1Introduction Compositionoperatorswithsurjectivesymbolandsmallapproximationnumbers

North-Western European Journal of Mathematics

W N

M

E J

Composition operators with surjective symbol and small approximation numbers

Daniel Li¹ Hervé Queffélec² Luis Rodríguez-Piazza³

Received: November 12, 2018/Accepted: January 17, 2019/Online: February 20, 2019

Abstract

We give a new proof of the existence of a surjective symbol whose associated composition operator onH²(D) is in all Schatten classes, with the improvement that its approximation numbers can be, in some sense, arbitrarily small. We show, as an application, that, contrary to the 1-dimensional case, forN ≥2, the behavior of the approximation numbersa_n =a_n(C_φ), or rather ofβ_N⁻ = lim inf_n→∞[a_n]^1/n1/N orβ⁺_N= lim sup_n→∞[a_n]^1/n1/N, of composition operators onH²(D^N) cannot be determined by the image of the symbol.

Keywords:Approximation numbers, cusp map, composition operator, Hardy space, lens map, polydisk.

msc: Primary 47B33; Secondary 32A35, 46B28.

1 Introduction

We start by recalling some notations and facts.

LetDbe the open unit disk,H²the Hardy space onD, andϕ:D→Da non- constant analytic self-map. It is well known⁴thatϕinduces a composition operator C_ϕ:H²→H²by the formula:

C_ϕ(f) =f ◦ϕ ,

and the connection between the “symbol”ϕ and the properties of the operator Cϕ:H²→H², in particular its compactness, can be further studied⁵.

1Univ. Artois, Laboratoire de Mathématiques de Lens (LML) EA 2462, Fédération CNRS Nord-Pas- de-Calais FR 2956, Faculté Jean Perrin, Rue Jean Souvraz, S.P. 18 F-62300 LENS, France

2Université de Lille, Laboratoire Paul Painlevé U.M.R. CNRS 8524, Fédération CNRS Nord-Pas-de- Calais FR 2956 F-59655 Villeneuve D’Ascq Cedex, France

3Universidad de Sevilla, Facultad de Matemáticas, Departamento de Análisis Matemático & IMUS, Calle Tarfia s/n, 41012 Sevilla, Spain

4J. Shapiro, 1993,Composition operators and classical function theory.

5Ibid.

We also recall that the nth approximation number a_n(T), n= 1,2, . . ., of an operatorT:H₁→H₂, between Hilbert spacesH₁andH₂, is defined as the distance ofT to operators of rank< n, for the operator-norm:

a_n(T) = inf

rankR <n

∥T−R∥. (1)

The p-Schatten class S_p(H₁, H₂), p >0 consists of all T:H₁ →H₂ such that a_n(T)

n∈ℓ^p. The approximation numbers have the ideal property:

an(AT B)≤ ∥A∥an(T)∥B∥.

Let now, forξ∈T=∂Dandh >0, the Carleson windowS(ξ, h) be defined as:

S(ξ, h) ={z∈D; |z−ξ| ≤h}. (2)

For a symbolϕ, we define mϕ =ϕ^∗(m) wherem is the Haar measure ofT and ϕ^∗:T→Dthe (almost everywhere defined) radial limit function associated withϕ, namely:

ϕ^∗(ξ) = lim

r→1⁻ϕ(rξ). Finally, we set forh >0:

ρ_ϕ(h) = sup

ξ∈T

m_ϕ[S(ξ, h)]. (3)

It is known⁶thatρϕ(h) = O (h) and thatCϕ7is compact if and only ifρϕ(h) = o (h) ash→0. Simpler criteria⁸exist whenϕis injective, or evenp-valent, meaning that for anyw∈D, the equationϕ(z) =whas at mostpsolutions.

A measureµonDis calledα-Carleson,α≥1, if sup|ξ|=1µ[S(ξ, h)] = O (h^α).

B. MacCluer and J. Shapiro showed in MacCluer and H. Shapiro (1986, Exam- ple 3.12) the following result, paradoxical at first glance.

Theorem 1 (MacCluer-Shapiro) – There exists asurjective and four-valentsymbol ϕ:D→Dsuch that the composition operatorC_ϕ:H²→H²is compact.

Observe that such a symbolϕcannot be one-valent (injective), because it would be an automorphism ofD, andCϕwould be invertible and therefore not compact.

In Lefèvre et al. (2012, Theorem 4.1), we gave the following improved statement.

6J. Shapiro, 1993,Composition operators and classical function theory.

7MacCluer, 1984, “Spectra of compact composition operators onH^p(B_N)”.

8J. Shapiro, 1993,Composition operators and classical function theory.

2. Background and preliminary results

Theorem 2 – For every non-decreasing functionδ: (0,1)→(0,1), there exists a two- valent symbol and nearly surjective (i.e.ϕ(D) =D\ {0}) symbolφ, and0< h₀<1, such that:

m({z∈T; |φ^∗(z)| ≥1−h})≤δ(h) for0< h≤h₀. (4) As a consequence, there exists asurjective and four-valentsymbolψ:D→Dsuch that the composition operatorC_ψ:H²→H²is in every Schatten classS_p(H²),p >0.

Our proof was rather technical and complicated, and based on arguments of barriers and harmonic measures.

The goal of this paper is to give a more precise statement of Theorem 2 in terms of approximation numbersa_n(C_ϕ), and not only in terms of Schatten classes, and with a simpler proof. We then apply this result to show that for the polydiskD^N,N ≥2, the nature (boundedness, compactness, asymptotic behavior of approximation numbers) of the composition operator cannot be determined by the geometry of the imageφ(D^N) of its symbolφ. For certain asymptotic behavior of approximation numbers, this is contrary to the 1-dimensional case (see Li, Queffélec, and Rodríguez- Piazza 2015, Theorem 3.1 and Theorem 3.14).

The notationA≲Bmeans thatA≤C Bfor some positive constantC, andA≈B thatA≲BandB≲A.

2 Background and preliminary results

We initiated the study of approximation numbers of composition operators onH²in Li, Queffélec, and Rodríguez-Piazza (2012), and proved the following basic results:

Theorem 3 – Ifϕis any symbol, then, for someδ >0andr >0, ora >0:

a_n(C_ϕ)≥δ rⁿ=δe^−an.

Moreover, as soon as∥ϕ∥_∞= 1, there exists some sequenceεn tending to0such that:

an(Cϕ)≥δe^−nεn.

We also proved in Li, Queffélec, and Rodríguez-Piazza (2012, Theorem 5.1) that:

Proposition 1 – For any symbolϕ, we have:

an(Cϕ)≲ inf

0<h<1

"

e^−nh+

rρ_ϕ(h) h

# .

We also recall (see Li, Queffélec, and Rodríguez-Piazza 2012) that, forγ >−1, the weighted Bergman spaceB_γis the space of functionsf(z) =P^∞

n=0a_nzⁿsuch that:

∥f∥²_γ:=

∞

X

n=0

|an|²

(n+ 1)^γ+1 <∞. (5)

Equivalently,B_γis the space of analytic functionsf:D→Csuch that:

Z

D

|f(z)|²(γ+ 1)(1− |z|²)^γdA(z)<∞, (6) wheredAis the normalized area measure onD, and then:

Z

D

|f(z)|²(γ+ 1)(1− |z|²)^γdA(z)≈ ∥f∥²_γ. (7) The caseγ= 0 corresponds to the usual Bergman spaceB², and the limiting case γ=−1 to the Hardy spaceH². We wish to note in passing (we will make use of that elsewhere) that the proof of Theorem 5.1 in Li, Queffélec, and Rodríguez-Piazza (2012) easily gives the following result.

Proposition 2 – Letγ >−1andϕa symbol inducing a bounded composition operator C_ϕ:B_γ→H². Then:

an(Cϕ:B_γ→H²)≲ inf

0<h<1







(n+ 1)^(γ+1)/2e^−nh+ sup

0<t≤h

s ρ_φ(t)

t^2+γ









·

Proof. TakeE=zⁿB_γ; this is a subspace ofB_γ of codimension≤n. Letf ∈Ewith

∥f∥_γ= 1. Writingf =zⁿgwith∥g∥²

γ≤(n+ 1)^γ+1and splitting the integral into two parts, we have, for 0< h <1:

∥C_ϕf∥²

H²= Z

D

|f|²dm_φ≤(1−h)²ⁿ Z

(1−h)D

|g|²dm_φ+ Z

D\(1−h)D

|f|²dm_φ.

For the first integral, we have:

Z

(1−h)D

|g|²dmφ≤ Z

D

|g|²dmφ=∥Cφg∥²

H²≤ ∥Cφ∥²

B_γ→H²∥g∥²_γ. (8) For the second integral, we have:

Z

D\(1−h)D

|f|²dmφ≤ ∥J:B_γ→L²(µh)∥²,

2. Background and preliminary results

whereµ_his the restriction ofm_ϕ to the annulus{z∈D; 1−h <|z|<1}andJ the canonical injection ofB_γ intoL²(µh). Hence Stegenga’s version of the Carleson embedding theorem forB_γ(Stegenga 1980, Theorem 1.2; see Hastings 1975 for the unweighted case; see also Duren and Schuster 2004, p. 62 or Zhu 2007, p. 167) gives us:

Z

D\(1−h)D

|f|²dm_φ≲ sup

0<t≤h

ρφ(t)

t^2+γ · (9)

Putting (8) and (9) together, that gives:

∥C_ϕf∥_H2≲e^−nh(n+ 1)^(γ+1)/2+ sup

0<t≤h

s ρφ(t)

t^2+γ

·

In other terms, using the Gelfand numbersc_k:

cn+1(C_φ:B_γ→H²)≲(n+ 1)^(γ+1)/2e^−nh+ sup

0<t≤h

s ρ_φ(t)

t^2+γ ·

Asan+1=cn+1and as we can ignore the difference betweena_nandan+1, that finishes

the proof. □

As an application, we mention the following result. We refer to Li, Queffélec, and Rodríguez-Piazza (2013, Section 4.1) for the definition of the cusp map, denotedχ.

Theorem 4 – Letχ: D→Dbe the cusp map andΦ:D^N →D^N the diagonal map defined by:

Φ(z₁, z₂, . . . , z_N) =

χ(z₁), χ(z₁), . . . , χ(z₁)

. (10)

Then, the composition operatorCΦ mapsH²(D^N)to itself and:

a_n(CΦ)≲e^−d

√

n (11)

wheredis a positive constant depending only onN.

Remark 1 – We have to compare with Bayart et al. (2018, Theorem 6.2) where, for:

Ψ(z₁, . . . , z_N) =

χ(z₁), . . . , χ(z_N) ,

it is shown that, for constantsb≥a >0 depending only onN: e^{−b(n1/N/lnn)}≲an(C_Ψ)≲e^{−a(n1/N/lnn)}.

Note also that forN = 1, the estimate of Theorem 4 is very crude.

Proof (Proof of Theorem 4 on the previous page). Takeγ=N−2. As in Li, Queffélec, and Rodríguez-Piazza (n.d.[a], Section 4), we have thanks to the Cauchy-Schwarz inequality, and the fact thatP

|α|=n1≈(n+ 1)^N−1, a factorization:

CΦ=JCχM ,

whereM:H²(D^N)→ B_γis defined byMf =gwith:

g(z) =f(z, z, . . . , z) =

∞

X

n=0







 X

|α|=n

a_α







zⁿ, z∈D, (12) for

f(z1, z2, . . . , zN) =X

α

aαz^α₁¹· · ·z^α_N^N,

and whereJ:H²(D)→H²(D^N) is the canonical injection given by:

(Jh)(z1, z2, . . . , zN) =h(z1). (13)

This corresponds to a diagram:

H²(D^N)−→ B^M _γ−→^Cχ H²(D)−→^J H²(D^N), (14) whereCχ:B_γ=B_N−2→H²(D) is a bounded operator. Indeed, we have the behav- ior⁹:

|1−χ^∗(e^iθ)| ≈ 1 ln(1/|θ|),

and this implies, withcan absolute constant:

mχ[S(ξ, h)]≲mχ[S(1, h)] =m({|χ^∗(e^iθ)−1|< h)

≲m[{c/ln(1/|θ|)< h}]≤e^−c/h; (15) in particular ρ_χ(h) ≤ e^−c/h = O (h^N), so m_χ is anN-Carleson measure and the Stegenga-Carleson theorem¹⁰says that the operatorCχ:B_N−2→H²(D) is bounded.

Now Proposition 2 on p. 4 with (15) give:

a_n(C_χ:B_γ→H²)≲ inf

0<h<1

h(n+ 1)^(N−1)/2e^−nh+ e^−c/hh^{−N /2}i . Adjustingh= 1/√

n, we getan(Cχ:B_γ→H²)≲e^−d

√

n for some positive constantd.

Finally, the factorization CΦ = JCχM and the ideal property of approximation

numbers give the result. □

2. Background and preliminary results

In the case of lens maps, Proposition 2 on p. 4 gives very poor estimates. We avoid using this theorem in Li, Queffélec, and Rodríguez-Piazza (n.d.[a], Section 4), whenN = 2, using the semi-group property of those lens maps. The same proof gives for arbitraryN≥2 the following result.

Theorem 5 – Letλ_θthe lens map with parameterθ,0< θ <1, and letΦ:D^N→D^N be the diagonal map defined by:

Φ(z₁, z₂, . . . , z_N) =

λ_θ(z₁), λ_θ(z₁), . . . , λ_θ(z₁)

. (16)

Then:

1) ifθ >1/N,CΦ is unbounded onH²(D^N);

2) ifθ= 1/N,CΦ is bounded and not compact onH²(D^N);

3) ifθ <1/N,C_Φ is compact onH²(D^N)and moreover:

a_n(CΦ)≲e^−d

√

n (17)

for a constantd >0depending only onθandN.

Remark 2 – In Bayart et al. (2018, Theorem 6.1), it is shown that, for:

Ψ(z₁, . . . , z_N) =

λ_θ(z₁), . . . , λ_θ(z_N) ,

we have, for constantsb≥a >0, depending only onθandN: e^{−b n1/(2N)}≲a_n(CΨ)≲e^{−a n1/(2N)}.

Proof (Proof of Theorem 5). That had been proved, forN = 2 in Li, Queffélec, and Rodríguez-Piazza (n.d.[a], Theorem 4.2 and Theorem 4.4). For convenience of the reader, we sketch the proof.

Assume firstθ≤1/N, and writeλθ=λN θ◦λ1/N, where we set, for convenience, λ1(z) =z, soCλ₁= Id. As in the proof of Theorem 4 on p. 5 (see Li, Queffélec, and Rodríguez-Piazza n.d.(a), Section 4), we have a factorization:

CΦ =JC_{λN θ}C_λ1/NM ,

whereMandJ are defined in (12) and (13).

9Li, Queffélec, and Rodríguez-Piazza, 2013, “Estimates for approximation numbers of some classes of composition operators on the Hardy space”, Lemma 4.2.

10Stegenga, 1980, “Multipliers of the Dirichlet space”, Theorem 1.2.

This corresponds to a diagram (recall thatγ=N−2):

H²(D^N)−→ B^M _γ

C_λ1/N

−→ H²(D)^C−→^λNθH²(D)−→^J H²(D^N).

The second arrow is bounded, since we know¹¹that the pullback measurem_λ1/N is N-Carleson, so thatC_λ1/NmapsB_N−2toH²(D) by the Stegenga-Carleson embedding theorem¹².

For θ <1/N, we have N θ <1 and Cλ_{N θ} is compact and, for some constant b=b(θ), we havean(Cλ_{N θ})≲e^−b

√

n13. HenceCΦis compact andan(CΦ)≲e^−b

√ n. Now, forθ≥1/N, we consider the reproducing kernels:

Ka₁,...,a_N(z₁, . . . , zN) =

N

Y

j=1

1 1−ajzj

·

We have:

∥Ka₁,...,a_N∥²=

N

Y

j=1

1 1− |a_j|² and:

C^∗_Φ(K_a1,...,aN) =K_{λθ(a1),...,λθ(a1)}, so:

∥C_Φ^∗(Ka₁,...,a_N)∥²= 1 1− |λ_θ(a₁)|²

!N

·

Since:

1− |λθ(a₁)|²≈1− |λθ(a₁)| ≈(1− |a1|)^θ,

we see that∥C_Φ^∗(K_a1,...,aN)∥/∥K_a1,...,aN∥is not bounded forθ >1/N, soC_φis then not bounded; and it does not converge to 0 forθ= 1/N, soCΦ is then not compact. □

11Lefèvre et al., 2013b, “Some new properties of composition operators associated with lens maps”, Lemma 3.3.

12Stegenga, 1980, “Multipliers of the Dirichlet space”, Theorem 1.2.

13Lefèvre et al., 2013b, “Some new properties of composition operators associated with lens maps”, Theorem 2.1.

3. Surjectivity

3 Surjectivity

Let us come back to our surjectivity issues.

Let us first remark that Theorem 2 on p. 3 gives the following result.

Theorem 6 – For every non-decreasing functionδ: (0,1)→(0,1), there exists asurjec- tive and four-valentsymbolψ, and0< h₀<1, such that, for0< h≤h₀:

m({z∈T; |φ^∗(z)| ≥1−h})≤δ(h). (18)

Proof. Just observe that the passage from “ϕtwo-valent and nearly surjective” to

“ψfour-valent and surjective” is harmless: for this, consider the Blaschke product:

B(z) = z−a 1−az

!2

,

where 0< a <1, and takeψ=B◦ϕ; we observe thatB(D\ {0}) =Dsincea²=B _2a

1+a²

, and, forz∈D:

1− |B(z)|

1− |z| ≥1− |^z−a

1−az|²

1− |z|² = 1−a²

|1−az|² ≥1−a² 4 , so that:

m(|ψ^∗|>1−h) =m(1− |B◦ϕ^∗|< h)≤m

1− |ϕ^∗| ≤κah ,

withκ_a= 4/(1−a²). Hence, this mapψis surjective, four-valent, and satisfies (18), as well, up to a change ofδ(h) toδ(h/κ_a) forϕat the beginning. □

3.1 A more precise statement

Our new statement is as follows.

Theorem 7 – For every positive sequence(ε_n)_n with limit0, there exists asurjective and four-valentsymbolϕsuch that:

a_n(C_ϕ)≲e^−nεn.

Consequently, there exists asurjective and four-valentsymbolϕ:D→Dsuch that the composition operatorCϕ:H²→H²is in every Schatten classSp(H²),p >0.

Proof. Observe first that∥ϕ∥_∞= 1 whenϕis surjective, so that, in view of Theorem 3 on p. 3, we cannot dispense with the numbersεn, even if they can tend to 0 arbitrarily slowly.

Now, we can chooseδ: (0,1)→(0,1) non-decreasing such thatδ(ε_n)≤e^−nεnfor alln, and then, using Theorem 6 on the previous page, we get a surjective and four-valent symbolϕ, satisfying for allhsmall enough:

ρϕ(h)≤h δ²(h). Proposition 1 on p. 3 gives:

a_n(C_ϕ)≲ inf

0<h<1

he^−nh+δ(h)i . Adjustingh=ε_n, we geta_n(C_ϕ)≲e^−nεn.

To get the second part of the theorem, just takeε_n=n^−1/2. □

3.2 A simplified proof of Theorem 2 on p. 3

We give here the announced simplified proof of Theorem 2 on p. 3. This proof is based on the following key lemma, in whichH(D) denotes the set of holomorphic functions onD.

Lemma 1 (Lefèvre et al. 2013a, Lemma 2.11) – There exists a numerical constantC such that, iff ∈ H(D)satisfies, for someα∈R:

(ℑ[f(0)]< α

f(D)⊆ {z∈C; 0<ℜz < π} ∪ {z∈C; ℑz < α}, then:

m({ℑf^∗> y})≤Ce^α−y, fory≥α .

Proof (Proof of Theorem 2 on p. 3). Letg: (0,∞)→(0,∞) be a continuous decreasing function such that:

tlim→0⁺g(t) = +∞, g(π) =π , lim

t→+∞g(t) = 0. Then letΩbe the simply connected region defined by:

Ω={x+iy; x >0, g(x)< y < g(x) + 4π},

andf:D→Ωbe a Riemann map such thatf(0) =π+ 3iπ. Observe that we can apply Lemma 1 tof withα= 5πsinceℑf(0) = 3πand iff(z) =x+iywithx≥π;

hence:

ℑf(z) =y < g(x) + 4π≤g(π) + 4π= 5π .

3. Surjectivity

Finally, consider the symbolϕ= e^−f. It is nearly surjective: φ(D) =D\ {0}, and two-valent, as easily checked.

For 0< h≤1/2, we have forξ∈T and|φ^∗(ξ)|>1−h:

e^−2h≤1−h <|φ^∗(ξ)|= exp

− ℜf^∗(ξ)

; henceℜf^∗(ξ)<2h.

But if 2h > x=ℜf^∗(ξ), we haveg(x)> g(2h). As f^∗(ξ) =x+iy∈ Ω, we get ℑf^∗(ξ) =y≥g(x)> g(2h). Lemma 1 on the preceding page now gives:

m({ξ;|ϕ^∗(ξ)|>1−h})≤m({ξ; ℑf^∗(ξ)> g(2h)})≤Ce^5π−g(2h). (19) It is now enough to adjustgso as to have e^g(t)≥Ce^5π/δ(t/2) fortsmall enough to

get (4) from (19). □

For sake of completeness, we give the proof of Lemma 1 on the preceding page.

Proof (Proof of Lemma 1 on the preceding page). We now prove Lemma 1 on the pre- ceding page. If e^y−α<2, there is nothing to prove, since then:

m(ℑf^∗> y)≤1≤2 e^α−y.

We can hence assume that e^y−α≥2. First, we make a comment. If the Riemann mapping theorem is very general and flexible, it gives very few informations on the parametrizationt7→f^∗(e^it) whenf:D→Ωis a conformal map, except in some specific cases (lens maps, cusps, etc.: see Li, Queffélec, and Rodríguez-Piazza 2013).

Here, the Kolmogorov weak type inequality provides a substitute. Write:

f =u+iv and set:

f1=−if +iπ

2−α=v−α+i π 2−u

!

and:

F₁= 1 + e^f1= (1 + e^v−αsinu) +ie^v−αcosu .

Ifv < α, thenℜF1>1− |sinu| ≥0. Ifv≥α, then 0< u < πandℜF1≥1. Hence F1mapsDto the right half-planeC₀={z;ℜz >0}. Finally, letF=U+iV:D→C₀ be defined by:

F=F₁−iℑF₁(0),

so thatV(0) = 0. By the Kolmogorov inequality for the conjugation mapU7→V, and the harmonicity ofU, we have, for allλ >0 (adesignating an absolute constant):

m(|F^∗|> λ)≤ a

λ∥U^∗∥₁= a λ

Z

T

U^∗dm= a

λU(0). (20)

Next, we claim that:

|ℑF₁(0)|<1 and U(0)<2. (21) Indeed,v(0)< αby hypothesis, so that|ℑF1(0)|= e^v(0)−α|cosu(0)|<1, andU(0) = 1 + e^v(0)−αsinu(0) < 2. Suppose now that, for some y > α andz∈ D, we have v(z)> y. Then, 0< u(z)< πby our second assumption, and this impliesℜe^f1(z)= e^v(z)−αsinu(z)>0, so that, using|1 +w| ≥ |w|ifℜw >0 and (21), and remembering that e^y−α≥2:

|F(z)|=

1 + e^f1(z)−iℑF1(0) ≥

1 + e^f1(z) −1

≥ e^f1(z)

−1 = e^v(z)−α−1>e^y−α−1≥1 2e^y−α. Taking radial limits and using (20) and (21), we get:

m(ℑf^∗> y)≤m(|F^∗|>e^y−α/2)≤4ae^α−y.

This ends the proof of Lemma 1 on p. 10 withC= max(2,4a). □

4 Application to the multidimensional case

In this section, we apply Theorem 6 on p. 9 and Theorem 7 on p. 9 to show that, forN≥2, the image of the symbol cannot determine the behavior of the approx- imation numbers, or rather of βN(Cφ), of the associated composition operator C_φ:H²(D^N)→H²(D^N).

Recall that for an operatorT:H₁→H₂, we set:

β_N⁻(T) = lim inf

n→∞ [a_n(T)]^1/n1/N and β_N⁺(T) = lim sup

n→∞

[a_n(T)]^1/n1/N, (22) and writeβN(T) whenβ_N⁻(T) =β_N⁺(T).

Theorem 8 – ForN ≥2, there exist pairs of symbolsΦ₁,Φ2:D^N →D^N, such that Φ₁(D^N) =Φ₂(D^N)and:

1) CΦ₁is not bounded, butCΦ₂is compact, and evenβ_N(CΦ₂) = 0;

2) CΦ₁ is bounded but not compact, so βN(CΦ₁) = 1, and CΦ₂ is compact, with βN(CΦ₂) = 0;

4. Application to the multidimensional case

3) C_Φ1 is compact, withβ⁻_N(C_Φ1)>0andβ_N⁺(C_Φ1)<1, andC_Φ2 is compact, with β_N(CΦ₂) = 0;

4) CΦ₁is compact, withβ_N(CΦ₁) = 1, andCΦ₂is compact, but withβ_N(CΦ₂) = 0.

Proof. Letσ:D→Dbe a surjective symbol such thatρ_σ(h)≤h^Ne^−2/h2 given by Theorem 6 on p. 9. By Proposition 2 on p. 4, we have, withγ=N−2:

a_n(C_σ:B_γ→H²)≲ inf

0<h<1(n^(N−1)/2e^−nh+ e^−1/h2), and, withh= 1/n^1/3, we getan(Cσ:B_γ→H²)≲e^{−d n2/3}.

We choose the exponent 2/3 for fixing the ideas, but every exponentα >1/2, withα <1, (i.e.an(Cσ:B_γ→H²)≲e^{−d nα}) would be suitable.

1) We takeΦ₁(z₁, z₂, z₃, . . . , z_N) = (z₁, z₁, . . . , z₁). The composition operatorCΦ₁

is not bounded because if f_n(z₁, . . . , z_N) = _z

1+z₂ 2

n

, then ∥f_n∥²

2 = 4⁻ⁿPn k=0 n

k

2

= 4^{−n 2n}_n≈1/√

n, though (C_Φ1f_n)(z₁, . . . , z_N) =zⁿ₁and∥C_Φ1f_n∥₂= 1.

We defineΦ₂by:

Φ₂(z1, z2, . . . , z_N) =

σ(z1), σ(z1), . . . , σ(z1) .

Sinceσ is surjective, we haveΦ₂(D^N) =Φ₁(D^N). Now, as in the proof of Theorem 4 on p. 5, we haveC_Φ2=JC_σM, so:

a_n(CΦ₂)≤a_n(C_σ:B_N−2→H²)≲e^{−d n2/3}, by the ideal property. Hence [a_n(CΦ₂)]^1/n1/N ≲e^{−d n}

23−1

N and thereforeβ_N(CΦ₂) = 0 since ²₃− ¹

N >0.

2) We consider the lens mapλ=λ_1/N of parameter 1/N. We define:











Φ₁(z₁, . . . , z_N) =

λ(z₁), λ(z₁), . . . , λ(z₁) Φ₂(z₁, . . . , z_N) =

λ[σ(z₁)], λ[σ(z₁)], . . . , λ[σ(z₁)]

.

Sinceσ is surjective, we haveΦ₁(D^N) =Φ₂(D^N) and we saw in Theorem 5 on p. 7 thatC_Φ1is bounded but not compact.

On the other hand, we have the factorization C_Φ2 =JC_σC_λM. Hence C_Φ2 is compact, and, as in 1),β_N(C_Φ2) = 0.

3) For this item, the mapσdoes not suffice, and we will use another surjective symbols:D→D. By Theorem 6 on p. 9, there exists such a mapswith:

ρ_s(t)≤t²e^−2/t2 (23)

and

ρ_s(t)≤t δ²(t) (24)

fortsmall enough, whereδ: (0,1)→(0,1) is a non-decreasing function such that δ(ε_n)≤e^−nεnand:

ε_n=n^−4N1−7. (25)

By the proof of Theorem 7 on p. 9, (24) implies that:

a_n(C_s)≤e^−nεn. (26)

We also consider a lens mapλ=λ_θ, with parameterθ <1/N, and we set:















Φ₁(z₁, . . . , z_N) =

λ(z₁), λ(z₁),z₃ 2, . . . ,z_N

2

Φ₂(z₁, . . . , z_N) =

λ[s(z₁)], λ[s(z₁)],s(z₃)

2 , . . . ,s(z_N) 2

. Sincesis surjective, we haveΦ₁(D^N) =Φ₂(D^N).

a) Let us prove thatβ_N⁻(C_Φ1)>0 andβ_N⁺(C_Φ1)<1.

Note that:

C_Φ1=C_u⊗C_v3⊗ · · · ⊗C_vN,

whereu:D²→D²is defined byu(z₁, z₂) =

λ(z₁), λ(z₁)

andv_j:D→Dis defined byvj(zj) =zj/2. In fact, iff ∈H²(D²) andgj∈H²(D), 3≤j≤N, we have:

[CΦ₁(f ⊗g3⊗ · · · ⊗gN)](z1, z2, z3, . . . , zN)

= (f⊗g₃⊗ · · · ⊗g_N)

u(z₁, z₂), v₃(z₃), . . . , v_N(z_N)

=f[λ(z1), λ(z1)]g3[v3(z3)]· · ·gN[vN(zN)]

= (C_uf)(z₁, z₂) (C_v3g₃)(z₃)· · ·(C_vNg_N)(z_N)

= [(C_u⊗C_v3⊗ · · · ⊗C_vN)(f⊗g₃⊗ · · · ⊗g_N)](z₁, z₂, z₃, . . . , z_N),

hence the result sinceH²(D²)⊗H²(D)⊗ · · · ⊗H²(D) is dense inH²(D^N). That proves in particular thatCΦ₁ is compact sinceC_u andC_v3, . . . , C_vN are (by Theorem 5 on p. 7 forC_u).

By the supermultiplicativity of singular numbers of tensor products (see Li, Queffélec, and Rodríguez-Piazza n.d.(a), Lemma 3.2), it ensues that:

a_nN(CΦ₁)≥a_n²(C_u) YN

j=3

a_n(C_vj) =a_n²(C_u) 1

2 n(N−2)

.

4. Application to the multidimensional case

By Li, Queffélec, and Rodríguez-Piazza (n.d.[a], Remark at the end of Section 4), we havea_n2(C_u)≳e^−bnfor some positive constantb=b(θ). Indeed, ifJ=J₂:H²(D)→ H²(D²) is the canonical injection defined by (Jh)(z₁, z₂) =h(z₁) andQ:H²(D²)→ H²(D) is defined by (Qf)(z₁) =f(z₁,0), we have C_λ = QC_uJ. Hence a_k(C_u) ≳ a_k(C_λ)≳e^−b

√ k. Therefore we get:

a_nN(CΦ₁)≳e^−cn

for some positive constant depending only onθandN. It follows thatβ_N⁻(CΦ₁)>0.

To see thatβ_N⁺(CΦ₁)<1, we need the following lemma, whose proof is postponed.

Lemma 2 – LetS:H₁→H₁andT:H₂→H₂be two operators between Hilbert spaces andA, Ba pair of positive numbers. Then, whenever:

a_[nA](S)≤e^−cn and a_[nB](T)≤e^−cn,

where[.]stands for the integer part, we have, for some constant integerM=M(A, B)>0:

a_M[nA+B](S⊗T)≤e^−cn.

LetS =C_u andT =C_v3⊗ · · · ⊗C_vN. Forc small enough, we havea_nN−2(T)≤ C(1/2)ⁿ≤e^−cnand, using (17),a_n2(S)≤e^−dn≤e^−cn. Hence, withA= 2,B=N−2, Lemma 2 gives:

a_MnN(C_Φ1)≲e^−cn.

Thereforeβ_N⁺(C_Φ1)≤e^−c/M1/N <1.

b) DefineΨ:D^N →D^Nby:

Ψ(z₁, z₂, z₃, . . . , z_N) =

s(z₁), s(z₁), s(z₃), . . . , s(z_N) . Ifτ₁:D²→D²is defined byτ₁(z₁, z₂) =

s(z₁), s(z₁)

and the mapτ₂:D^N−2→ D^N−2byτ₂(z₃, . . . , z_N) =

s(z₃), . . . , s(z_N)

, we have:

CΨ =C_τ1⊗C_τ2.

As in the proof of Theorem 4 on p. 5, we have the factorization:

τ1:H²(D²)−→ B^M ₀=B²−→^Cs H²(D)−→^J H²(D²). Hencean(Cτ₁)≤ ∥M∥ ∥J∥an(Cs:B²→H²).

By Proposition 2 on p. 4, we have:

a_n(C_s:B²→H²)≲ inf

0<h<1

√

ne^−nh+ sup

0<t≤h

rρ_s(t) t²

!

; so (23) implies that a_n(C_s: B² → H²) ≲ inf0<h<1(√

ne^−nh+ e^−1/h2) and, taking h=n^−1/3, we get, with somecsmall enough:

an(Cs:B²→H²)≲e^−cn2/3.

It follows thata_n(C_τ1)≲e^{−c n2/3}and hence:

a_[n3/2](C_τ1)≲e^{−c n}. (27)

On the other hand, Bayart et al. (2018, Theorem 5.5) says that:

an(Cτ₂)≤2^N−3∥Cs∥^N−2 inf

n₃···n_N≤n

an₃(Cs) +· · ·+an_N(Cs) .

Takingn3=· · ·=nN=n^N1−2, we get, using (26):

a_n(C_τ2)≤K^NNexp

−n^N1−2ε

n^N1−2

.

Using (25), that gives:

an(Cτ₂)≲exp

−n^{N1−2(1−4N1−7)}

= exp

−n^4N4−7 , or:

ah

n^N−74

i(C_τ2)≲e⁻ⁿ≤e^−cn. (28) Now, (27) and (28) allow to use Lemma 2 on the previous page withA= 3/2 and B=N−7/4, and we get:

aMh

n^N−14

i(CΨ)≲e^−cn. Equivalently:

a_k(CΨ)≲exp

−c^′k^4N4−1 and:

a_k(CΨ)1/k^1/N

≲exp

−c^′k^4N4−1−N1

= exp

−c^′k^N(4N1−1) ,

4. Application to the multidimensional case which givesβ_N(C_Ψ) = 0.

To end the proof, it suffices to remark thatCΦ₂=CΨ ◦CΦ₁, sinceΦ₂=Φ₁◦Ψ, and henceβ_N⁺(CΦ₂)≤β⁺_N(CΨ) = 0, soβ_N(CΦ₂) = 0.

4) We use a Shapiro-Taylor map. This one-parameter map ς_θ, θ >0, was introduced by J. Shapiro and P. Taylor in 1973¹⁴ and was further studied, with a slightly different definition, in Lefèvre et al. (2008, Section 5). J. Shapiro and P. Taylor proved thatC_ςθ:H²→H²is always compact, but is Hilbert-Schmidt if and only ifθ >2. Let us recall their definition.

For 0< ε <1, we setV_ε ={z∈C; ℜz >0 and|z|< ε}. Forε=ε_θ >0 small enough, one can define:

f_θ(z) =z(−lnz)^θ,

forz∈Vε, where lnzwill be the principal determination of the logarithm. Let nowgθbe the conformal mapping fromDontoVε, which mapsT=∂Donto∂Vε, defined byg_θ(z) =ε φ₀(z), whereφ₀is given by:

φ0(z) =

z−i iz−1

1/2

−i

−i z−i

iz−1 1/2

+ 1

·

Then, we define:

ς_θ= exp(−f_θ◦g_θ).

We proved in Li, Queffélec, and Rodríguez-Piazza (2013, Section 4.2) (though it is not sharp) that:

a_n(C_ςθ)≳ 1

n^θ/2· (29)

We defineΦ₁:D^N →D^Nas:

Φ₁(z₁, z₂, . . . , z_N) =

ς_θ(z₁),0, . . . ,0

. (30)

IfJ=J_N:H²(D)→H²(D^N) is the canonical injection defined by (Jh)(z₁, . . . , z_N) = h(z1) andQ=Q_N:H²(D^N)→H²(D) is defined by (Qf)(z1) =f(z1,0, . . . ,0), then CΦ₁=JCς_θQ; henceCΦ₁is compact. On the other hand, we also haveQCΦ₁J=Cς_θ, which implies thata_n(CΦ₁)≳a_n(C_ςθ)≳n^−θ/2. It follows that:

β_N(CΦ₁)≥ lim

n→∞(n^−θ/2)^1/n1/N = 1, and henceβN(CΦ₁) = 1.