Approximation numbers of composition operators on Hp

We give estimates for the approximation numbers of composition operators on the Hp spaces, 1 $\le$ p \textless{} $\infty$.


Introduction
Recently, the study of approximation numbers of composition operators on H 2 has been initiated (see [10], [11], [8], [18], [12]), and (upper and lower) estimates have been given. However, most of the techniques used there are specifically Hilbertian (in particular Weyl's inequality; see [10]). Here, we consider the case of composition operators on H p for 1 ≤ p < ∞. We focus essentially on lower estimates, because the upper ones are similar, with similar proofs, as in the Hilbertian case. We give in Theorem 2.4 a minoration involving the uniform separation constant of finite sequences in the unit disk and the interpolation constant of their images by the symbol. We finish with some upper estimates.

Preliminary
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 .
For every a ∈ D, we denote by e a ∈ (H p ) * the evaluation map at a, namely: We know that ( [22], p. 253): (1.2) e a = 1 1 − |a| 2 1/p and the mapping equation 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 ∞. As usual, the notation A B means that there is a constant c such that A ≤ c B and A ≈ B means that A B and B A.

Singular numbers
For an operator T : X → Y between Banach spaces X and Y , its approximation numbers are defined, for n ≥ 1, as: One has T = a 1 (T ) ≥ a 2 (T ) ≥ · · · ≥ a n (T ) ≥ a n+1 (T ) ≥ · · · , and (assuming that Y has the Approximation Property), T is compact if and only if a n (T ) −→ n→∞ 0.
The n-th Bernstein number b n (T ) of T , defined as: where S E = {x ∈ E ; x = 1} is the unit sphere of E. When these numbers tend to 0, T is said to be superstrictly singular, or finitely strictly singular (see [17]).
The n-th Gelfand number of T , defined as: One always has: (1.7) a n (T ) ≥ c n (T ) and a n (T ) ≥ b n (T ) , and, when X and Y are Hilbert spaces, one has a n (T ) = b n (T ) = c n (T ) ( [16], Theorem 2.1).

Sub-geometrical decay
We first show that, as in the Hilbertian case H 2 ([10], Theorem 3.1), the approximation numbers of the composition operators on H p cannot decrease faster than geometrically.
Though we cannot longer appeal to the Hilbertian techniques of [10], Weyl's inequality has the following generalization ( [3], Proposition 2).
Proof. If C ϕ is not compact, the result is trivial, with r = 1; so we assume that C ϕ is compact. Before carrying on, we first recall some notation used in [10]. For every z ∈ D, let be the pseudo-hyperbolic derivative of ϕ at z, and By the Schwarz-Pick inequality, one has [ϕ] ≤ 1. Moreover, since ϕ is not constant, one has [ϕ] > 0. We also set, for every operator T : H p → H p : For every a ∈ D, we are going to show that β − (C ϕ ) ≥ ϕ ♯ (a) 2 , which will give β − (C ϕ ) ≥ [ϕ] 2 , by taking the supremum for a ∈ D, and the stated result, with 0 < r < [ϕ] 2 .

Main result
In this section, we use the fortunate fact that, though the evaluation maps at well-chosen points of D can no longer be said to constitute a Riesz sequence, they will still constitute an unconditional sequence in H p with good constants, as we are going to see, which will be sufficient for our purposes.
We now prove the following lower estimate.
Then, for some constant c p depending only on p, we have: For the proof, we need to know some precisions on the constant in Carleson's embedding theorem. Recall that the uniform separation constant δ σ of a finite sequence σ = (z 1 , . . . , z n ) in the unit disk D, is defined by: . . , z n ) be a finite sequence of distinct points in D with uniform separation constant δ σ . Then: Proof. For a ∈ D, let k a (z) = √ 1−|a| 2 1−az be the normalized reproducing kernel. For every positive Borel measure µ on D, let: The so-called Reproducing Kernel Thesis (see [14], Lecture VII, pp. 151-158) says that there is an absolute positive constant A 1 such that: for every f ∈ H p (that follows from the case p = 2 in writing f = Bh 2/p where B is a Blaschke product and h ∈ H 2 ). Actually, one can take A 1 = 2 e (see [15], Theorem 0.2). But when µ is the discrete measure n j=1 (1 − |z j | 2 ) δ zj , it is not difficult to check (see [4], Lemma 1, p. 150, or [6], p. 201) that: That gives the result since 4 e ≤ 12.
We will separate three cases.
One has C * ϕ (L) = n j=1 λ j e vj . Using Lemma 2.3, we obtain for any choice of complex signs ω 1 , . . . , ω n : Let now q be the conjugate exponent of p. We know that the space H p is of type p as a subspace of L p ([9], p. 169) and therefore its dual (H p ) * is of cotype q ([9], p. 165), with cotype constant ≤ τ p , the type p constant of L p (let us note that we might use that (H p ) * is isomorphic to the subspace H q of L q , but we have then to introduce the constant of this isomorphism). Hence, by averaging (2.7) over all independent choices of signs and using the cotype q property of (H p ) * , we get: where: It remains to give a lower bound for Λ q . But, by Hölder's inequality:

Lemma 2.5 gives
: Taking the supremum over all f with f p ≤ 1, we get, taking into account that L = 1: By combining (2.8) and (2.9), we get: We follow the same route, but in this case, H p is of type 2 and hence (H p ) * is of cotype 2. Therefore, we get: and, using Cauchy-Schwarz inequality: (2.11) Λ 2 ≥ 12 1 + log 1 δ u −1/2 ; so: In this case (H 1 ) * (which is isomorphic to the space BM OA) has no finite cotype. But, for each k = 1, . . . , n, one has, using Lemma 2.3: hence: we get, as above, using Lemma 2.5: (2.14) and therefore: and that finishes the proof of Theorem 2.4.
Example. We will now apply this result to lens maps. We refer to [19] or [8] for their definition. For θ ∈ (0, 1), we denote: Proposition 2.6 Let λ θ be the lens map of parameter θ acting on H p , with 1 ≤ p < ∞. Then, for positive constants a and b, depending only on θ and p: Actually, this estimate is valid for polygonal maps as well.

A minoration depending on the radial behaviour of ϕ
We are using Theorem 2.4 to give, as in [11], Theorem 3.2, a lower bound for a n (C ϕ ) which depends on the behaviour of ϕ near ∂D.
We recall first (see [11], Section 3) that an analytic self-map ϕ : D → D is said to be real if it takes real values on ] − 1, 1[. If ω : [0, 1] → [0, 2] is a modulus of continuity (meaning that ω is continuous, increasing, sub-additive, vanishing at 0, and concave), ϕ is said to be an ω-radial symbol if it is real and: We have the following result.
Theorem 2.7 Let ϕ be an ω-radial symbol. Then, for 1 ≤ p < ∞, the approximation numbers of the composition operator C ϕ : H p → H p satisfy: where c ′ p is a constant depending only on p, p * is the conjugate exponent of p, and a = 1 − ϕ(0) > 0.
Proof. As in [11], p. 556, we fix 0 < σ < 1 and define inductively u j ∈ [0, 1) by u 0 = 0 and, using the intermediate value theorem: We set v j = ϕ(u j ). We have −1 < v j < 1 and 1 − v n = a σ n . We proved in [11], p. 556, that: Moreover, we proved in [11], p. 557, that the uniform separation constant of v = (v 1 , . . . , v n ) is such that: Since δ u ≥ δ v , we get, from (2.17), that: √ n) and hence: Note that the coefficient of √ n in the exponential is slightly different of that in (2.21), but of the same order.

Upper bound
For upper bounds, there is essentially no change with regard to the case p = 2. Hence we essentially only state some results.
We have the following upper bound, which can be obtained with the same proof as in [8].
Theorem 3.1 Let C ϕ : H p → H p , 1 ≤ p < ∞, a composition operator, and n ≥ 1. Then, for every Blaschke product B with (strictly) less than n zeros, each counted with its multiplicity, one has: where m ϕ is the pullback measure of m, the normalized Lebesgue measure on T, under ϕ and S(ξ, h) = D ∩ D(ξ, h) is the Carleson window of size h centered at ξ ∈ T.
Proof. We first estimate the Gelfand number c n (C ϕ ) by restricting to the subspace BH p which is of codimension < n. As in [8], Lemma 2.4: . Now (see [2], Proposition 2.4.3), one has a n (C ϕ ) ≤ √ 2n c n (C ϕ ), hence the result.
We can then deduce, with the same proof, the following version of [11], Theorem 2.3.
Recall ( [11], Definition 2.2) that a symbol ϕ ∈ A(D) (i.e. ϕ : D → D is continuous and analytic in D) is said to be globally regular if ϕ(D) ∩ ∂D = {ξ 1 , . . . , ξ l } and there exists a modulus of continuity ω (i.e. a continuous, increasing and sub-additive function ω : [0, A] → R + , which vanishes at zero, and that we may assume to be concave), such that, writing E ξj = {t ; γ(t) = ξ j }, one has T = l j=1 E ξj + [−r j , r j ] for some r 1 , . . . , r l > 0, and for some positive constants C, c > 0: for j = 1, . . . , l, all t j ∈ E ξj with |t − t j | ≤ r j . Theorem 3.2 Let ϕ be a symbol in A(D) whose image touches ∂D exactly at the points ξ 1 , . . . , ξ l and which is globally-regular. Then there are constants κ, K, L > 0, depending only on ϕ, such that, for every k ≥ 1: where N k is the largest integer such that lN d N < k and d N is the integer part of log κ 2 −N ω −1 (κ 2 −N ) log(χ −p ) + 1, with 0 < χ < 1 an absolute constant. As a corollary, we get for lens maps λ θ (as well as for polygonal maps), in the same way as Theorem 2.4 in [11], p. 550 (recall that then ω(h) ≈ h θ ), the following upper bound. For the cusp map, we also have as in [11], Theorem 4.3 (here, ω(h) ≈ 1/ log(1/h)). a n (C χ ) ≤ c e −b n/ log n .