Some examples of composition operators and their approximation numbers on the Hardy space of the bi-disk

(1)

HAL Id: hal-01536919

https://hal-univ-artois.archives-ouvertes.fr/hal-01536919v2

Preprint submitted on 1 Mar 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Some examples of composition operators and their approximation numbers on the Hardy space of the

bi-disk

Daniel Li, Hervé Queffélec, Luis Rodríguez-Piazza

To cite this version:

Daniel Li, Hervé Queffélec, Luis Rodríguez-Piazza. Some examples of composition operators and their

approximation numbers on the Hardy space of the bi-disk. 2018. �hal-01536919v2�

(2)

Some examples of composition operators and their approximation numbers on the

Hardy space of the bi-disk

Daniel Li, Hervé Queffélec, L. Rodríguez-Piazza

March 1, 2018

Abstract. We give examples of composition operators C

Φ

on H

²

( D

²

) showing that the condition k Φ k

∞

= 1 is not sufficient for their approximation numbers a

n

(C

Φ

) to satisfy lim

n→∞

[a

n

(C

Φ

)]

^1/^√ⁿ

= 1, contrary to the 1-dimensional case.

We also give a situation where this implication holds. We make a link with the Monge-Ampère capacity of the image of Φ.

Key-words: approximation numbers; Bergman space; bidisk; composition operator; Green capacity; Hardy space; Monge-Ampère capacity; weighted composition operator.

MSC 2010 numbers – Primary : 47B33 – Secondary : 30H10 – 30H20 – 31B15 – 32A35 – 32U20 – 41A35 – 46B28

1 Introduction and notation

1.1 Introduction

The purpose of this paper is to continue the study of composition operators on the polydisk initiated in [2], and in particular to examine to what extent one of the main results of [21] still holds.

Let H be a Hilbert space and T : H → H a bounded operator. Recall that the approximation numbers of T are defined as:

a

n

(T ) = inf

rankR<n

k T − R k , n ≥ 1 , and we have:

k T k = a

1

(T ) ≥ a

2

(T ) ≥ · · · ≥ a

n

(T ) ≥ · · ·

The operator T is compact if and only if a

n

(T )

_n

−→

_→∞

0.

(3)

For d ≥ 1, we define:

( β

_d⁻

(T ) = lim inf

n→∞

a

n^d

(T )

^1/n

β

_d⁺

(T ) = lim sup

n→∞

a

_n^d

(T )

1/n

We have:

0 ≤ β

_d⁻

(T ) ≤ β

⁺_d

(T ) ≤ 1 , and we simply write β

d

(T ) in case of equality.

It may well happen in general (consider diagonal operators) that β

⁻_d

(T ) = 0 and β

_d⁺

(T ) = 1.

When H = H

²

( D ) is the Hardy space on the open unit disk D of C , and T = C

Φ

is a composition operator, with Φ : D → D a non-constant analytic function, we always have ([19]):

β

₁⁻

(C

Φ

) > 0 , and one of the main results of [19] is the equivalence:

(1.1) β

⁺₁

(C

Φ

) < 1 ⇐⇒ k Φ k

∞

< 1 .

An alternative proof was given in [21], as a consequence of a so-called “spectral radius formula”, which moreover shows that:

β

₁⁻

(C

Φ

) = β

⁺₁

(C

Φ

) .

In [2], for d ≥ 2, it is proved that, for a bounded symmetric domain Ω ⊆ C

^d

, if Φ : Ω → Ω is analytic, such that Φ(Ω) has a non-void interior, and the composition operator C

Φ

: H

²

(Ω) → H

²

(Ω) is compact, then:

β

_d⁻

(C

Φ

) > 0 . On the other hand, if Ω is a product of balls, then:

k Φ k

∞

< 1 = ⇒ β

⁺_d

(C

Φ

) < 1 .

We do not know whether the converse holds and the purpose of this paper is to study some examples towards an answer.

The paper is organized as follows. Section 1 is this short introduction, as well as some notations and definitions on singular numbers of operators and Hardy spaces of the polydisk to follow. Section 2 contains preliminary results on weighted composition operators in one variable, which surprisingly play an important role in the study of non-weighted composition operators in two variables. Section 3 studies the case of symbols with “separated” variables. Our main one variable result extends in this case. Section 4 studies the “glued case”

Φ(z

1

, z

2

) = φ(z

1

), φ(z

1

)

for which even boundedness is an issue. Here, the

(4)

Bergman space B

²

( D ) enters the picture. Section 5 studies the case of “triangularly separated” variables. This section lets direct Hilbertian sums of weighted composition operators in one variable appear, and it contains our main result:

an example of a symbol Φ satisfying k Φ k

∞

= 1 and yet β

₂⁺

(C

Φ

) < 1. The final Section 6 discusses the role of the Monge-Ampère pluricapacity, which is a multivariate extension of the Green capacity in the disk. Even though, as evidenced by our counterexample of Section 5, this capacity will not capture all the behavior of the parameter β

m

(C

Φ

), some partial results are obtained, relying on theorems of S. Nivoche and V. Zakharyuta.

1.2 Notation

We denote by D the open unit disk of the complex plane and by T its boundary, the 1-dimensional torus.

The Hardy space H

²

( D

^d

) is the space of holomorphic functions f : D

^d

→ C whose boundary values f

^∗

on T

^d

are square-integrable with respect to the Haar measure m

d

of T

^d

, and normed with:

k f k

²2

= k f k

²H²(D^d)

= Z

T^d

| f

^∗

(ξ

1

, . . . , ξ

d

) |

²

dm

d

(ξ

1

, . . . , ξ

d

) . If f (z

1

, . . . , z

d

) = P

α1,...,αd≥0

a

α1,...,αd

z

₁^α¹

· · · z

^α_d^d

, then:

k f k

²2

= X

α1,...,αd≥0

| a

α1,...,αd

|

²

.

We say that an analytic map Φ : D

^d

→ D

^d

is a symbol if its associated composition operator C

Φ

: H

²

( D

^d

) → H

²

( D

^d

), defined by C

Φ

(f ) = f ◦ Φ, is bounded.

We say that Φ is truly d-dimensional if Φ( D

^d

) has a non-void interior.

We will make use of two kinds of symbols defined on D . The lens map λ

θ

: D → D is defined, for θ ∈ (0, 1), by:

(1.2) λ

θ

(z) = (1 + z)

^θ

− (1 − z)

^θ

(1 + z)

^θ

+ (1 − z)

^θ

(see [26], p. 27, or [16], for more information), and corresponds to u 7→ u

^θ

in the right half-plane.

The cusp map χ : D → D was first defined in [15] and in a slightly different form in [20]; we actually use here the modified form introduced in [17], and then used in [18]. We first define:

χ

0

(z) =

z − i iz − 1

1/2

− i

− i z − i iz − 1

1/2

+ 1

;

(5)

we note that χ

0

(1) = 0, χ

0

( − 1) = 1, χ

0

(i) = − i, χ

0

( − i) = i, and χ

0

(0) = √ 2 − 1.

Then we set:

χ

1

(z) = log χ

0

(z), χ

2

(z) = − 2

π χ

1

(z) + 1, χ

3

(z) = a χ

2

(z) , and finally:

χ(z) = 1 − χ

3

(z) , where:

(1.3) a = 1 − 2

π log( √

2 − 1) ∈ (1, 2)

is chosen in order that χ(0) = 0. The image Ω of the (univalent) cusp map is formed by the intersection of the inside of the disk D 1 −

^a2

,

^a₂

and the outside of the two disks D 1 +

^ia₂

,

^a₂

and D 1 −

^ia2

,

^a₂

· Besides the approximation numbers, we need other singular numbers for an operator S : X → Y between Banach spaces X and Y .

The Bernstein numbers b

n

(S), n ≥ 1, which are defined by:

(1.4) b

n

(S) = sup

E

x

min

∈SE

k Sx k ,

where the supremum is taken over all n-dimensional subspaces of X and S

E

is the unit sphere of E.

The Gelfand numbers c

n

(S), n ≥ 1, which are defined by:

(1.5) c

n

(S) = inf {k S

_|M

k ; codim M < n } . The Kolmogorov numbers d

n

(S), n ≥ 1, which are defined by:

(1.6) d

n

(S) = inf

dimE<n

sup

x∈BX

dist (Sx, E)

.

Pietsch showed that all s-numbers on Hilbert spaces are equal (see [24], § 2, Corollary, or [25], Theorem 11.3.4); hence:

(1.7) a

n

(S) = b

n

(S) = c

n

(S ) = d

n

(S) .

We denote m the normalized Lebesgue measure on T = ∂ D . If ϕ: D → D , m

ϕ

is the pull-back measure on D defined by m

ϕ

(E) = m[ϕ

^∗−¹

(E)], where ϕ

^∗

stands for the non-tangential boundary values of ϕ.

The notation A . B means that A ≤ C B for some positive constant C and

we write A ≈ B if we have both A . B and B . A.

(6)

2 Preliminary results on weighted composition operators on H ² ( D )

We see in this section that the presence of a “rapidly decaying” weight allows simpler estimates for the approximation numbers of a corresponding weighted composition operator. Such a study, but a bit different, is made in [14].

Let ϕ: D → D a non-constant analytic self-map in the disk algebra A( D ) such that, for some constant C > 1 and for all z ∈ D :

(2.1) ϕ(1) = 1 , | 1 − ϕ(z) | ≤ 1 , | 1 − ϕ(z) | ≤ C (1 − | ϕ(z) | )

as well as ϕ(z) 6 = 1 for z 6 = 1. We can take for example ϕ =

^1+λ₂^θ

where λ

θ

is the lens map with parameter θ.

Let w ∈ H

^∞

and let T be the weighted composition operator T = M

w◦ϕ

C

ϕ

: H

²

→ H

²

.

Note that M

w◦ϕ

C

ϕ

= C

ϕ

M

w

. We first show that:

Theorem 2.1. Let T = M

w◦ϕ

C

ϕ

: H

²

→ H

²

be as above and let B be a Blaschke product with length < N . Then, with the implied constant depending only on the number C in (2.1) (and of ϕ):

a

N

(T ) . sup

|z−1|≤1, z∈ϕ(D)

| B(z) | | w(z) | .

Proof. The following preliminary observation (see also [16], p. 809), in which we denote by S(ξ, h) = { z ∈ D ; | z − ξ | ≤ h } the Carleson window with center ξ ∈ T and size h, and by K

ϕ

the support of the pull-back measure m

ϕ

, will be useful.

(2.2) u ∈ S(ξ, h) ∩ K

ϕ

= ⇒ u ∈ S(1, Ch) ∩ K

ϕ

. Indeed, if | u − ξ | ≤ h and u ∈ K

ϕ

, (2.1) implies:

1 − | u | ≤ | u − ξ | ≤ h and | u − 1 | ≤ C(1 − | u | ) ≤ Ch .

Set E = BH

²

. This is a subspace of codimension < N. If f = Bg ∈ E, with k g k = k f k (isometric division by B in BH

²

), we have T f = (wBg) ◦ ϕ, whence:

k T (f ) k

²

= Z

D

| B |

²

| w |

²

| g |

²

dm

ϕ

,

implying k T (f ) k

²

≤ k f k

²

k J k

²

where J : H

²

→ L

²

(σ) is the natural embedding and where

σ = | B |

²

| w |

²

dm

ϕ

.

(7)

Now, Carleson’s embedding theorem for the measure σ and (2.2) show that (the implied constants being absolute):

k J k

²

. sup

ξ∈T,0<h<1

1 h

Z

S(ξ,h)∩Kϕ

| B |

²

| w |

²

dm

ϕ

. sup

0<h<1

1 h

Z

S(1,Ch)∩Kϕ

| B |

²

| w |

²

dm

ϕ

.

sup

|z−1|≤1, z∈ϕ(D)

| B(z) |

²

| w(z) |

²

sup

0<h<1

1 h

Z

S(1,Ch)∩Kϕ

dm

ϕ

. sup

|z−1|≤1, z∈ϕ(D)

| B(z) |

²

| w(z) |

²

,

since m

ϕ

is a Carleson measure for H

²

and where we used that, according to (2.1):

K

ϕ

⊆ ϕ( D ) ⊆ { z ∈ D ; | z − 1 | ≤ 1 } .

This ends the proof of Theorem 2.1 with help of the equality of a

N

(T ) with the Gelfand number c

N

(T ) recalled in (1.7).

In order to specialize efficiently the general Theorem 2.1, we recall the following simple Lemma 2.3 of [16], where:

(2.3) ρ(a, b) =

a − b 1 − ¯ ab

, a, b ∈ D ,

is the pseudo-hyperbolic distance:

Lemma 2.2 ([16]). Let a, b ∈ D such that | a − b | ≤ L min(1 − | a | , 1 − | b | ). Then:

ρ(a, b) ≤ L

√ L

²

+ 1 =: κ < 1 . We can now state:

Theorem 2.3. Assume that ϕ is as in (2.1) and that the weight w satisfies, for some parameters 0 < θ ≤ 1 and R > 0:

| w(z) | ≤ exp

− R

| 1 − z |

^θ

, ∀ z ∈ D with R e z ≥ 0 .

Then, the approximation numbers of T = M

w◦ϕ

C

ϕ

satisfy:

a

nm+1

(T ) . max

exp( − an), exp( − R 2

^mθ

) , for all integers n, m ≥ 1, where a = log[ √

16C

²

+ 1/(4C)] > 0 and C is as in

(2.1).

(8)

Proof. Let p

l

= 1 − 2

⁻^l

, 0 ≤ l < m and let B be the Blaschke product:

B(z) = Y

0≤l<m

z − p

l

1 − p

l

z

n

.

Let z ∈ K

ϕ

∩ D so that 0 < | z − 1 | ≤ 1. Let l be the non-negative integer such that 2

⁻^l⁻¹

< | z − 1 | ≤ 2

⁻^l

. We separate two cases:

Case 1: l ≥ m. Then, the weight does the job. Indeed, majorizing | B(z) | by 1 and using the assumption on w, we get:

| B(z) |

²

| w(z) |

²

≤ | w(z) |

²

≤ exp

− 2R

| 1 − z |

^θ

≤ exp( − 2R 2

^lθ

) ≤ exp( − 2R 2

^mθ

) .

Case 2: l < m. Then, the Blaschke product does the job. Indeed, majorize | w(z) | by 1 and estimate | B(z)) | more accurately with help of Lemma 2.2; we observe that

| z − p

l

| ≤ | z − 1 | + 1 − p

l

≤ 2 × 2

⁻^l

= 2(1 − p

l

) ≤ 4C(1 − p

l

) and then, since z ∈ K

ϕ

, we can write with C ≥ 1 as in (2.1):

1 − | z | ≥ 1

C | 1 − z | ≥ 1

2C 2

⁻^l

≥ 1

4C | z − p

l

| ,

so that the assumptions of Lemma 2.2 are verified with L = 4C, giving:

ρ(z, p

l

) ≤ 4C

√ 16C

²

+ 1 = exp( − a) < 1 .

Hence, by definition, since l < m:

| B(z) | ≤ [ρ(z, p

l

)]

ⁿ

≤ exp( − an) .

Putting both cases together, and observing that our Blaschke product has length nm < nm + 1, we get the result by applying Theorem 2.1 with N = nm + 1.

2.1 Some remarks

1. Twisting a composition operator by a weight may improve the compact- ness of this composition operator, or even may make this weighted composition operator compact though the non-weighted was not (see [8] or [14]). However, this is not possible for all symbols, as seen in the following proposition.

Proposition 2.4. Let w ∈ H

^∞

. If ϕ is inner, or more generally if | ϕ | = 1 on

a subset of T of positive measure, then M

w

C

ϕ

is never compact (unless w ≡ 0).

(9)

Proof. Indeed, suppose T = M

w

C

ϕ

compact. Since (z

ⁿ

)

n

converges weakly to 0 in H

²

and since T (z

ⁿ

) = w ϕ

ⁿ

, we should have, since | ϕ | = 1 on E, with m(E) > 0:

Z

E

| w |

²

dm = Z

E

| w |

²

| ϕ |

²ⁿ

dm ≤ Z

T

| w |

²

| ϕ |

²ⁿ

dm = k T (z

ⁿ

) k

²_n

−→

→∞

0 , but this would imply that w is null a.e. on E and hence w ≡ 0 (see [7], Theorem 2.2), which was excluded.

Note that É. Amar and A. Lederer proved in [1] that | ϕ | = 1 on a set of positive measure if and only if ϕ is an exposed point of of the unit ball of H

^∞

; hence the following proposition can be viewed as the (almost) opposite case.

Proposition 2.5. Let ϕ: D → D such that k ϕ k

∞

= 1. Assume that:

Z

T

log(1 − | ϕ | ) dm > −∞

(meaning that ϕ is not an extreme point of the unit ball of H

^∞

: see [7], Theo- rem 7.9). Then, if w is an outer function such that | w | = 1 − | ϕ | , the weighted composition operator T = M

w

C

ϕ

is Hilbert-Schmidt.

Proof. We have:

∞

X

n=0

k T (z

ⁿ

) k

²

=

∞

X

n=0

Z

T

(1 − | ϕ | )

²

| ϕ |

²ⁿ

dm = Z

T

1 − | ϕ |

1 + | ϕ | dm < + ∞ ,

and T is Hilbert-Schmidt, as claimed.

2. In [14], Theorem 2.5, it is proved that we always have, for some constants δ, ρ > 0:

(2.4) a

n

(M

w

C

ϕ

) ≥ δ ρ

ⁿ

, n = 1, 2, . . .

(if w 6≡ 0). We give here an alternative proof, based on a result of Gunatillake ([9]), this result holding in a wider context.

Theorem 2.6 (Gunatillake). Let T = M

w

C

ϕ

be a compact weighted composition operator on H

²

and assume that ϕ has a fixed point a ∈ D . Then the spectrum of T is the set:

σ(T ) = { 0, w(a), w(a) ϕ

^′

(a), w(a) [ϕ

^′

(a)]

²

, . . . , w(a) [ϕ

^′

(a)]

ⁿ

, . . . }

Proof of (2.4). First observe that, in view of Proposition 2.4, ϕ cannot be an

automorphism of D so that the point a is the Denjoy-Wolff point of ϕ and is

attractive. Theorem 2.6 is interesting only when w(a) ϕ

^′

(a) 6 = 0.

(10)

Now, we can give a new proof Theorem 2.5 of [14] as follows. Let a ∈ D be such that w(a) ϕ

^′

(a) 6 = 0 (H( D ) is a division ring and ϕ

^′

6≡ 0, w 6≡ 0). Let b = ϕ(a) and τ ∈ Aut D with τ(b) = a. We set:

ψ = τ ◦ ϕ and S = M

w

C

ψ

= T C

τ

. This operator S is compact because T is.

Since ψ(a) = a and ψ

^′

(a) = τ

^′

(b)ϕ

^′

(a) 6 = 0, Theorem 2.6 says that the non-zero eigenvalues of S , arranged in non-increasing order, are the numbers λ

n

= w(a) [ψ

^′

(a)]

ⁿ⁻¹

, n ≥ 1. As a consequence of Weyl’s inequalities, we know that:

a

1

(S) a

n

(S) ≥ | λ

2n

|

²

≥ δ ρ

ⁿ

, with:

δ = | w(a) |

²

> 0 and ρ = | ψ

^′

(a) |

⁴

> 0 .

To finish, it is enough to observe that a

n

(S) ≤ a

n

(T ) k C

τ

k by the ideal property of approximation numbers.

3 The splitted case

Theorem 3.1. Let Φ = (φ, ψ) : D

^d

→ D

^d

be a truly d-dimensional symbol with φ: D → D depending only on z

1

and ψ : D

^d⁻¹

→ D

^d⁻¹

only on z

2

, . . . , z

d

, i.e.

Φ(z

1

, z

2

, . . . , z

d

) = φ(z

1

), ψ(z

2

, . . . , z

d

)

. Then, whatever ψ behaves:

k φ k

∞

= 1 = ⇒ β

d

(C

Φ

) = 1 .

Proof. The proof is based on the following simple lemma, certainly well-known.

Lemma 3.2. Let S : H

1

→ H

1

and T : H

2

→ H

2

be two compact linear operators, where H

1

and H

2

are Hilbert spaces. Let S ⊗ T be their tensor product, acting on the tensor product H

1

⊗ H

2

. Then:

a

mn

(S ⊗ T ) ≥ a

m

(S) a

n

(T ) for all positive integers m, n.

We postpone the proof of the lemma and show how to conclude.

We can assume C

Φ

to be compact, so that C

φ

is compact as well. Since k φ k

∞

= 1, we have, thanks to (1.1) :

a

m

(C

φ

) ≥ e

⁻^{m ε}^m

with ε

m_m

−→

→∞

0 .

Replacing ε

m

by δ

m

:= sup

_p_≥_m

ε

p

, we can assume that (ε

m

)

m

is non-increasing.

Moreover,

m ε

m

→ ∞

(11)

since C

φ

is compact and hence a

m

(C

φ

)

_m

−→

_→∞

0. We next observe that, due to the separation of variables in the definition of φ and ψ, we can write:

(3.1) C

Φ

= C

φ

⊗ C

ψ

.

Indeed, write z = (z

1

, w) with z

1

∈ D and w ∈ D

^d⁻¹

. If f ∈ H

²

( D ) and g ∈ H

²

( D

^d⁻¹

), we see that:

C

Φ

(f ⊗ g)(z) = (f ⊗ g) φ(z

1

), ψ(w)

= f φ(z

1

)

g ψ(w)

= [C

φ

f (z

1

)] [C

ψ

g(w)] = (C

φ

f ⊗ C

ψ

g)(z) .

Since tensor products f ⊗ g generate H

²

( D

^d

) = H

²

( D ) ⊗ H

²

(D

^d⁻¹

), this proves (3.1).

Let now m be a large positive integer. Set ([ . ] stands for the integer part):

(3.2) n

m

= [mε

m

]

^d⁻¹

and N

m

= m n

m

.

From what we know in dimension d − 1 (see [2], Theorem 3.1) and from the preceding, we can write (observe that ψ has to be truly (d − 1)-dimensional since Φ is truly d-dimensional):

a

m

(C

φ

) ≥ exp( − m ε

m

) and a

n

(C

ψ

) ≥ a exp( − C n

^1/(d⁻¹⁾

) ,

for some positive constant C, which will be allowed to vary from one formula to another. Lemma 3.2 implies:

a

Nm

(C

Φ

) ≥ a exp[ − C (m ε

m

+ n

m1/(d−1)

)] . Since n

m

. (mε

m

)

^d⁻¹

, we get:

a

Nm

(C

Φ

) ≥ a exp( − C m ε

m

) . Observe that N

m

= m n

m

∼ m

^d

ε

md−1

and so N

m1/d

∼ m ε

m1−1/d

. As a consequence:

a

Nm

(C

Φ

) ≥ a exp( − C m ε

m

) = a exp

− (C ε

m1/d

) (m ε

m1−1/d

)

≥ a exp( − η

m

N

m1/d

) with η

m

:= C ε

m1/d

.

Now, for N > N

1

, let m be the smallest integer satisfying N

m

≥ N (so that N

m−1

< N ≤ N

m

), and set δ

N

= η

m

. We have lim

N→∞

δ

N

= 0. Next, we note that lim

m→∞

N

m

/N

m−1

= 1, because N

m

≥ N

m−1

and:

N

m

N

m−1

≤ m m − 1

m ε

m

+ 1 (m − 1) ε

m−1

d−1

∼ ε

m

ε

m−1

d−1

≤ 1 . Finally, if N is an arbitrary integer and N

m−1

< N ≤ N

m

, we obtain:

a

N

(C

Φ

) ≥ a

Nm

(C

Φ

) ≥ a exp( − η

m

N

m1/d

) ≥ a exp( − C δ

N

^1/d

) , since we observed that lim

m→∞

N

m

/N

m−1

= 1.

This amounts to say that β

d

(C

Φ

) = 1.

(12)

Proof of Lemma 3.2. It is rather formal. Start from the Schmidt decomposi- tions of S and T respectively (recall that Hilbert spaces, the approximation numbers are equal to the singular ones):

S =

∞

X

m=1

a

m

(S) u

m

⊙ v

m

, T =

∞

X

n=1

a

n

(T ) u

^′_n

⊙ v

_n^′

,

where (u

m

), (v

m

) are two orthonormal sequences of H

1

, (u

^′_n

), (v

_n^′

) two orthonormal sequences of H

2

, and u

m

⊙ v

m

and u

^′_n

⊙ v

^′_n

denote the rank one operators defined by (u

m

⊙ v

m

)(x) = h x, v

m

i u

m

, x ∈ H

1

, and (u

^′_n

⊙ v

_n^′

)(x) = h x, v

^′_n

i u

^′_n

, x ∈ H

2

.

We clearly have:

(u

m

⊙ v

m

) ⊗ (u

^′_n

⊙ v

_n^′

) = (u

m

⊗ u

^′_n

) ⊙ (v

m

⊗ v

^′_n

) ,

so that the Schmidt decomposition of S ⊗ T is (with SOT-convergence):

S ⊗ T = X

m,n≥1

a

m

(S) a

n

(T ) (u

m

⊗ u

^′_n

) ⊙ (v

m

⊗ v

^′_n

) ,

since the two sequences (u

m

⊗ u

^′_n

)

m,n

and (v

m

⊗ v

^′_n

)

m,n

are orthonormal: for instance, we have by definition:

h u

m1

⊗ u

^′_n₁

, u

m2

⊗ u

^′_n₂

i = h u

m1

, u

m2

, i h u

^′_n₁

, u

^′_n₂

i .

This shows that the singular values of S ⊗ T are the non-increasing rear- rangement of the positive numbers a

m

(S) a

n

(T ) and ends the proof of the lemma: the mn numbers a

k

(S) a

l

(T ), for 1 ≤ k ≤ m, 1 ≤ l ≤ n all satisfy a

k

(S) a

l

(T ) ≥ a

m

(S) a

n

(T ), so that a

mn

(S ⊗ T ) ≥ a

m

(S) a

n

(T ).

4 The glued case

Here we consider symbols of the form:

(4.1) Φ(z

1

, z

2

) = φ(z

1

), φ(z

1

) , where φ: D → D is a non-constant analytic map.

Note that such maps Φ are not truly 2-dimensional.

4.1 Preliminary

We begin by remarking the following fact.

Let B

²

( D ) be the Bergman space of all analytic functions f : D → C such that:

k f k

²B²

:=

Z

D

| f (z) |

²

dA(z) < ∞ ,

where dA is the normalized area measure on D .

(13)

Proposition 4.1. Assume that the composition operator C

φ

maps boundedly B

²

( D ) into H

²

( D ). Then C

Φ

: H

²

( D

²

) → H

²

( D

²

), defined by (4.1), is bounded.

Proof. If we write f ∈ H

²

( D

²

) as:

f (z

1

, z

2

) = X

j,k≥0

c

j,k

z

^j₁

z

^k₂

, with X

j,k≥0

| c

j,k

|

²

= k f k

²H²

,

we formally (or assuming that f is a polynomial) have:

[C

Φ

f ](z

1

, z

2

) = X

j,k≥0

c

j,k

[φ(z

1

)]

^j

[φ(z

1

)]

^k

=

∞

X

n=0

X

j+k=n

c

j,k

[φ(z

1

)]

ⁿ

. Hence, if we set g(z) = P

_∞

n=0

P

j+k=n

c

j,k

z

ⁿ

, we get:

[C

Φ

(f )](z

1

, z

2

) = [C

φ

(g)](z

1

) , so that, by integrating:

k C

Φ

(f ) k

^H²⁽D²)

= k C

φ

(g) k

^H²⁽D)

. By hypothesis, there is a positive constant M such that:

k C

φ

(g) k

^H²⁽D)

≤ M k g k

^B²⁽D)

. But, by the Cauchy-Schwarz inequality:

k g k

²B²(D)

=

∞

X

n=0

1 n + 1

X

j+k=n

c

j,k

2

≤

∞

X

n=0

X

j+k=n

| c

j,k

|

²

= X

j,k≥0

| c

j,k

|

²

= k f k

²H²(D²)

,

and we obtain k C

Φ

(f ) k

H²(D²)

≤ M k f k

H²(D²)

.

4.2 Lens maps

Let λ

θ

be a lens map of parameter θ, 0 < θ < 1. We consider Φ

θ

: D

²

→ D

²

defined by:

(4.2) Φ

θ

(z

1

, z

2

) = λ

θ

(z

1

), λ

θ

(z

1

) . We have the following result.

Theorem 4.2. The composition operator C

Φθ

: H

²

( D

²

) → H

²

( D

²

) is:

1) not bounded for θ > 1/2;

2) bounded, but not compact for θ = 1/2;

3) compact, and even Hilbert-Schmidt, for 0 < θ < 1/2.

(14)

Proof. The reproducing kernel of H

²

( D

²

) is, for (a, b) ∈ D

²

: (4.3) K

a,b

(z

1

, z

2

) = 1

1 − ¯ az

1

1 1 − ¯ bz

2

, (z

1

, z

2

) ∈ D

²

, and:

k K

a,b

k

²

= 1

(1 − | a |

²

)(1 − | b |

²

) ·

1) If C

Φθ

were bounded, we should have, for some M < ∞ : k C

_Φ^∗_θ

(K

a,b

) k

^H²

≤ M k K

a,b

k

^H²

, for all a, b ∈ D . Since C

_Φ^∗_θ

(K

a,b

) = K

Φθ(a,b)

= K

λθ(a),λθ(a)

, we would get, with b = 0:

1 1 − | λ

θ

(a) |

²

2

≤ M

²

1 1 − | a |

²

;

but this is not possible for θ > 1/2, since 1 − | λ

θ

(a) |

²

≈ 1 − | λ

θ

(a) | ∼ (1 − a)

^θ

when a goes to 1, with 0 < a < 1.

For 2) and 3), let us consider the pull-back measure m

θ

of the normalized Lebesgue measure on T = ∂ D by λ

θ

. It is easy to see that:

(4.4) sup

ξ∈T

m

θ

[D(ξ, h) ∩ D )] = m

θ

[D(1, h) ∩ D ] ≈ h

^1/θ

.

In particular, for θ ≤ 1/2, m

θ

is a 2-Carleson measure, and hence (see [15], The- orem 2.1, for example) the canonical injection j : B

²

( D ) → L

²

(m

θ

) is bounded, meaning that, for some positive constant M < ∞ :

Z

D

| f (z) |

²

dm

θ

(z) ≤ M

²

k f k

²B²

.

Since Z

D

| f (z) |

²

dm

θ

(z) = Z

T

| f [λ

θ

(u)] |

²

dm(u) = k C

λθ

(f ) k

²H²

, we get that C

λθ

maps boundedly B

²

( D ) into H

²

( D ).

It follows from Proposition 4.1 that C

Φθ

: H

²

( D

²

) → H

²

( D

²

) is bounded.

However, C

Φ1/2

is not compact since C

_Φ^∗_1/2

(K

a,b

)/ k K

a,b

k does not converge to 0 as a, b → 1, by the calculations made in 1).

For 3), let e

j,k

(z

1

, z

2

) = z

₁^j

z

₂^k

, j, k ≥ 0, be the canonical orthonormal basis of H

²

( D

²

); we have [C

φθ

(e

j,k

)](z

1

, z

2

) = [λ

θ

(z

1

)]

^j+k

. Hence:

X

j,k≥0

k C

φθ

(e

j,k

) k

²H²(D²)

≤

∞

X

n=0

(2n + 1) Z

T

| λ

θ

|

²ⁿ

dm ≤ Z

T

2 (1 − | λ

θ

|

²

)

²

dm .

Since, by Lemma 4.3 below, 1 − | λ

θ

(e

^it

) |

²

& | 1 − e

^it

|

^θ

≥ t

^θ

for | t | ≤ π/2, we get:

X

j,k≥0

k C

φθ

(e

j,k

) k

²H²(D²)

. Z

π/2

0

dt

t

^2θ

< ∞ ,

since θ < 1/2. Therefore C

φθ

is Hilbert-Schmidt for θ < 1/2.

(15)

For sake of completeness, we recall the following elementary fact (see [26], p. 28, or also [16], Lemma 2.5)).

Lemma 4.3. With δ = cos(θπ/2), we have, for | z | ≤ 1 and R e z ≥ 0:

1 − | λ

θ

(z) |

²

≥ δ

2 | 1 − z |

^θ

. Proof. We can write:

λ

θ

(z) = 1 − w

1 + w with w = 1 − z

1 + z

^θ

and | w | ≤ 1 . Then:

R e w ≥ δ | w | ≥ δ

2 | 1 − z |

^θ

. Hence:

1 − | λ

θ

(z) |

²

= 4 R e w

| 1 + w |

²

≥ δ | w | ≥ δ

2 | 1 − z |

^θ

, as announced

We now improve the result 3) of Theorem 4.2 by estimating the approximation numbers of C

Φθ

and get that C

Φθ

is in all Schatten classes of H

²

( D

²

) when θ < 1/2.

Theorem 4.4. For 0 < θ < 1/2, there exists b = b

θ

> 0 such that:

(4.5) a

n

(C

Φθ

) . e

⁻^b^√ⁿ

.

In particular β

₂⁺

(C

Φθ

) ≤ e

⁻^b

< 1, though k Φ

θ

k

∞

= 1, and even Φ

θ

( T

²

) ∩ T

²

6 = ∅ . Proof. Proposition 4.1 (and its proof) can be rephrased in the following way: if C

φ

maps boundedly B

²

( D ) into H

²

( D ), then, we have the following factorization:

(4.6) C

Φ

: H

²

( D

²

) −→

^J

B

²

( D ) −−→

^C^φ

H

²

( D ) −→

^I

H

²

( D

²

) ,

where I : H

²

( D ) → H

²

( D

²

) is the canonical injection given by (If )(z

1

, z

2

) = f (z

1

) for f ∈ H

²

( D ), and J : H

²

( D

²

) → B

²

( D ) is the contractive map defined by:

(Jf )(z) =

∞

X

n=0

X

j+k=n

c

j,k

z

ⁿ

, for f ∈ H

²

( D

²

) with f (z

1

, z

2

) = P

j,k≥0

c

j,k

z

₁^j

z

₂^k

.

In the proof of Theorem 4.2, we have seen that, for 0 < θ ≤ 1/2, the composition operator C

λθ

is bounded from B

²

( D ) into H

²

( D ); we get hence the factorization:

C

Φθ

: H

²

( D

²

) −→

^J

B

²

( D ) −−−→

^C^λθ

H

²

( D ) −→

^I

H

²

( D

²

) ,

(16)

Now, the lens maps have a semi-group property:

(4.7) λ

θ1θ2

= λ

θ1

λ

θ2

, giving C

λθ1θ2

= C

λθ1

◦ C

λθ2

.

For 0 < θ < 1/2, we therefore can write C

λθ

= C

λ2θ

◦ C

λ1/2

(note that 2θ < 1, so C

λ2θ

: H

²

( D ) → H

²

( D ) is bounded), and we get:

C

Φθ

= I C

λ2θ

C

λ1/2

J . Consequently:

a

n

(C

Φθ

) ≤ k I k k J k k C

λ1/2

k

B²→H²

a

n

(C

λ2θ

) .

Now, we know ([16], Theorem 2.1) that a

n

(C

λ2θ

) . e

⁻^b^√ⁿ

, so we get that a

n

(C

Φθ

) . e

⁻^b^√ⁿ

.

Remark. In [2], we saw that for a truly 2-dimensional symbol Φ, we have β

₂⁻

(C

φ

) > 0. Here the symbol Φ

θ

is not truly 2-dimensional, but we nevertheless have β

2

(C

Φθ

) > 0. In fact, let E = { f ∈ H

²

( D

²

) ;

_∂z^∂f₂

≡ 0 } ; E is isometrically isomorphic to H

²

( D ) and the restriction of C

Φθ

to E behaves as the 1-dimensional composition operator C

λθ

: H

²

( D ) → H

²

( D ); hence ([19], Proposition 6.3):

e

⁻^b⁰^√ⁿ

. a

n

(C

λθ

) = a

n

(C

Φθ|E

) ≤ a

n

(C

Φθ

) ,

and β

₂⁻

(C

Φθ

) ≥ e

⁻^b⁰

> 0.

5 Triangularly separated variables

In this section, we consider symbols of the form:

(5.1) Φ(z

1

, z

2

) = φ(z

1

), ψ(z

1

) z

2

, where φ, ψ : D → D are non-constant analytic maps.

Such maps Φ are truly 2-dimensional.

More generally, if h ∈ H

^∞

, with h(0) = 0 and k h k

∞

≤ 1, has its powers h

^k

, k ≥ 0, orthogonal in H

²

(for convenience, we shall say that h is a Rudin function), we can consider:

(5.2) Φ(z

1

, z

2

) = φ(z

1

), ψ(z

1

) h(z

2

)

For such h we can take for example an inner function vanishing at the origin,

but there are other such functions, as shown by C. Bishop:

(17)

Theorem (Bishop [4]). The function h is a Rudin function if and only if the pull-back measure µ = µ

h

is radial and Jensen, i.e for every Borel set E:

µ(e

^iθ

E) = µ(E) and Z

D

log(1/ | z | ) dµ(z) < ∞ .

Conversely, for every probability measure µ supported by D , which is radial and Jensen, there exists h in the unit ball of H

^∞

, with h(0) = 0, such that µ = µ

h

.

If we take for µ the Lebesgue measure of T , we get an inner function. But, as remarked in [4], we can take for µ the Lebesgue measure on the union T ∪ (1/2) T , normalized in order that µ(T ) = µ (1/2) T

= 1/2. Then the corresponding h is not inner since | h | = 1/2 on a subset of T of positive measure. He also showed that h(z)/z may be a non-constant outer function. Also, P. Bourdon ([6]) showed that the powers of h are orthogonal if and only if its Nevanlinna counting function is almost everywhere constant on each circle centered on the origin.

5.1 General facts

We first observe that if f ∈ H

²

( D

²

) and:

f (z

1

, z

2

) = X

j,k≥0

c

j,k

z

^j₁

z

^k₂

,

then we can write:

f (z

1

, z

2

) = X

k≥0

f

k

(z

1

)

z

₂^k

with:

f

k

(z

1

) = X

j≥0

c

j,k

z

^j₁

, and:

k f k

²H²(D²)

= X

j,k≥0

| c

j,k

|

²

= X

k≥0

k f

k

²H²(D)

.

That means that we have an isometric isomorphism:

J : H

²

( D

²

) −→

∞

M

k=0

H

²

( D ) , defined by Jf = (f

k

)

k≥0

.

Now, for symbols Φ as in (5.1), we have:

(C

Φ

f )(z

1

, z

2

) = X

j,k≥0

c

j,k

[φ(z

1

)]

^j

[ψ(z

1

)]

^k

z

₂^k

,

(18)

so that J C

Φ

J

⁻¹

appears as the operator L

k

M

ψ^k

C

φ

on L

k

H

²

( D ), where M

ψ^k

is the multiplication operator by ψ

^k

:

[(M

ψ^k

C

φ

)f

k

](z

1

) = [ψ(z

1

)]

^k

[(f

k

◦ φ)(z

1

)] . When Φ is as in (5.2), we have:

(C

Φ

f )(z

1

, z

2

) = X

j,k≥0

c

j,k

[φ(z

1

)]

^j

[ψ(z

1

)]

^k

[h(z

2

)]

^k

,

with:

k C

Φ

f k

²

≤

∞

X

k=0

k T

k

f

k

²

and:

T

k

= M

ψ^k

C

φ

;

hence J C

Φ

J

⁻¹

appears as pointwise dominated by the operator T = ⊕

^k

T

k

on L

k

H

²

( D ). This implies a factorization C

Φ

= AT with k A k ≤ 1, so that a

n

(C

Φ

) ≤ a

n

(T ) for all n ≥ 1.

We recall the following elementary fact.

Lemma 5.1. Let (H

k

)

k≥0

be a sequence of Hilbert spaces and T

k

: H

k

→ H

k

be bounded operators. Let H = L

∞

k=0

H

k

and T : H → H defined by T x = (T

k

x

k

)

k

. Then:

1) T is bounded on H if and only if sup

_k

k T

k

k < ∞ ;

2) T is compact on H if and only if each T

k

is compact and k T

k

k −→

_k

→∞

0. Going back to the symbols of the form (5.1), we have k M

_ψ^k

k ≤ k ψ

^k

k

∞

≤ 1, since k ψ k

∞

≤ 1; hence k M

_ψ^k

C

φ

k ≤ k C

φ

k and the operator (M

_ψ^k

C

φ

)

k

is bounded on L

k

H

²

( D ). Therefore C

Φ

is bounded on H

²

( D

²

).

For approximation numbers, we have the following two facts.

Lemma 5.2. Let T

k

: H

k

→ H

k

be bounded linear operators between Hilbert spaces H

k

, k ≥ 0. Let H = L

k

H

k

and T = (T

k

)

k

: H → H , assumed to be compact. Then, for every n

1

, . . . , n

K

≥ 1, and 0 ≤ m

1

< · · · < m

K

, K ≥ 1, we have:

(5.3) a

N

(T ) ≥ inf

1≤k≤K

a

nk

(T

mk

) , where N = n

1

+ · · · + n

K

.

Proof. We use the Bernstein numbers b

n

(see (1.4)), which are equal to the approximation numbers (see (1.7)).

For k = 1, . . . , K, there is an n

k

-dimensional subspace E

k

of H

mk

such that:

b

nk

(T

mk

) ≤ k T

mk

x k , for all x ∈ S

Ek

.

(19)

Then E = L

K

k=1

E

k

is an N-dimensional subspace of H and for every x = (x

1

, x

2

, . . .) ∈ E, we have:

k T x k

²

= X

k≤K

k T

mk

x

mk

k

²

≥ X

k≤K

[b

nk

(T

mk

)]

²

k x

mk

k

²

≥ inf

k≤K

[b

nk

(T

mk

)]

²

X

k≤K

k x

mk

k

²

= inf

k≤K

[b

nk

(T

mk

)]

²

k x k

²

; hence b

N

(T ) ≥ inf

k≤K

b

nk

(T

mk

), and we get the announced result.

Lemma 5.3. Let T = L

_∞

k=0

T

k

acting on a Hilbertian sum H = L

_∞

k=0

H

k

. Let n

0

, . . . , n

K

be positive integers and N = n

0

+ · · · + n

K

− K. Then, the approximation numbers of T satisfy:

(5.4) a

N

(T ) ≤ max

0≤

max

k≤K

a

nk

(T

k

), sup

k>K

k T

k

k .

Proof. Denote by S the right-hand side of (5.4). Let R

k

, 0 ≤ k ≤ K be operators on H

k

of respective rank < n

k

such that k T

k

− R

k

k = a

nk

(T

k

) and let R = L

K

k=0

R

k

. Then R is an operator of rank ≤ n

0

+ · · · + n

K

− K − 1 < N . If f = P

∞

k=0

f

k

∈ H , we see that:

k T f − Rf k

²

=

K

X

k=0

k T

k

f

k

− R

k

f

k

²

+ X

k>K

k T

k

f

k

²

≤

K

X

k=0

a

nk

(T

k

)

²

k f

k

²

+ X

k>K

k T

k

f

k

²

≤ S

²

∞

X

k=0

k f

k

²

= S

²

k f k

²

, hence the result.

We give now two corollaries of Lemma 5.3.

Example 1. We first use lens maps. We get:

Theorem 5.4. Let λ

θ

the lens map of parameter θ and let ψ : D → D such that k ψ k

∞

:= c < 1 and h a Rudin function. We consider:

Φ(z

1

, z

2

) = λ

θ

(z

1

), ψ(z

1

) h(z

2

) . Then, for some positive constant β, we have, for all N ≥ 1:

(5.5) a

N

(C

Φ

) . e

⁻^{β N}^1/3

.

Proof. Let T

k

= M

_ψ^k

C

λθ

. We have k T

k

k ≤ c

^k

, so sup

_k>K

k T

k

k ≤ c

^K

. On the other hand, we have a

n

(T

k

) ≤ c

^k

a

n

(C

λθ

) ≤ a

n

(C

λθ

) . e

⁻^β^θ^√ⁿ

([16], Theo- rem 2.1). Taking n

0

= n

1

= · · · = n

K

= K

²

in Lemma 5.3, we get:

0≤

max

k≤K

a

nk

(T

k

) . e

⁻^β^θ^K

.

Since n

0

+ · · · + n

K

− K ≈ K

³

, we obtain a

K³

. e

⁻^βK

, which gives the claimed result, by taking β = max β

θ

, log(1/c)

.

(20)

Example 2. We consider the cusp map χ. We have:

Theorem 5.5. Let χ be the cusp map, h a Rudin function, and ψ in the unit ball of H

^∞

, with k ψ k

∞

:= c < 1. Let:

Φ(z

1

, z

2

) = χ(z

1

), ψ(z

1

) h(z

2

) . Then, for positive constant β , we have, for all N ≥ 1:

a

N

(C

Φ

) . e

⁻^β^√^{N /}^√^log^N

.

Proof. Let T

k

= M

_ψ^k

C

χ

. As above, we have sup

_k>K

k T

k

k ≤ c

^K

. For the cusp map, we have a

n

(C

χ

) . e

⁻^αn/^logⁿ

([20], Theorem 4.3); hence a

n

(T

k

) . e

⁻^αn/^logⁿ

. We take n

0

= n

1

= · · · = n

K

= K [log K] (where [log K] is the integer part of log K). Since n

0

+ · · · + n

K

≈ K

²

[log K], we get, for another α > 0:

a

K²[logK]

(C

Φ

) . e

⁻^αK

, which reads: a

N

(C

Φ

) . e

⁻^β

√

N/logN

, as claimed.

5.2 Lower bounds

In this subsection, we give lower bounds for approximation numbers of composition operators on H

²

of the bidisk, attached to a symbol Φ of the previous form Φ(z

1

, z

2

) = φ(z

1

), ψ(z

1

) h(z

2

) where h is a Rudin function. The sharp- ness of those estimates will be discussed in the next subsection. We first need some lemmas in dimension one.

Lemma 5.6. Let u, v : D → D be two non-constant analytic self-maps and T = M

v

C

u

: H

²

( D ) → H

²

( D ) be the associated weighted composition operator.

For 0 < r < 1, we set A = u(r D ) and Γ = exp − 1/Cap (A)

. Then, for 0 < δ ≤ inf

_|z|=r

| v(z) | , we have:

(5.6) a

n

(T ) & √

1 − r δ Γ

ⁿ

.

In this lemma, Cap (A) denotes the Green capacity of the compact subset A ⊆ D (see [21], § 2.3 for the definition).

For the proof, we need the following result ([27], Theorem 7, p. 353).

Theorem 5.7 (Widom) . Let A be a compact subset of D and C (A) be the space of continuous functions on A with its natural norm. Set:

d ˜

n

(A) = inf

E

sup

f∈BH∞

dist (f, E)

,

where E runs over all (n − 1)-dimensional subspaces of C (A) and dist (f, E) = inf

h∈E

k f − h k

C(A)

. Then

(5.7) d ˜

n

(A) ≥ α e

⁻^{n/Cap (A)}

for some positive constant α.

(21)

Proof of Lemma 5.6. We apply Theorem 5.7 to the compact set A = u(r D ).

Let E be an (n − 1)-dimensional subspace of H

²

= H

²

( D ); it can be viewed as a subspace of C (A), so, by Theorem 5.7, there exists f ∈ H

^∞

⊆ H

²

with k f k

²

≤ k f k

∞

≤ 1 such that:

k f − h k

C(A)

≥ α Γ

ⁿ

, ∀ h ∈ E . Then:

k v (f ◦ u − h ◦ u) k

C(rT)

≥ δ k (f − h) ◦ u k

C(rT)

= δ k f − h k

C(A)

≥ α δ Γ

ⁿ

. But:

k v (f ◦ u − h ◦ u) k

C(rT)

≤ 1

√ 1 − r

²

k v (f ◦ u − h ◦ u) k

^H²

; Hence:

k T f − T h k

^H²

≥ α p

1 − r

²

δ Γ

ⁿ

≥ α √

1 − r δ Γ

ⁿ

.

Since h is an arbitrary function of E, we get (B

H²

being the unit ball of H

²

):

dim

inf

E<n

sup

f∈B_H2

dist T f, T (E)

≥ α √

1 − r δ Γ

ⁿ

.

But the left-hand side is equal to the Kolmogorov number d

n

(T ) of T (see [21], Lemma 3.12), and, as recalled in (1.7), in Hilbert spaces, the Kolmogorov numbers are equal to the approximation numbers; hence we obtain:

(5.8) a

n

(T ) ≥ α √

1 − r δ Γ

ⁿ

, n = 1, 2, . . . , as announced.

The next lemma shows that some Blaschke products are far away from 0 on some circles centered at 0.

We consider a strongly interpolating sequence (z

j

)

j≥1

of D in the sense that, if ε

j

:= 1 − | z

j

| , then:

(5.9) ε

j+1

≤ σ ε

j

and so ε

j

≤ σ

^j⁻¹

ε

1

, where 0 < σ < 1 is fixed. Equivalently, the sequence ( | z

j

| )

j≥1

is interpolating. We consider the corresponding interpolating Blaschke product:

(5.10) B(z) =

∞

Y

j=1

| z

j

| z

j

z

j

− z 1 − z

j

z ·

The following lemma is probably well-known, but we could find no satisfac-

tory reference (see yet [10] for related estimates) and provide a simple proof.

(22)

Lemma 5.8. Let (z

j

)

j≥1

be a strongly interpolating sequence as in (5.9) and B the associated Blaschke product (5.10).

Then there exists a sequence r

l

:= 1 − ρ

l

such that:

(5.11) C

1

σ

^l

≤ ρ

l

≤ C

2

σ

^l

, where C

1

, C

2

are positive constants, and for which:

(5.12) | z | = r

l

= ⇒ | B(z) | ≥ δ , where δ > 0 does not depend on l.

Proof. Let us denote by p

l

, 1 ≤ p

l

≤ l, the biggest integer such that ε

pl

≥ σ

^l⁻¹

ε

1

. We separate two cases.

Case 1: ε

pl

≥ 2 σ

^l⁻¹

ε

1

.

Then, we choose ρ

l

= α σ

^l⁻¹

ε

1

with α fixed, 1 < α < 2. Since ρ(ξ, ζ) ≥ ρ( | ξ | , | ζ | ) for all ξ, ζ ∈ D (recall that ρ is the pseudo-hyperbolic distance on D ), we have the following lower bound for | z | = r

l

:

| B(z) | =

∞

Y

j=1

ρ(z, z

j

) ≥

∞

Y

j=1

ρ(r

l

, | z

j

| ) = Y

j≤pl

ρ(r

l

, | z

j

| ) × Y

j>pl

ρ(r

l

, | z

j

| ) := P

1

× P

2

,

and we estimate P

1

and P

2

separately.

We first observe that ρ

l

ε

pl

≤ α σ

^l⁻¹

ε

1

2 σ

^l⁻¹

ε

1

≤ α

2 , and then:

ρ

l

ε

j

= ρ

l

ε

pl

ε

pl

ε

j

≤ α 2 σ

^p^l⁻^j

.

The inequality ρ(1 − u, 1 − v) ≥

(u+v)^|^u⁻^v^|

for 0 < u, v ≤ 1 now gives us:

(5.13) ρ(r

l

, | z

j

| ) ≥ ε

j

− ρ

l

ε

j

+ ρ

l

= 1 − ρ

l

/ε

j

1 + ρ

l

/ε

j

≥ 1 − (α/2) σ

^p^l⁻^j

1 + (α/2) σ

^p^l⁻^j

, for j ≤ p

l

, and:

(5.14) P

1

≥

Y

∞

k=0

1 − (α/2) σ

^k

1 + (α/2) σ

^k

· Similarly:

ε

pl+1

ρ

l

≤ σ

^l⁻¹

ε

1

α σ

^l⁻¹

ε

1

≤ 1 α

and: ε

j

ρ

l

≤ 1

α σ

^j⁻^p^l⁻¹

for j > p

l

; so that:

(5.15) ρ(r

l

, | z

j

| ) ≥ ρ

l

− ε

j

ρ

l

+ ε

j

= 1 − ε

j

/ρ

l

1 + ε

j

/ρ

l

≥ 1 − α

⁻¹

σ

^j⁻^p^l⁻¹

1 + α

⁻¹

σ

^j⁻^p^l⁻¹

, for j > p

l

,

(23)

and

(5.16) P

2

≥

∞

Y

k=0

1 − α

⁻¹

σ

^k

1 + α

⁻¹

σ

^k

· Finally, the condition of lower and upper bound for ρ

l

is fulfilled by construction.

Case 2: ε

pl

≤ 2 σ

^l⁻¹

ε

1

.

Then, we choose ρ

l

= a ε

pl

with σ < a < 1 fixed. Computations exactly similar to those of Case 1 give us:

(5.17) | B(z) | ≥

∞

Y

k=0

1 − a σ

^k

1 + a σ

^k

×

∞

Y

k=0

1 − a

⁻¹

σ

^k

1 + a

⁻¹

σ

^k

=: δ > 0 , for | z | = r

l

. Moreover, in this case:

a σ

^l⁻¹

ε

1

≤ ρ

l

≤ 2 a σ

^l⁻¹

ε

1

, and the proof is ended.

Now, we have the following estimation.

Theorem 5.9. Let φ, ψ : D → D be two non-constant analytic self-maps and Φ(z

1

, z

2

) = φ(z

1

), ψ(z

1

) h(z

2

)

, where h is inner.

Let (r

l

)

l≥1

be an increasing sequence of positive numbers with limit 1 such that:

|z

inf

|=rl

| ψ(z) | ≥ δ

l

> 0 , with δ

l

≤ e

⁻^{1/Cap (A}^l⁾

, where A

l

= φ r

l

D

.

Then the approximation numbers a

N

(C

Φ

), N ≥ 1, of the composition operator C

Φ

: H

²

( D

²

) → H

²

( D

²

) satisfy:

(5.18) a

N

(C

Φ

) & sup

l≥1

h √

1 − r

l

exp − 8 √ N p

log(1/δ

l

) p

log(1/Γ

l

i ,

where:

(5.19) Γ

l

= e

⁻^{1/Cap (A}^l⁾

.

Proof. Since h is inner, the sequence (h

^k

)

k≥0

is orthonormal in H

²

and hence a

n

(C

Φ

) = a

n

(T ) for all n ≥ 1, where T = L

_∞

k=0

T

k

and T

k

= M

ψ^k

C

φ

. Then Lemma 5.6 gives:

(5.20) a

n

(T

k

) & √

1 − r

l

δ

^k_l

Γ

ⁿ_l

for all n ≥ 1 and all k ≥ 0.

Let now:

(5.21) p

l

=

log(1/δ

l

) log(1/Γ

l

)

,

(24)

where [ . ] stands for the integer part, and:

(5.22) n

k

= p

l

k , for k = 1, . . . , K .

By Lemma 5.2, applied with m

k

= k (i.e. to H

1

, . . . , H

K

), we have, if N = n

1

+ · · · + n

K

:

a

N

(T ) ≥ inf

1≤k≤K

α √

1 − r

l

δ

_l^k

Γ

ⁿ_l

= α √

1 − r

l

δ

^K_l

Γ

ⁿ_l^K

. But, since p

l

≤ log(1/δ

l

)/ log(1/Γ

l

):

δ

_l^K

Γ

ⁿ_l^K

= exp

− K log(1/δ

l

) + p

l

K log(1/Γ

l

)

≥ exp[ − 2K log(1/δ

l

)] . Since:

N = p

l

K(K + 1) 2 ≥ p

l

K

²

4 ≥ K

²

16 log(1/δ

l

) log(1/Γ

l

) , we get:

δ

^K_l

Γ

ⁿ_l^K

≥ exp

− 8 √ N p

log(1/δ

l

) p

log(1/Γ

l

) , and the result ensues.

Example 1. We take φ = λ

θ

, a lens map, and ψ = B, a Blaschke product associated to a strongly regular sequence, as defined in (5.10); then we get:

Theorem 5.10. Let Φ : D

²

→ D

²

be defined by:

Φ(z

1

, z

2

) = λ

θ

(z

1

), c B(z

1

) h(z

2

) ,

where B is a Blaschke product as in (5.10), 0 < c < 1, and h is an arbitrary inner function, we have, for some positive constant b, for all N ≥ 1:

(5.23) a

N

(C

Φ

) & exp( − b N

^1/3

) = exp( − b √

N /N

^1/6

) . In particular β

2

(C

Φ

) = β

^±₂

(C

Φ

) = 1.

Remark. We saw in Theorem 5.4 that this is the exact size, since we have:

a

N

(C

Φ

) . e

⁻^{β N}^1/3

.

Proof. By Lemma 5.8, there is a sequence of numbers r

l

≈ σ

^l

such that | B(z) | ≥ δ for | z | = r

l

, where δ is a positive constant (depending on σ). Since λ

θ

(0) = 0, we have:

diam

ρ

(A

l

) ≥ λ

θ

(r

l

) & 1 − (1 − r

l

)

^θ

; hence, by [21], Theorem 3.13, we have:

Cap (A

l

) & log 1 1 − r

l

& l ,

or, equivalently: Γ

l

≥ e

⁻^b/l

, some some b > 0. Then (5.18) gives, for all l ≥ 1 (with another b):

a

N

(C

Φ

) & exp

− b

l +

√ N

√ l

.

Taking l = N

^1/3

, we get the result.

(25)

Example 2. By taking the cusp instead of a lens map, we obtain a better result, close to the extremal one.

Theorem 5.11. Let Φ(z

1

, z

2

) = χ(z

1

), c B(z

1

) h(z

2

)

, where χ is the cusp map, B a Blaschke product as in (5.10), 0 < c < 1, and h an arbitrary inner function. Then, for all N ≥ 1:

a

N

(C

Φ

) & e

⁻^b^√^{N /}^√^log^N

. In particular β

2

(C

Φ

) = 1.

Remark. We saw in Theorem 5.5 that this is the exact size, since we have:

a

N

(C

φ

) . e

⁻^β

√

_N/_log_N

.

Proof. The proof is the same as that of Proposition 5.10, except that, for the cusp map, we have (note that χ(0) = 0):

diam

ρ

(A

l

) ≥ χ(r

l

) . But when r goes to 1:

1 − χ(r) ∼ π ( √ 2 − 1) 2

1 log 1/(1 − r)

(see [20], Lemma 4.2). Hence, by [21], Theorem 3.13, again, we have:

Cap (A

l

) & log log 1/(1 − r

l

) ,

so Γ

l

≥ e

⁻^b/^log^l

. Then, (5.18) gives (with another b):

a

N

(C

Φ

) & exp

− b

l +

√ N

√ log l

. In taking l = p

N/ log N , we get the announced result.

5.3 Upper bounds

All previous results point in the direction that, if k Φ k

∞

= 1, then however small a

n

(C

Φ

) is, it will always be larger than α e

⁻^βεⁿ^√ⁿ

with ε

n

→ 0

⁺

, as this is the case in dimension one (with n instead of √ n). But Theorem 5.12 to follow shows that we cannot hope, in full generality, to get the same result in dimension d ≥ 2, and that other phenomena await to be understood. Here is our main result. It shows that, even for a truly 2-dimensional symbol Φ, we can have k Φ k

∞

= 1 and nevertheless β

₂⁺

(C

Φ

) < 1, in contrast to the 1-dimensional case where (1.1) holds.

Theorem 5.12. There exist a map Φ : D

²

→ D

²

such that:

1) the composition operator C

Φ

: H

²

( D

²

) → H

²

( D

²

) is bounded and compact;

2) we have k Φ k

∞

= 1 and Φ is truly 2-dimensional, so that β

₂⁻

(C

Φ

) > 0;

3) the singular numbers satisfy a

n

(C

Φ

) ≤ α e

⁻^β^√ⁿ

for some positive con-

stants α, β; in particular β

⁺₂

(C

Φ

) < 1.