Comparison of singular numbers of composition operators on different Hilbert spaces of analytic functions
Abstract
We compare the rate of decay of singular numbers of a given composition operator acting on various Hilbert spaces of analytic functions on the unit disk D. We show that for the Hardy and Bergman spaces, our results are sharp. We also give lower and upper estimates of the singular numbers of the composition operator with symbol the “cusp map” and the lens maps, acting on weighted Dirichlet spaces.
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Journal of Functional Analysis 280 (2021) 108834 Contents lists available at ScienceDirect Journal of Functional Analysis www.elsevier.com/locate/jfa Comparison of singular numbers of composition operators on different Hilbert spaces of analytic functions Pascal Lefèvre a, Daniel Li a,∗, Hervé Queffélec b, Luis Rodríguez-Piazza c aUniv. Artois, Laboratoire de Mathématiques de Lens (LML) UR 2462, & Fédération Mathématique des Hauts-de-France FR 2037 CNRS, Faculté Jean Perrin, Rue Jean Souvraz, S.P. 18, F-62 300 Lens, France bUniv. Lille Nord de France, USTL, Laboratoire Paul Painlevé U.M.R. CNRS 8524 & Fédération Mathématique des Hauts-de-France FR 2037 CNRS, F-59 655 Villeneuve d’Ascq Cedex, France cUniversidad de Sevilla, Facultad de Matemáticas, Departamento de Análisis Matemático & IMUS, Calle Tarfia s/n, 41 012 Sevilla, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 13 January 2020 Accepted 16 October 2020 Available online 28 October 2020 Communicated by K. Seip MSC: primary 47B33 secondary 46B28, 30H10, 30H20 Keywords: Approximation numbers Composition operator Hilbert spaces of analytic functions Schatten classes We compare the rate of decay of singular numbers of a given composition operator acting on various Hilbert spaces of analytic functions on the unit disk D. We show that for the Hardy and Bergman spaces, our results are sharp. We also give lower and upper estimates of the singular numbers of the composition operator with symbol the “cusp map” and the lens maps, acting on weighted Dirichlet spaces. © 2020 Elsevier Inc. All rights reserved. *Corresponding author. E-mail addresses: [email protected] (P. Lefèvre), [email protected] (D. Li), Herve.Queff[email protected] (H. Queffélec), [email protected] (L. Rodríguez-Piazza). https://doi.org/10.1016/j.jfa.2020.108834 0022-1236/© 2020 Elsevier Inc. All rights reserved.
2P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 1. Introduction Composition operators are mainly studied on Hilbert spaces of analytic functions, and more specifically on the Hardy space H2, the Bergman space B2, and the Dirichlet space D=D2. It is well known, thanks to the Littlewood subordination principle, that every analytic self-map ϕ:D→Dinduces a bounded composition operator Cϕon H2and on B2, but not necessarily on D2([36, Chapter 1 and Exercises]; see also [11, Section 6.2]). There exist even composition operators which are not bounded on D2but which are in all Schatten classes Sp(H2)and Sp(B2)with p >0, of both the Hardy space and the Bergman space ([24, Theorem 2.10]). Nevertheless, for compact composition operators, the following results hold: 1) every composition operator which is compact on H2is compact on B2(see [36, Theorem 3.5] and [32, Theorem 3.5]); 2) every composition operator that is compact on D2is in all the Schatten classes Sp(H2)for p >0([24, Theorem 2.9]); for every p >0, every composition operator that is in Sp(H2)is in Sp(B2). Since the membership in a Schatten class Spof an operator on a Hilbert space means that its approximation numbers are p-summable, that suggests that there is a strong link between the approximation numbers aD2 n(Cϕ), aH2 n(Cϕ)and aB2 n(Cϕ)of the composition operator Cϕon D2, H2and B2respectively. The aim of this paper is to prove that, indeed, in some sense aD2 n(Cϕ)is “greater” than aH2 n(Cϕ), which is “greater” than aB2 n(Cϕ). We recover then that Cϕ∈Sp(H2) implies that Cϕ∈Sp(B2) (Section 3). In Section 3.6, we also give some results about conditional multipliers. In Section 4we give an example with Cϕcompact on H2but not in any Schatten class Sp(B2)for p <∞. We prove that Cϕ∈Sp(H2) implies that Cϕ∈Sp/2(B2)and give an example with Cϕ∈Sp(H2) but Cϕ/∈Sq(B2)for any q<p/2. However, our result is not sufficient to explain why the compactness of Cϕon D2 implies that Cϕ∈Sp(H2)for all p >0. A more subtle relationship should exist between aD2 n(Cϕ)and aH2 n(Cϕ). In fact, for every composition operator Cϕthat is compact on D2, we have limn→∞ aD2 n(Cϕ)1/n = limn→∞ aH2 n(Cϕ)1/n ([28, Theorem 3.1 and Theorem 3.14]); in particular, for symbols ϕsuch that ϕ∞<1, the numbers aD2 n(Cϕ)and aH2 n(Cϕ)behave like rn, with r=exp(−1/Cap[ϕ(D)]), and where Cap[ϕ(D)] is the Green capacity of ϕ(D). On the other hand, for the so-called cusp map χ, we have, for some constants c1>c 1>0([27, Theorem 4.3]): e−c1n/ log naH2 n(Cχ)e−c 1n/ log n(1.1) and, for some constants c2>c 2>0([25, Theorem 3.1]): e−c2√naD2 n(Cχ)e−c 2√n,(1.2) which is much greater. In Section 5.2, we show that the behavior of an(Cχ)in (1.1)holds in all weighted Dirichlet spaces D2 αfor α>0(with other constants), and hence (1.2) shows that a jump happens for α=0. We also look at the lens maps.
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 3 2. Notation and background Let Dbe the open unit disk in C. We denote dA =dx dy/π the normalized area measure on D. The normalized Lebesgue measure dt/2πon T=∂Dis denoted dm. 2.1. Hilbert spaces of analytic functions Recall that the Hardy space H2is the space of analytic functions f:D→Csuch that: f2 H2:= sup 0<r<1 T |f(rξ)|2dm(ξ)<∞. If f(z) =∞ k=0 ckzk, we have f2 H2=∞ k=0 |ck|2. The Bergman space B2is the space of analytic functions f:D→Csuch that: f2 B2:= D |f(z)|2dA(z)<∞. If f(z) =∞ k=0 ckzk, we have f2 B2=∞ k=0 |ck|2 k+1 . More generally, for γ>−1, the weighted Bergman space B2 γis the space of analytic functions f:D→Csuch that: f2 B2 γ=(γ+1) D |f(z)|2(1 −|z|2)γdA(z)<∞, and if f(z) =∞ k=0 ckzk, we have: f2 B2 γ=∞ k=0 βk|ck|2, with: βk=k!Γ(γ+2) Γ(k+γ+2) ≈1 (k+1) γ+1 (the equivalence depends on γ). Hence B2=B2 0and H2corresponds to the degenerate case γ=−1. The Dirichlet space D2is the space of analytic functions f:D→Csuch that: f2 D2=|f(0)|2+ D |f(z)|2dA(z)<∞.
4P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 If f(z) =∞ k=0 ckzk, we have f2 D2=|c0|2+∞ k=0 k|ck|2. With the equivalent norm |f|2 D2=f2 H2+D|f(z)|2dA(z), we have the more pleasant form |f|2 D2=∞ k=0(k+1) |ck|2. More generally, for α>−1, the weighted Dirichlet space D2 αis the space of analytic functions f:D→Csuch that: f2 D2 α=|f(0)|2+(α+1) D |f(z)|2(1 −|z|2)αdA(z)<∞, and if f(z) =∞ k=0 ckzk, we have: f2 D2 α=∞ k=0 βk|ck|2, with β0=1and for k≥1: βk=k.k!Γ(α+2) Γ(k+α+1) ≈1 (k+1) α−1 (the equivalence depending on α). Another equivalent expression is: βk=(k+1)!Γ(α+2) Γ(k+α+1) ; we have βk≤ βk≤2βk. In particular, for γ>−1: D2=D2 0,H 2=D2 1and B2 γ=D2 γ+2 .(2.1) 2.2. Composition operators Any analytic self-map ϕ:D→Ddefines a bounded composition operator Cϕon the Hardy space H2(see [32, Section 2.2]) and on every weighted Bergman space B2 γfor γ>−1([32, Proposition 3.4]), hence on every weighted Dirichlet space D2 αwith α≥1. However, this is not always the case on the weighted Dirichlet spaces D2 αfor α<1([32, Proposition 3.12]). For convenience, we assume that ϕis not constant and we say that ϕis a symbol. We denote ϕ∗the boundary values function of ϕ. The Carleson window of size hcentered at ξ∈Tis: W(ξ,h)={z∈D;|z|≥1−hand −πh ≤arg(¯ ξz)<πh}.(2.2) For every integer n ≥1and for j=0, ..., 2n−1, we set:
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 5 Wn,j =W(e2jiπ/2n,2−n).(2.3) We also use the Carleson boxes S(ξ,h)={z∈D;|ξ−z|<h},(2.4) which satisfy S(ξ, h) ⊆W(ξ, h) ⊆S(ξ, 2πh). The Hastings-Luecking boxes are defined, for every integer n ≥1and for 0 ≤j≤ 2n−1, as: Rn,j =z∈D;1−1 2n−1≤|z|<1−1 2nand 2jπ 2n≤arg z<2(j+1)π 2n(2.5) A measure μon Dis a Carleson measure if supξ∈TμW(ξ,h)=O (h). By the Carleson embedding theorem, μis a Carleson measure if and only if the inclusion map Jμ:H2→L2(μ)is bounded. The automatic boundedness of Cϕon H2implies that the pull-back measure mϕ, defined as mϕ(B) =m[ϕ∗−1(B)] for all Borel sets B⊆D, is a Carleson measure. This composition operator is compact on H2if and only if mϕis supported by Dand supξ∈Tmϕ[W(ξ, h)] =o (h)([31]). Similar results hold for composition operators on the weighted Bergman spaces ([32, Theorem 4.3]). 2.3. Singular numbers, approximation numbers and Schatten classes Let Hbe a separable complex Hilbert space and T:H→Hbe a compact operator. There exist two orthonormal sequences (un)and (vn)and a non-increasing sequence (sn) of non-negative numbers with sn−→ n→∞0such that, for all x ∈H: T(x)= ∞ n=1 snx|vnun.(2.6) This representation T=∞ n=1 snun⊗vnis called the Schmidt decomposition of Tand the numbers sn=sn(T)the singular numbers of T. They are actually the eigenvalues of |T| =√T∗Trearranged in non-increasing order. In particular s1=T. These numbers have the important “ideal property”: sn(ATB)≤Asn(T)B. The nth approximation number of Tdefined as: an(T)= inf rank R<n T−R.(2.7) It is known that, for all n ≥1, we have sn(T) =an(T)(see [5, p. 155]).
6P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 For p >0, the Schatten class Sp(H)is the set of all compact operators T:H→H for which Tp p:= ∞ n=1[an(T)]p<∞. We have Sq(H) ⊆Sp(H)for 0 <q≤p. For composition operators Cϕ, D. Luecking ([29, Corollary 2]) characterized their membership in the Schatten classes. For γ>−1, let dAγ(z) =(γ+1)(1 −|z|2)γdA(z). For γ=−1, we set H2=B2 −1and dm =dA−1. Then, for γ≥−1, the composition operator Cϕbelongs to Sp(B2 γ)if and only if: ∞ n=1 2n−1 j=0 2n(γ+2)Aγ,ϕ(Rn,j)p/2<∞,(2.8) where Aγ,ϕ is the pull-back measure of Aγby ϕ(by ϕ∗for γ=−1). As usual, the notation A Bmeans that A ≤CB for some positive constant C, which may depend on some parameters, and A ≈Bmeans that A Band A B. 3. Comparison of approximation numbers 3.1. Main result In the introduction, we said that, in some sense, aD2 n(Cϕ)is “greater” than aH2 n(Cϕ), which is “greater” than aB2 n(Cϕ). This vague statement is made more precise in the following result. Theorem 3.1. For any symbol ϕ, we have, for every n ≥1: n j=1 aB2 j(Cϕ)≤ n j=1 aH2 j(Cϕ)≤ n j=1 aD2 j(Cϕ).(3.1) It is understood that if Cϕis not bounded on D2, then aD2 j(Cϕ) =+∞. As a consequence, we recover a previous result ([24, Corollary 3.2]; see also [7, Theorem 2.5]). Corollary 3.2. For any symbol ϕ, we have: 1) a)if Cϕis compact on D2, then Cϕis compact on H2; b)if Cϕis compact on H2, then Cϕis compact on B2. Moreover, for every p >0, we have: 2) a)if Cϕ∈Sp(D2), then Cϕ∈Sp(H2);
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 7 b)if Cϕ∈Sp(H2), then Cϕ∈Sp(B2). Items 1) a)and 2) a)are not sharp, since we proved in [24, Theorem 2.9] that if Cϕis compact on D2, then Cϕbelongs to all Schatten classes Sp(H2), with p >0. However, we will see in Section 4that the item 1) b)is sharp, but 2) b)is not. We will prove these results in a more general setting in Theorem 3.12. 3.2. Subordination of sequences Let Sbe the set of non-increasing sequences u =(uj)j≥1of real numbers. If u, v∈S, the sequence uis said to be subordinate to the sequence v, and we write u ≺v, if: n j=1 uj≤ n j=1 vjfor all n≥1.(3.2) For example, if u =(1, 1, 0, 0, ...)and v=(2, 0, 0, ...), we have u ≺v. We have a basic stability property of this notion (see [38, Theorem 1.16, p. 13]). Proposition 3.3. Let Ibe an interval of Rand h:I→Rbe increasing and convex. Then, if u, v∈Sare sequences of numbers in I, we have: u≺v=⇒h(u)≺h(v). Proof. We may assume that his C2. We fix n ≥1and set a =min{un, vn}. Then, for x ∈Iand x >a: h(x)=h(a)+(x−a)h(a)+ +∞ a (x−t)+h(t)dt . One easily checks, using (3.2), that n j=1(uj−t)+≤n j=1(vj−t)+for all t ≥a. Hence, thanks to the positivity of h(a)and h: n j=1 h(uj)=nh(a)+h(a) n j=1 (uj−a)+ +∞ a n j=1 (uj−t)+h(t)dt ≤nh(a)+h(a) n j=1 (vj−a)+ +∞ a n j=1 (vj−t)+h(t)dt = n j=1 h(vj). A stronger notion is that of log-subordination.
8P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 Definition 3.4. We say that the sequence u ∈Sof positive numbers is log-subordinate to the sequence v∈Sof positive numbers if log u ≺log v. In other terms, if: n j=1 uj≤ n j=1 vjfor all n≥1. The following result will be useful. Proposition 3.5. For sequences of positive numbers u, v∈S, the following two conditions are equivalent: 1) log u ≺log v; 2) up≺vpfor all p >0. Proof. If log u ≺log v, it suffices to apply Proposition 3.3 to the sequences log uand log vand to the function h(x) =e px to get up≺vp. Conversely, if up≺vpfor all p >0, we have: 1 n n j=1 up j1/p ≤1 n n j=1 vp j1/p , and letting pgoing to 0, we get: n j=1 uj1/n ≤n j=1 vj1/n , i.e. log u ≺log v. Corollary 3.6. Let u, v∈Sbe two sequences of positive numbers such that uis logsubordinate to v. Then for N≥n: uN≤vn/N 1v1−n/N n.(3.3) In particular, for any n ≥1: u2n≤√v1vn.(3.4) Proof. We have: uN N≤ N j=1 uj≤ N j=1 vj= n j=1 vj N j=n+1 vj≤vn 1vN−n n, and (3.3) follows. Now, the choice N=2ngives (3.4).
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 9 Note that the choice N=[n log n], the integer part of n log n, can be useful (see [4]). 3.3. Singular numbers The following Weyl type result is crucial for the proof of our main result. It is certainly known by specialists, but we have not found any reference. Proposition 3.7. Let Tbe a compact operator on a separable complex Hilbert space H and T=∞ j=1 sjuj⊗vjits Schmidt decomposition. Then, for every integer n ≥1: s1···sn=maxdetTfj|gii,j, where the supremum is taken over all pairs (fj)1≤j≤nand (gi)1≤i≤nof orthonormal systems of length nin H. Proof. First, assume that His n-dimensional. We may assume that H=n 2and we denote (ei)1≤i≤nits canonical basis. Since T(vj) =sjuj, we have detTvj|uii,j =s1···sn. Now, if (fj)1≤j≤nand (gi)1≤i≤nare two orthonormal systems, we consider the following diagram: n 2 U −→ n 2 T −→ n 2 V −→ n 2 where U, Vare the unitary operators defined by: Un j=1 tjej= n j=1 tjfjand Vx= n j=1x|gjej. We observe that VTUe j|ei=TUej|V∗ei=Tfj|gi, so that: detTfj|gii,j=|detV||detT||detU|=|detT|=s1···sn. In the general case, denote by Pnand Qnthe orthogonal projections onto Fn:= [f1, ..., fn]and Gn:= [g1, ..., gn] respectively. We can see Fnand Gnas isometric copies of n 2. Observe that Tfj|gi =QnTPnfj|gi. By the above special case, we get, using the ideal property of singular numbers: detTfj|gii,j= n j=1 sj(QnTPn)≤ n j=1 sj(T).
16 P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 As an application of the general principles of Section 3.4 we have the following result, whose first items were previously obtained by I. Chalendar and J. Partington in [6]and [7] (actually (3.b) is also proved in [7], but for values p ≥1). Theorem 3.12. Let H2(β)and H2(γ)be two weighted Hilbert spaces. Assume that γis dominated by βin the sense that the sequence (βk/γk)is increasing, so that the continuous inclusion H2(β) ⊆H2(γ)holds. Then, for ϕ:D→Dwith ϕ(0) =0: 1) if Cβ ϕis bounded, Cγ ϕis bounded as well, and Cγ ϕ ≤Cβ ϕ; 2) if Cβ ϕis compact, so is Cγ ϕ; 3) the sequence sγ=sn(Cγ ϕ)n≥1is log-subordinate to the sequence sβ=sn(Cβ ϕ)n≥1, so that: a)s2n(Cγ ϕ) ≤s1(Cβ ϕ)sn(Cβ ϕ), for all n ≥1; b)Cβ ϕ∈SpH2(β)=⇒Cγ ϕ∈SpH2(γ), for any p >0. Remark. Let us mention that we can apply the previous theorem in the framework of weighted Dirichlet spaces. Indeed, let 0 <β<γand consider the two weights: βk=k.k!Γ(β+2) Γ(k+β+1) and γk=k.k!Γ(γ+2) Γ(k+γ+1) associated with the weighted Dirichlet spaces D2 βand D2 γrespectively, with γ>β, so that D2 β⊂D2 γ. In order to apply our comparison Theorem 3.12, we have to show that the sequence (βk/γk) increases. But βk γk =Γ(β+2) Γ(γ+2) Γ(γ+k+1) Γ(β+k+1) =: Γ(β+2) Γ(γ+2)Ak, and, setting h =γ−β>0and xk=β+k+1, we see that: Ak=Γ(xk+h) Γ(xk)· Since the function Γis log-convex, the map x → Γ(x+h) Γ(x)increases on (0, ∞), and we get that the sequence (βk/γk) increases. Proof of Theorem 3.12.We set dk=βk/γkand ek(z) =zk. Let J:H2(β) →H2(γ)the unitary (onto isometry) and diagonal operator defined by J(ek) =dkek, for all k≥0. The operator A =JCβ ϕJ−1maps H2(γ)into itself and sn(A) =sn(Cβ ϕ)for all n ≥1 (in particular AL(H2(γ)) =Cβ ϕL(H2(β))). Moreover, Ahas a lower-triangular matrix. Now we consider the diagonal operator D:H2(γ) →H2(γ) defined by D(ek) =dkek. In general, it is an unbounded operator. It is plain that D−1J:H2(β) →H2(γ)is the
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 17 canonical inclusion, since (D−1J)(ek) =ekfor all k≥0. Hence (D−1J)Cβ ϕ=Cγ ϕ(D−1J), and since AJ =JCβ ϕ, we have the following commutative diagram: H2(β)Cβ ϕ J H2(β) J H2(γ)DH2(γ)AH2(γ) D−1H2(γ) By Theorem 3.9, we get: log s(D−1AD)≺log s(A) (so we have, in particular, D−1ADL(H2(γ)) ≤AL(H2(γ)) =Cβ ϕL(H2(β))). But D−1AD =Cγ ϕ, and this proves Theorem 3.12, using Proposition 3.5 and Corollary 3.6. Remark. Actually, the same proof gives the following generalization of Theorem 3.12. Theorem 3.13. With the hypothesis of Theorem 3.12, let T:Hol(D) →Hol(D)be a linear map such that its restriction Tβto H2(β)is bounded from H2(β)into H2(β)and has a matrix in the canonical basis of H2(β)which is lower-triangular. Then: 1) Tγis bounded as well, and Tγ ≤Tβ; 2) if Tβis compact, so is Tγ; 3) the sequence of singular numbers s(Tγ) =sn(Tγ)n≥1is log-subordinate to the sequence s(Tβ) =sn(Tβ)n≥1, so that: a)s2n(Tγ) ≤s1(Tβ)sn(Tβ), for all n ≥1; b)Tβ∈SpH2(β)=⇒Tγ∈SpH2(γ), for any p >0. 3.6. Application to conditional multipliers We first recall the following well-known proposition (and give a short proof, for sake of completeness). Note that this result does not hold for the Dirichlet spaces D2 αwhen α≤0 ([40, Theorem 10]; see also [39, Theorem 2.7], [17, Theorem A], and [41, Theorem 4.2]). Recall that it is well-known that the space M(H2)of multipliers of H2is isometric to H∞. Proposition 3.14. For every γ>−1, the space M(B2 γ)of multipliers of B2 γis isometric to the space H∞. If His a Hilbert space of analytic functions on D, containing the constants, and with reproducing kernels Ka, a ∈D, then the space M(H)of multipliers of His contained contractively into the space H∞.
18 P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 Proof. If hf ∈Hfor all f∈H, then, taking f=1, we have h ∈H, so his analytic. The same proof as in [1, Proposition 3.1] shows that h ∈H∞. For sake of completeness we give a short different proof. In fact, we have, for all a ∈D: M∗ h(Ka)=h(a)Kafor all a∈D; (3.17) hence |h(a)| Ka ≤M∗ h Ka, and, since Kais not null, that proves that h ∈H∞ and h∞≤Mh. Hence M(H) H∞, contractively. When H=B2 γ, we have the reverse inclusion. Indeed, for every h ∈H∞, one clearly has hf ∈B2 γand hfB2 γ≤h∞fB2 γfor all f∈B2 γ, so the multiplication operator Mh:B2 γ→B2 γis bounded with norm no greater than h∞. Now let ϕbe an analytic self-map of Dand H=H2(β)be a weighted Hilbert space of analytic functions on D, with reproducing kernel Ka, a ∈D, on which Cϕacts boundedly. We denote its multiplier set, respectively multiplier set conditionally to ϕ, by: M(H)={w∈H;wf ∈Hfor each f∈H}(3.18) and M(H,ϕ)={w∈H;w(f◦ϕ)∈Hfor all f∈H}.(3.19) We have M(H) ⊆M(H, ϕ). The set M(H, ϕ)plays an important role in the study of weighted composition operators. Definition 3.15. A Hilbert space Hof analytic functions on D, containing the constants, and with reproducing kernels Ka, a ∈D, is said to be admissible if: (i)H2is continuously embedded in H; (ii)M(H) =H∞; (iii)the automorphisms of Dinduce bounded composition operators on H; (iv)KaH KbH≤h1−|b| 1−|a|for a, b ∈D, where h:R+→R+is an non-decreasing function. Note that (i) implies that fH≤CfH2for all f∈H2, for some positive constant C, and so (BHand BH2being the unit ball of Hand H2respectively): KaH=sup f∈BH|f(a)|≥C−1sup f∈BH2|f(a)|=C−1(1 −|a|2)−1/2,
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 19 implying that: lim |a|→1−KaH=∞. Examples. 1) The weighted Bergman space B2 γ, with γ>−1is admissible. Indeed, we know that it is continuously embedded in H2=B2 −1; condition (ii) is Proposition 3.14; condition (iii)is satisfied according to the Remark before Theorem 3.12, and Ka2=1 (1−|a|2)γ+2 , giving (iv). 2) More generally, we have the following result. Proposition 3.16. For any decreasing sequence βsuch that the automorphisms of Dinduce bounded composition operators on H2(β), the space H2(β)is admissible. Recall that H2(β)is defined in (3.10). A particular case is obtained as follows. Let ω:(0, 1) →R+be an integrable function such that, for some positive and locally bounded function ρ:R+→R+, we have: ω(y) ω(x)≤ρy xfor all x, y ∈(0,1),(3.20) and let H2 ωbe the space of analytic functions f:D→Csuch that: f2 H2 ω:= D |f(z)|2ω(1 −|z|2)dA(z)<∞.(3.21) Such spaces are used in [16]and in [28]. We have H2 ω=H2(β)with: βn=2 1 0 r2n+1ω(1 −r2)dr = 1 0 tnω(1 −t)dt . (3.22) Indeed, since βn=1 0(1 −t)nω(t) dt, the sequence β=(βn)nis decreasing. Moreover, the fact that the automorphisms of Dinduce bounded composition operators on H2 ωis proved as in the Remark before Theorem 3.12, namely: f◦ϕa2 H2 ω= D |f(w)|2|ϕ a(w)|2ω(1 −|ϕa(w)|2)dA(w) ≤1+|a| 1−|a|2 D |f(w)|2ρ1−|ϕa(w)|2 1−|w|2ω(1 −|w|2)dA(w)
20 P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 ≤1+|a| 1−|a|2 cρ,a D |f(w)|2ω(1 −|w|2)dA(w) =: κaf2 H2 ω, where we used that |ϕ a(w)| ≤1+|a| 1−|a|, that 1−|ϕa(w)|2 1−|w|2≤1+|a| 1−|a|, and that ρis locally bounded. Note that B2 γ, γ>−1, corresponds to ω(t) =(γ+1) tγ. Proof of Proposition 3.16.Condition (i)is satisfied because βis decreasing. Moreover, since βis decreasing, Theorem 3.13, applied to T=Mw, with w∈H∞, ensures that H∞=M(H2) ⊆M H2(β), and, for all w∈H∞: Mw:H2(β)→H2(β)≤Mw:H2→H2=w∞. Now, Proposition 3.14 implies that H∞=MH2(β)and Mw:H2(β)→H2(β)=w∞ for all w∈H∞. It remains to show that, for H=H2(β), the condition (iii) implies the condition (iv). Since H2(β)is isometrically rotation invariant, it is clear that Ka =K|a|; hence Kx ≤Ky, for 0 ≤x ≤y<1. Assume now that 0 <y<x <1. Let Tbe the disk automorphism: T(z)=2z+1 z+2 ,z∈D.(3.23) The fixed points of Tare 1and −1, and T(0) =1/2. We define the sequence (an)n≥0by induction, with: a0=0,a n+1 =T(an). We see that: 1−an+1 = 1 an T(x)dx = 1 an 3 (x+2) 2dx ≤3 4(1 −an); (3.24) so (an)nis increasing and converges to 1. In the same way, we see that: 1−an+1 = 1 an T(x)dx = 1 an 3 (x+2) 2dx ≥1 3(1 −an).(3.25)
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 21 Since 0 <y<x <1, we can find m ≤nsuch that: am−1<y<a m,and an−1<x<a n. We have Kx ≤Kanand Ky ≥Kam−1. Since C∗ TKz=KT(z)for all z∈D, we have: Kx Ky≤Kan Kam−1≤C∗ Tn−m+1 =αn−m+1 , with α=CT ≥1. Applying (3.24)and (3.25), we get: 1−y 1−x≥1−am 1−an−1≥1−am 3(1 −an)≥1 34 3n−m. It suffices now to take s ≥0such that (4/3)s=α, and A >0 large enough in order that, with the increasing function h(t) =max{Ats, 1}, t >0, we have: h1−y 1−x≥A 3sαn−m≥A 3sαKx Ky≥Kx Ky· Let us come back to the conditional multipliers. In general, we obviously have: H∞⊆M(H,ϕ)⊆H. (3.26) The extreme cases were characterized by Attele ([2]) when H=H2=B2 −1(and Contreras and Hernández-Díaz in [8]for the spaces Hp) as follows. Theorem 3.17 (Attele). We have: 1) M(H2, ϕ) =H2if and only if ϕ∞<1. 2) M(H2, ϕ) =H∞if and only if ϕis a finite Blaschke product. A key tool for the most delicate second necessary condition is the use of inner and outer functions. We no longer have this tool at our disposal for the admissible spaces H=H2(β), but we can nevertheless state the following analogous result. Theorem 3.18. Let ϕan analytic self-map of Dand Hbe an admissible Hilbert space on which Cϕacts boundedly. We have: 1) M(H2, ϕ) ⊆M(H, ϕ); 2) M(H, ϕ) =Hif and only if ϕ∞<1; 3) M(H, ϕ) =H∞if and only if ϕis a finite Blaschke product.
22 P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 Note that the assumption that Cϕacts boundedly on His automatically satisfied when H=H2(β)with βdecreasing, by Theorem 3.12. Proof. 1) Suppose first that ϕ(0) =0. Let w∈M(H2, ϕ). The weighted composition operator MwCϕis bounded on H2, and moreover lower triangular on the canonical basis; applying Theorem 3.13,1), we get that MwCϕis bounded on Has well; that is, w∈M(H, ϕ). In the general case, let ϕ(0) =a, so that (ϕa◦ϕ)(0) =0. Property (iii) implies that M(H2,ϕ)=M(H2,ϕ a◦ϕ)⊆M(H,ϕa◦ϕ)=M(H, ϕ), since f∈H2if and only if f◦ϕa∈H2and f∈Hif and only if f◦ϕa∈H. 2) The necessary condition is proved as in [2]for H2; we recall some details. We start from the (obvious, but useful) mapping equation: (MwCϕ)∗(Kz)=w(z)Kϕ(z).(3.27) The assumption implies the existence of a constant Csuch that: MwCϕL(H)≤CwHfor all w∈H. As a consequence, for given z∈D: (MwCϕ)∗(Kz)H≤CwHKzH, that is, in view of (3.27): |w(z)|Kϕ(z)H≤CwHKzH.(3.28) Testing this inequality with w=Kzand simplifying by Kz2 H, we get that Kϕ(z)H≤ C. Since lim|a|→1−KaH=∞, as a consequence of (i), this implies that ϕ∞<1, by this same consequence. For the sufficient condition, observe that if ϕ∞<1, then f◦ϕ ∈H∞for all f∈H. Since M(H) =H∞, according to (iv), we get that f◦ϕ ∈M(H)and therefore w(f◦ϕ) =Mf◦ϕw∈Hfor all w∈H. That means that H⊆M(H, ϕ). Therefore, by (3.26), we have M(H, ϕ) =H. 3) The sufficient condition goes as follows: finite Blaschke products ϕclearly satisfy (and actually are characterized by: see [2]): Rϕ:= sup z∈D 1−|ϕ(z)| 1−|z|<∞. Now let w∈M(H, ϕ), so that C:= MwCϕ <∞. We may assume that wH≤1. The mapping equation (3.28)gives, for z∈D:
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 23 |w(z)|Kϕ(z)H≤CKzH. By (ii), this implies that, for |z|close enough to 1: |w(z)|≤CKzH Kϕ(z)H≤Ch 1−|ϕ(z)| 1−|z|≤Ch(Rϕ). This means that w∈H∞. Finally, for the necessary condition, assume that M(H, ϕ) =H∞. Then M(H2, ϕ) = H∞, by 1), and then ϕis a finite Blaschke product by Attele’s theorem (Theorem 3.17). Remark. In Proposition 3.16, we assume that the automorphisms of Dinduce bounded composition operators on H2(β). It is known ([18, Theorem 1]) that this is not always the case. Let us give a simpler proof, for a particular case. Let βn=exp(−√n), and consider the space H2(β). We then have, for 0 <r=e −ε<1: Kr2=∞ n=0 r2nexp(√n)= ∞ n=0 e−2nε exp(√n)=:S(ε). We easily see (using e.g. the Euler-MacLaurin formula) that, when ε →0+: S(ε)∼I(ε):= ∞ 0 exp(√t−2εt)dt =(4ε2)−1∞ 0 exp√x−x 2εdx . We use the Laplace theorem ([10] p. 125) on the equivalence of integrals: ∞ 0 eAϕ(x)dx ∼2π(|ϕ(x0)|)−1A−1/2eAϕ(x0),as A→∞, and apply it to A =1/2εand to the function ϕ(x) =√x−x, which takes its maximum at x0=1/4, with ϕ(x0) =1/4. We get that: S(ε)≈ε−3/2exp1 8ε≈(1 −r)−3/2exp1 8(1 −r). Now, consider the automorphism Tof Dgiven by (3.23). For r<1, we have 1 −T(r) ∼ (1 −r)T(1) =(1 −r)/3; so: 1 1−T(r)−1 1−r=1 1−r1−r 1−T(r)−1∼1 1−r1 T(1) −1=2 1−r, and that implies that:
24 P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 KT(r)2 Kr2≈exp1 81 1−T(r)−1 1−r−→ r→1−∞. Since KT(r)=C∗ T(Kr), this implies that C∗ T, and hence CT, is not bounded on H2(β). 4. Schatten classes for Hardy spaces and Bergman spaces We know that if a composition operator Cϕis compact on the Hardy space H2, then it is compact on the Bergman space B2(see [32, Proposition 2.7 and Theorem 3.5]). Theorem 4.3 below shows that we cannot expect better. Let us begin by a preliminary result. Recall that the 2-Carleson function of the analytic map ϕ:D→Dis: ρϕ,2(h)=sup ξ∈T AϕW(ξ,h),(4.1) where Ais the normalized area measure on D, Aϕis the pull-back measure of Aby ϕ, i.e. Aϕ(B) =A[ϕ−1(B)] for all Borel sets B⊆D. It is well-known (see [13]) that ρϕ,2(h) =O (h2), due to the fact that all composition operators Cϕare bounded on B2, and that Cϕis compact on B2if and only if ρϕ,2(h) =o (h2). For Schatten classes, we have the following result. Proposition 4.1. If the composition operator Cϕ:B2→B2is in the Schatten class Sp(B2)for some p ∈(0, ∞), then: ρϕ,2(h)=oh2log 1 h−2/p.(4.2) Proof. We follow the proof of [20, Proposition 3.4]. By [29, Corollary 2] and [20, Proposition 3.3], Cϕ∈Sp(B2)if and only if: ∞ n=1 2n−1 j=0 4np/2[Aϕ(Wn,j)]p/2<∞,(4.3) where Wn,j =W(e2jiπ/2n, 2−n). Observing that, for h =2 −n, we have: ρϕ,2(2−n)p/2≤ 2n−1 j=0 Aϕ(Wn,j)p/2, (4.3) yields: ∞ n=1 ρϕ,2(2−n)p/24np/2<+∞.
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 25 By [22, Theorem 3.1], we have a constant C0>0such that: ρϕ,2(εh)≤C0ε2ρϕ,2(h) for 0 <ε ≤1and 0 <h <1. Hence, if we set: un=ρϕ,2(2−n) 4−np/2 , we have, for n ≥k: un≤Cp/2 0uk. The following lemma, whose proof is postponed, then shows that: nρϕ,2(2−n) 4−np/2 −→ n→∞0.(4.4) Lemma 4.2. Let unbe a convergent series of positive numbers such that un≤Cu k for n ≥k, for some positive constant C. Then un=o (1/n). To finish the proof, it remains to consider, for every h ∈(0, 1/2), the integer nsuch that 2−n−1<h ≤2−n; then (4.4)gives: lim h→0+ρϕ,2(h) h2p/2 log(1/h)=0, as announced. Proof of Lemma 4.2.Let: vn= n/2<k≤n uk. Since the series unconverges, on the one hand, we have vn−→ n→∞0, and on the other hand: vn≥n 2C−1un. Theorem 4.3. There exists a symbol ϕfor which the composition operator Cϕis compact on the Hardy space H2, but is not in any Schatten class Sp(B2)of the Bergman space B2with p <∞.
32 P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 In order to apply Theorem 5.2 for weighted Dirichlet spaces, we will use the following process. First, it suffices to prove the lower estimate with (D2 α)∗instead of D2 α, where: (D2 α)∗={f∈D2 α;f(0) = 0}(5.1) is the hyperplane of D2 αof functions vanishing at 0. The derivation Δis by definition a unitary operator from (D2 α)∗onto B2 α. For any symbol ϕvanishing at 0, we have the following diagram, where w=ϕ: (D2 α)∗Δ −−−−→B2 α MwCϕ −−−−−→B2 α Δ−1 −−−−→(D2 α)∗,(5.2) with obviously (since ϕ(f◦ϕ) =(f◦ϕ)): C(D2 α)∗ ϕ=Δ −1(MwCϕ)Δ, which shows that C(D2 α)∗ ϕ, acting on the delicate space (D2 α)∗is unitarily equivalent to the weighted composition operator MwCϕacting on the more robust space B2 α(in that M(B2 α) =H∞). Moreover, C(D2 α)∗ ϕand MwCϕ:B2 α→B2 αhave the same approximation numbers. 5.2. The cusp map on weighted Dirichlet spaces First, we recall the definition of the cusp map χ. We begin by defining: χ0(z)= z−i iz −11/2−i −iz−i iz −11/2+1·(5.3) That defines a conformal mapping from Donto the right half-disk D={z∈D;Rez>0} such that χ0(1) =0, χ0(−1) =1, χ0(i) =−i, χ0(−i) =i, and χ0(0) =√2−1. Then we set: χ1(z)=logχ0(z),χ 2(z)=−2 πχ1(z)+1,χ 3(z)= a χ2(z), (5.4) and finally: χ(z)=1−χ3(z),(5.5)
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 33 where: a=1−2 πlog(√2−1) = 1.56 ... ∈(1,2) (5.6) is chosen in order that χ(0) =0. The image Ωof the (univalent) cusp map χis formed by the intersection of the inside of the disk D1 −a 2,a 2and the outside of the two closed disks D1+ia 2,a 2and D1−ia 2,a 2. Since χis injective, it follows from Zorboska’s characterization in [43](see also [35, Section 6.12]) that the composition operator Cχis bounded on D2 αfor α≥0. In particular χ ∈D2 α. Theorem 5.3. Let χbe the cusp map acting on the Dirichlet space D2 α. Then, for some positive constants b α>b α>0, depending only on α, we have, for all n ≥1: e−b αn/ log nan(Cχ)e−bαn/ log nfor α>0,(5.7) and e−b 0√nan(Cχ)e−b0√nfor α=0.(5.8) Actually, the proof shows that, for α>0, the constant bαcan be chosen as c min(1, α), and b αcan be chosen as cmax(1, α), where c, care absolute positive constants. Note that the case α<0is not relevant, since then composition operators Cϕthat are compact on D2 αmust satisfy ϕ∞<1([34]). The estimates (5.8)was first proved in [25, Theorem 3.1], with ad hoc methods. We give here a more transparent proof of the lower bound, based on weighted composition operators acting on B2 α, and that works for all α≥0. Here the weight is χ. Since χ ∈D2 α, we have χ∈B2 α. We have the following estimates (the first one was given in [27, Lemma 4.2]). Lemma 5.4. When r→1−, it holds: 1−χ(r)≈1 log[1/(1 −r)] and χ(r)≈1 (1 −r)log2[1/(1 −r)] ·(5.9) Proof. For r∈(0, 1), we have ([27, Lemma 4.2]): χ0(r)=tan1 2arctan1−r 1+r=tanπ 8−1 2arctanr; hence χ0(r) ≈1 −rand: χ 0(r)=−1+[χ0(r)]2 2(1 + r2)·
34 P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 Using the definitions (5.4)and (5.5), we get: χ 2=−2 π χ 0 χ0 and χ 3=−aχ 2 χ2 2· So we have, when r→1−: χ2(r)≈log[1/(1 −r)] and: χ 2(r)=1+[χ0(r)]2 π(1 + r2)1 χ0(r)≈1 χ0(r)≈1 1−r· The result follows. Proof of Theorem 5.3. Proof of the lower bound. We choose 0 <u j<1so as to get via (5.9), with vj=χ(uj)and ε >0to be adjusted later: 1−vj=e −jε ; (5.10) hence: log 1 1−uj≈ejε .(5.11) The interpolation constant Ivof the sequence v=(vj)jsatisfies Iv1/δ2 v, where δv is its Carleson constant (see [12, Chapter VII, Theorem 1.1]). Since 1−vj+1 1−vj=e −ε, we have ([26, Lemma 6.4]), for some positive constants c1, c2: δv≥exp−c1 1−e−ε≥e−c2/ε ; hence, with c0=2 c2: Ivec0/ε (where the implicit constant does not depend on ε). The reproducing kernel of the Bergman space B2 αsatisfies: Ka=1 (1 −|a|2)1+α/2≈1 (1 −|a|)1+α/2· Using (5.9)and (5.10), we get, for 1 ≤j≤n:
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 35 |χ(uj)|Kvj Kuj1 (1 −uj)log 2[1/(1 −uj)] (1 −uj)1+α/2 (1 −vj)1+α/2 =1 log2[1/(1 −uj)] (1 −uj)α/2 (1 −vj)1+α/2 ≈1 e2jε exp(−α 2ejε) e−jε(1+α/2) =exp−α 2ejε +jε1−α 2 exp−α 2enε +nε1−α 2, since the function t → α 2et+t 1 −α 2is increasing; in fact, its derivative is positive. Theorem 5.2 with w=χgives: aB2 α n(MχCχ)exp−α 2enε +nε1−α 2+2c0 ε.(5.12) Case α=0. In this case, we have: aB2 0 n(MχCχ)exp−nε +2c0 ε. Taking ε =1/√n, we get: aD2 0 n(Cχ)=aB2 0 n(MχCχ)e−c√n for some positive absolute constant c. Case α>0. We take ε =1 nlog n log nand we get: aB2 α n(MχCχ)exp−α 2 n log(n)+logn log(n)+2c0 n log(n)−log(log(n)). Since log n log(n)+2 c0n log(n)−log(log(n)) =O n log(n), we get: aD2 α n(Cχ)exp(−b αn/ log n), with b α=cmax(α, 1), where cis some positive absolute constant. Proof of the upper bound. For α=0, the upper bound is proved in [25, Theorem 3.1]; for α>0, we will follow that proof, with the same notation, but with the weighted Nevanlinna counting function Nχ,α instead of the counting function nχ. Recall that the weighted Nevanlinna counting function of the analytic function ϕ:D→Dis: Nϕ,α(w)= ϕ(z)=w (1 −|z|2)α.
36 P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 Note that this definition is slightly different from, although equivalent to, that given in (4.18), but it is more convenient here. Since the cusp map χis univalent, we have: Nχ,α(w)=(1 −|χ−1(w)|2)αfor w∈χ(D), 0otherwise. (5.13) The Schwarz lemma gives Nχ,α(w) ≤(1 −|w|2)α, but the following lemma gives the better estimate: Nχ,α(w)e−c0α/(1−|w|),for w∈χ(D) (5.14) Lemma 5.5. We have: 1−|χ(z)|1 log[1/|1−z|]for all z∈D, so that, for some positive constant c0: 1−|χ−1(w)|≤|1−χ−1(w)|exp−c0 1−|w|for all w∈χ(D). Proof of the lemma. It suffices to look at the neighborhood of 1, since outside the two functions are continuous and do not vanish. Setting u =1 −z, an easy computation with Taylor expansions gives that z−i iz−1=−1 +iu +o (u), as u →0, so z−i iz−11/2= i[1 −iu/2 +o (u)] =i +u/2 +o (u), and (recall (5.3)and (5.4)): χ0(1 −u)=u 4+o(u)asu→0; and |1 −χ(z)| =|χ3(z)| ≈1 |log(χ0(z))|1 log(1/|1−z|)· Finally, since the cusp is contained in an angular sector, there exists some δ>0such that 1 −|χ(z)| ≥δ|1 −χ(z)|for every z∈D. The result follows. Thus we obtain the following estimate. Lemma 5.6. We have: χn2 D2 αn2e−√2c0αn. Proof. We have, since χ(0) =0: χn2 D2 α= D |nχn−1(z)χ(z)|2dAα(z)
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 37 = χ(D) n2|w|2n−2Nχ,α(w)dA(w) ≤ χ(D) n2|w|2n−2e−c0α/(1−|w|)dA(w) ≤n2(1 −h)2n−2+n2 χ(D)∩{|w|≥1−h} e−c0α/(1−|w|)dA(w) n2e−2nh +n2e−c0α/h . Choosing h =c0α/2ngives: χn2 D2 αn2e−√2c0αn. For f(z) =∞ n=0 cnzn, we define: (SNf)(z)= N n=0 cnzn. As a consequence of Lemma 5.6, we have the following majorization. Lemma 5.7. We have: Cχ−CχSND2 αN3+2α 4e−√2c0αN .(5.15) Proof. It suffices to use the Hilbert-Schmidt norm and Lemma 5.6: Cχ−CχSN2 D2 α≤Cχ−CχSN2 HS ≈ n>N χn2 D2 α n1−α n>N n1+αe−√2c0αn ≈N3 2+αe−√2c0αN . Lemma 5.8. Let Jbe the canonical injection J:H2→L2(μ)with dμ =Nχ,α dA. Then: an(CχSN)N1+α 2an(J).(5.16) Proof. Let f∈(D2 α)∗and write f(z) =∞ j=1 cjzj, we have: CχSNf2 D2 α= D N j=1 jcjwj−1 2 Nχ,α(w)dA(w)= D |(ΔNf)(w)|2dμ(w), where ΔN:(D2 α)∗→H2is defined by:
38 P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 (ΔNf)(w)= N j=1 jcjwj−1. We hence have CχSNfD2 α=JΔNfL2(μ)for all f∈(D2 α)∗. It follows that there exists a contraction TN:L2(μ) →(D2 α)∗such that: CχSN=TNJΔN. Now: ΔNf2 H2= N j=0 j1+αj1−α|cj|2≤N1+α N j=0 j1−α|cj|2≤N1+αf2 D2 α; hence, by the ideal property of approximation numbers: an(CχSN)≈an(CχSN)|(D2 α)∗≤TNΔNan(J)Nα+1 2an(J). Proposition 5.9. Let α>0and Jbe the canonical injection J:H2→L2(μ)with dμ = Nχ,α dA. Then, for some absolute positive constant c: an(J)exp−cmin(α, 1) n log n.(5.17) Proof. We use a modification of the Blaschke product of [25, page 168], as follows. Let r=[log 2n]be the greatest integer <log2n, where log2is the binary logarithm, and B0 be the Blaschke product with simple zeros at the points: zj=1−2−j,1≤j≤r, and we consider the Blaschke product B=Bn 0. Let E=BH2, which is a subspace of H2of codimension n [log2n]. We have, by the Carleson embedding theorem for H2(see [23, Lemma 2.4]): J|E2sup 0<h<1 ξ∈T 1 h S(ξ,h)∩Ω |B|2Nχ,α dA , (5.18) where S(ξ, h) ={z∈D; |z−ξ| <h}and: Ω=χ(D).(5.19) Note that A[S(ξ, h) ∩Ω] h3since the area of χ(D)∩{|w| ≥1 −h}is ≈h3; in fact this set is delimited at the cuspidal point 1by two circular arcs.
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 39 Now we majorize the right-hand side of (5.18). For that, we first note that, since Ω is contained in an angular sector, there is an absolute positive constant δ0such that 1 −|w| ≥δ0|1 −w|for all w∈Ω. Hence if w∈S(ξ, h) ∩Ω, we have: δ0|1−w|≤1−|w|≤|ξ−w|<h, and w∈S(1, h/δ0). Hence S(ξ, h) ∩Ω ⊆S(1, h/δ0) ∩Ω. Moreover, we may assume that h =δ02−land we separate two cases. •l≥r. We simply majorize |B|by 1. Lemma 5.5 and (5.14)lead to: 1 h S(ξ,h)∩Ω |B|2Nχ,α dA 1 h S(ξ,h)∩Ω exp(−αc 0/h)dA 1 he−αc0/h A[S(ξ,h)∩Ω] ≈h2e−αc0/h ≤e−α(c0/δ0)2 l≤e−α(c0/δ0)2 r≤e−α(c0/2δ0)n. •l<r. We write: S(ξ,h)∩Ω |B|2Nχ,α dA ≤ S(1,h/δ0)∩Ω |B|2Nχ,α dA = S(1,2−r)∩Ω |B|2Nχ,α dA + {w∈Ω; 2 −r≤|w−1|<2−l} |B|2Nχ,α dA . By the previous case, the first integral in the right-hand side, divided by h, is e−α(c0/2δ0)n. Now, if 2−r≤|w−1| <2−l, there is some j∈{l+1, ..., r}such that 2−j≤|w−1| < 2−j+1. Then: |w−zj|≤|w−1|+|1−zj|≤2−j+1 +2 −j=3.2−j; hence, since 2−j≤(1/δ0) (1 −|w|)and 1 −|zj| =1 −zj=2 −j, we have, with M= max(3, 1/δ0): |w−zj|≤Mmin(1 −|w|,1−|zj|). Therefore [23, Lemma 2.3] shows that the modulus of the j-th factor of B0(w)is less or equal than κ =M/√M2+1 <1. It follows that |B(w)| =|B0(w)|n≤κn. Using Nχ,α(w) ≤1we obtain:
40 P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 1 h {w∈Ω; 2 −r≤|w−1|<2−l} |B|2Nχ,α(w)dA 1 hA[S(ξ,h)∩Ω] κ2nκ2n=e −ε0n for some absolute constant ε0>0. We get: 1 h S(ξ,h)∩Ω |B|2Nχ,α dA e−α(c0/2δ0)n+e −ε0ne−c1min(α,1) n,(5.20) where c1>0does not depend on α. In either case, we obtain, J|Ee−c2min(α,1) n. That means that the Gelfand number cn[log2n](J)is e−c2min(α,1) n. Since the Gelfand numbers are the same as the approximation numbers on Hilbert spaces, we get: an[log2n](J)e−c2min(α,1) n, or, making a change of variables in the indices: an(J)exp−c3min(α, 1) n log n, as claimed. End of the proof of the upper bound. For every operator R:D2 α→D2 αwith rank <n, we have: Cχ−R≤Cχ−CχSN+CχSN−R; so: an(Cχ)≤Cχ−CχSN+an(CχSN). Using Lemma 5.7, Lemma 5.8 and Proposition 5.9, we obtain: an(Cχ)N3+2α 4e−√2c0αN +Nα+1 2e−c3min(α,1) n/ log n N3+2α 4e−√2c0αN +e −c3min(α,1) n/ log n. a) Suppose first that α≤1. Then: an(Cχ)N3+2α 4e−√2c0αN +e −c3αn/log n.
P. Lefèvre et al. / Journal of Functional Analysis 280 (2021) 108834 41 Choosing Nas the integral part of α(c2 3/2c0) (n/ log n)2, we get: an(Cχ)n log n3+2α 2e−c3αn/ log nn log n5 2e−c3αn/ log ne−c 3αn/ log n, for another absolute constant c 3<c 3. b) For α>1, we have: an(Cχ)N3+2α 4e−√2c0αN +e −c3n/ log n. Choosing for Nthe integral part of (1/α)(c2 3/2c0)(n/ log n)2, we get: an(Cχ)n log n3+2α 2e−c3n/ log n. However, the term (n/ logn)3+2α 2tends to infinity when αtends to infinity (even if the implicit constants in the inequalities depend on α). In order to have a better estimate, we need a different strategy. We recall that D2 1=H2. We now use Theorem 3.12, which is licit as indicated in the remarks following the statement of this theorem. Using the previously treated case, we obtain that: aD2 α 2n(Cχ)≤aH2 1(Cχ)aH2 n(Cχ)exp−cn log n· Hence, rescaling on one hand and using the monotony of the sequence of the approximation numbers on the other hand, we get: aD2 α n(Cχ)exp−cn log n, where the underlying constants do not depend on α. That ends the proof of Theorem 5.3. Remark 1. Actually, for α>0, the formula (5.12)gives that: aD2 α n(Cχ)exp−α 2enε +nε1−α 2+2c0 ε.(5.21) Taking ε =1/√n, we get, for 0 <α<2, with c1=1 +2 c0, this “bad” estimate: aD2 α n(Cχ)exp−α 2e√n+c1√n.