Approximation and entropy numbers of composition operators

Abstract We give a survey on approximation numbers of composition operators on the Hardy space, on the disk and on the polydisk, and add corresponding new results on their entropy numbers, revealing how they are different.


Introduction
This paper surveys results on approximation numbers of composition operator on the Hardy space, and gives new results, on their entropy numbers, in one or several dimensions. In various papers (see [3,[19][20][21][22]), pretty sharp estimates are obtained for them, either for classes of examples like the lens maps or the cusp maps, or in the general case. In particular, a few properties are investigated, related with the so-called "spectral radius type formula", obtained, in dimension one through a result of Widom in [21], and, partially in dimension N ≥ [22,23], through a result of Nivoche [26] and Zakharyuta [33]. One of our main results (quoted in dimension one) was the implication (using Green capacity considerations): [an(Cφ)] /n = , (1.1) where an(Cφ) is the n-th approximation number of Cφ. Note that it is straightforward that if φ ∞ = r < , then limn→∞[an(Cφ)] /n ≤ r < .
Another way of measuring the compactness of operators is using the entropy numbers instead of the approximation numbers. Those numbers stand a little apart in the jungle of "s-numbers", even though they seem to be the most natural for the study of compactness, since their membership in c characterizes compactness, even in the general framework of arbitrary Banach spaces.
Given a compact operator T : H → H between Hilbert spaces, the relation between its entropy numbers en(T) and its approximation (if one prefers singular) numbers an(T) is theoretically known, through a general result on diagonal operators on , recalled in Theorem 2.2 to follow, and through the Schmidt decomposition of T. This comparison can be thought useless, since in principle we do not know better the numbers an(T) than the numbers en(T). But in our case, with T = Cφ, namely T(f ) = f • φ where φ is an analytic self-map of the polydisk D N , the situation is slightly di erent. Answering a question of J. Wengenroth [30] about the behavior of the entropy numbers of composition operators, we give in this paper estimates for these numbers, analog to that on the approximation numbers. In particular, we have: (1. 2) The proofs are not di cult, but as indicated for example by the comparison between (1.1) and (1.2), the statements feature a very di erent behavior of those entropy numbers, which deserves attention. This di erence is actually more transparent in the polydisk D N , and the main interest of this paper is to point out how the dependence of the entropy numbers with respect to the dimension N di ers from that of the approximation numbers.
The paper is organized as follows. Section 1 is this introduction. In Section 2 we recall the necessary background. In Section 3, we survey results on approximation numbers and give the corresponding new results on entropy numbers. We rst begin with general facts, and then give speci c results, with a particular interest to the examples of the lens maps and the cusp map. For any non-constant analytic map φ : D → D, we have: whereas: For the cusp map χ, we have, for absolute constants: α e −Cn/ log n ≤ an(Cχ) ≤ β e −cn/ log n , (1.7) and (1.8) Section 4 is concerned by the multivariate case, again rst in the general case, then on speci c cases. We are in particular interested by the multi-lens map Λ θ , de ned by: and the multi-cusp map Ξ de ned by: We prove that: (1.12) Section 5 is more speci cally devoted to the multidimensional case, in connection with the notion of Monge-Ampère (or Bedford-Taylor) pluricapacity, which recently turned out to play an important role in connection with composition operators [23].

Lens maps
For < θ < , the lens map λ θ with parameter θ is obtained by sending conformally the unit disk D onto the right half-plane Π = {z ∈ C ; Re z > }; then making u → u θ , and coming back to D (see [29, page 27]). Namely: It is a conformal map from D onto the domain represented on Figure 1.

Cusp map
The cusp map is a conformal mapping χ sending the unit disk D onto the domain represented on Figure 2. This map was rst introduced in [14].
To obtain it, we rst map D onto the half-disk D + = {z ∈ D ; Re z > }. To do that, map D onto itself by z → iz; then map D onto the upper half-plane H = {z ∈ C ; Im z > } by Take the square root to map H in the rst quadrant Q = {z ∈ H ; Re z > }, and go back to the half-disk {z ∈ D ; Im z < } by T − : T − (s) = +is is− ; nally, make a rotation by i to go onto D + . We get ; |θ| ≤ π/ } is sent onto the two circular arcs tangent at to the real axis.

. Approximation and entropy numbers
Given an operator T : X → Y between Banach spaces, recall (see [6]) that we can attach to this operator ve non-increasing sequences (an), (bn), (cn), (dn), (en) of non-negative numbers (depending on T), respectively the sequences of approximation, Bernstein, Gelfand, Kolmogorov, and entropy numbers of T. We only de ne here the rst one and the last one. The approximation numbers are de ned as: The entropy numbers are de ned for n ≥ as: where B Recall (see [ the smallest such constant C is denoted T (X). Every Hilbert space has type , thanks to the parallelogram identity.
Those inequalities indicate that entropy numbers are always bigger than singular numbers, up to a constant, and that, as far as the scale of powers n α is implied, they are dominated by approximation numbers in a weak sense. But it turns out that, individually, they can be much bigger than the latter for composition operators, as we shall see.
We will rely on the following estimate ([6, Proposition 1.3.2, p. 17]), in which denotes the space of square-summable sequences x = (x k ) k≥ of complex numbers. This estimate is given for the sequence (εn) , but en = ε n− , by de nition. (2.10) A useful corollary of Theorem 2.1 is the following.

Theorem 2.2. Let T : H → H be a compact operator between the complex Hilbert spaces H and H , and let
(an) n≥ be its sequence of approximation numbers. Then, for all n ≥ : This theorem might be thought useless, because we do not know better the numbers an than the numbers en! In our situation, this is not the case, since we made a more or less systematic study of the approximation numbers an for composition operators in [3,[19][20][21] for example.

The -dimensional case . General results
In [21], we coined the parameter β (T) = lim n→∞ an(T) /n (3.1) and its versions β + (T), β − (T) with an upper limit and a lower limit respectively. The following result, proved in [21, Theorem 3.1] for φ ∞ < and in [21,Theorem 3.14] for φ ∞ = , shows in particular that no lower or upper limit is needed for β = β , and provides a simpler proof of the second item in Theorem 3.1 below than in our initial proof of [  ) In particular, one has the equivalence The rst item says in particular that we always have:

. Estimates for approximation numbers
Estimates of approximation numbers of composition operators can be obtained by using the boundary behavior of the symbol and Blaschke products for the upper estimates, and the radial behavior of this symbol, reproducing kernels, and interpolating sequences, for the lower estimates.
In order to treat simultaneously the two cases of the lens maps and of the cusp map, we will put ourselves in a more general situation.

Upper estimates
Let φ be a symbol in the disk algebra, i.e. φ : D → D is continuous on D and analytic in D, and such that φ(∂D) ∩ ∂D = {ξ , . . . , ξp}. We say that this symbol is boundary regular if, writing ξ j = e it j , we have: 1) for some positive constant C, we have |φ(e it ) − φ(e it j )| ≤ C − |φ(e it )| , for t in a neighborhood of t j and for j = , . . . , p; 2) for some modulus of continuity ω and for some positive constant c, we have c ω(|t − t j |) ≤ |φ(e it ) − φ(e it j )|, for t in a neighborhood of t j and for j = , . . . , p.
The following result is proved in [20, Theorem 2.3].

Theorem 3.3.
Let φ be a symbol in the disk algebra whose image touches ∂D at the points ξ , . . . , ξp, and nowhere else, and such that φ is boundary regular. Then, there are constants κ, K, L > , depending only on φ, such that, for every q ≥ : where Nq is the largest integer such that p Nd N < q, with d N the integer part of σ This theorem allows to give an upper estimate for all approximation numbers an(Cφ), n ≥ when we can interpolate between the integers Nd N and (N + ) d N+ , but this is not the case in general. Nevertheless, this is the case in the examples below.

Theorem 3.4.
) For the lens map λ θ with parameter θ, we have, for some positive constants α and β, depending on θ:   (see [9], Remark after the proof of Theorem 9.3, at the top of page 163; actually, in that book, Carleson's windows W(ξ , h) are used instead of pseudo-Carleson's windows S(ξ , h), but that does not matter, since

Lower estimates
We consider symbols φ taking real values in the real axis (i.e. its Taylor series has real coe cients) and such that lim r→ − φ(r) = , with a given speed.
We Theorem 3.6. Let φ be a radially regular symbol. Then, for the approximation numbers an(Cφ) of the composition operator Cφ of symbol φ, one has the following lower bound.
where a = − φ( ) > and c is another constant depending only on φ.
For our examples, we get: ) For the lens map λ θ of parameter θ, we have, for some positive constants c and C, depending on θ: ) For the cusp map χ, we have, for some positive constants c and C: an(Cφ) ≥ c exp(−C n/ log n). (3.14) Proof. 1) The lens map λ θ satis es the conditions of Theorem 3.6 with ω − (h) ≈ h /θ . We get the result by adjusting σ = − / √ n.
2) The cusp map χ satis es the conditions of Theorem 3.6 with ω − (h) ≈ e −C /h , and by taking σ = exp(− log n/ n), we get the result.
The tools for proving Theorem 3.6 are the following.
Recall (see [13, pages 194-195], or [25, pages 302-303]) that if (z j ) is a Blaschke sequence, its Carleson constant δ is de ned as where B is the Blaschke product whose zeros are the points z j . Recall also (see [7], [13, pages 194-195], or [25, pages 302-303]) that an interpolation sequence (zn) with (best) interpolation constant C is a sequence (zn) (necessarily Blaschke, i.e. ∞ n= ( − |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: Now (see [11, Chapter VII, Theorem 1.1]), every H ∞ -interpolation sequence (z j ) is a Blaschke sequence and its Carleson constant δ is connected to its interpolation constant C by the inequalities where κ is an absolute constant (actually C ≤ κ ( /δ)( +log /δ)). Now, if (z j ) is a H ∞ -interpolation sequence with constant C, the sequence of the normalized reproducing kernels

. Entropy numbers
In this section, we give the behavior of the entropy numbers of the lens maps and the cusp map, seeing that they signi cantly di er from that of the approximation numbers. Proof. We use Theorem 3.9. It follows, using Theorem 2.2, writing a k = a k (C λ θ ), that (a · · · a k ) /k ≤ a e −b √ k and that, for some positive constant C: Taking k = n / gives the claimed upper bound, with a di erent value of b. The lower bound is proved similarly, using Theorem 3.13 and the left inequality in Theorem 2.2. The lower bound is obtained similarly, using now Theorem 3.14 and Theorem 2.2.

The multidimensional case . General results
Let φ : D N → D N be an analytic map. We will say that φ is non-degenerate if φ(D N ) has non-empty interior, equivalently if det φ (z) ≠ for at least one point z ∈ D N . Let now φ : D N → D N be a non-degenerate analytic map inducing a bounded composition operator Cφ : H (D N ) → H (D N ) (this is not always the case as soon as N > , even if φ is injective and hence nondegenerate, see for example [8, p. 246], when the polydisk is replaced by the ball; but similar examples exist for the polydisk). Assume moreover that Cφ is a compact operator. ) if φ ∞ < , then en(Cφ) ≤ C exp − c n N+ , with C > c > depending on φ.
Proof. ) It is proved in [3, Theorem 3.1] that, for a non-degenerate map φ, it holds: As in the previous section, it follows from Theorem 2.2, that (a · · · a k ) /k ≥ e −b k /N , and then, taking k = n N/ (N+ ) , that: en(Cφ) ≥ c e −Cn /(N+ ) . We de ne similarly γ ± N (Cφ), and will say more on it in next section.

. Speci c results
When Φ : D N → D N is a non degenerate analytic map inducing a compact composition operator , a version of Theorem 3.6 is proved in [3,Theorem 4.2]. For upper bounds, a result was obtained in [3, Theorem 5.5] when Φ = φ ⊗ · · · ⊗ φ N , where φ , . . . , φ N : D → D are symbols inducing compact composition operators on H (D). We do not give here the precise statements and refer to the paper [3]. However, we recall the result that can be obtained from that for the multi-lens maps and the multi-cusp map ([3, Theorem 6.1 and Theorem 6.2]). First, we de ne the multi-lens map Λ θ and the multi-cusp map.
Let λ θ be lens maps with parameter θ. The multi-lens map Λ θ with parameter θ the multi-cusp map on the polydisk D N are de ned respectively as: and: Ξ (z , . . . , z N ) = χ(z ), χ(z ), . . . , χ(z N ) . With the same proof as in Theorem 3.9 and Theorem 3.10, we obtain estimates for the entropy numbers.

Links with pluricapacity and Zakharyuta's results
Here, in dimension N ≥ , the situation is satisfactory for upper bounds (see [22]); for lower bounds, see [23]. The notion involved is now that of pluricapacity, or Monge-Ampère capacity, coined by Bedford and Taylor in [4]. More precisely, if A is a Borel subset of D N , we refer to [22] or [23] for the de nition of its pluricapacity Cap N (A), belonging to [ , +∞], and set: We temporarily assume that φ ∞ < so that K = φ(D N ) is a compact subset of D N . We proved in [ We have the following result, which extends the previous result in dimension . The supremum is essentially attained for k the integral part of (log / ρα) α n α and then, in view of (5.7) and α/N = − α, as n goes to in nity: n k log + ρ α k /N ∼ (log ) −α ( ρ α) α n −α .
This clearly ends the proof of Theorem 5.2.

Remark.
We have so far no sharp lower bound for entropy numbers, at least when φ ∞ = , since we already fail to have one in general for approximation numbers (see however [23]). In dimension 1, we used the fact that every connected Borel subset E of D such thatĒ ⊆ D, the Green capacity of E is equal to that ofĒ ([21, Theorem 2.3]).
Besides, let J : H ∞ (D N ) → C(K) be the canonical embedding, when K ⊆ D N is a "condenser", namely a compact subset of D N such that any bounded analytic function on D N which vanishes on K vanishes identically, which is moreover "regular". The positive solution to the Kolmogorov conjecture can be expressed in terms of the Kolmogorov numbers dn(J) of J or equivalently, in terms of the entropy numbers en(J) of J ([32, Theorem 5], generalizing Erokhin's result in dimension appearing in his posthumous paper [10] and methods due to Mityagin [24] and Levin and Tikhomirov [16]; see also [33,Lemma 2.2]). The result is that, taking K = φ(D N ), one has, with sharp constants c K , c K depending on the pluricapacity of K in D N : dn(J) ≈ e −c K n /N and en(J) ≈ e −c K n /(N+ ) . (5.8) This jump from the exponent /N to the exponent /(N + ) is re ected in our Theorem 5.2, through the new parameter γ + N .