On approximation numbers of composition operators

arXiv:1104.4451v1 [math.FA] 22 Apr 2011 On approximation numbers of composition operators Daniel Li, Hervé Queffélec, Luis Rodríguez-Piazza April 25, 2011 Abstract. We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces Bαof the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they grow at least exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bounds and explicit an example. Mathematics Subject Classification. Primary: 47B06 – Secondary: 47B33; 47B10 Key-words. approximation number – Bergman space – Carleson measure – composition operator – Hardy space – interpolation sequence – reproducing kernel – weighted Bergman space – weighted shift 1 Introduction Let Dbe the open unit disk of the complex plane, equipped with its normalized area measure dA(z) = dxdy π. For α > −1, let Bαbe the weighted Bergman space of analytic functions f(z) = P∞ n=0 anznon Dsuch that kfk2 α= (α+ 1) ZD|f(z)|2(1 −|z|2)αdA(z) = ∞ X n=0 n!Γ(2 + α) Γ(n+ 2 + α)|an|2<∞. The limiting case, as α> −→−1, of those spaces is the usual Hardy space H2 (indeed, if fis a polynomial, we have limα> −→−1kfk2 α=P∞ n=0 |an|2=kfk2 H2), which we shall treat as B−1. Note that kfk2 α≈P∞ n=0 |an|2 (n+1)α+1 and that dAα(z) = (α+ 1)(1 −|z|2)αdA(z) is a probability measure on D. Bergman spaces ([46] page 75, page 78) are Hilbert spaces of analytic functions on Dwith reproducing kernel Ka∈Bα, given by Ka(z) = ( 1 1−az )α+2, namely, for every a∈D: (1.1) f(a) = hf, Kai,∀f∈Bα;and kKak2=Ka(a) = 1 1−|a|2α+2. 1 An important common feature of those spaces is that the multipliers of Bαcan be (isometrically) identified with the space H∞of bounded analytic functions on D, that is: (1.2) ∀g∈H∞,kgk∞= sup f∈Bα,kfkα≤1kfgkα. Indeed, kfgkα≤ kgk∞kfkαis obvious, and if kfgkα≤Ckfkαfor all f∈Bα, testing this inequality successively on f= 1,g,...,gn,... easily gives g∈H∞ and kgk∞≤C. Let now ϕbe a non-constant analytic self-map (a so-called Schur function) of Dand let Cϕ:Bα→H(D)the associated composition operator: Cϕ(f) = f◦ϕ. It is well-known ([9] page 30) that such an operator is always bounded from Bα into itself, and we are interested in its approximation numbers. Also recall that the approximation (or singular)numbers an(T)of an operator T∈ L(H1, H2), between two Hilbert spaces H1and H2, are defined, for n= 1,2,..., by: an(T) = inf{kT−Rk; rank (R)< n}. We have: an(T) = cn(T) = dn(T), where the numbers cn(resp. dn) are the Gelfand (resp. Kolmogorov)numbers of T([6], page 59 and page 51 respectively). In the sequel we shall need the following quantity: (1.3) β(T) = lim inf n→∞ an(T)1/n. Those approximation numbers form a non-increasing sequence such that a1(T) = kTk, an(T) = an(T∗) = pan(T∗T) and verify the so-called “ideal” and “subadditivity” properties ([17] page 57 and page 68): (1.4) an(AT B)≤ kAkan(T)kBk;an+m−1(S+T)≤an(S) + am(T). Moreover, the sequence (an(T)) tends to 0iff Tis compact. If (an(T)) ∈ℓp, we say that Tbelongs to the Schatten class Spof index p,0< p < ∞. Taking for Ta compact diagonal operator, we see that this sequence is non-increasing with limit 0, but otherwise arbitrary. But if we restrict ourselves to a specified class of operators, the answer is far from being so simple, although in some cases the situation is completely elucidated. For example, for the class of Hankel operators on H2(those operators Hφwhose matrix (ai,j )on the canonical basis 2 of H2is of the form ai,j =b φ(i+j)for some function φ∈L∞), it is known that Hφis compact if and only if the conjugate ¯ φof the symbol φbelongs to H∞+C, where Cdenotes the space of continuous, 2π-periodic functions (Hartman’s theorem, [32] page 214). For those Hankel operators, the following theorem, due to A. V. Megretskii, V. V. Peller, and S. R. Treil ([31] and [37], Theorem 0.1, page 490), shows that the approximation numbers are absolutely arbitrary, under the following form. Theorem 1.1 (Megretskii-Peller-Treil) Let (εn)n≥1be a non-increasing sequence of positive numbers. Then there exists a Hankel operator Hφsatisfying: an(Hφ) = εn,∀n≥1. Indeed, if we take a positive self-adjoint operator Awhose eigenvalues sncoincide with the εn’s and whose kernel is infinite-dimensional, it is easily checked that this operator Averifies the three necessary and sufficient conditions of Theorem 0.1, page 490 in [37] and is therefore unitarily equivalent to a Hankel operator Hφwhich will verify, in view of (1.4): an(Hφ) = an(A) = εn, n = 1,2,... In particular, if εn→0, the above Hankel operator will be compact, and in no Schatten class if εn= 1/log(n+ 1) for example. We also refer to [16] for the following slightly weaker form due to S. V. Khruscëv and V. Peller, but with a more elementary proof based on interpolation sequences in the Carleson sense: for any δ > 0, there exists a Hankel operator Hφsuch that 1 1 + δεn≤an(Hφ)≤(1 + δ)εn, n = 1,2,... Now, the aim of this work is to prove analogous theorems for the class of composition operators (whose compactness was characterized in [29] and [42]). But if we are able to obtain the Khruscëv-Peller analogue for the lower bounds, we will only obtain subexponential estimates for the upper bounds, a fact which is explained by our second result: the speed of convergence to 0of the approximation numbers of a composition operator cannot be greater than geometric (and is geometric for symbols ϕverifying kϕk∞<1). Our first result involves a constant <1and is not as precise as the result of Megretskii-Peller-Treil or even that of Khruscëv-Peller; this is apparently due to the non-linearity of the dependence with respect to the symbol for the class of composition operators, contrary to the case of the Hankel class.This latter lower bound improves several previously known results on “non-Schattenness” of those operators (see Corollary 4.2 below) and also answers in the positive to a question which was first asked to us by C. Le Merdy ([26]) in the OT Conference 2008 of Timisoara, concerning the bad rate of approximation of compact composition operators. Those theorems are, to our knowledge, the first individual results on approximation numbers anof composition operators (in the work of Parfenov [35], some good estimates are given for the approximation numbers of the Carleson embedding 3 operator in the case of the space H2=B−1, but they remain fairly implicit, and are not connected with composition operators), whereas all previous results where in terms of symmetric norms of the sequence (an), not on the behaviour of each an. Before describing our results, let us recall two definitions. For every ξwith |ξ|= 1 and 0< h < 1, the Carleson window W(ξ, h)centered at ξand of size h is the set W(ξ, h) = {z∈D;|z| ≥ 1−hand |arg(zξ)| ≤ πh}. Let µbe a positive, finite, measure on D; the associated maximal function ρµis defined by: (1.5) ρµ(h) = sup |ξ|=1 µW(ξ, h). The measure µis called a Carleson measure for the Bergman space Bα, or an (α+ 2)-Carleson measure (including the case B−1=H2), if ρµ(h) = O(h2+α) as h→0. For any Schur function ϕ, we shall denote by mϕthe image ϕ∗(m)of the Haar measure mof the unit circle under the radial limits function ϕ∗(u) = limr→1−ϕ(ru)of ϕ,|u|= 1, and by Aϕ,α+2 the image of the probability measure (α+ 1)(1 −|z|2)αdA(z)under ϕ. The corresponding maximal function will be denoted by ρϕ,α+2. This notation is justified by the fact that mϕ def =Aϕ,1is a 1Carleson measure and Aϕ,α an (α+ 2)-Carleson measure for α > −1, in view of the famous Carleson embedding theorem which, expressed under a quantitative and generalized form, states the following, implicit as concerns kjkand with different notations, but fully proved in [44], Theorem 1.2, for the case α > −1 (see [32], page 153). Theorem 1.2 (Carleson’s theorem) For any (α+ 2)-Carleson measure µ, the canonical inclusion mapping j:Bα→L2(µ)is defined and continuous, and its norm satisfies (1.6) C−1sup 0<h<1rρµ(h) h2+α≤ kjk ≤ Csup 0<h<1rρµ(h) h2+α· The paper is organized as follows. Section 1 is this introduction. In Section 2, we prove some preliminary lemmas. Our first theorems concern lower bounds. In Section 3, we prove (Theorem 3.1) that the convergence of the approximation numbers an(Cϕ)of a composition operator Cϕ:Bα→Bαcannot exceed an exponential speed: for some r∈(0,1) and some constant c > 0, one has an(Cϕ)≥c rn. More precisely, with the notations (1.3) and (3.1), one has β(Cϕ)≥[ϕ]2. Moreover, this speed of convergence is only attained if the values of ϕdo not approach the boundary of the unit disk: kϕk∞<1(Theorem 3.4). On the other hand, the speed of convergence to 0of an(Cϕ)can be arbitrarily slow; this is proved in Section 4. The proof is mainly an adaptation of the 4 one in [7], but is fairly technical at some points, and will require several additional explanations. In Section 5, we prove an upper estimate (Theorem 5.1), and give three applications of this theorem. In the final Section 6, we test our general results against the example of lens maps, which are known to generate composition operators belonging to all Schatten classes. 2 Preliminary lemmas In this Section, we shall state several lemmas, which are either already known or quite elementary, but turn out to be necessary for the proofs of our Theorem 3.1 and Theorem 4.1. For the proof of Theorem 3.1, we shall need the Weyl lemma ([6] Proposition 4.4.2, page 157). Lemma 2.1 (Weyl lemma) Let T:H→Hbe a compact operator. Suppose that (λn)n≥1is the sequence of eigenvalues of Trearranged in non-increasing order. Then, we have: n Y k=1 ak(T)≥ n Y k=1 |λk|. We recall ([3], [13] pages 194–195, [33] pages 302–303) that an interpolation sequence (zn)with (best) interpolation constant Cis a sequence (zn)(necessarily Blaschke, i.e. P∞ n=1(1 −|zn|)<∞) in the unit disk such that, for any bounded sequence (wn)of scalars, there exists a bounded analytic function f(i.e. f∈ H∞) such that: f(zn) = wn,∀n≥1,and kfk∞≤Csupn≥1|wn|. The Carleson constant δof a Blaschke sequence (zn)is defined as follows: (2.1) δn=Y j6=n ρ(zn, zj) ; δ= inf δn= inf n≥1(1 −|zn|2)|B′(zn)|, where Bis the Blaschke product with zeroes zn,n≥1. The interpolation constant Cis related to the Carleson constant δby the following inequality ([10] page 278), in which λis a positive numerical constant: (2.2) 1 δ≤C≤λ δ1 + log 1 δ· This latter inequality can be viewed as a quantitative form of the Carleson interpolation theorem. Interpolation sequences and reproducing kernels of Bα are related as follows ([33] pages 302–303). Lemma 2.2 Let (zn)n≥1be an H∞-interpolation sequence of the unit disk, with interpolation constant C. Then, the sequence (fn) = (Kzn/kKznk)of 5 normalized reproducing kernels at znis C-equivalent to an orthonormal basis in Bα, namely we have for any finite sequence (λn)of scalars: (2.3) C−1Xn|λn|21/2≤Xnλnfnα≤CXn|λn|21/2. The proof in [33] is only for H2, therefore we indicate a simple proof valid for Bergman spaces Bαas well. Let S=PλnKznbe a finite linear combination of the kernels Kzn,ω= (ωn)be a sequence of complex signs, Sω=PωnλnKzn and g∈H∞an interpolating function for the sequence (ωn), i.e. g(zn) = ωn and kgk∞≤C. If f∈Bαand kfkα≤1, we see that: hSω, fi=Xωnλnf(zn) = Xλn(fg)(zn) = XλnhKzn, fgi=hS, fgi, so that using (1.2): |hSω, fi| ≤ kSkαkfgkα≤ kSkαkgk∞kfkα≤CkSkα and passing to the supremum on f, we get kSωkα≤CkSkα. Since the coefficients λnare arbitrary, this implies that (fn)is C-unconditional, namely: C−1Xωnλnfnα≤Xλnfnα≤CXωnλnfnα. Now, squaring and integrating with respect to random, independent, choices of signs ωn’s, we get (2.3).  We also recall ([13] pages 203–204) that an increasing sequence (rn)of numbers such that 0< rn<1and 1−rn+1 1−rn≤ρ < 1(i.e. verifying the so-called Hayman-Newman condition) is an interpolation sequence (see also [32]). In the following, let (rn)be such a sequence verifying moreover the backward induction relation: (2.4) ϕ(rn+1) = rn. Set fn=Krn/kKrnkand W=span(fn). Let (en)n≥1be the canonical basis of ℓ2,ϕa Schur function and h∈H∞a function vanishing at r1. Denote by Mh:Bα→Bαthe operator of multiplication by h. Then, we have the following basic lemma, which shows that some compression of C∗ ϕis a backward shift with controlled weights ([7]). Lemma 2.3 Let J:ℓ2→Wbe the isomorphism given by J(en) = fn. Then, the operator B=J−1C∗ ϕM∗ hJ:ℓ2→ℓ2is the weighted backward shift given by: (2.5) B(en+1) = wnenand B(e1) = 0,where wn=h(rn+1)kKrnk kKrn+1 k· 6 To exploit Lemma 2.3, we shall need the following simple fact on approximation numbers of weighted backward shifts. Lemma 2.4 Let (en)n≥1be an orthonormal basis of the Hilbert space Hand B∈ L(H)the weighted backward shift defined by B(e1) = 0 and B(en+1) = wnen,where wn→0. Assume that |wn| ≥ εnfor all n≥1, where (εn)is a non-increasing sequence of positive numbers. Then Bis compact, and satisfies: (2.6) an(B)≥εn,∀n≥1. Proof. The compactness of Bis obvious. Let Rbe an operator of rank < n. Then ker Ris of codimension < n, and therefore intersects the n-dimensional space generated by e2,...,en+1 in a vector x=Pn j=1 xjej+1 of norm one. We then have: kB−Rk2≥ kBx−Rxk2=kBxk2=Xn j=1 |wj|2|xj|2 ≥Xn j=1 ε2 j|xj|2≥ε2 nXn j=1 |xj|2=ε2 n. This ends the proof of Lemma 2.4.  Now, in view of (1.1) and (2.5), the weight wnroughly behaves as q1−rn+1 1−rn, so we shall need good estimates on that quotient, before defining the sequence (rn)explicitly. We first connect this estimate with the hyperbolic distance din D. We denote (see [12] or [15] for the definition) by d(z, w;U)the hyperbolic distance of two points z, w of a simply connected domain U. It follows from the generalized Schwarz-Pick lemma ([15] Theorem 7.3.1, page 130) applied to the canonical injection U→Vthat the bigger the domain the smaller the hyperbolic distance, namely: (2.7) U⊂Vand z, w ∈U=⇒d(z, w;V)≤d(z, w;U). Moreover, as is well-known, 0≤r < 1 =⇒d(0, r;D) = 1 2log 1 + r 1−r· Recall that the pseudo-hyperbolic and hyperbolic distances ρand don Dare defined by: ρ(a, b) = a−b 1−ab,d(a, b) = 1 2log 1 + ρ(a, b) 1−ρ(a, b),a, b ∈D, In the sequel, we shall omit the symbol Das far as the open unit disk is concerned. For this unit disk, we have the following simple inequality ([7]) . Lemma 2.5 Let a, b ∈Dwith 0< a < b < 1. Then: (2.8) e−2d(a,b)≤1−b 1−a≤2 e−2d(a,b). 7 Finally, before proceeding to the construction of our Schur function ϕin Section 4, it will be useful to note the following simple technical lemma. Lemma 2.6 Let (εn)be a non-increasing sequence of positive numbers of limit 0. Then there exists a decreasing and logarithmically convex sequence (δn)of positive numbers, with limit 0, such that δn≥εnfor all n≥1. Proof. Provided that we replace εnby εn+1 n, we may assume that (εn)is decreasing. Let us define our new sequence by the inductive relation: δ1=ε1;δ2=ε2;δn+1 = max εn+1, δ2 n/δn−1. This sequence is log-convex by definition, i.e. δ2 n≤δn+1δn−1. By induction, it is seen to be decreasing. Therefore, it has a limit l≥0. If δn=εnfor infinitely many indices, l= 0. Otherwise, for nlarge enough, we have the inductive relation δn+1 =δ2 n/δn−1, which implies that δn= exp(λn +µ)for some constants λ, µ. Since (δn)is decreasing, we must have λ < 0and again we get l= 0. In the sequel, we may and will thus assume, without loss of generality, that (εn)is decreasing and logarithmically convex. 3 Lower bounds We first introduce a notation. If ϕ#(z) = lim w→z ρ(ϕ(w), ϕ(z)) ρ(w, z)=|ϕ′(z)|(1 −|z|2) 1−|ϕ(z)|2 is the pseudo-hyperbolic derivative of ϕ, we set: (3.1) [ϕ] = sup z∈D ϕ#(z) = kϕ#k∞. In our first theorem, we get that the approximation numbers cannot supersede a geometric speed. Theorem 3.1 For any Schur function ϕ, there exist positive constants c > 0 and 0< r < 1such that, for Cϕ:Bα→Bα, we have: (3.2) an(Cϕ)≥c rn, n = 1,2,... More precisely, one has β(Cϕ)≥[ϕ]2and hence, for each κ < [ϕ], there exists a constant cκ>0such that: (3.3) an(Cφ)≥cκκ2n. 8 For the proof, we need the following lemma. Lemma 3.2 Let T:H→Hbe a compact operator. Suppose that (λn)n≥1, the sequence of eigenvalues of Trearranged in non-increasing order, satisfies, for some δ > 0and r∈(0,1): |λn| ≥ δrn, n = 1,2,... Then there exists δ1>0such that an(T)≥δ1r2n, n = 1,2,... In particular β(T)≥r2. Proof. By Weyl’s inequality (Lemma 2.1), we have n Y k=1 ak(T)≥ n Y k=1 |λk| ≥ δnrn(n+1)/2. Since ak(T)is non-increasing and ak(T)≤ kTkfor every k, changing ninto 2n, we get: kTknan(T)n≥ 2n Y k=1 ak(T)≥δ2nrn(2n+1) ≥δ2nr2n2 and therefore an(T)≥δ2 kTkr2n=δ1r2n, as claimed.  By applying this lemma to composition operators, we get the following result, which ends the proof of Theorem 3.1. Proposition 3.3 For every composition operator Cϕ:Bα→Bαof symbol ϕ:D→D, we have β(Cϕ)≥[ϕ]2. Proof. For every a∈D, let Φabe the (involutive) automorphism of the unit disk defined by Φa(z) = a−z 1−az ,z∈D. Observe that we have Φa(a) = 0,Φa(0) = a, Φ′ a(a) = 1 |a|2−1,Φ′ a(0) = |a|2−1. Define now ψ= Φϕ(a)◦ϕ◦Φa. We have that 0is a fixed point of ψ, whose derivative is, by the chain rule: (3.4) ψ′(0) = Φ′ ϕ(a)(φ(a))ϕ′(a)Φ′ a(0) = ϕ′(a)(1 −|a|2) 1−|ϕ(a)|2 def =ϕ#(a). By Schwarz’s lemma, we know that |ψ′(0)| ≤ 1and so |ϕ′(a)|(1−|a|2) 1−|ϕ(a)|2≤1 (Schwarz-Pick’s inequality). 9 where a1< a2and b > 0. Then, we have the upper estimate: (4.2) d(a1, a2;U)≤π 4b(a2−a1) + π 2· Proposition 4.4 Suppose that Ucontains the rectangle R={z∈C;a1−c < Rez < a2+c, |Imz|< c}, where a1< a2and c > 0, but that the horizontal sides {z∈C;a1−c≤Rez≤a2+c, |Imz|=c} of that rectangle are disjoint from U. Then, we have the lower estimate: (4.3) d(a1, a2;U)≥π 4c(a2−a1)−π 2· We now proceed to the construction of our Schur function ϕ. We first define a continuous map ψ:R→Ras follows: ψ(t) = (K(1 + |t|) if |t| ≤ 1 |t|/A(|t|) if |t|>1, where Kis a positive constant adjusted below and A: [0,∞[→[0,∞[an increasing piecewise linear function on the intervals (0,1) and (en−1,en)such that A(0) def =A0= 0, A(en−1)def =Anfor n≥1,and 2K= 1/A(1), the increasing sequence (An)being positive and concave for n≥1, and tending to ∞. It then follows that the sequence of slopes An−An−1 en−en−1is decreasing, since An+1 −An≤An−An−1≤e (An−An−1), that the function Ais increasing and concave on (0,∞)and vanishing at 0, implying that A(t)/t is decreasing on (0,∞), and that in particular ψis increasing on (1,∞). We then define a domain Ωof the complex plane by: (4.4) Ω = {w∈C;|Imw|< ψ(|Rew|)}. Let σ:D→Ωbe the unique Riemann map such that σ(0) = 0 and σ′(0) >0. This map exists in view of the following simple fact. Lemma 4.5 The domain Ωdefined by (4.4) is star-shaped with respect to the origin and σ: (−1,1) →Ris an increasing bijection such that σ(−1) = −∞ and σ(1) = ∞. Proof. The star-shaped character of Ωwill follow from the implication: |Imw|< ψ(|Rew|)and 0< λ < 1 =⇒ |Im (λw)|< ψ(|Re (λw)|). 16 We may assume that both Rew, Imware positive, and it is enough to prove: (4.5) λψ(x)≤ψ(λx),∀λ∈[0,1],∀x > 0. This is easy to check separating three cases: 1) x≤1; then λψ(x) = λK(1 + x)≤K(1 + λx) = ψ(λx); 2) λx ≤1< x; then, since A(x)> A(1), λψ(x) = λx A(x)<2Kλx ≤K(1 + λx) = ψ(λx); 3) λx > 1; we then have, since ψincreases, λψ(x) = λx A(x)≤λx A(λx)=ψ(λx) and this ends the proof of (4.5). Now, since σis determined by the value of σ(0) and the sign of σ′(0), we have σ(z) = σ(z)for all z∈D, so that σ[(−1,1)] ⊂R. And since the derivative of an injective analytic function does not vanish and σ′(0) >0, we get that σis increasing on (−1,1). Finally, if w∈Rand w=σ(z), we have w=w, so that σ(z) = σ(z)and z=z, which proves the surjectivity of σ: (−1,1) →R. We now choose Anas follows, η > 0denoting a positive numerical constant to be specified later. (4.6) An=ηlog 1 εn ,n≥1. Observe that this is an increasing, concave sequence tending to ∞since we assumed that (εn)is log-convex and decreasing to 0. Finally, we define our Schur function ϕand our sequence (rn)under the form of the following lemma, in which the increasing character of ψis important. Lemma 4.6 Let ϕbe defined by ϕ(z) = σ−1(e−1σ(z)), and let rn=σ−1(en). Then we have: 1. ϕis univalent and maps Dto D,(rn)increases, and ϕ(0) = 0; 2. ϕ(rn+1) = rn; 3. 1−rn+1 1−rn→0and therefore (rn)is an interpolation sequence; 4. Cϕ:Bα→Bαis compact. Proof. 1. Since Ωis star-shaped, e−1σ(z)∈Ωwhen z∈D, so ϕis well-defined and maps Dto itself in a univalent way. Moreover, ϕ(0) = σ−1(0) = 0, and (rn) increases since σ−1increases on R. 2. We have ϕ(rn+1) = σ−11 eσ(rn+1)=σ−11 een+1=σ−1(en) = rn. 17 3. This assertion is more delicate and relies on Proposition 4.4 as follows. Set dn=ψ(en). We have clearly en+1 +dn+2 <en+2 for large n(recall that ψ(t) = o(t)as t→ ∞), so that ψ(en+1 +dn+2)< ψ(en+2) = dn+2 since ψis increasing. By the intermediate value theorem for the function ψ(en+1 +x)−x, we can therefore find a positive number cn< dn+2 such that ψ(en+1 +cn) = cn. Now, consider the open sets: Rn={z∈C; en−cn<Rez < en+1 +cnand |Imz|< cn}, Un=Rn∪Ω. Those sets Unsatisfy the assumptions of Proposition 4.4 in view of (4.4). Indeed, if zbelongs to the horizontal sides of Rn, we have z /∈Unsince en−cn≤Rez≤en+1 +cn=⇒ψ(Rez)≤ψ(en+1 +cn) = cn=|Imz|. This proposition then gives, since Ω⊂Unand cn< dn+2, and since the hyperbolic metric is conformally invariant, d(rn, rn+1) = d(en,en+1; Ω) ≥d(en,en+1;Un)≥π 4cn (en+1 −en)−π 2 ≥cen+2 ψ(en+2)=cA(en+2)≥cAn, where cis a positive constant. Now, we use Lemma 2.5 to obtain: 1−rn+1 1−rn≤2 e−2d(rn,rn+1)≤2 e−2cAn, which proves that 1−rn+1 1−rn→0, and implies that (rn)is an interpolation sequence. 4. Since ϕis univalent, the compactness of Cϕ:Bα→Bαamounts to proving that lim|z|→11−|ϕ(z)| 1−|z|=∞. For α > −1, this follows from [30], Theorem 3.5 and for α=−1from [41], page 39. By the Julia-Carathéodory Theorem ([41], page 57), this in turn is equivalent to proving that for any u, v on the unit circle, the quotient ϕ(z)−v z−uhas no finite limit as ztends to uradially. This latter fact requires some precise justification. First, we notice that σextends continuously to an injective map of the open upper half of the unit circle onto the upper part of the boundary of Ω (and similarly for lower parts). This follows from the Carathéodory extension theorem ([39], page 290), applied to the restriction of σ−1to the Jordan region limited by ∂Ωand two vertical lines Rew=±Rwhere R > 0is arbitrarily large. Now, let u∈∂Dwith u6=±1. Then, σ(ru)→w∈∂Ωas r→1−, so that e−1σ(ru)→e−1w=w′∈Ωand that ϕ(ru)→σ−1(w′)∈D. Therefore the image of ϕtouches the unit circle only at ±1, and the assumption of the Julia-Carathéodory Theorem is fulfilled if u6=±1. By symmetry, it remains to test the point u= 1 for which we have: lim sup r< −→1 1−ϕ(r) 1−r≥lim sup n→∞ 1−ϕ(rn+1) 1−rn+1 = lim sup n→∞ 1−rn 1−rn+1 =∞ 18 by the preceding point 3. Since |v−ϕ(r)| ≥ 1−ϕ(r), this ends the proof of Lemma 4.6.  We now want a good lower bound for the weights wnappearing in (2.5). To that effect, we apply Proposition 4.3 with U= Ω, a1= en, a2= en+1 and bn=ψ(en−1), as well as R′ n={z∈C; en−bn<Rez < en+1 +bnand |Imz|< bn}. We have en−bn>en−1for large n, since this amounts to en−en−1> bn=en−1 A(en−1),or e−1>1 A(en−1), which holds for large nsince A(t)tends to ∞with t. We then observe that R′ n⊂Ω. Indeed, z∈R′ n=⇒Rez > en−bn>en−1and, since ψis increasing, we have ψ(Rez)> ψ(en−1) = bn>|Imz|. Therefore, we can apply (4.2) and get, for all n≥1: d(en,en+1; Ω) ≤π 4ψ(en−1)(en+1 −en) + π 2≤C0A(en−1) = C0An, where C0is a numerical constant. By conformal invariance, we have as well d(rn, rn+1)≤C0An. It then follows from (2.8) that: (4.7) 1−rn+1 1−rn≥exp −2d(rn, rn+1)≥exp(−2C0An). Now, we take h(z) = z−r1in Lemma 2.4 and use the ideal property (1.4) of the approximation numbers. We get, denoting by Cthe interpolation constant of the sequence (rn), and using the fact that kMhk=khk∞≤2: (4.8) an(B)≤ kJ−1kan(Cϕ)kMhkkJk ≤ 2C2an(Cϕ). Next, we choose η= 1/C0in (4.6) and we set d= (r2−r1)/√2. Using Lemma 2.3 and relations (1.1), (2.5) and (4.7), we see that the weights wnassociated with Bverify: |wn|=h(rn+1)kKrnk kKrn+1 k=h(rn+1)s1−r2 n+1 1−r2 n≥r2−r1 √2r1−rn+1 1−rn ≥dexp(−C0An)≥dεnfor all n≥1. (4.9) Finally, using Lemma 2.4, (4.8) and (4.9): an(Cϕ)≥1 2C2an(B)≥1 2C2d εn def =δεnfor all n≥1. We thus get the desired conclusion (4.1) of Theorem 4.1.  19 5 An upper bound We do not obtain a fairly good upper bound, and we shall content ourselves with the following result, whose proof is quite simple and, for the case α=−1, partly contained in [35], but under a very cryptic form which is not easy to decipher. Theorem 5.1 Let ϕbe a Schur function and α≥ −1.Then, we have for the approximation numbers of Cϕ:Bα→Bαthe upper bound: (5.1) an(Cϕ)≤Cinf 0<h<1nα+1 2(1 −h)n+rρϕ,α+2(h) h2+α, n = 1,2,... where Cis a constant. In particular, if ρϕ,α+2(h) h2+α≤e−h/A(h), where the function A: [0,1] →[0,1] is increasing, with A(0) = 0 and with inverse function A−1, we have: (5.2) an(Cϕ)≤Cnα+1 2e−nA−1(1/2n), n = 1,2,.... The proof of (5.1) uses a contraction principle which was first proved for α=−1([18]) and α= 0 ([23]), but is also valid for any α≥ −1, as follows from the forthcoming work [25]. To prove Theorem 5.1, it will be convenient to prove first the following simple lemma. Lemma 5.2 Let nbe a positive integer, g∈Bαand f(z) = zng(z). Then, we have: (5.3) kgkα≤Cnα+1 2kfkα. Proof. Let wn=n!Γ(2+α) Γ(n+2+α)and f(z) = P∞ n=0 anzn. We first observe that (5.4) wk wk+n≤Cnα+1,∀k≥0,∀n≥1. Indeed, we have: wk wk+n =k! (k+n)! Γ(k+α+ 2 + n) Γ(k+α+ 2) = n Y j=1 (k+j+α+ 1) (k+j)≤ n Y j=1 j+α+ 1 j = n Y j=1 1 + α+ 1 j≤exp (α+ 1) n X j=1 1 j≤Cnα+1, which proves (5.4). Now, if f(z) = P∞ k=nakzk, we have g(z) = P∞ k=0 ak+nzkso that, using (5.4): kgk2 α=∞ X k=0 |ak+n|2wk=∞ X l=n|al|2wl−n≤Cnα+1 ∞ X l=n|al|2wl=Cnα+1kfk2 α, 20 proving (5.3).  We shall now majorize an+1(Cϕ), but provided that we change the constant C, this makes no difference with majorizing an(Cϕ). The choice of the approximating operator Rof rank ≤nfor Cϕis quite primitive, but in counterpart we shall estimate kCϕ−Rkrather sharply. We denote by Pnthe projection operator defined by Pnf=Pn−1 k=0 ˆ f(k)zkand we take R=Cϕ◦Pn, i.e. if we have f(z) = P∞ k=0 ˆ f(k)zk∈Bα, then R(f) = Pn−1 k=0 ˆ f(k)ϕk, so that (Cϕ−R)f=Cϕ(r), with, making use of (5.3): (5.5) r(z) = ∞ X k=nb f(k)zk=zns(z),with ksk2 α≤Cnα+1krk2 α,krkα≤ kfkα. Assume that kfkα≤1, fix 0< h < 1and denote by µhthe restriction of the measure Aϕ,α+2 to the annulus 1−h < |z| ≤ 1. Then, we have: k(Cϕ−R)fk2 α=kCϕ(r)k2 α=ZD|r(z)|2dAϕ,α+2(z) ≤(1 −h)2nZ|z|≤1−h|s(z)|2dAϕ,α+2(z) +Z1−h<|z|≤1|r(z)|2dAϕ,α+2(z) ≤(1 −h)2nZD|s(z)|2dAϕ,α+2(z) + ZD|r(z)|2dµh(z) = (1 −h)2nkCϕ(s)k2 α+ZD|r(z)|2dµh(z) ≤C(1 −h)2nksk2 α+ZD|r(z)|2dµh(z) ≤Cnα+1(1 −h)2n+ sup 0<t≤h ρϕ,α+2(t) t2+α if we use (5.5), as well as (1.6) under the form ZD|r(z)|2dµh(z)≤Csup 0<t≤h ρϕ,α+2(t) t2+αkrk2 α, and we know that krkα≤ kfkα≤1. To get rid of the supremum with respect to t, we make use of the following inequality, which holds for h≤1−|ϕ(0)|and 0< ε ≤1: (5.6) ρϕ,α+2(εh)≤Cεα+2ρϕ,α+2(h). For α= 0 or α=−1, this follows respectively from [18], Theorem 4.19, p. 55, and from [23], Theorem 3.1. The general case is proved in [25]. Setting t=εh for 0< t ≤h, this also reads ρϕ,α+2(t) tα+2 ≤Cρϕ,α+2(h) hα+2 , and we can forget the 21 supremum in tin the previous inequalities. Taking square roots, we get the relation (5.1). When ρϕ,α+2(h) h2+α≤e−h/A(h), let us take for hthe nearly optimal value h= A−1(1/2n), so that h/A(h) = 2nh. We then have from (5.1), since (1 −h)2n≤ e−2nh: an+1(Cϕ)2≤ kCϕ−Rk2 α≤Cnα+1[e−2nh + e−h/A(h)]≤2Cnα+1e−2nA−1(1/2n), proving (5.2), and ending the proof of Theorem 5.1.  Let us now indicate three corollaries, which improve results of [19], [22], and [23] respectively. Corollary 5.3 Suppose that ρϕ,α+2(h)≤Ch(2+α)βfor some β > 1. Then: an(Cϕ)≤Cn−(β−1)(α+2) 2(log n)(β−1)(α+2) 2. In particular, Cϕbelongs to the Schatten class Sp=Sp(Bα)for each p > 2 (β−1)(α+2) · Proof. Set γ= (β−1)(α+ 2)/2,a= (α+ 1)/2, and c=a+γ. If we apply (5.1) of Theorem 5.1 with the value h=clog n/n which satisfies nae−nh =n−γ, as well as the inequality (1 −h)n≤e−nh, we get: an(Cϕ)≤Cn−γ+log n nγ≤Clog n nγ, ending the proof.  In [19], we had only the assertion on Schatten classes, for the single value α=−1, and not the upper bound for the individual approximation numbers an(Cϕ). Corollary 5.4 Let (εn)a sequence of positive numbers which tends to 0. Then, there exists a Schur function ϕwith the following properties: 1. ϕ:D→Dis surjective and 4-valent; 2. an(Cϕ)≤Ce−nεn,n= 1,2,... In particular, we can get an(Cϕ)≤Ce−n log(n+1) and Cϕis in every Schatten class Sp(Bα),p > 0. Notice that the sequence (εn)in the statement cannot be dispensed with. Indeed, if ϕis surjective, we surely have kϕk∞= 1! And we know from Theorem 3.4 that β(Cϕ) = 1 in that case. We begin with a lemma of independent interest. 22 Lemma 5.5 Let δ: (0,1] →Rbe a positive and non-decreasing function. Then there exists a Schur function ϕwith the following properties: 1. ϕ:D→Dis surjective and 4-valent; 2. ρϕ,α+2(h)≤δ(h), for h > 0small enough. Proof. We begin with the case α=−1. Set, for a= 1/2: Φa(z) = a−z 1−az ,B= Φ2 a, and C=1+a 2(1−a)= 3/2. Note that B2a a+1 =B(0). Let now bn=1 4nπ,ε(h) = 1 2δ(h/C), εn=ε(bn+1). In the proof of Theorem 4.1 of [22], using an argument of harmonic measure and of barrier, we have found a 2-valent symbol ϕ1with ϕ1(D) = D∗such that, noting ρϕfor ρϕ,1: (5.7) bn+1 < h ≤bn=⇒ρϕ1(h)≤εn. This gives ρϕ1(h)≤ε(bn+1)≤ε(h). Let now, as in [22], ϕ=B◦ϕ1. This Schur function is surjective (since ϕ(D) = B(D∗) = B(D) = D), and 4-valent. Moreover, if I= (u, v)is an arc of Tof length h < 1/2and J= (u 2,v 2), we have B−1(I)⊂Φa(J)∪Φa(−J) = I1∪I2, where I1, I2are two arcs of Tof length at most kPak∞(h/2) = Ch, since Φabeing an inner function, we have ([34]), Pa being the Poisson kernel at a: mΦa=Pam. Hence, using (5.7), we obtain: mϕ(I) = mϕ1(B−1(I)) ≤mϕ1(I1) + mϕ1(I2)≤2ρϕ1(Ch)≤2ε(Ch) = δ(h), and ρϕ(h)≤δ(h)for small h, by passing to the supremum on all I’s. For the general case α≥ −1, we use the following extension of an inequality from [23] (which treats the case α= 0, see Remark before Corollary 3.11): Lemma 5.6 For small h, namely 0< h < (1 − |ϕ(0)|)/4, we have, for every α > −1: (5.8) ρϕ,α+2(h)≤C[ρϕ(Ch)]α+2. Proof. Let us define, as in [42], the generalized Nevanlinna counting function Nϕ,α+2 by the formula Nϕ,α+2(w) = X ϕ(z)=w [log(1/|z|)]α+2, w ∈D\{ϕ(0)}. 23 The case α=−1corresponds to the usual Nevanlinna counting function, which will be denoted by Nϕ. The partial Nevanlinna counting function Nϕ(r, w)is defined, for 0≤r≤1, by: Nϕ(r, w) = X ϕ(z)=w log+(r/|z|), so that Nϕ(1, w) = Nϕ(w). Since α+ 2 ≥1, we have the obvious but useful inequality: (5.9) Nϕ,α+2(w)≤[Nϕ(w)]α+2. We shall also make use of the following identity, due to J. Shapiro ([42], Proposition 6.6, where a weight 1/r is missing), and which can easily be checked after two integrations by parts: (5.10) Nϕ,α+2(w) = (α+ 2)(α+ 1) Z1 0 Nϕ(r, w)[log(1/r)]αdr r· As it was noticed in ([23], Theorem 3.10), this formula reads, for wclose to the boundary, as follows, for 0< h < (1 −|ϕ(0)|)/4and |w|>1−h: (5.11) Nϕ,α+2(w) = (α+ 2)(α+ 1) Z1 1/3 Nϕ(r, w)[log(1/r)]αdr r· Under the same conditions on hand w, this obviously implies: Nϕ,α+2(w)≥1 CZ1 1/3 Nϕ(r, w)(1 −r2)αr dr =1 CZ1 0 Nϕ(r, w)(1 −r2)αr dr. Now, using the same arguments as in [23], Theorem 3.10 and in particular using (5.11) for ϕr(z) = ϕ(rz), the identity Nϕ(r, w) = Nϕr(w)and an integration in polar coordinates, we get: (5.12) sup |w|≥1−h Nϕ,α+2(w)≥1 Cρϕ,α+2(h/C). The end of the proof is easy: changing hinto Ch and using successively (5.12) and (5.9), we get for small h, depending on ϕ: ρϕ,α+2(h)≤Csup |w|≥1−Ch Nϕ,α+2(w)≤Csup |w|≥1−Ch [Nϕ(w)]α+2 ≤C[ρϕ(Ch)]α+2, the last inequality coming from [21], Theorem 3.1. This ends the proof of (5.8). Going back to the proof of Lemma 5.5, if we apply the already settled case α=−1to the function ˜ δ(h) = [δ(h/C)/C]1 α+2 , we obtain a surjective and 4-valent Schur function ϕsuch that: ρϕ,α+2(h)≤C[ρϕ(Ch)]α+2 ≤C[˜ δ(Ch)]α+2 =δ(h), 24 for hsmall enough.  Proof of Corollary 5.4. Set a= (α+ 1)/2. Provided that we replace (εn) by the decreasing sequence (ε′ n)with ε′ n=1 n+ supk≥nεk≥εn, we can assume that (εn)decreases. Let A: [0,1] →[0,1] be a function such that A(0) = 0, and which increases (as well as A(t)/t) so slowly that A(εn+a(log n/n)) ≤1/2n; therefore A−1(1/2n)≥εn+a(log n/n)and nae−nA−1(1/2n)≤e−nεn. We now apply Lemma 5.5 to the non-decreasing function δ(h) = h2+αe−h/A(h) to get the result, in view of (5.2) of Theorem 5.1.  Our last corollary involves Hardy-Orlicz spaces Hψand Bergman-Orlicz spaces Bψ. For the definitions, we refer to [18]. Corollary 5.7 There exists a Schur function ϕand an Orlicz function ψsuch that Cϕ:Hψ→Hψis compact whereas Cϕ:Bψ→Bψis not compact. Moreover, the approximation numbers an(Cϕ)of Cϕ:Bα→Bαsatisfy the upper estimate an(Cϕ)≤ae−b√nwhere a,bare positive constants independent of n, and therefore Cϕbelongs to Tp>0Sp(Bα). Proof. Let α≥ −1be fixed. The Schur function constructed in the proof of Theorem 4.2 of [23] satisfies the two first assertions, as well as ρϕ(h)/h ≤e−d/h for some positive constant d > 0. We now apply (5.8) to get for small h: ρϕ,α+2(h) hα+2 ≤C[ρϕ(Ch)]α+2 hα+2 ≤Cα+3e−(α+2)d/Ch ≤ae−b/h for positive constants aand b. We can thus apply (5.2) of Theorem 5.1, for some δ > 0, with the increasing function A(h) = h2/δ (hence A−1(x) = √δx) to get the result, diminishing slightly bto absorb the power factor nα+1 2. Remark. Let us alternatively consider the entropy numbers en(Cϕ)(see [4] or [17], page 69 for the definition) of composition operators. Those numbers are also a very good indicator of the “degree of compactness” of general operators T:X→Ywhere X, Y are Banach spaces and are smaller than the approximation numbers, in the following weak sense ([38], page 64). sup 1≤k≤n [kαek(T)] ≤Cαsup 1≤k≤n [kαak(T)],∀α > 0.(5.13) (an(T)) ∈ℓq=⇒(en(T)) ∈ℓq,∀q > 0.(5.14) The converse of (5.14) does not hold in Banach spaces, but it does for operators between Hilbert spaces, by polar decomposition. More precisely, we have ([38], page 68) an(T)≤4en(T)and, in particular, (en(T)) ∈ℓqif and only if (an(T)) ∈ℓq. We now have the following improved version of Theorem 3.1. 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