as u i is strictly concave, if W i is useful and

W i 6 = 0, then E ⇡ (u i (X ⇤i + tW i )) > E ⇡ (u i (X ⇤i )), 8 ⇡ 2 P i , hence p ⇤ • W i > 0.
Hence if there exists an equilibrium, there exists a no-arbitrage price and 5 implies 1. Conversely if 1 holds true, then from Dana and Le Van [START_REF] Dana | Overlapping risk adjusted sets of priors and the existence of e cient allocations and equilibria with short-selling[END_REF], the economy with variational utilities (V E i ) i2I has an equilibrium (X ⇤ , p ⇤ ). Let us show that (X ⇤ , p ⇤ ) is a weak B-equilibrium with inertia. Indeed if

E ⇡ (u i (X i )) > E ⇡ (u i (X ⇤i )), 8 ⇡ 2 P i , then V E i (X) > V E i (X ⇤i ), hence p ⇤ • X i > p ⇤ • X ⇤i . Furthermore as the family of utilities (V E i ) i2I is strictly concave, if X ⇤i 6 = E i , then V E i (X ⇤i ) > V E i (E i ), hence X ⇤i i E i proving that (X ⇤ , p ⇤
) is a weak Bewley equilibrium with inertia, hence a Bewley equilibrium with inertia.

Remark 1 When stating theorem 1, we have assumed that all agents had Bewley's incomplete preferences. From Dana and Le Van [START_REF] Dana | Overlapping risk adjusted sets of priors and the existence of e cient allocations and equilibria with short-selling[END_REF], we could have assumed that all agents had Gilboa-Schmeidler's utilities. A more general result is true : agent i may either have a Bewley's incomplete preference or a Gilboa-Schmeidler's utility or an RAA utility with any reference point with utility index u i and sets of priors P i or any utility with useful vectors R i . Indeed if agent i has a Bewley's incomplete preference, one can consider the fictitious agent with an RAA utility with reference point E i , utility index u i and priors P i . From Dana and Le Van [START_REF] Dana | Overlapping risk adjusted sets of priors and the existence of e cient allocations and equilibria with short-selling[END_REF], we obtain the existence of an equilibrium for the ficti-tious economy. As in the proof of 1 implies 4, any equilibrium of the fictitious economy is an equilibrium of the original economy.

Remark 2

The strict concavity of the utility functions plays an important and subtle role. First, for most purposes, when X Y , we may assume that E ⇡ (u(X)) > E ⇡ (u(Y )), 8 ⇡ 2 P. Second, strict concavity is used to prove the equivalence between weak B-equilibrium and B-equilibrium and weak B-e ciency and B-e ciency. Third, as u is strictly concave, for any useful vector W 6 = 0 and any ⇡ 2 P , the map t ! E ⇡ (u(X + tW )) is strictly increasing. In theorem 1, this is used in 4 implies 2 and in the assertion that an equilibrium price is a no-arbitrage price.

Remark 3 In our model an e cient allocation exists i↵ \ i int e P i 6 = ; while a given attainable allocation (X i ) m

i=1 is e cient i↵ the sets of risk-adjusted probabilities e P i (X i ) at X i , i = 1, . . . , m intersect (see Rigotti and Shannon [START_REF] Rigotti | Uncertainty and Risk in Financial Markets[END_REF]).

3 An abstract economy with incomplete preferences

The economy

Many agents in financial markets are not single individuals but groups of individuals. We learn from social choice that it is not possible to assign a complete preference satisfying reasonable assumptions to a group of individuals. A natural way to resolve this paradox is to allow for the possibility of incomplete preferences. In this section, we assume that each agent has a set of utilities, one for each member of the group. Two goods may be compared by the agent only if there is unanimity in the group on the choice.

More precisely, we consider an economy with m agents and d goods. Agent i has consumption space IR d , endowment E i 2 IR d and incomplete (or complete) convex preferences over IR d , defined by a family U i : IR d ! IR of concave utility functions :

X i 2 IR d is preferred to Y i 2 IR d by agent i, denoted X i ⌫ i Y i if u i (X i ) u i (Y i ) for every u i 2 U i . The associated strict preference is X i Y if X ⌫ i Y and u i (X) > u i (Y ) for some u i 2 U i . Let E = P m i=1 E i be the aggregate endowment.
We assume that for every i, the utilities in U i are everywhere finite, concave, and there is a topology on U i which makes it compact and such that the evaluation map u 2 U i 7 ! u(X) is continuous for every X 2 IR d . We assume that u(0) = 0 for all u 2 U i and all i. Let us give two examples of such families.

For notational simplicity, we drop the subscript i.

Example 1

Let T be a compact subset of IR p which can be interpreted as a subset of parameters and U = {u : IR d ⇥ IR p ! IR}, u continuous on IR d ⇥ T and u(., t) concave for every t 2 T .

Example 2

U is a set of concave functions IR d ! IR, closed for the topology of uniform convergence on compact subsets and uniformly bounded on each ball of IR d . U is then k R Lipschitz on the ball of radius R for some constant k R which depends on the radius. From Ascoli's theorem, U is then compact for the topology of uniform convergence on comp2act subsets.

An allocation (X

i ) i2I 2 IR dm is feasible if P i X i = E. The set of individu- ally rational attainable allocations A((E i ) m i=1
) is defined as in 2.3

Arbitrage concepts

We briefly redefine for abstract incomplete preferences, the concepts which were defined in section 2.2. Let

c P i (X) = {Y 2 R k | u(Y ) u(X), 8 u 2 U i } = \ u2U i {Y 2 R k | u(Y ) u(X)}
be the preferred set at X by i and R i (X) be its asymptotic cone. As the utilities

u 2 U i are concave, R i (X)
is independent of X and denoted R i . It is called the set of useful vectors for ⌫ i . From Rockafellar's [START_REF] Rockafellar | Convex Analysis[END_REF] corollary 8.3.3,

R i = {W 2 IR d | u( W ) 0, 8 2 IR + , u 2 U i }.
As discussed in the previous section, for any C 2 IR d , R i is also the set of useful vectors for any complete preference represented by a utility of the form

V E i (X) = min u2U i [u(X) u(E i )].
Note that this utility is well defined under our assumptions.

A price vector p 2 R d is a " no-arbitrage price" for agent i if p • W > 0, for all W 2 R i \{0}. Let S i denote the set of no-arbitrage prices for i. Then

S i = int(R i ) 0 (where (R i ) 0 = {p 2 R d | p • X  0, for all X 2 R i }). A price vector p 2 R k
is a " no-arbitrage price" for the economy if it is a noarbitrage price for each agent. A price vector p 2 R k is a no-arbitrage price for the economy if and only if p 2 \ i S i = \ i int(R i ) 0 . An m tuple (W 1 , . . . , W m ) 2 R dm is a feasible trade if

P i W i = 0. A trade W 2 R d \{0} is a half-line for a utility u : R d ! R if there exists X 2 R d such that u(X + W ) = u(X) for all 0.
As u is concave, when it has no half-line, then u(X + W ) > u(X) for every X 2 R d and > 0 and W 6 = 0.

E ciency and equilibrium concepts

Definition 5

1. A feasible allocation (X i ) m i=1 is e cient if there does not exist (X 0i ) m i=1 feasible such that X 0i ⌫ i X i for all i with a strict inequality for some i. It is weakly e cient if there does not exist (X 0i ) m i=1 feasible such that u i (X 0i ) > u i (X i ), 8 u i 2 U i , 8 i. It is individually rational (weakly) e cient if it is (weakly) e cient and X i ⌫ i E i for all i.

A pair

(X ⇤ , p ⇤ ) 2 A((E i ) m i=1 ) ⇥ IR d \{0} is an (weak) equilibrium with inertia if (a) for any i and X i 2 IR d , X i i X ⇤i (u i (X i ) > u i (X ⇤i ) for every u i 2 U i ) implies p ⇤ • X i > p ⇤ • X ⇤i , (b) for any i, p ⇤ • X ⇤i = p ⇤ • E i .
A common increasing direction is a trade e 2 IR d such that e • p > 0 for every (X, u) 2 IR d ⇥ U i , p 2 @u(X) and every i. When e is a common increasing direction, u(X te) < u(X), u(X +te) > u(X) for every t 0, X 2 IR d , u 2 U i and i.

Lemma 4 Let u be strictly concave for every u 2 U i and i.

A pair (X ⇤

, p ⇤ ) 2 A((E i ) m i=1 ) ⇥ IR d \{0} is an equilibrium with inertia if and only if it is a weak equilibrium with inertia 2. If agents have a common increasing direction, then an attainable allocation (X i ) m i=1 is e cient if and only if it is weakly e cient.

Proof : The proof of the first assertion is omitted. To prove the second, clearly, if (X i ) m i=1 is e cient, it is weakly e cient. Let us show that weak e ciency implies e ciency. Assume that (X i ) m i=1 is a weakly e cient allocation and that it is not e cient. W.l.o.g. assume that there exists a feasible allocation

(Y i ) m i=1 such that Y 1 1 X 1 and Y i ⌫ i X i , i 6 = 1. By considering (Y 1 +X 1 )
2 instead of Y 1 , we may assume that u(Y 1 ) > u(X 1 ), 8 u 2 U 1 . For a given " > 0, let

V " = u 2 U 1 | u(Y 1 e") > u(X 1 )
V " is open, since the evaluation maps are continuous. Let us show that [ " V " = U 1 . Let u 2 U 1 . Since u is continuous, there exists " u such that u(Y 1 e") > u(X 1 ) for every "  " u . Hence [ " V " = U 1 . As U 1 is compact, there exists a finite subcovering of U 1 by (V " i ). Let " = min i " i and " 0 = " m 1 . We then have

Y 1 e" 1 X 1 and Y i + e" 0 i X i , i 6 = 1
contradicting the weak e ciency of X.

Remark 4 It is easy to verify that if (X ⇤ , p ⇤ ) is a weak equilibrium with inertia, then X ⇤ is weakly e cient and that, if agents have a common increasing direction and (X ⇤ , p ⇤ ) is an equilibrium with inertia, then X ⇤ is e cient. In other words, the first welfare theorem holds true and is straightforward. Carlier and Dana [START_REF] Carlier | Pareto optima and equilibria when preferences are incompletely known[END_REF] provide general conditions on the families of utilities under which a weakly e cient allocation is a weak equilibrium with transfer. Hence a weak form of the second welfare theorem holds true. Under these conditions, when the utilities are strictly concave and have a common increasing direction (as it is the case in the Bewley model of section 2), weak equilibria with inertia coincide with equilibria with inertia, weakly e cient allocations are e cient and any e cient allocation is an equilibrium with transfer.

Existence results

Proposition 1

The following assertions are equivalent:

1. there exists a no-arbitrage price for the economy (\ i int (R i ) 0 6 = ;),

NUBA:

P i W i = 0 and W i 2 R i for all i implies W i = 0 for all i, 3. the set of individually rational attainable allocations is compact.

Proof : As the set of useful vectors of ⌫ i and V E i coincide and the set of individually rational attainable allocations for the economy with utilities (V E i ) i coincides with the set of individually rational attainable allocations for the economy with preferences (⌫ i ) i2I , the equivalence between 1, 2 and 3 follows from standard results on arbitrage with complete preferences.

Theorem 2 Let any assertion of Proposition 1 hold true.Then 1. there exists an individually rational weakly e cient allocation, 2. there exists a weak equilibrium with inertia.

Proof : It is also standard that any assertion of Proposition 1 implies the existence of an individually rational e cient allocation ( Xi ) m i=1 or of an equilibrium (X ⇤ , p ⇤ ) for the economy with utilities (V E i ) i . By the same proofs as in theorem 1, ( Xi ) m i=1 is an individually rational weakly e cient allocation for the economy with preferences (⌫ i ) i and (X ⇤ , p ⇤ ) is a weak equilibrium with inertia.

Theorem 3 1. If u has no half-line for every u 2 U i and i, then the assertions of Proposition 1 and Theorem 2 are equivalent and any weak equilibrium price is a no-arbitrage price.

2. If u is strictly concave for every u 2 U i and i, then Proof : Let us first remark that if u has no half-line for every u 2 U i and i, any weak equilibrium price is a no-arbitrage price. Indeed let (X ⇤ , p ⇤ ) be a weak equilibrium with inertia. Then for any useful vector W i 6 = 0, u(X ⇤i + tW i ) > u(X ⇤i ), 8 u 2 U i , for any t > 0. Hence p ⇤ • W i > 0 which proves that p ⇤ is a no-arbitrage price for the economy. This shows that assertion 2 of Theorem 2 implies assertion 1 of Proposition 1.

As mentioned in remark 4, it is easy to verify that a weak equilibrium with inertia is weakly e cient. Hence assertion 2 of Theorem 2 implies assertion 1 of 2. To complete the proof that the assertions of Proposition 1 and Theorem 2 are equivalent, let us show that assertion 1 of Theorem 2 implies assertion 2 of Proposition 1. Let ( Xi ) m i=1 be an individually rational weakly e cient allocation and suppose that there exists a feasible trade W 1 , . . . , W m with W i 2 R i for all i and W i 6 = 0 for some i. We have u( Xi + tW i ) u( Xi ), 8 u 2 U i , for all i, and u( Xi + tW i ) > u( Xi ), 8u 2 U i , for any i such that W i 6 = 0. The allocation ( Xi +tW i ) i2I being feasible, this contradicts the weak e ciency of ( Xi ) m i=1 .

Finally, if u is strictly concave, then u has no half-line, hence the assertions of Proposition 1 and Theorem 2 are equivalent. From lemma 4, any weak equilibrium with inertia is an equilibrium with inertia. If furthermore, agents have a common increasing direction, from lemma 4, any weakly e cient allocation is e cient .

  (a) the assertions of Proposition 1 and Theorem 2 are equivalent to the existence of an equilibrium (b) If furthermore, agents have a common increasing direction, then the assertions of Proposition 1 and of Theorem 2 are equivalent to the existence of an individually rational e cient allocation.

Documents de Travail du Centre d'Economie de la Sorbonne -2014.41

A Dana acknowledges the support of the Fondation du Risque, chaire Dauphine-ENSAE-Groupama, and the ANR project "Risk". 1 Documents de Travail du Centre d'Economie de la Sorbonne -2014.41 Documents de Travail du Centre d'Economie


Daniel Li

Hervé Queffélec

Luis Rodríguez-Piazza

A spectral radius type formula for approximation numbers of composition operators

Keywords: Mathematics Subject Classification 2010. Primary: 47B06 -Secondary: 30H10, 30H20, 31A15, 47B32, 47B33 approximation numbers, Bergman space, composition operator, Dirichlet space, Green capacity, Hardy space, weighted analytic Hilbert space

For approximation numbers a n (C ϕ ) of composition operators C ϕ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol ϕ of uniform norm < 1, we prove that lim n→∞ [a n (C ϕ )] 1/n = e -1/Cap [ϕ(D)] , where Cap [ϕ(D)] is the Green capacity of ϕ(D) in D. This formula holds also for H p with 1 ≤ p < ∞.

Introduction

The determination of the approximation numbers of composition operators on Hilbert spaces of analytic functions on the unit disk (Hardy space, weighted Bergman space, Dirichlet space) is a difficult problem. Some partial results (see [START_REF] Li | On approximation numbers of composition operators[END_REF], [START_REF] Lefèvre | Some new properties of composition operators associated to lens maps[END_REF], [START_REF] Li | Estimates for approximation numbers of some classes of composition operators on the Hardy space[END_REF], [START_REF] Lefèvre | Approximation numbers of composition operators on the Dirichlet space[END_REF], [START_REF] Queffélec | Decay rates for approximation numbers of composition operators[END_REF]) show that no simple answer may be expected. However, we proved in [START_REF] Li | On approximation numbers of composition operators[END_REF] and [START_REF] Lefèvre | Approximation numbers of composition operators on the Dirichlet space[END_REF] that these approximation numbers cannot decay faster than geometrically: we always have a n (C ϕ ) ≥ c r n for some constant c > 0 and some 0 < r < 1. Moreover, we showed in those papers that lim n→∞ [a n (C ϕ ] 1/n = 1 if and only if ϕ ∞ = 1.

The quantity lim n→∞ [a n (C ϕ )] 1/n looks like a spectral radius formula for the approximation numbers. Recall that if T is a bounded operator on a complex Hilbert space H, with spectrum σ(T ), the classical spectral radius formula tells that for the spectral radius r(T ) := sup λ∈σ(T ) |λ|, one has the formula:

r(T ) = lim n→∞ T n 1/n
(the existence of the limit being part of the conclusion). * Supported by a Spanish research project MTM 2012-05622.

1 Now, if a n = a n (T ) is the n-th approximation number of a bounded operator T on a Hilbert space H, it was shown ( [START_REF] König | Eigenvalue distribution of Compact Operators, Operator theory[END_REF], p. 133), by taking a rank-one perturbation of an n-dimensional shift, that, given 0 < σ < 1, we can have a 1 = • • • = a n-1 = 1, and a n = σ. Using orthogonal blocks of such normalized operators, one easily builds examples of compact operators T for which the quantity [a n (T )] 1/n has no limit as n goes to infinity, and indeed satisfies:

lim inf n→∞ [a n (T )] 1/n = 0 , lim sup n→∞ [a n (T )] 1/n = 1 .
We might as well use a diagonal operator with non-increasing positive diagonal entries ε n such that lim inf n ε which satisfy 0 ≤ β -(T ) ≤ β + (T ) ≤ 1 are similar to the term lim n→∞ T n 1/n in the spectral radius formula. When the limit exists we will denote it by:

(1.2)

β(T ) = lim n→∞ [a n (T )] 1/n .
These parameters were shown to play an important role in the study of composition operators (see [START_REF] Li | On approximation numbers of composition operators[END_REF] and [START_REF] Lefèvre | Approximation numbers of composition operators on the Dirichlet space[END_REF]). As said above, the following was proved in these papers.

Theorem 1.1 Let H be a weighted Bergman space B α (in particular the Hardy space H 2 ) or the Dirichlet space D and ϕ : D → D inducing a composition operator C ϕ : H → H. Then:

1) if 0 < ϕ ∞ < 1, one has 0 < β -(C ϕ ) ≤ β + (C ϕ ) < 1; 2) if ϕ ∞ = 1, one has β (C ϕ ) = 1.
The aim of this work is to complete this result by showing that β(C ϕ ) exists as well when ϕ ∞ < 1 and to give a closed formula for this β(C ϕ ) in terms of a Green capacity, relying on a basic work of [START_REF] Widom | Rational approximation and n-dimensional diameter[END_REF] (see also [START_REF] Fisher | The n-width of sets of analytic functions[END_REF]). We thus get another proof of 2) in the above theorem.

We end the paper with some words on the H p case for 1 ≤ p < ∞.

We begin by giving notations, definitions and facts which will be used throughout this work.

Background, framework, and notations

Recall that if X and Y are two Banach spaces of analytic functions on the unit disk D, and ϕ : D → D is an analytic self-map of D, one says that ϕ induces a composition operator C ϕ : X → Y if f • ϕ ∈ Y for every f ∈ X; ϕ is then called the symbol of the composition operator. One also says that ϕ is a symbol for X and Y if it induces a composition operator C ϕ : X → Y .

Singular numbers

For an operator T : X → Y between Banach spaces X and Y , its approximation numbers are defined, for n ≥ 0, as: The n-th Kolmogorov number d n (T ) of T is defined as (see [START_REF] Carl | Entropy, Compactness and the Approximation of Operators[END_REF], p. 49):

(2.1) a n (T ) = inf rank R<n T -R . One has T = a 1 (T ) ≥ a 2 (T ) ≥ • • • ≥ a n (T ) ≥ a n+1 (T ) ≥ •
(2.2)

d n (T ) = inf E⊆Y dim E<n sup x∈BX dist (T x, E) = inf E⊆Y dim E<n Q E T Y /E ,
where Q E : Y → Y /E is the quotient map. One always has a n (T ) ≥ d n (T ) and, when X and Y are Hilbert spaces, one has a n (T ) = d n (T ) (see [START_REF] Carl | Entropy, Compactness and the Approximation of Operators[END_REF], p. 51).

As usual, the notation A B means that there is a constant c such that A ≤ C B.

Weighted analytic Hilbert spaces

An analytic Hilbert space H on D is a Hilbert space H ⊂ Hol(D), the analytic functions on the unit disk D, for which the evaluations f → f (a) are continuous on H for all a ∈ D and therefore given by a scalar product:

f (a) = f, K a , K a ∈ H.
Since weakly convergent sequences of H are norm-bounded, the reproducing kernels K a are automatically norm-bounded on compact subsets of D, that is:

(2.3) L r := sup |a|≤r K a < ∞, for all r < 1.
We will be slightly less general here, and adopt the framework of [START_REF] Kellay | Compact composition operators on weighted Hilbert spaces of analytic functions[END_REF]. Let ω : [0, 1) → (0, ∞) be a continuous, positive, and Lebesgue-integrable function. We extend this function to a radial weight on D by setting ω(z) = ω(|z|). We denote by H ω the space of analytic functions on D such that

f 2 ω := |f (0)| 2 + D |f ′ (z)| 2 ω(z) dA(z) < +∞,
where dA stands for the normalized area measure on D. We will often omit the subscript ω and write . for

. ω . If f (z) = ∞ n=0 b n z n ,
a computation in polar coordinates shows that:

(2.4)

f 2 = ∞ n=0 |b n | 2 w n
where:

(2.5)

w 0 = 1 and w n = 2n 2 1 0 r 2n-1 ω(r) dr , n ≥ 1.
Observe that there is a constant C = C(ω) ≥ 1 and, for each ε > 0, a δ ε > 0 such that:

(2.6)

δ ε e -εn ≤ w n ≤ C n 2 , n ≥ 1.
Indeed, in one side, one has w n ≤ 2n 2 1 0 ω(r) dr, and, on the other side, for each 0 < δ < 1, setting c δ = inf 0≤r≤δ ω(r), we have c δ > 0 and:

w n ≥ 2n 2 c δ δ 0 r 2n-1 dr = c δ n δ 2n ,
giving (2.6). This shows in passing that H ω is an analytic Hilbert space, and we call it a weighted analytic Hilbert space. This framework is sufficiently general for our purposes and includes for example the case of (weighted) Bergman, Hardy, and Dirichlet spaces, corresponding to ω(r) = (1 -r 2 ) α , α > -1, that is w n ≈ n 1-α . The standard Bergman, Hardy, Dirichlet spaces correspond to the respective values α = 2, 1, 0.

The following simple fact will be used. Let a ∈ D and j ≥ 0; then:

(2.7) f → f (j) (a) is a continuous linear form on H.
This holds for any analytic Hilbert space on D, thanks to (2.3), and here can also be viewed as a consequence of (2.6).

An analytic self-map ϕ : D → D which induces a composition operator C ϕ : H → H will be called a symbol for H = H ω . In our space H, we have a quite easy case for deciding if some ϕ is a symbol. Lemma 2.1 If ϕ ∞ < 1, then ϕ is a symbol if and only if ϕ ∈ H. Equivalently, if and only if the positive measure µ = |ϕ ′ | 2 ω dA is finite. In that case, we moreover have

ϕ k ≤ C k ϕ k ∞ ϕ for every k ≥ 1. Proof. If ϕ is a symbol, then ϕ = C ϕ (z) ∈ H. Conversely, let ρ = ϕ ∞ < 1.
We first note that, if ϕ ∈ H, we have for any integer k ≥ 1:

ϕ k 2 = |ϕ(0)| 2k + D ω(z) k 2 |ϕ(z)| 2(k-1) |ϕ ′ (z)| 2 dA(z) ≤ ρ 2k (1 + k 2 ρ -2 ) ϕ 2 .
(2.8)

Now, let ε > 0 be such that ρ e ε < 1. If f (z) = b k z k ∈ B H , the unit ball of H, we have by (2.6): |b k | ≤ w -1/2 k ≤ C ε e kε ,
so that, using (2.8), we see that the series b k ϕ k = f • ϕ converges absolutely in H, which proves that C ϕ is compact (and even nuclear).

Green capacity

The Green function g : D × D → (0, ∞] of the unit disk D is defined as:

(2.9) g(z, w) = log 1wz z -w .

If µ is a finite positive Borel measure on D with compact support in D, its Green potential is:

(2.10) G µ (z) = D g(z, w) dµ(w)
and its energy integral is:

(2.11)

I(µ) = D×D g(z, w) dµ(z) dµ(w) .
Of course, (2.12)

I(µ) = D G µ (z) dµ(z) .
For any subset E of D, one sets:

(2.13)

V (E) = inf µ I(µ) ,
where the infimum is taken over all probability measures µ supported by a compact subset of E. Then the Green capacity1 of E in D is:

(2.14)

Cap (E) = 1/V (E) .
If K ⊆ D is compact, the infimum in (2.13) is attained for a probability measure µ 0 . If moreover V (K) < ∞ (i.e. Cap (K) > 0), this measure is unique and is called the equilibrium measure of K. One always has V (K) < ∞ when K has non-empty interior, since then I(λ) < ∞ where λ is the normalized planar measure on some open disk ∆ ⊆ K. It is clear that we have:

K ⊆ L ⇒ V (K) ≥ V (L) ⇒ Cap (K) ≤ Cap (L) ,
i.e. Cap (K) increases with K and:

Cap (E) = sup K⊆E,K compact Cap (K) .
We refer to [START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF] and [START_REF] Conway | Functions of one Complex Variable II[END_REF] and to the clear presentation of [START_REF] Nieminen | Boundary correspondence of Nevanlinna counting function for self-maps of the unit disk[END_REF] for the definition of the Green capacity and of its basic properties. Actually, in [START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF], the capacity is defined by another way (see [START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF], Chapitre V, pp. 52-55), as follows.

Lemma 2.2 For every compact set K ⊆ D, one has:

Cap (K)
= sup{ µ ; µ positive Borel measure supported by K and G µ ≤ 1 on D} This is the definition of de la Vallée-Poussin. Since our main result is based on H. Widom's paper [START_REF] Widom | Rational approximation and n-dimensional diameter[END_REF], it must be specified that he also used this definition in [START_REF] Widom | Rational approximation and n-dimensional diameter[END_REF].

Let us note, though we will not use that, that we also have: where q.e. means: out of a set of null capacity. The equivalence between these two definitions is shown in [START_REF] Nieminen | Boundary correspondence of Nevanlinna counting function for self-maps of the unit disk[END_REF], Lemma 4.1 (see also [START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF], Chapitre XI, p. 140 and pp. 144-145).

Cap (K) =
An important fact for this paper is well-known to specialists on the (Green) capacity. This fact, kindly communicated to us with its proof by A. Ancona ( [1]), is as follows.

Theorem 2.3 For every connected Borel subset E of D whose closure E is contained in D, one has:

(2.15) Cap (E) = Cap (E) .
For sake of completeness, we provide details for the reader. We begin with a definition: a subset E of D is said to be thin (in French: "effilé") at u ∈ E if there exists a function s which is superharmonic in a neighbourhood of u and such that s(u) < lim inf

v→u v∈E s(v) .
We denote by Ẽ the union of E and of points in E at which E is not thin (it is known that Ẽ is the closure of E for the fine topology: see [START_REF] Conway | Functions of one Complex Variable II[END_REF], Proposition 21.13.10). Then:

Lemma 2.4 If E is a connected Borel subset of D whose closure E is contained in D, one has: Ẽ = E .
Proof. Lemma 2.4 is an immediate consequence of the following result (see [START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF], Chapitre VII, Corollaire, p. 89). Cap (E) = Cap ( Ẽ) .

Proof. We know (Cartan's Theorem) that Cap ( Ẽ \ E) = 0 (see [START_REF] Conway | Functions of one Complex Variable II[END_REF], Theorem 21.12.14, and Proposition 21.13.10, or see [START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF], Chapitre VII, p. 86 and Chapitre V, p. 57, or [START_REF] Papadimitrakis | Notes on classical potential theory[END_REF], Proposition 8.2 and Proposition 8.3). Since the capacity of Borel sets is easily seen (see [START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF], Chapitre V, p. 62, or [START_REF] Landkof | Foundations of Modern Potential Theory[END_REF], Chap. II, § 1, p. 145) to be a subadditive set function, one gets

Cap (E) ≤ Cap ( Ẽ) ≤ Cap (E) + Cap ( Ẽ \ E) = Cap (E).
Throughout this paper, for convenience, we sometimes use the notation:

(2.16) M (E) := e -1/Cap (E) = e -V (E) .

Main result

The goal of this paper is to prove the following result. exists and the value of this limit is:

(3.1) β(C ϕ ) = e -1/Cap [ϕ(D)] .
Note that, by Theorem 2.3, Cap [ϕ(D)] = Cap [ϕ(D)], so Theorem 3.1 will follow immediately from Theorem 3.8 and Theorem 3.11 below.

The proof is based on two results of H. Widom ([24]). Though those theorems are in the H ∞ setting, we will be able to transfer them to our Hilbertian setting. Before giving this proof, we will check the result "by hand" with an explicit example.

A very special test case

Before going into the proof of Theorem 3.1, we are going to illustrate it in a simple situation.

Let ϕ be a symbol acting on H = H 2 with ϕ ∞ < 1. We know from [START_REF] Li | On approximation numbers of composition operators[END_REF] that β + (C ϕ ) < 1, and for very special ϕ's we will show directly, without appealing to Widom's results, that (3.1) holds. Theorem 3.2 Let ϕ(z) = az+b cz+d be a fractional linear function mapping D into D, i.e. :

|a| 2 + |b| 2 + 2 |ab -cd| ≤ |c| 2 + |d| 2 and |c| ≤ |d| . Then β(C ϕ ) = exp - 1 Cap (K) .
The example ϕ(z) = z/(2z + 1) shows that one cannot omit the condition |c| ≤ |d|.

Recall that the pseudo-hyperbolic distance on D is defined by:

(3.2) ρ(z, w) = z -w 1 -zw , z, w ∈ D .
We denote by ∆(w, r) = {z ∈ D ; ρ(z, w) < r} the open pseudo-hyperbolic disk of center w and radius r.

We have the following two facts ( [START_REF] Nieminen | Boundary correspondence of Nevanlinna counting function for self-maps of the unit disk[END_REF], p. 3173 for the first one).

Lemma 3.3 Let L = ∆(w, r) be a closed pseudo-hyperbolic disk of pseudohyperbolic radius r. Then:

(3.3) Cap (L) = 1 log(1/r) • Lemma 3.4 Let u, v : D → D be univalent analytic maps such that u(D) = v(D). Then, u = v • ψ where ψ ∈ Aut (D).
Indeed , by hypothesis u = v •ψ with ψ well-defined and holomorphic for v is injective. Moreover,

u(D) = v[ψ(D)] = v(D), whence ψ(D) = D, again because v is injective. Finally ψ is injective since u is.
Proof of Theorem 3.2. We may assume ϕ ∞ < 1. We first consider the case ϕ(z) = az, with |a| < 1. In that case, it is clear that a n (C ϕ ) = |a| n-1 , and hence β(C ϕ ) = |a| and ϕ(D) = D(0, |a|) = ∆(0, |a|). So that (3.1) holds in view of (3.3).

In the general case, one might say that the conformal invariance of Cap and β does the rest. Let us provide some details.

In general, ϕ(D) is an euclidean disk, therefore a pseudo-hyperbolic disk ∆(w, r) := {z ; ρ (z, w) < r} = ψ 1 [∆(0, r)], where ρ is the pseudo-hyperbolic distance and ψ 1 ∈ Aut (D); one has the same radius since automorphisms preserve ρ. If h(z) = rz, one therefore has ϕ(D) = ψ 1 [h(D)] (since ∆(0, r) and the euclidean disk D(0, r) coincide). From Lemma 3.4

, ϕ = ψ 1 • h • ψ 2 with ψ 2 ∈ Aut (D), and so C ϕ = C ψ2 C h C ψ1 , implying β(C ϕ ) = β(C h ) = r ,
by the ideal property. Moreover,

Cap [ϕ(D)] = Cap [h(D)]
by conformal invariance. Since we know that the desired equality between β and Cap holds for h, we get the result.

Let us numerically test the claimed value of β(C ϕ ) on the affine example ϕ(z) = ϕ a,b (z) = az + b with a, b > 0 and a + b < 1 (note that C ϕ a,b and C ϕ |a|,|b| are unitarily equivalent and have the same approximation numbers a n , so that there is no loss of generality by assuming a, b > 0). In that case, the a n (C ϕ ) = a n were computed exactly by Clifford and Dabkowski ( [START_REF] Clifford | Singular values and Schmidt pairs of composition operators on the Hardy space[END_REF]). Their result is as follows. One sets:

(3.4) ∆ = (a 2 -b 2 -1) 2 -4b 2 and Q = 1 + a 2 -b 2 - √ ∆ 2a 2 •
Then, one has a n = a n-1 Q n-1/2 , and so:

(3.5) β(C ϕ ) = aQ .
The result of the theorem can be tested on that example. Indeed, we have K := ϕ(D) = D(b, a), so that ( [START_REF] Landkof | Foundations of Modern Potential Theory[END_REF], p. 175-177):

Cap (K) = 1 log λ ,
where λ > 1 is the biggest root of the quadratic polynomial

P (z) = az 2 -(1 + a 2 -b 2 )z + a .
In explicit terms:

e -1/Cap (K) = 1 λ = 1 + a 2 -b 2 - √ ∆ 0 2a , with: (3.6) ∆ 0 = (1 + a 2 -b 2 ) 2 -4a 2 .
To get β(C ϕ ) = e -1/Cap (K) , it remains to compare (3.5) and (3.1), using (3.4) and (3.6), and to observe that

∆ = ∆ 0 = (1 + a + b)(1 + a -b)(1 -a + b)(1 -a -b) .

Widom's results reformulated

We are going to state Widom's results in a form suitable for us. We first quote the following lemma from [START_REF] Widom | Rational approximation and n-dimensional diameter[END_REF]. Lemma 3.5 (Widom) Let K ⊆ D be compact. Then, given ε > 0, there exists a cycle γ, which is a finite union of disjoint Jordan curves contained in D, and whose interior U contains K, and a rational function R of degree < n, having no zero on γ and all poles on ∂D, such that, for n large enough:

1) |R(z)| ≥ e -εn for z / ∈ U ;

2) |R(z)| ≤ e εn e -n/Cap (K) for z ∈ K.

The first theorem of Widom ([24], Theorem 2, p. 348), in which C(K) denotes the space of complex, continuous functions on K with the sup-norm, can now be rephrased as follows.

Theorem 3.6 (Widom) Let K ⊆ D be a compact set, and ε > 0. Then, there exist a constant C ε > 0 and, for every integer n large enough, a rational function R with poles on ∂D and points ζ i ∈ D \ K such that for every g ∈ H ∞ , one has:

(3.7) g -h C(K) ≤ C ε e εn e -n/Cap (K) g ∞ ,
where:

h(w) = R(w) i,k 1≤k≤mi c i,k (g) (w -ζ i ) -k with i m i < n
and the maps g ∈ H ∞ → c i,k (g) are linear.

Moreover, if H is a weighted analytic Hilbert space, these maps, restricted to H ∞ ∩ H, extend to continuous linear forms on H.

Widom's theorem precisely says the following. If R and γ are the rational function and cycle of Lemma 3.5, let ζ i be the zeros of R inside γ. Consider, for w ∈ K, the function

G(w) = R(w) 1 2πi γ g(ζ) R(ζ) (ζ -w) dζ ;
then, by the residues theorem,

G(w) = g(w) -R(w) i,k c i,k (g) (w -ζ i ) -k = g(w) -h(w) , and
Widom's theorem says that G C(K) ≤ C ε e 2 εn [M (K)] n g ∞ .
The only additional remark made here is that the c i,k are of the form

c i,k (g) = j≤mi-k λ i,j,k g (j) (ζ i )
where λ i,j,k are fixed scalars, so that by (2.7) they extend to continuous linear forms on H.

Observe that the linear forms g → c i,k (g ′ ) are also continuous on H since

(3.8) c i,k (g ′ ) = j≤mi-k λ i,j,k g (j+1) (ζ i ) .
This observation will be useful later.

Remark. The rational function h above is analytic in D. Indeed, since the ζ i are zeros of R, the polar factors (w -ζ i ) -k are compensated by R(w) with the right multiplicity, so that the only poles of R have modulus ≥ 1. However (see [START_REF] Widom | Rational approximation and n-dimensional diameter[END_REF], Lemma 1, p. 346), the poles of R are located on ∂D, but we cannot ensure that h ∈ H. Fortunately, we will see that h • ϕ ∈ H, and this will be sufficient for our purposes.

We will need a second theorem of H. Widom ([24], Theorem 7, p. 353), which goes as follows.

Theorem 3.7 (Widom) Let K be a compact subset of D and C(K) be the space of continuous functions on K with its natural norm. Set:

δ n (K) = inf E sup f ∈B H ∞ dist (f, E) ,
where E runs over all (n -1)-dimensional subspaces of C(K) and dist (f, E) = inf h∈E f -h C(K) . Then

(3.9) δ n (K) ≥ α e -n/Cap (K)
for some positive constant α.

The upper bound

Theorem 3.8 Let H be an analytic weighted Hilbert space with norm . . Let ϕ : D → D be a symbol for H, such that ϕ ∞ = ρ < 1. Then:

β + (C ϕ ) := lim sup n→∞ [a n (C ϕ )] 1/n ≤ e -1/Cap [ϕ(D)] . Proof. Fix ε > 0 such that ρ e ε < 1. If f (z) = ∞ k=0 b k z k ∈ H, let g(z) := S l f (z) = l-1 k=0 b k z k
, with l = l(n) be an integer to be adjusted. Lemma 3.9 We have:

f • ϕ -g • ϕ ≤ K ε ρ l e εl .
Proof. For f (z) = k b k z k , we have:

f • ϕ -g • ϕ = ∞ k=l b k ϕ k ≤ ∞ k=l |b k | ϕ k ≤ ∞ k=l |b k | 2 w k 1/2 ∞ k=l ϕ k 2 w -1 k 1/2 ≤ K ε ρ l e εl ,
by using Cauchy-Schwarz inequality, the fact that f ≤ 1, the inequalities (2.6), and a geometric progression.

Also, remark that we have, by the Cauchy-Schwarz inequality:

(S l f ) ′ ∞ ≤ l-1 k=0 k |b k | ≤ l-1 k=0 |b k | 2 w k 1/2 l-1 k=0 k 2 w -1 k 1/2 ≤ f l-1 k=0 k 2 w -1 k 1/2 .
Therefore, using (2.6), we see that the linear map S ′ l : H → H ∞ , defined by S ′ l (f ) = (S l f ) ′ , is continuous with a norm less than ( l-1 k=0 k 2 w -1 k ) 1/2 ≤ K ε e εl . We now use Theorem 3.6, with K = ϕ(D) ⊆ D (and for n -1 instead of n). Set, for n ≥ 2, large enough:

h 1 (w) = R(w) i,k 1≤k≤mi c i,k (g ′ )(w -ζ i ) -k with i m i < n -1 .
Recall that h 1 is analytic in D. Remark that h 1 depends linearly on f and the map f → h 1 has a rank < n -1. We denote by I 1 ∈ Hol (D) the primitive of h 1 taking the value g[ϕ(0)] at ϕ(0):

I 1 (z) = z ϕ(0) h 1 (u) du + g[ϕ(0)] .
Next, define an operator A of rank < n on H (the continuity of A being justified by (3.8)) by the formula:

(3.10) A(f ) = I 1 • ϕ .
Note that, even if I 1 / ∈ H, we easily check on the integral representation of the norm that I 1 • ϕ ∈ H since we assumed ϕ ∈ H, i.e. (see Lemma 2.1) that ϕ is a symbol.

Assuming for the rest of the proof that f ≤ 1, we have the following lemma. Lemma 3.10 We have:

g • ϕ -I 1 • ϕ ≤ K ε e ε(n-1)
e εl e -(n-1)/Cap (K) .

Proof. Since ϕ ∈ H and since h 1 = I ′ 1 approximates g ′ uniformly on K and g ′ ∞ = (S l f ) ′ ∞ ≤ K ε e εl , we have, by Theorem 3.6:

g • ϕ -I 1 • ϕ 2 = D g ′ [ϕ(z)] -h 1 [ϕ(z)] 2 |ϕ ′ (z)| 2 ω(z) dA(z) ≤ K 2 ε e 2ε(n-1) [M (K)] 2(n-1) g ′ 2 ∞ D |ϕ ′ (z)| 2 ω(z) dA(z) ≤ C K 3 ε e 2εl e 2ε(n-1) [M (K)] 2(n-1) ,
(with C = ϕ 2 ω ), hence the lemma, provided that we increase K ε .

We can now end the proof of Theorem 3.8. Writing:

C ϕ (f ) -A(f ) = f • ϕ -I 1 • ϕ ≤ f • ϕ -g • ϕ + g • ϕ -I 1 • ϕ , we have: 1) f • ϕ -g • ϕ ≤ K ε ρ l e εl by Lemma 3.9; 2) g • ϕ -I 1 • ϕ ≤ K ε e ε(n-1) [M (K)]
n-1 e εl by Lemma 3.10.

We therefore get, since a n := a n (C ϕ ) ≤ C ϕ -A :

a n ≤ K ε ρ l e εl + K ε e εl e ε(n-1) [M (K)] n-1 .
Next, since (a + b) 1/n ≤ a 1/n + b 1/n , we infer that:

(3.11) a 1/n n ≤ (K ε ) 1/n (ρ e ε ) l/n + K 1/n ε e εl/n e ε(n-1)/n M (K) (n-1)/n .

We now adjust l = N n, where N is a fixed positive integer, and pass to the upper limit with respect to n in (3.11). We get:

L := lim sup a 1/n n ≤ [ρ e ε ] N + e ε e εN M (K).
Letting ε go to 0, we get L ≤ ρ N + M (K). Finally, letting N tend to infinity, we get L ≤ M (K) as claimed, and that ends the proof of Theorem 3.8.

The lower bound

Theorem 3.11 Let H be a weighted analytic Hilbert space and ϕ ∈ H such that ϕ ∞ < 1. Then:

β -(C ϕ ) := lim inf n→∞ [a n (C ϕ )] 1/n ≥ e -1/Cap [ϕ(D)] .
It will be convenient to work with the Kolmogorov numbers d n (C ϕ ) instead of the approximation numbers a n (C ϕ ). Recall that, for Hilbert spaces, one has d n (C ϕ ) = a n (C ϕ ). We begin with a simple lemma, undoubtedly well known to experts, on approximation numbers of an operator T on a Hilbert space H. Lemma 3.12 For every Hilbert space H and every compact operator T: H → H, one has, B H denoting the unit ball of H:

(3.12) d n (T ) = inf dim E<n sup f ∈BH dist T f, T (E) .
Proof. Indeed, if ε n (T ) denotes the right hand side in (3.12), we clearly have d n (T ) ≤ ε n (T ). Now, let:

T f = ∞ j=1 a j (T ) f, v j u j ,
with (u j ) and (v j ) two orthonormal sequences, be the Schmidt decomposition of T . Let E 0 be the span of v 1 , . . . , v n-1 . Observe that

u j = T (a -1 j v j ) ∈ T (E 0 ) for j < n. Now, if f ∈ B H , one has: dist T f, T (E 0 ) 2 = ∞ j=n a j (T ) f, v j u j 2 = ∞ j=n [a j (T )] 2 | f, v j | 2 ≤ [a n (T )] 2 ∞ j=n | f, v j | 2 ≤ [a n (T )] 2 ; so that ε n (T ) ≤ sup f ∈BH dist T f, T (E 0 ) ≤ a n (T ) = d n (T ).
Proof of Theorem 3.11. Let 0 < r j < 1, r j → 1 and ψ j : D → D be given by ψ j (z) = r j z. Set K j = ϕ • ψ j (D) = ϕ(r j D). Let E be a subspace of H of dimension < n. By restriction, E can be viewed as a subspace of C(K j ). By the second result of Widom (Theorem 3.7), we can find

f ∈ B H ∞ , f (z) = k≥0 b k z k , such that: f -h C(Kj) ≥ 2α [M (K j )] n , ∀h ∈ E ,
where α > 0 is an absolute constant. If H ∞ contractively embeds into H, we can continue with this f . In the general case, we have to correct f in order to be in B H , the unit ball of H. To that effect, we simply consider a partial sum:

g(z) = l-1 k=0 b k z k
and we note that, setting ρ j = sup w∈Kj |w|, one has ρ j < 1 and:

f -g C(Kj) ≤ ρ l j (1 -ρ 2 j ) 1/2 (3.13) g H ≤ C l , (3.14)
where C = C(ω) ≥ 1 is the constant appearing in (2.6).

Indeed, we have

f -g C(Kj) ≤ ∞ k=l |b k | ρ k
j and then (3.13) follows from Cauchy-Schwarz's inequality and the fact that

k≥0 |b k | 2 ≤ 1 since f ∈ B H ∞ .
For (3.14), we simply use that, by (2.6), the weight w satisfies w k ≤ C (k + 1) 2 and get:

g 2 H = l-1 k=0 |b k | 2 w k ≤ C l 2 l-1 k=0 |b k | 2 ≤ C l 2 ≤ C 2 l 2 .
We then notice that (3.13) gives, for every h ∈ E:

g -h C(Kj) ≥ f -h C(Kj) -f -g C(Kj) ≥ 2 α [M (K j )] n - ρ l j (1 -ρ 2 j ) 1/2 ≥ α [M (K j )] n , (3.15)
if we take l = A j n where A j is a large positive integer depending only on j. Explicitly:

A j > log 1/ α (1 -ρ 2 j ) 1/2 log(1/ρ j ) + log[1/M (K j )] log(1/ρ j ) • Finally, set F = g/C l. Then F ∈ B H
. Since E is a vector space, (3.14) and (3.15) imply:

F -h C(Kj) = 1 C l g -C l h C(Kj) ≥ 1 C l α [M (K j )] n .
But we also know that:

F -h C(Kj) = F • ϕ • ψ j -h • ϕ • ψ j ∞ ≤ L rj F • ϕ -h • ϕ H ,
so we are left with (recall that l = A j n):

C ϕ F -C ϕ h H ≥ α C L rj A j M (K j ) n n , ∀h ∈ E,
implying by Lemma 3.12:

a n (C ϕ ) = d n (C ϕ ) ≥ α C L rj A j [M (K j )] n n •
Now, taking n-th roots and passing to the lower limit, we get:

(3.16) β -(C ϕ ) ≥ M (K j ).
It remains now to let j → ∞. Observe that the compact subsets K j ⊆ ϕ(D) form an exhaustive sequence of compact subsets of ϕ(D). Let then L ⊆ ϕ(D) be compact; we have L ⊆ K j0 for some j 0 , and using (3.16), we get

β -(C ϕ ) ≥ M (K j0 ) ≥ M (L). Passing to the supremum on L, we get β -(C ϕ ) ≥ M [ϕ(D)],
and this ends the proof of Theorem 3.11.

The case ϕ ∞ = 1

As said in the Introduction, for weighted Bergman spaces (including the Hardy space), and for the Dirichlet space, we proved in [START_REF] Li | On approximation numbers of composition operators[END_REF] and [START_REF] Lefèvre | Approximation numbers of composition operators on the Dirichlet space[END_REF], respectively, that β(C ϕ ) = 1 if ϕ ∞ = 1 for every ϕ inducing a composition operator on one of those spaces.

In this section, we use Theorem 3.1 to generalize this result to all composition operators C ϕ on weighted analytic Hilbert spaces, with another, and simpler, proof.

For that, it suffices to use the following result, which is certainly well-known to specialists. The pseudo-hyperbolic metric ρ on D is defined in (3.2) and we denote by diam ρ the diameter for this metric. Theorem 3.13 Let K be a compact and connected subset of D. Then, for 0 < ε < 1:

diam ρ K > 1 -ε =⇒ Cap (K) ≥ c log 1/ε ,
for some absolute positive constant c. Hence, the Green capacity of K tends to ∞ as its pseudo-hyperbolic diameter tends to 1.

Before proving that, let us give two suggestive examples, borrowed from [START_REF] Landkof | Foundations of Modern Potential Theory[END_REF], p. 175-177.

1) Let K = D(0, r); then:

diam ρ K = 2r 1 + r 2 and Cap (K) = 1 log 1/r •
One sees that r goes to 1 when diam ρ K goes to 1, and hence Cap (K) tends to infinity.

2) Let K = [0, h], with 0 < h < 1. Then:

diam ρ K = h and Cap (K) = 1 π I ′ I ,
where I and I ′ are the elliptic integrals:

I = 1 0 1 (1 -t 2 )(1 -k 2 t 2 ) dt and I ′ = 1 0 1 (1 -t 2 )(1 -k ′2 t 2 ) dt , with k = 1-h 1+h and k ′ 2 = 1 -k 2 . If 0 ≤ a < b ≤ h, then b -a + hab ≤ h -a + ah 2 = h -a(1 -h 2 ) ≤ h, so that ρ(a, b) ≤ h. Therefore, in this example again, the assumption diam ρ K -→ 1 implies successively that h → 1, k → 0, k ′ → 1, I → π/2, I ′ → ∞, and at last Cap (K) → ∞.
This example shows that Theorem 3.13 is optimal since

1 0 dt (1 -t 2 )(1 -k ′2 t 2 ) ≈ log 1 1 -k ′ 2 ≈ log 1 1 -h as h (and hence k ′ ) goes to 1.
The following proof of Theorem 3.13 was kindly shown to the second-named author by E. Saksman ([23]).

It make use of the following alternative definition of Green capacity, where C ∞ 0 (D) is the space of infinitely differentiable functions on D which are null on ∂D, and dz = dxdy is the usual 2-dimensional Lebesgue measure.

Lemma 3.14 For every compact subset K of D, one has:

Cap (K) = inf 1 2π D |∇u(z)| 2 dz ; u ∈ C ∞ 0 (D) and u ≥ 1 on K .
Proof of Theorem 3.13. If diam ρ K > 1 -ε and K is connected, it contains two points z 1 and z 2 such that ρ(z 1 , z 2 ) = 1 -ε. By the invariance of the Green capacity and of ρ under automorphisms of the disk, we can assume that z 1 = 0 and z 2 = 1 -ε. Take ε < r < 1. Denote by ∆ r the intersection of the closed disk with center 1 and radius r with the closed unit disk. We observe that K meets the exterior of ∆ r at 0 and its interior at 1 -ε. The connectedness of K implies that K meets the boundary of ∆ r : there is b ∈ K such that |b -1| = r. Write b = 1 + re iϑ . Take now a = 1 + re iθ with |a| = 1 and 0 ≤ θ ≤ ϑ ≤ 2π.

Since u(a) = 0 and u(b) ≥ 1, we get, by the fundamental theorem of calculus, that:

1 ≤ u(b) -u(a) = ϑ θ ir e it ∇u(1 + re it ) dt = ϑ θ ir e it ∇u(1 + re it ) dt ≤ r ϑ θ |∇u(1 + re it )| dt ≤ r 2π 0 |∇u(1 + re it )| dt.
Now, Cauchy-Schwarz inequality gives:

2π 0 |∇u(1 + re it )| 2 dt ≥ 1 2πr 2 •
Integrating in polar coordinates centered at 1 and remembering that u = 0 outside D, we get:

D |∇u(z)| 2 dz ≥ ε<|z-1|<1 |∇u(z)| 2 dz = 1 ε 2π 0 |∇u(1 + re it )| 2 dt r dr ≥ 1 2π 1 ε dr r = 1 2π log 1 ε •
In view of (3.14), this ends the proof of Theorem 3.13.

Proof of Lemma 3.14. Though this result is often considered as "well-known", we were not able to find anywhere an explicit reference. Since the average reader (if any!) of this paper will not be a specialist in Potential theory, we give a proof here. 1) We first prove that the capacity of the compact K is less than the righthand side (though we only need that it is greater). We shall use Lemma 2.2.

We know ( [START_REF] Conway | Functions of one Complex Variable II[END_REF], Corollary 21.4.7, or [START_REF] Landkof | Foundations of Modern Potential Theory[END_REF], p. 91) that for every measure µ on D supported by K, one has ∆G µ = -2πµ, where G µ is seen as a distribution. Hence, for every function u ∈ C ∞ 0 (D) such that u ≥ 1 on K and every positive measure µ supported by K such that G µ ≤ 1 on D, one has:

µ(K) = K dµ ≤ D u dµ = - 1 2π D u(z) ∆G µ (z) dz .
Then, by definition of the Laplacian of a distribution, we get:

µ(K) ≤ - 1 2π D ∆u(z) G µ (z) dz .
But (see [START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF], Chapitre XI, p. 132 and pp. 144-145, or [START_REF] Landkof | Foundations of Modern Potential Theory[END_REF], Chap. IV, § 1, p. 215), for every real Borel measures ν 1 and ν 2 with finite energy (meaning that their positive and negative parts have finite energy), this energy is positive and one has the Cauchy-Schwarz inequality for the Dirichlet space :

D G ν1 dν 2 ≤ D G ν1 dν 1 1/2 D G ν2 dν 2 1/2
.

Applying this to the measures ν 1 = µ and ν 2 = ν = ∆u.dz, we get, since G µ ≤ 1:

µ(K) ≤ 1 2π D G µ (z) dµ(z) 1/2 D G ν (z)∆u(z) dz 1/2 ≤ 1 2π [µ(K)] 1/2 D G ν (z)∆u(z) dz 1/2 = 1 2π [µ(K)] 1/2 D G ν dν 1/2
. Now, since u ∈ C ∞ 0 (D), one has G. C. Evans' theorem [START_REF] Evans | On potentials of positive mass I[END_REF] (see [START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF], Chapitre XI, Lemme 1, p. 141, or [START_REF] Landkof | Foundations of Modern Potential Theory[END_REF], Theorem 1.20, p. 97):

D G ν dν = 2π D |∇u(z)| 2 dz .
Therefore, we get:

µ(K) ≤ 1 2π D |∇u(z)| 2 dz .
Taking the supremum on µ of the left-hand side and the infimum on u of the right-hand side, we obtain:

Cap (K) ≤ inf 1 2π D |∇u(z)| 2 dz ; u ∈ C ∞ 0 (D) and u ≥ 1 on K .
2) Let ε > 0.

Let K j = {z ∈ C ; dist (z, K) ≤ 1/j}, j ≥ 1. Each K j is compact and is contained in D for j large enough, say j ≥ j 0 . Since K = j≥j0 K j (and the sequence is decreasing), one has Cap (K j ) -→ j→∞ Cap (K) ( [START_REF] Conway | Functions of one Complex Variable II[END_REF], Proposition 21.7.15; note that though this proposition is stated for the logarithmic capacity, the proof clearly works also for the Green capacity). Hence, there is some j ≥ j 0 such that, for K ′ = K j , one has (1 + ε) Cap (K) ≥ Cap (K ′ ).

Let µ 0 be an equilibrium measure of K ′ . One has µ 0 (K ′ ) = 1, I(µ 0 ) = V (K ′ ), G µ0 ≤ V (K ′ ) on D. Moreover, by [START_REF] Conway | Functions of one Complex Variable II[END_REF], Lemma 21.10.1 (based on Frostman's theorem: see [START_REF] Conway | Functions of one Complex Variable II[END_REF], Theorem 21.7.12, whose proof works also for the Green capacity), one has

G µ0 = V (K ′ ) on int (K ′ ), hence on K. Let µ = Cap (K ′ ) µ 0 . Then µ(K ′ ) = Cap (K ′ ), I(µ) = [Cap (K ′ )] 2 I(µ 0 ) = Cap (K ′ ), and, since G µ = Cap (K ′ ) G µ0 , one has also G µ ≤ 1 on D and G µ = 1 on K.
By a theorem of G. Choquet [START_REF] Choquet | Sur les fondements de la théorie fine du potentiel[END_REF], we can find, by regularization ( [START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF], p. 26 and Lemma, p. 135 and pp. 142-145, or [START_REF] Landkof | Foundations of Modern Potential Theory[END_REF], Theorem 1.9, p. 70, which applies since G µ -U µ 2 is a harmonic function) an increasing sequence of positive infinitely differentiable functions v n on D which converges pointwise to G µ and such that:

D |∇v n (z)| 2 dz -→ n→∞ D |∇G µ (z)| 2 dz .
Since (v n ) n is increasing and converges pointwise to 1 on the compact set K, Dini's theorem tells that one has uniform convergence. Hence, we can find some

v = v n such that v ≥ (1 + ε) -1 on K and D |∇v(z)| 2 dz ≤ (1 + ε) D |∇G µ (z)| 2 dz . Note that v = 0 on ∂D since 0 ≤ v ≤ G µ , which is equal to 0 on ∂D. Putting u = (1 + ε)v, one has u ∈ C ∞ 0 (D), u ≥ 1 on K and D |∇u(z)| 2 dz ≤ (1 + ε) 3 D |∇G µ (z)| 2 dz .
But we know by G. C. Evans's theorem (see [START_REF] Papadimitrakis | Notes on classical potential theory[END_REF], Proposition 7.3, or [START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF], Chapitre XI, p. 142 and pp. 144-145, or [START_REF] Landkof | Foundations of Modern Potential Theory[END_REF], Theorem 1.20, p. 97) that:

I(µ) = 1 2π D |∇G µ (z)| 2 dz .
We get hence:

(1 + ε) Cap (K) ≥ Cap (K ′ ) = I(µ) = 1 2π D |∇G µ (z)| 2 dz ≥ 1 (1 + ε) 3 1 2π D |∇u(z)| 2 dz .
Since ε > 0 was arbitrary, we get:

Cap (K) ≥ inf 1 2π D |∇u(z)| 2 dz ; u ∈ C ∞ 0 (D) and u ≥ 1 on K ,
and that ends the proof.

Remark. After this paper was completed, we have found an alternative proof of Theorem 3.13. We sketch it here. As in the above proof, we may assume that 0 and 1-ε belong to K. Consider K * = {|z| ; z ∈ K}. Since K is connected, the same holds for K * . Hence the interval [0, 1 -ε] is contained in K * . It follows that Cap ([0, 1 -ε]) ≤ Cap (K * ). But we saw in Example 2 that Cap ([0, 1 -ε]) ≈ log(1/ε); hence Cap (K * ) log(1/ε). It remains to use that the map α : z → |z| is a contraction for the pseudo-hyperbolic metric and hence Cap (K * ) ≤ Cap (K) (see [START_REF] Landkof | Foundations of Modern Potential Theory[END_REF], Chap. II, Theorem 2.9, and the comment p. 175 for the Green capacity). In fact, if ν is any probability measure supported by K * , there exists (see [START_REF] Garnett | Harmonic measure[END_REF], Chap. III, Lemma 4.6) a probability measure µ on K such that α(µ) = ν. Hence:

V (K) ≤ I K (µ) = D×D g(z, w) dµ(z) dµ(w) = D×D log 1 ρ(z, w) dµ(z) dµ(w) ≤ D×D log 1 ρ(|z|, |w|) dµ(z) dµ(w) = D×D log 1 ρ(z, w) dν(z) dν(w) = I K * (ν).
Taking the infimum over all ν, we get V (K) ≤ V (K * ).

As a corollary of Theorem 3.13, we get a new proof of [START_REF] Li | On approximation numbers of composition operators[END_REF], Theorem 3.4 and of [START_REF] Lefèvre | Approximation numbers of composition operators on the Dirichlet space[END_REF], Theorem 2.2. Theorem 3.15 There exists an absolute constant c > 0 such that, for any symbol ϕ on a weighted analytic space H, one has:

diam ρ [ϕ(D)] > r =⇒ β(C ϕ ) ≥ exp - c log 1/(1 -r) .
In particular:

ϕ ∞ = 1 =⇒ β(C ϕ ) = 1 .
Proof. The first statement is a direct consequence of Theorem 3.1, modulo Theorem 2.3 and Theorem 3.13, applied to ϕ(D) and its closure. One cannot replace diam ρ [ϕ(D)] > r by ϕ ∞ > r in this first statement as indicated by the following example:

ϕ(z) = a -(z/2) 1 -a(z/2) = Φ a [h(z)] ,
where Φ a (z) = a-z 1-az with a ∈ D and h(z) = z/2 is the dilation with ratio 1/2.

Then ϕ ∞ ≥ |Φ a (0)| = |a| and β(C ϕ ) = β(C h ) = 1/2.
However, one can do so if moreover ϕ(0) = 0 because then, clearly:

ϕ ∞ > r =⇒ diam ρ [ϕ(D)] > r .
This is enough for the second statement since, putting a = ϕ(0), we have, due to the fact that Φ a is unimodular on the whole unit circle:

Φ a • ϕ ∞ = ϕ ∞ = 1, (Φ a • ϕ)(0) = 0 and β(C ϕ ) = β(C Φa •ϕ ).

A remark

We proved in [START_REF] Lefèvre | Compact composition operators on the Dirichlet space and capacity of sets of contact points[END_REF] that every composition operator C ϕ which is bounded on the Dirichlet space D is compact on the Hardy space H 2 (and hence on the Bergman space B 2 ), and even in all Schatten classes on H 2 and B 2 . So one may expect that the approximation numbers of composition operators on the Dirichlet space are bigger than those on the Hardy space (and bigger than those on the Bergman space). Since Theorem 3.1 and Theorem 1.1 show that β(C ϕ ) is the same for these three spaces, it follows that the answer will be certainly quite subtle and cannot only involve log a n (C ϕ ).

4 The H p case, 1 ≤ p < ∞ Here, we consider the case of composition operators on H p for 1 ≤ p < ∞.

For every a ∈ D, we denote by e a ∈ (H p ) * the evaluation map at a, namely: still holds.

Throughout this section we denote by . , without any subscript, the norm in the dual space (H p ) * .

Let us stress that this dual norm of (H p ) * is, for 1 < p < ∞, equivalent, but not equal, to the norm . q of H q , and the equivalence constant tends to infinity when p goes to 1 or to ∞.

With this preliminaries, we are going to see that Theorem 3.1 remains true. 2) One has:

ϕ ∞ = 1 =⇒ β(C ϕ ) = 1 .
We begin with the following lemma, which extends Lemma 3.12.

Lemma 4.2 Let X be a Banach space, and T : X → X be a compact operator. Let us set: Then ε n (T ) ≤ 2 √ n c n (T ).

Proof. Let ε > 0, and let F be a subspace of X of codimension < n such that T |F ≤ c n (T ) + ε. Let Q : X → F be an onto projection of norm Q ≤ √ n + 1 ≤ 2 √ n (see [START_REF] Li | Introduction à l'étude des espaces de Banach[END_REF], Chapitre 5, Théorème III. 4, 2), or [START_REF] Wojtaszczyk | Banach Spaces for Analysts[END_REF], III.B.11) and let R = T (I -Q). Then E = (I -Q)X satisfies dim E < n. If x ∈ B X , the closed unit ball of X, then:

dist (T x, T E) ≤ T x -Rx = T Qx ≤ T |F Qx ≤ (c n (T ) + ε) 2 √ n .
This implies ε n (T ) ≤ 2 √ n (c n (T ) + ε). The result follows since ε was arbitrary. Let Lr = sup |a|≤r e a = 1 1-r 2 1/p , for 0 < r < 1. Using the same notations and estimations as in Theorem 3.11, up to the replacement of L r by Lr , we get:

ε n (T ) ≥ (1 -ε) L-1 rj α [M (K j )] n .
Lemma 4.2 now implies:

a n (T ) ≥ c n (T ) ≥ α 1 -ε 2 √ n L-1 rj [M (K j )] n .
The rest of the proof is unchanged, since the presence of the factor 1/ √ n does not affect the result.

b) The upper bound is even simpler since H ∞ ⊆ H p . For example, with the notations of Section 3.2.1, setting A(f ) = h • ϕ as in (3.10), we can replace Lemma 3.10 by

g • ϕ -h • ϕ p ≤ g • ϕ -h • ϕ ∞ = g -h C(K) ,
where K = ϕ(D).

2) That follows from Theorem 3.13, as in Section 3.13.

Luis Rodríguez-Piazza, Universidad de Sevilla, Facultad de Matemáticas, Departamento de Análisis Matemático & IMUS, Apartado de Correos 1160, 41 080 SEVILLA, SPAIN piazza@us.es

= 1 .

 1 Nevertheless, the parameters(1.1) β -(T ) = lim inf n→∞ [a n (T )] 1/n , β + (T ) = lim sup n→∞ [a n (T )] 1/n

  inf{ µ ; µ positive Borel measure on D and G µ ≥ 1 on K} = inf{ µ ; µ positive Borel measure on D and G µ ≥ 1 q.e. on K} ,

Theorem 2 . 5 (Lemma 2 . 6

 2526 Beurling-Brelot) Let E ⊆ D and u ∈ E. If E is thin at u, there exist circles with center u and arbitrarily small radius > 0 which do not intersect E.Indeed, taking the previous result for granted, suppose that E is thin at u ∈ E, u / ∈ E, and let v 0 ∈ E, with |v 0 -u| = d > 0. The function ρ : E → R defined by ρ(v) = |v -u| takes the value d as well as arbitrarily small values since u ∈ E. By the intermediate value theorem, it takes every value in (0, d], contradicting Theorem 2.5. This contradiction shows that E ⊆ Ẽ, thereby ending the proof of Theorem 2.3. Now, One has:

Theorem 3 . 1

 31 Let H be a weighted analytic Hilbert space with norm . . Let ϕ : D → D be a symbol for H, with ϕ(D) ⊆ D. Then lim n→∞ [a n (C ϕ )] 1/n =: β(C ϕ )

(4. 1 )

 1 e a (f ) = f (a) , f ∈ H p .We know that ([START_REF] Zhu | Operator Theory in Function Spaces, Second Edition[END_REF], p. 253):

( 4

 4 .3) C *ϕ (e a ) = e ϕ(a)

Theorem 4 . 1 1 )

 411 Let 1 ≤ p < ∞ and C ϕ : H p → H p . If ϕ(D) ⊆ D, then: β(C ϕ ) = e -1/Cap [ϕ(D)] .

  Proof of Theorem 4.1. 1) a) We first prove that β -(C ϕ ) ≥ e -1/Cap [ϕ(D)] .

  • • , and (assuming that Y has the Approximation Property), T is compact if and only if a

n (T ) -→ n→∞ 0.

Actually the inner capacity, but for open and compact sets, it it is equal to the outer capacity and hence, is the capacity: see[START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF], Chapitre V, p. 63. Choquet's Theorem ([START_REF] Choquet | Theory of capacities[END_REF]; see also[START_REF] Brelot | Éléments de la théorie classique du potentiel, 4 e édition, Les cours de Sorbonne, 3 e cycle[END_REF], Chapitre V, p. 66), asserts that the inner capacity is equal to the outer capacity for all Borel sets.

Ackowledgements. We thanks A. Ancona and E. Saksman for crucial informations on the Green capacity.