HYPERBOLIC COMPOSITION OPERATORS ON THE BALL 1

Transcription

HYPERBOLIC COMPOSITION OPERATORS ON THE BALL
F. BAYART, S. CHARPENTIER
Abstract. We give a classification, up to automorphisms, of hyperbolic linear fractional
maps of the ball. We then show that this classification is very convenient to study the geometric properties of these maps, as well as the spectrum and the dynamics of the associated
composition operators. We conclude by showing how these properties can be tranfered to
composition operators associated to hyperbolic self-maps of the ball which are not linearfractional maps.
1. Introduction
If X is a Banach space of holomorphic functions on a domain U and if φ is a (holomorphic)
self-map of U, the composition operator of symbol φ is defined by Cφ (f ) = f ◦ φ for any
f ∈ X. The study of composition operators consists in the comparison of the properties of
the operator Cφ with that of the function φ itself, which is called the symbol of Cφ . For
instance, if X is the Hardy space H 2 (D) on the unit disk D, it is well-known that every selfmap φ induces a bounded composition operator on H 2 (D); one can characterize compactness
of Cφ and many other properties. We refer to [11] and to [21] to learn much more on this
subject.
Even on the disk, some properties are not easily managed. A strategy to study composition
operators is to use the so-called linear fractional model. Recall that a linear fractional map
of the Riemann sphere Ĉ is a map u : Ĉ → Ĉ which can be written u(z) = az+b
cz+d . The linear
fractional model theorem asserts that every univalent self-map of the disk is conjugate to a
linear fractional self-map of some, usually more complicated, plane domain. It is particularly
useful for composition operators because, if φ : D → D is holomorphic and u is its model, one
can often deduce properties of Cφ from properties of Cu . And of course, it is easier to study
directly Cu .
On the other hand, the theory of composition operators on the (euclidean) ball Bd := {Z =
(z, w) ∈ C × Cd−1 ; kZk22 < 1} of Cd , d ≥ 2 is much more involved. There are simple examples
of self-maps of Bd which do not induce a bounded composition operator on H 2 (Bd ). In this
context, the introduction of the class of linear fractional maps of the ball seems to be crucial,
at least for two reasons. First, to give a source of examples that are easily managed, yet still
rich enough to exhibit surprisingly diverse behaviors. Second, to hope to produce a tool like
the linear fractional model. This motivated Cowen and MacCluer to introduce in [12] the
following definition.
Definition 1.1. An application φ : Cd → Cd is said to be a linear fractional map if it can
be written
AZ + B
φ (Z) =
,
hZ, Ci + D
where A ∈ Md (C) is a matrix, B and C are two vectors in Cd , and D ∈ C.
If U is a domain of Cd , we say that φ is a linear fractional map of U if it is defined in a
neighbourhood of U and if φ(U) ⊂ U. We denote the set of such maps by LF M (U).
In order to classify these linear fractional maps according to their fixed points (which
play an essential role in the study of the associated composition operators), we recall the
Denjoy-Wolff theorem in the ball Bd :
1
2
Theorem 1.2. Let φ ∈ LF M (Bd ) without fixed points in Bd . Then there exists a unique
fixed point τ ∈ ∂Bd such that φ (τ ) = τ and α (φ) = hdφτ (τ ) , τ i ∈ (0, 1].
τ is called the Denjoy-Wolff point of φ and α(φ) is called the boundary dilation coefficient
of φ.
Looking at this theorem, we may distinguish three types of linear fractional map φ of Bd :
• φ is said elliptic if it has a fixed point in Bd ;
• φ is said parabolic if it has no fixed point in Bd and if α (φ) = 1.
• φ is said hyperbolic if it has no fixed point in Bd and if α (φ) ∈ (0, 1).
From now on, if φ ∈ LF M (Bd ) has no fixed points in Bd then, up to conjugation by
an automorphism of the ball, we may assume that the Denjoy-Wolff point of φ is e1 =
(1, 0, . . . , 0).
In this paper, we intend to begin a systematic study of hyperbolic linear fractional maps
and of their associated composition operators. Our first task is to classify the hyperbolic
linear fractional maps. We move to the Siegel upper half-space
n
o
Hd = Z = (z, w) ∈ C × Cd−1 , =m(z) > |w1 |2 + . . . + |wd−1 |2 = kwk2
and we show that any hyperbolic linear fractional map of Bd is conjugated, via the Cayley
transform and automorphisms of Bd , to a self-map of Hd which may be written
ψ(z, u, v) = (λz + b, Du, Av + c)
where z ∈ C, (u,√v) ∈ Cd−1 , =m(b) ≥ 0, D is a diagonal matrix with diagonal coefficients
having modulus λ and A and c satisfy several technical conditions. We call this map ψ a
normal form of φ, and this rewriting of φ will allow us to simplify the computations in the
next parts of the paper.
Although the normal form is not unique up to automorphisms, we shall identify three
properties of φ which may be easily read on any of its normal forms and which are invariant
by automorphisms. We will call these properties the signature of φ, and we shall see that
the properties of both φ and Cφ do really depend on the signature. This will be for instance
the case for the geometric properties of φ, like the shape of its characteristic domain which
is entirely determined by the signature of φ.
Next, we will focus on the composition operators associated to hyperbolic linear fractional
maps, and first of all, on their spectra. The study of the spectrum of composition operators
has drawn the attention of many mathematicians, see for instance [11, Chapter 7]. In our
context of linear fractional maps, the spectrum of Cφ has been computed when φ is parabolic
in [2] and when φ is an automorphism (see [18]). The spectral radius is also known (see [17]).
We achieve the computation for hyperbolic linear fractional maps.
Theorem 1.3. Let φ ∈ LF M (Bd ) be hyperbolic and not an automorphism. Then the spectrum of Cφ is
S
• n Cn ∪ {0} where (Cn ) is a sequence of coronae centered at 0 whose radii go to zero,
if φ acts as an automorphism on a slice of Bd ;
• a disk centered at 0 otherwise.
In Section 4, we do a complete study of the dynamics of the composition operator associated
to a hyperbolic linear fractional map. Recall that an operator T acting on a Banach space
X is called hypercyclic if there exists a vector x ∈ X such that its orbit {T n x; n ≥ 0} is
dense in X. It is called supercyclic if there exists a vector x ∈ X such that its projective
orbit C · {T n x; n ≥ 0} is dense in X. It is called cyclic if there exists x ∈ X such that
{P (T )x; P polynomial} is dense in X. For a complete and up-to-date account on linear
dynamics, we refer to [3].
3
The dynamics of composition operators is an important feature of both linear dynamics and
the theory of composition operators (see for instance [2], [6], [9], [15], [16], [21]). In particular,
the dynamics of a composition operator associated to a hyperbolic linear fractional map of
the disk is already well-known (see [6]):
Theorem 1.4 (Disk theorem). Let φ ∈ LF M (D) be hyperbolic. Then Cφ is hypercyclic.
The corresponding theorem for Bd , d ≥ 2, is much more complicated.
Theorem 1.5. Let φ ∈ LF M (Bd ) be hyperbolic and univalent and let ψ(z, w) = (λz +
b, Du, Av + c) be a normal form of φ.
(1) If dim(v) = 0, then Cφ is hypercyclic;
(2) If dim(v) > 0, then
(a) Cφ is never hypercyclic;
(b) Cφ is supercyclic iff =m(b) > 0;
(c) Cφ is always cyclic.
This result is, for at least two reasons, rather surprizing. First of all, we get examples of
supercyclic composition operators which are not hypercyclic. This seems to be the first such
examples and indeed, it has been proved in [4] that in some cases (of course, not in this one!),
compositional supercyclicity implies compositional hypercyclicity. Second, that Cφ has better
dynamical properties when =m(b) > 0 is not intuitive. The condition =m(b) = 0 means that
z 7→ λz + b is an automorphism of the upper half-plane (that the restriction of φ to some slice
of Bd is an automorphism). Generally, the dynamical properties of a composition operator
are better when φ is an automorphism in contradiction which what happens here. We shall
prove Theorem 1.5 in Section 4. This will depend heavily on the work done before.
Finally, we end up the paper by applying the results on hyperbolic linear fractional composition operators to get results on general hyperbolic composition operators via the linear
fractional model. As mentioned before, this technics was very efficient in the disk setting,
especially for the dynamics of composition operators, see [6]. A linear fractional model for
hyperbolic maps of the ball has been developped in [1] and in [8]. Due to new difficulties
which arise in the several-variables setting, we will not obtain a result as general as that of
[6]. However, the theorem that we will get is hopeful because it shows that the study of
composition operators associated to linear fractional maps do have applications to general
composition operators, even on Bd with d ≥ 2. Thus the idea of Cowen and MacCluer was
justified.
Notations. In what follows, the notation f (x) . g(x) means that there exists some C > 0
such that, for any x, f (x) ≤ Cg(x). P+ denotes the upper half-plane, P+ = {z ∈ C; =m(z) >
0}.
2. Geometric study
2.1. Moving to the Siegel half-space. Let φ ∈ LF M (Bd ) without fixed points in Bd and
whose Denjoy-Wolff point is e1 . The geometric properties of Cφ will be easier to understand
if we move to the Siegel upper half-space, defined as follows:
Definition 2.1. The Siegel upper half-space Hd of Cd is defined by
n
o
Hd = (z, w) ∈ C × Cd−1 , =m(z) > |w1 |2 + . . . + |wd−1 |2 = kwk2 .
As in the complex plane, there exists a biholomorphism σc from Bd onto Hd called the
Cayley transform and given by the following formula
1 + z iw
,
, (z, w) ∈ C × Cd−1 .
(1)
σc (z, w) = i
1−z 1−z
4
Its reciprocal satisfies
σc−1 (z, w) =
z − i 2w
,
z+i z+i
.
This map may be extended to Bd and it is then onto Hd ∪ ∂Hd ∪ {∞}, where σc (1) = ∞.
Since the Cayley map is itself a linear fractional map, it is clear that LF M (Hd ) = σc−1 ◦
LF M (Bd )◦σc . Moreover, if φ ∈ LF M (Bd ) admits e1 as Denjoy-Wolff point, the Denjoy-Wolff
point of ψ = σc−1 ◦ φ ◦ σc is ∞.
It will be technically easier to deal with linear fractional maps of Hd fixing ∞ because
the fractional part disappears for these maps. Precisely, F. Bracci, M. D. Contreras, S.
Diaz-Madrigal in [7] have computed which maps of LF M (Hd ) fix ∞.
Theorem 2.2. Let ψ = σc−1 ◦ φ ◦ σc ∈ LF M (Hd ) be non-elliptic with Denjoy-Wolff point ∞
and let λ = 1/α(φ). Then there exist a, c ∈ Cd−1 , b ∈ C and M = Mφ ∈ Md−1 (C), such
that
(2)
ψ (z, w) = (λz + 2i hw, ai + b, M w + c) .
Moreover, an application ψ satisfying (2) maps Hd into itself if and only if
(a) Q = λI − M M ∗ is a hermitian positive semi-definite matrix;
(b) =m(b) − kck2 ≥ hQ+ (M ∗ c − a) , M ∗ c − ai, where Q+ is the pseudo-inverse of Q;
(c) M ∗ c − a belongs to the space spanned by the columns of Q.
In the present paper, we intend to study more specifically the hyperbolic linear fractional
maps. We begin by simplifying, up to conjugation by automorphisms, formula (2).
2.2. Normal form of a hyperbolic linear fractional map. Let φ ∈ LF M (Bd ) be of
hyperbolic type (with Denjoy-Wolff point at e1 ) and let ψ = σc−1 ◦ φ ◦ σc ∈ LF M (Hd ) be
the conjugated map in the Siegel half-space.
√ Let us remind that λ = 1/α(φ) > 1. Property
λ, so that the spectrum of M is contained in the
(a) of Theorem 2.2 entails
that
kM
k
≤
√
closed disk of radius λ. Moreover, the same property√also yields that the characteristic
subspaces of M associated to the eigenvalues of modulus λ coincide with the corresponding
eigenvectorspaces, and that their orthogonal subspace is invariant by M . As a consequence,
conjugating M by an unitary map U , which means that we conjugate ψ by the automorphism
of Hd (z, w) 7→ (z, U w), we may assume that the matrix M which appears in (2) may be
written
D 0
(3)
M=
,
0 A
√
where D is a diagonal matrix whose diagonal coefficients all have modulus equal to λ, and
where the matrix A is such that λI − AA∗ is a hermitian positive definite matrix.
Keeping on this way, we now conjugate ψ by an automorphic Heisenberg translation τ of
the Siegel half-space,
τ (z, w) = (z + 2i hw, γi + β, w + γ)
with =m(β) = kγk2 . We have (see [2])
τ −1 (z, w) = z − 2i hw, γi − β + i kγk2 , w − γ .
After a small computation, one obtains
τ −1 ◦ ψ ◦ τ (z, w) = λz + 2ihw, (λ̄ − M ∗ )γ + ai + b + 2ihM γ, γi + λβ, M (w + γ) + c .
√
Since kM ∗ k ≤ λ and λ > 1, λ̄ − M ∗ is invertible and we choose γ such that (λ − M ∗ ) (γ) =
−a. Next, adjusting the value of <e(β), which is independent of kγk2 , we may be sure that
b + 2ihM γ, γi + λβ is a pure imaginary.
5
Therefore, up to conjugation by automorphisms, we get the following reduced formula for
ψ ∈ LF M (Hd ) hyperbolic:
ψ (z, w) = (λz + b, Du + c1 , Av + c2 ) ,
where b is a pure imaginary, w = (u, v) and c = (c1 , c2 ) are respectively the decompositions
of w and of c along the orthogonal subspaces of Cd−1 involved in the decomposition of M in
(3). Besides, as ψ verifies
condition (c), M ∗ c belongs to the space spanned by the columns of
0
0
Q = λI − M ∗ M =
, so that we must have c1 = 0 because D∗ is one-to-one.
0 λI − A∗ A
Moreover, if =m(b) = 0, then c2 is also equal to 0 by (b).
We now conjugate ψ by a non-isotropic dilation hµ ,
√
hµ (z, w) = (µz, µw), µ > 0.
We get
hµ ◦ ψ ◦ h1/µ (z, w) = (λz + µb, Du, Av +
√
µc).
In particular, when =m(b) > 0, we may choose µ such that =m(µb) is very close to 0. For
our applications, we will need =m(µb) ∈ (0, λ − 1).
Finally, any matrix is always unitarily conjugated to an upper-triangular matrix. This
means that, up to automorphisms, we may assume that the matrix A is upper-triangular.
We summarize the work done until now in the following proposition:
Proposition 2.3. Let φ ∈ LF M (Bd ) be hyperbolic and let ψ = σc−1 ◦ φ ◦ σc . Then ψ is
conjugated to a map
(4)
(z, w) 7→ (λz + b, Du, Av + c)
where
√
(a) D is diagonal with diagonal coefficients of modulus λ;
(b) Q = λI − A∗ A is hermitian positive definite and A is upper-triangular;
(c) b is a pure imaginary and kck2 + hQ−1 A∗ c, A∗ ci ≤ =m(b) < λ − 1. In particular, if b is
equal to zero, then c is equal to zero.
The map appearing in (4) will be called a normal form of φ.
A linear fractional map of Bd does not admit an unique normal form. However, several
parameters of it are invariant by conjugation by automorphisms fixing ∞:
• λφ := λ = 1/α(φ), which is the inverse of the boundary dilation coefficient of φ;
• the fact that =m(b) = 0 or not. This determines whether ψ1 is, or is not, an automorphism of {=m(z) > 0}. Coming back to φ, =m(b) = 0 if and only if φ|D acts as an
automorphism on some slice D of Bd . We set εφ = 1 if =m(b) = 0, εφ = 0 otherwise;
• the
√ number of eigenvalues of M , counted with their multiplicity, which have modulus
λ. We shall denote by pφ this number of eigenvalues, namely the dimension of u.
Definition 2.4. The triple (λφ , εφ , pφ ) is called the signature of φ.
The signature classifies, up to conjugation by automorphisms, the linear fractional maps of
Bd . Our intention is to convince the reader that, as soon as we know the signature of φ,
we know many things on the behavior of φ and on its associated composition operator. We
begin by some geometrical considerations.
2.3. Characteristic domain of a hyperbolic type linear fractional map. We recall
that the characteristic domain of φ ∈ LF M (Bd ) is the smallest domain containing the ball
and invariant by φ. When φ is univalent (when Mφ is one-to-one), this is the smallest domain
containing Bd on which φ acts as an automorphism.
6
Theorem 2.5. Let φ ∈ LF M (Bd ) be hyperbolic and let ψ(z, w) = (λz + b, Du, Av + c) be a
normal form of φ. Then the characteristic domain of φ is, up to conjugation by the Cayley
transform and by automorphisms of Hd , equal to
=m(b)
2
d
Ω0 := (z, u, v) ∈ C , =m(z) > kuk −
.
λ−1
S
Proof. The characteristic domain of ψ is Ω = n≥0 ψ −n (Hd ). It suffices to prove that Ω is
equal to Ω0 . Let ψn denotes the nth iterated of ψ. A straightforward computation gives
λn − 1
n
n
n
n−1
ψn (z, w) = λ z +
b, D u, A v + A
c + ... + c
λ−1
so that (z, w) ∈ ∪n≥0 ψ −n (Hd ) iff there exists an integer n such that
2
1 1 λn − 1
=m(b) > kuk2 + n An v + An−1 c + . . . + c
=m(z) + n
λ
λ−1
λ
=m(b)
=m(b)
1
+ n
+
kAn v + An−1 c + · · · + ck2 .
λ−1
λ (λ − 1) λn
This shows immediately that Ω ⊂ Ω0 . The converse inclusion is also satisfied since
(
=m(b)
λn (λ−1) → 0
⇐⇒ =m(z) > kuk2 −
√
(recall that kAk ≤ λ).
1
n
λn kA v
+ An−1 c + · · · + ck2 → 0
We notice that, if we want to know the shape of the characteristic domain, the important
part of the signature of φ is the value of pφ .
2.4. Convergence to infinity. Let ψ be a self-map of Hd with Denjoy-Wolff point at ∞.
For each (z, w) ∈ Hd , ψn (z, w) goes to ∞. Two types of convergence (which are invariant by
conjugation by an automorphism) are of particular interest:
Definition 2.6. Given (zn , wn ) a sequence of points in Hd convergent to infinity. We say
that this convergence is
2
kwn k
tends to 0;
(a) special if =m(z
n)
(b) restrictive if (zn )n converges non-tangentially to infinity, that is to say there exists C > 0
such that |<e(zn )| ≤ C=m(zn ) for any n ≥ 0.
The classification of hyperbolic linear fractional maps gives the following result:
Theorem 2.7. Let φ ∈ LF M (Bd ) be hyperbolic and let ψ = σc−1 ◦φ◦σc be its conjugated map
in Hd . For any (z, w) ∈ Hd , ψn (z, w) converges to its Denjoy-Wolff point ∞ in a restrictive
way, and this convergence is special if and only if pφ = 0.
Proof. Since special and restrictive convergence are invariant by conjugation by an automorphism, we may assume that ψ is a normal form of φ. Put (zn , wn ) := ψn (z, w) ∈ Hd . Since
the Denjoy-Wolff point of ψ is ∞, (zn ) must tend to infinity. Moreover
λn <e(z)
<e(z)
|<e(zn )|
=
−−−→
,
n −1
λ
n
=m(zn )
λ =m(z) + λ−1 =m(b) n→∞ =m(z) + =m(b)
λ−1
proving the restrictive convergence.
Besides,
2
kDn uk2 + An v + An−1 c + . . . + c
kwn k2
kuk2
=
−
−
−
→
,
n
−1
n→∞ =m (z) + =m(b)
=m(zn )
λn =m(z) + λλ−1
=m(b)
λ−1
so that the convergence√to infinity is special if and only if the subspace associated to the
eigenvalues of modulus λ in the normal form of ψ is reduced to 0.
7
3. Spectrum of hyperbolic composition operators
Let φ ∈ LF M (Bd ) be hyperbolic. We suppose that φ is not an automorphism of Bd and
we intend to compute the spectrum of Cφ . We first recall the following result of M. Jury,
which gives the spectral radius:
Theorem 3.1. [17, Corollary 13] If φ is a hyperbolic linear fractional map of the ball Bd with
dilation coefficient α, then the spectral radius of Cφ acting on H 2 (Bd ) is α−d/2 .
As one can guess, our main tool to prove Theorem 1.3 will be the normal form of φ.
Precisely, if ψ denotes a normal form of φ, we will compute the spectrum of Cψ . To do that,
we have to define the space where Cψ acts. The image of H 2 (Bd ) by the Cayley transform is
denoted by H2 (Hd ) :
H2 (Hd ) = F : Hd → C holomorphic; F ◦ σc ∈ H 2 (Bd ) .
H2 (Hd ) is endowed with the norm inherited from H 2 (Bd ), kF kH2 = kF ◦ σc kH 2 so that Cσc
is an unitary map from H2 (Hd ) onto H 2 (Bd ). As one easily sees by computing the jacobian
of σc , the norm on H2 (Hd ) satisfies
Z
|F (z, w)|2
2
2
kF kH2 = κ
dσ∂Hd ,
2d
∂Hd |z + i|
where κ is a constant we will not try to compute and dσ∂Hd is the Lebesgue measure on ∂Hd .
Of course, Cφ acting on H 2 (Bd ) and Cψ acting on H2 (Hd ) are unitarily similar, so that
they share the same properties. We are now ready for the proof.
3.1. Spectrum when φ does not act as an automorphism on a slice. In this subsection, we compute the spectrum of Cφ when εφ = 0, namely when a normal form of φ is
ψ(z, w) = (λz + b, Du, Av + c) with =m(b) > 0.
Theorem 3.2. With the notations above, σ(Cφ ) = D(0, λd/2 ).
Proof. We already know by Theorem 3.1 that σ(Cφ ) is contained in the closed disk D(0, λd/2 ).
We need only to show that any element of D(0, λd/2 )\{0} is an eigenvalue of Cφ . Now, given
s ∈ C, let us define
s
b
Fs (z) = z +
.
λ−1
It is clear that Fs ◦ ψ = λs Fs , and so we are interested in the values of s such that Fs belong
to H2 (Hd ). Now, setting it = b/(λ − 1) with t > 0, one gets
Z
Z
|(z + it)s |2
1
2
kFs k =
dσ∂Hd .
dσ∂Hd
2d
2(d−<e(s))
∂Hd |z + i|
∂Hd |z + i|
and this last integral if finite iff <e(s) < d/2. Hence, any λs with <e(s) < d/2 is an eigenvalue,
achieving the proof of the theorem.
3.2. Spectrum when φ acts as an automorphism on a slice - Introduction. We
are now in the situation where a normal form of φ is given by ψ(z, w) = (λz, Du, Av).
The previous proof does not work as well, because the functions Fs (z) = z s which satisfy
Fs ◦ ψ = λs Fs belongs to H2 (Hd ) if and only if −d/2 < <e(s) < d/2, showing only that
the corona {λ−d/2 < <e(s) < λd/2 } is contained in σp (Cφ ). This is sufficient to determine
the spectrum of Cφ when φ is an automorphism of Bd . Otherwise, this just give the first
corona which appears in Theorem 1.3. To get the remaining part of the spectrum, we have
to introduce a nice decomposion of H2 (Hd ).
8
For α ∈ Ns (here s = d − 1 − pφ ), we denote by Hα the set of allQfunctions F of H2 (Hd )
which can be written F (z, u, v) = Fα (z, u) v α , where v α stands for si=1 viαi . It is clear that
s
H2 (Hd ) = ⊕⊥
α∈Ns Hα . Moreover, let us order N as follows:


|α| < |β|
α ≺ β iff or


|α| = |β| and there exists j such that α1 = β1 , . . . , αj−1 = βj−1 and αj < βj .
If we define
KN :=
⊥
M
Hα ,
α∈Ns
|α|≥N
then Cφ is upper triangular under the decomposition
2
H (Hd ) =
⊥
M
Hα ⊕⊥ KN
α∈Ns
|α|<N
Ns
with respect to the order ≺ on
(remind that A has been supposed to be upper triangular)
and is diagonal if A itself is diagonal. Let finally Tα be the diagonal block of Cφ corresponding
to the subspace Hα . Tα is also the diagonal block corresponding to Hα of the simpler
composition operator associated to the application ψ̃ defined by
ψ̃ (z, u, v) = λz, Du, Ãv
where Ã is the diagonal matrix composed of the eigenvalues of A. That ψ̃ is a selfmap of Hd
follows from the easily verified fact kÃk ≤ kAk.
With this in mind, a general result concerning the spectrum of triangular or diagonal
operators on a finite sum of subspaces of a given vector space (see for instance [2, Lemma
5.3]) directly yields the following lemma:
Lemma 3.3. With the previous notations, we have:
[
σ(Cψ ) ⊆
σ (Tα ) ∪ D 0, Cψ|KN .
α∈Ns
|α|<N
In view of Lemma 3.3, our strategy is now rather clear:
Step 1: Show that kCψ|KN k goes to zero as N goes to ∞. By the above lemma, this
will yield
[
σ(Cψ ) ⊂
σ(Tα ) ∪ {0}.
α∈Ns
Step 2: Using the operator Cψ̃ , which is diagonal with diagonal blocks Tα , we will prove
that
(
αi
αi )
s s Y
Y
|µ
|
|µ
|
√i
√i
≤ |z| ≤ λd/2
,
σ(Tα ) ⊂ z ∈ C, λ−d/2
λ
λ
i=1
i=1
where µ1 , . . . , µs are the eigenvalues of A, written with multiplicity.
Step 3: For each α ∈ Ns and each z ∈ C belonging to
(
αi
αi )
s s Y
Y
|µ
|
|µ
|
i
i
√
√
z ∈ C, λ−d/2
< |z| < λd/2
,
λ
λ
i=1
i=1
we shall exhibit an eigenvector of Cψ corresponding to the eigenvalue z.
Provided that we are able to do that, we obtain the following precised version of Theorem
1.3:
9
Theorem 3.4. Let φ ∈ LF M (Bd ) be such that εφ = 1 and let ψ(z, w) = (λz, Du, Av) be a
normal form of φ. Let also µ1 , . . . , µs be the eigenvalues of A. Then
(
)
s s [
Y
Y
|µi | αi
|µi | αi
−d/2
d/2
√
√
σ(Cφ ) =
z ∈ C, λ
≤ |z| ≤ λ
∪ {0}.
λ
λ
s
i=1
i=1
α∈N
Let us proceed with the details of the proof.
3.3. Spectrum when φ acts as an automorphism on a slice - Step 1. The aim of this
step is the following lemma.
Lemma 3.5. Under the above conditions and notations,
Cψ|K −−−−→ 0.
N
N →∞
Proof. Let A = U ΣV be a singular value decomposition of A; U and V are two unitary maps
and Σ is the diagonal matrix whose diagonal terms are the square roots of the eigenvalues of
AA∗ :


|µ1 | . . .
0
h

..  with in particular |µ | ∈ 0, √λ .
..
Σ =  ...
i
.
. 
0 . . . |µs |
Then we factorize Cψ as Cψ = Cτ1 ◦ CψΣ ◦ Cτ2 = Cτ2 ◦ψΣ ◦τ1 where

 τ1 : (z, w) 7→ (z, u, V v)
ψΣ : (z, w) 7→ (λz, Du, Σv) .

τ2 : (z, w) 7→ (z, u, U v)
⊥ , C
−−−→
As τ1 and τ2 are automorphic self-maps of Hd which preserve both KN and KN
ψ|KN −
N →∞
0 will stand as soon as
Cψ |K −−−−→ 0.
Σ
N
N →∞
√ We introduce ψ0 = λz, Du, λv ; ψ0 is an automorphism of Hd , hence Cψ0 is invertible
on H2 (Hd ). For F ∈ Hα , F (z, u, v) = Fα (z, u) v α , we have both
√ |α|
Cψ0 (F ) = Fα (λz, Du) λ v α
!
s
Y
αi
vα.
CψΣ (F ) = Fα (λz, Du)
|µi |
i=1
This yields
CψΣ |KN = A ◦ Cψ0
where A is the diagonal operator on KN = ⊕⊥
|α|≥N Hα whose diagonal block corresponding
Qs |µi | αi
to Hα is i=1 √λ
I. Therefore
Cψ |K ≤ kAk · kCψ k .
0
Σ
N
Moreover,
s Y
|µi | αi
max1≤i≤s |µi | N
√
√
kAk = maxs
≤
.
α∈N
λ
λ
i=1
|α|≥N
Consequently,
Cψ |K ≤ kCψ k
0
Σ
N
since |µi | <
√
λ for all i.
max1≤i≤s |µi |
√
λ
N
−−−−→ 0
N →∞
10
3.4. Spectrum when φ acts as an automorphism on a slice - Step 2. The next
proposition provides an upper bound for the spectrum of Tα :
Proposition 3.6. With the above conditions and notations,
(
αi
αi )
s s Y
Y
|µ
|
|µ
|
i
i
√
√
σ (Tα ) ⊂ z ∈ C, λ−d/2
≤ |z| ≤ λd/2
.
λ
λ
i=1
i=1
Proof. Let Pα denote the orthogonal projection on Hα . As Tα = Cψ̃ ◦ Pα , one has to prove
that
(
αi
αi )
s s Y
Y
|µ
|
|µ
|
√i
√i
σ Cψ̃|Hα ⊂ z ∈ C, λ−d/2
≤ |z| ≤ λd/2
,
λ
λ
i=1
i=1
where we recall that ψ̃ (z, u, v) = λz, Du, Ãv with


µ1 . . . 0


Ã =  ... . . . ...  .
0 . . . µs
√ As before, we introduce the automorphism of H2 (Hd ) defined by ψ0 = λz, Du, λv . It is
already well-known (see [18]) that
o
n
σ (Cψ0 ) = z ∈ C, λ−d/2 ≤ |z| ≤ λd/2 .
Therefore, as Hα and Hα⊥ are stable under Cψ0 ,
o
n
σ Cψ0|Hα ⊂ σ (Cψ0 ) = z ∈ C, λ−d/2 ≤ |z| ≤ λd/2 .
Moreover, for F ∈ Hα , F (z, u, v) = Fα (z, u) v α , we have both
√ |α|
Cψ0 (F ) = Fα (λz, Du) λ v α
!
s
Y
Cψ̃ (F ) = Fα (λz, Du)
µαi i v α ,
i=1
Qs
so that Cψ̃|Hα =
αi
i=1 µi
√ |α| Cψ0|Hα .
This yields
λ
σ Cψ̃|H
=
Qs
αi i=1 µi
σ
C
ψ0|Hα
√ |α|
α
λ
(
⊂
z ∈ C, λ
−d/2
s s Y
Y
|µi | αi
|µi | αi
d/2
√
√
≤ |z| ≤ λ
λ
λ
i=1
i=1
)
.
In fact, the inclusion appearing in the above proposition is an equality, as this will be clear
in the next subsection.
3.5. Spectrum when φ acts as an automorphism on a slice - Step 3. To end up the
proof of Theorem 3.4, we just need the following proposition:
Proposition 3.7. With the above conditions and notations,
(
)
s s Y
Y
|µi | αi
|µi | αi
−d/2
d/2
√
√
σp Cψ|Hα ⊃ z ∈ C, λ
< |z| < λ
for every α ∈ Ns .
λ
λ
i=1
i=1
11
In particular, we have
(
[
σp (Cψ ) ⊃
z ∈ C, λ
−d/2
α∈Ns
s s Y
Y
|µi | αi
|µi | αi
√
√
< |z| < λd/2
λ
λ
i=1
i=1
)
.
Proof. For t + il ∈ C, let F : (z, u, v) 7−→ z t+il ṽ(v)α with ṽ = (v(1), . . . , v(s)) where, for
1 ≤ i ≤ s,Qv(i) ∈ (Cs )∗ is the eigenvector of AT associated to the eigenvalue µi and ṽ α stands
for ṽ α = si=1 v(i)αi . We get
!
s
Y
α
Cψ (F ) = λt+il
µi i F.
i=1
Now, let us see for which values of t + il the function F is in H2 (Hd ). Writing v(i)(v) =
ai,1 v1 + · · · + ai,s vs and expanding the product, it suffices to know for which values of t + il
and α the function z t+il v α belongs to H2 (Hd ). Let us admit for a while that this is true if
|α|
|α|
d
t+il
and only if −d
2 − 2 < t < 2 − 2 . When t moves in this interval and l moves in R, λ
describes the corona
)
(
λd/2
λ−d/2
< |z| < |α|/2
λ|α|/2
λ
which gives the proposition. So, let us study kz t+il v α kH2 (Hd ) :
Z
t+il α z v H2 (Hd )
=
|z|2t |v α |2
dσ∂Hd
|z + i|2d
t Q
s
αi 2
Z Z
x2 + k(u, v)k4
i=1 |vi |
=
2 d dudvdx,
R Cd−1
2
2
x + 1 + k(u, v)k
∂Hd
where we put z = x + i k(u, v)k2 in the last integral. We use polar coordinates in the integral
depending on (u, v) in Cd−1 and we put
(u, v) = r (ξ2,1 + iξ2,2 , . . . , ξd−s,1 + iξd−s,2 , ξd−s+1,1 + iξd−s+1,2 , . . . , ξd,1 + iξd,2 )
with r ∈ R+ and ξ = (ξ2,1 , ξ2,2 , . . . , ξd−s,1 , ξd−s,2 , ξd−s+1,1 , ξd−s+1,2 , . . . , ξd,1 , ξd,2 ) ∈ S2(d−1)−1 .
Next, we can write
s
s
Y
Y
|viαi |2 = r2|α| |ξd−s+i,1 + iξd−s+i,2 |2αi ,
i=1
|i=1
{z
:=Cα (ξ)
}
and then
kF kH2 (Hd ) =
Z Z
R
0
+∞ Z
S2(d−1)−1
Z Z
R
0
+∞
t
x2 + r4 r2|α|+2(d−1)−1 Cα (ξ)
dσS2(d−1)−1 (ξ) drdx =
d
2
2
2
x + (1 + r )
t
Z
x2 + r4 r2|α|+2(d−1)−1
Cα (ξ) dσS2(d−1)−1 (ξ) drdx.
d
2
S
2
2
2(d−1)−1
x + (1 + r )
Q
As S2(d−1)−1 is bounded and as si=1 |ξd−s+i,1 + iξd−s+i,2 |2αi is also bounded and bounded
away from 0 on S2(d−1)−1 , and using the change of variables r 7−→ r2 , we deduce that F is in
12
H2 (Hd ) if and only if
Z
+∞ Z +∞
It,α,d :=
0
0
t
x2 + r2 r|α|+d−2
d drdx < +∞.
2
2
x + (1 + r)
We shall use the following lemma:
Lemma 3.8. Let e, f, g ∈ R. The integral
Z 1Z 1
g
xe y f x2 + y 2 dxdy
0
0
is finite if and only if e + f + 2g > −2.
Proof. This is a straightforward computation:
Z 1Z 1
g
xe y f x2 + y 2 dxdy < ∞
0
0
m
Z
1
xe
0
1
1
2
x2 + y f +1
g
by the change of variable y 7−→ y f +1
dydx < ∞
0
0
Z
Z
m
y 2 g
f +1
xe+2g
1+
dydx < ∞
f +1
x
0
m change of variable y 7−→
Z 1
Z 1
xe+f +2g+1
(1 + y)g dydx < ∞
Z
1
y
xf +1
0
0
which is equivalent to e + f + 2g + 1 > −1 as we wished.
Wecome back to the study of the integrability of It,α,d . On the compact set [0, 1] × [0, 1],
since x2 + (1 + r)2 is greater than 1, we just have to study the integrability of (r, x) 7−→
t
rd+|α|−2 x2 + r2 . This follows directly from the lemma, and the integrability on [0, 1]×[0, 1]
d |α|
is equivalent to the condition t > − −
.
2
2
On [1, +∞] × [1, +∞], we are reduced to study the integrability of
t−d
(r, x) 7−→ rd+|α|−2 x2 + r2
.
1
1
Using the changes of variables x 7−→ and r 7−→ , we come back to the situation of Lemma
x
r
3.8 for the function
t−d
(x, r) 7−→ rd−|α|−2t x−2t+2d−2 x2 + r2
.
d |α|
After a small computation, we find that It,α,d is finite if and only if t < −
, which was
2
2
the missing inequality.
3.6. Examples. Let ψ1 (z, w) = (4z, w) and ψ2 (z, w) = (4z, w/10), acting on H2 . We know
that
n [ 1 1 n
[
1
σ(Cψ1 ) =
≤ |z| ≤ 4
∪ {0} :=
Cn ∪ {0}.
4 2
2
n≥0
n≥0
Now, the coronae Cn and Cn+1 do intersect, and apparently the spectrum of Cψ1 does not
look like an union of coronae: it is the disk D(0, 4). This phenomenon does not occur for
Cψ2 , because in that case the coronae are disjoint.
13
Thus, even when φ acts as an automorphism on a slice, its spectrum can be a disk. However,
it is better to think that it is an union of coronae which intersect rather than to think that
it is a disk. This point of view will be crucial for the study of the dynamics of the associated
composition operator.
4. Dynamics of hyperbolic linear fractional composition operators
4.1. Introduction. The study of their dynamics is an important aspect of the theory of
composition operators (see for instance the books [6], [15]). Regarding linear fractional maps
of the ball, several partial results have been obtained (in [9] for the automorphism case, in
[16] for a special case of hyperbolic maps) and a complete study of the parabolic case has
been done in [2]1. We shall completely solve here the study of the hyperbolic case, closing the
subject at least for composition operators on the Hardy space H 2 (Bd ). The result that we
will prove is more precise than the announced Theorem 1.5. To state it, we need a definition:
Definition 4.1. Let X be a separable complex Banach space and let T ∈ L (X). T is called
chaotic if T is hypercyclic and has a dense set of periodic points.
In other words, T is chaotic if and only if it is hypercyclic and
span ker (T − z) , z ∈ e2πiQ
is dense in X.
Let us now give the complete statement that we shall prove. Observe that the dynamical
properties of Cφ are uniquely determined by the signature of φ.
Theorem 4.2. Let φ ∈ LF M (Bd ) be hyperbolic and univalent.
(1) If pφ = d − 1, then
(a) µCφ is chaotic when εφ = 1 and µ belongs to the corona {λ−d/2 < |µ| < λd/2 };
(b) µCφ is chaotic when εφ = 0 and µ belongs to C\D 0, λ−d/2 .
(2) If pφ < d − 1, then
(a) Cφ is never hypercyclic;
(b) Cφ is supercyclic iff εφ = 0;
(c) Cφ is always cyclic.
We will first prove part (1) of the above theorem. Following a method of M. Taniguchi
[22], we will not prove directly that span ker (T − z) , z ∈ e2πiQ is dense. We will rather
apply an indirect method based on the following chaoticity criterion (see [3, Theorem 6.10]).
Theorem 4.3 (Chaoticity Criterion). Let X be a separable complex Banach space and let
T ∈ L (X). Assume that there exist a dense set D ⊂ X and a sequence of maps Sn : D → D
such that
P n
P
(1)
T (x) and
Sn (x) are unconditionnaly convergent, for each x ∈ D;
(2) T n Sn = I on D for each n.
Then T is chaotic.
The most difficult part of Theorem 4.2 is the proof of (2)(b). The basic tool will be the
outer supercyclicity criterion given by N. Feldman, V. Miller and T. Miller in [14]:
Theorem 4.4 (Outer supercyclicity criterion). Let X be a Banach space and let T ∈ L (X).
Suppose that there exists a dense linear subspace Y and that, for every y ∈ Y , there exists a
dense linear subspace Xy such that:
(1) there exist functions Sn : Y → X such that T n Sn y = y for all y ∈ Y , and
1A result for hyperbolic linear fractional maps was announced in [5]. Unfortunately, there is a mistake in
the proof and the result which is announced is not correct.
14
(2) for any y ∈ Y and any x ∈ Xy , then kT n xk · kSn yk → 0 as n → ∞;
Then T is supercyclic.
The main difficulty consists in exhibiting the subspaces Y and Xy . At this point, in a
surprizing fashion, the result of part (1) will come into play. Indeed, with our method, we
obtain a posteriori that, underthe assumptions of part (1)(b) of Theorem 4.2, the subspace
span ker (µCφ − λ) , λ ∈ e2πiQ is dense. Now, it turns out that these vectors also have a
good behavior under the action of Cφ , even when φ satisfies the assumptions of part (2)(b).
This will allow the construction of Y and Xy .
Finally, the result concerning cyclicity will be a consequence of the following Theorem (see
[2, Corollary 6.3]):
S
Theorem 4.5. Let T ∈ L(X) be such that, for any µ ∈ C, P ∈Pµ ker P (T ) is dense where
Pµ = {P ∈ C[z], P (µ) 6= 0}. Then T is cyclic.
4.2. Proof of chaoticity. In this section, we will prove part (1) of Theorem 4.2 and we will
give some interesting consequences of it. We begin with a technical lemma.
Lemma 4.6. Let a ∈ R+ . The following estimates hold true.
Z
Z
dσ
1
dσ
I := (z,w)∈∂H
. d and (z,w)∈∂H
. ad
2d
d |z + i|2d
d
a
|z
+
i|
|z|≥a
|z|≤a
Proof. The proof consists in a straightforward computation. Successively passing to polar
u
coordinates in I, making the change of variable y = r2 , then putting u = y d−1 and v = d−1 ,
x
we get
Z +∞ Z +∞
r2d−3
I ≤
drdx
d
0
max(0,(a−x)1/2 ) (x2 + r 4 )
Z +∞ Z +∞
y d−2
≤
dydx
d
0
max(0,(a−x)) (x2 + y 2 )
Z +∞ Z +∞
du
.
dx
u 2/(d−1) d
0
max(0,(a−x)d−1 ) 2d
x
1+
xd−1
Z +∞ Z +∞
dv
.
“
”
d dx.
d−1
0
max 0,( a−x
xd+1 1 + v 2/(d−1)
x )
Now we cut the last integral into two parts. First on the part x ≥ a/2, we write
Z +∞ Z +∞
Z +∞
Z +∞
dv
dx
dv
1
dx
=
d . d .
d
d+1
2/(d−1)
2/(d−1)
d+1
x
a
a/2
0
a/2
0
x
1+v
1+v
Next, on the part 0 ≤ x ≤ a/2, using
Z +∞
c
1
(1 + xα )β
dx . min 1,
1
cαβ−1
,
we get
Z
0
a/2 Z +∞
( a−x
x )
dv
d−1
xd+1 1 + v 2/(d−1)
Z
d dx .
x( d−1 −1)(d−1)
dx
2d
xd+1 (a − x)( d−1 −1)(d−1)
a/2
dx
(a − x)d+1
0
Z
.
0
.
2d
a/2
1
.
ad
The proof of the second estimate is easier and just follows from a volume argument.
15
Let us start now with a normal form of φ which may be written
ψ(z, w) = (λz + b, Dw)
with λ−1/2 D unitary, =m(b) ≥ 0. We need to study the chaoticity of µCψ acting on H2 (Hd ).
The main idea of the proof is that ψ induces an automorphism of its characteristic domain,
with an attractive fixed point, equal to ∞, for ψ and an attractive fixed point, denoted by z0 ,
for ψ −1 . Choosing for D functions vanishing both in ∞ and in z0 , we will be able to apply
Theorem 4.3.
Let us proceed with the details. Let α = 1 and β = (z0 − i)/(z0 + i) with z0 = −b/(λ − 1).
Notice that β ∈
/ D since =m (b) ≥ 0. For k ≥ 0, we define the subspace D0,k of H 2 (Bd ) as
follows
n
o
D0,k := (z − α)k (z − β)k Q (z, w) , Q polynomial in (z, w) ∈ C × Cd−1 .
Since α, β ∈
/ D, using the series representation of functions in H 2 (Bd ) and the Hilbertian
properties of this space, it is easy to prove that D0,k is dense in H 2 (Bd ). Denoting by Dk the
analogous of D0,k in the upper half-space,
Dk = P ◦ σc−1 , P ∈ D0,k
where σc : Bd → Hd is the Cayley map, it remains true that Dk is dense in H2 (Hd ) and it is
sufficient to find some k (large enough...) such that (1) and (2) of Theorem 4.3 are verified
for T = µCψ and for some convenient operator S.
We need another lemma.
Lemma 4.7. Let Ω be the characteristic domain of ψ. Then σc−1 (Ω) is bounded.
Proof. Recall that
Ω=
=m(b)
(z, w) ∈ C , =m(z) > kwk −
λ−1
2
d
and that
z − i 2w
,
.
=
z+i z+i
First, (z, w) 7→ (z − i)/(z + i) is bounded on Ω because
z − i 2 <e (z)2 + (=m (z) − 1)2
z + i = <e (z)2 + (=m (z) + 1)2
σc−1 (z, w)
and
(5)
=m (z) + 1 ≥ 1 −
=m (b)
> 0.
λ−1
This is the place where we need to impose, in the normal form, the condition =m(b) ∈
(0, λ − 1). Next, (z, w) 7→ 2w/(z + i) is also bounded on Ω since
2w 2
2 kwk2
2=m (z)
=
z + i
2
2 ≤
2
<e (z) + (=m (z) + 1)
<e (z) + (=m (z) + 1)2
and we conclude using (5) again.
The previous lemma has the following interesting consequence. Let R ∈ Dk . There is some
polynomial Q (z, w) such that
h
i
R (z, w) = (z − α)k (z − β)k Q (z, w) ◦ σc−1 .
16
Because of Lemma 4.7, there exist M, C1, C2 ∈ R such that Q, (z − α)k and (z − β)k are
respectively bounded by M , C1 and C2 on σc−1 (Ω). Then, for every (z, w) ∈ Ω, R(z, w) is
well-defined and we get the following estimations:
z − i k
C
≤
|R (z, w)| ≤ C2 M (6)
z+i
|z + i|k
z − i z0 − i k
≤ C |z − z0 |k .
(7)
|R (z, w)| ≤ C1 M −
z + i z0 + i Let C = C\D 0, λ−d/2 if =m(b) > 0 and C = {λ−d/2 < |µ| < λd/2 } when =m(b) = 0. For
µ ∈ C, n ≥ 1 and R ∈ Dk , we define
λ−n − 1
−n
−n
−n
Sn (R) = µ R λ z − −1
λb, D w
λ −1
= µ−n R ◦ ψn−1 (z, w).
This is a well-defined formula since, for any (z, w) ∈ Hd , ψn−1 (z, w) ∈ Ω. Moreover, (µCψ )n ◦
Sn = I. Therefore, it remains to show that condition (1) of Theorem 4.3 is satisfied.
P
Step 1- n≥0 kSn (R)k < +∞ when k is large enough.
Let ϑ ∈ (0, 1) such that |µ|2 > λ−ϑd . We cut the integral into two parts:
kSn (R)k2H2
2
λ−n − 1
−n
−n
Z
λb,
D
w
R
λ
z
−
λ−1 − 1
−2n
dσ∂Hd
≤ |µ|
2d
(z,w)∈∂Hd
|z + i|
ϑn
|z|≤λ
2
−n − 1
λ
−n
−n
R λ z −
Z
λb, D w −1
λ −1
+ |µ|−2n (z,w)∈∂H
dσ∂Hd .
d
|z + i|2d
ϑn
|z|≥λ
First, since R is bounded on Ω, Lemma 4.6 implies
2
−n − 1
λ
−n
−n
R λ z −
Z
λb, D w −1
λ −1
|µ|−2n (z,w)∈∂H
dσ∂Hd .
d
|z + i|2d
ϑn
|z|≥λ
|µ|−2
λϑd
!n
Next, the inequality (7) above yields
2
λ−n − 1
−n
−n
Z
R
λ
z
−
λb,
D
w
λ−1 − 1
−2n
|µ|
dσ∂Hd
2d
(z,w)∈∂Hd
|z + i|
ϑn
|z|≤λ
2k
Z
−n − 1
−n
λ
−2n
λ z −
dσ∂H
. |µ|
λb
−
z
0
d
−1
(z,w)∈∂Hd
λ −1
ϑn
|z|≤λ
Z
2k
. |µ|−2n (z,w)∈∂H λ−n (z − z0 ) dσ∂Hd
d
|z|≤λϑn
.
Thus,
P
|µ|−2n
λ2(1−ϑ)nk
n≥0 kSn (R)k
.
is convergent, assuming |µ| > λ−d/2 and k is large enough.
Step 2-
P
n≥0 k(µCψ )
n (R)k
17
< +∞ when k is large enough.
For this step, we have to distinguish the cases =m(b) = 0 and =m(b) > 0. When =m(b) >
0, observe that for any (z, w) ∈ ∂Hd ,
n
n
n
λ z + λ − 1 b + i ≥ =m λ − 1 b & λn .
λ−1
λ−1
This yields
k(µCψ )n (R)k2H2
Z
= |µ|2n
.
|µ|
λk
2
n
R λn z + λ − 1 b, Dn w λ−1
|z + i|2d
∂Hd
2n
dσ∂Hd
P
by inequality (6) above. Thus, for k large enough, n≥0 k(µCψ )n (R)kH2 is convergent, and
this property does not depend on the value of |µ|.
When =m(b) = 0 (in fact, b = 0 by our simplifications), we proceed as in Step 1, splitting
the integral into two parts:
Z
n
C (R) 2 = |µ|2n
|R (λn z, Dn w)|2 dσ∂Hd
ψ
H
(z,w)∈∂H
d
|z|≤λ−ϑn
2n
+ |µ|
Z
(z,w)∈∂Hd
|z|≥λ−ϑn
|R (λn z, Dn w)|2 dσ∂Hd .
where ϑ ∈ (0, 1) is such that |µ|2 < λϑd . For the second integral, we may use inequality (6)
satisfied by R above and fix k large enough so that this part becomes the term of a convergent
series. For the first one, we appeal to (the easy part of) Lemma 4.6 to conclude (this is the
point where the condition |µ|2 < λd comes into play).
Application- We conclude this subsection by an application to the density
√ of eigenvectors.
Suppose that =m(b) > 0 and let r > λ−d/2 . Let also ψ0 (z, w) = (λz + b, λw). Since rCψ0
is chaotic, we know that
span ker rCψ0 − eiθ , θ ∈ R is dense.
Let us introduce a decomposition of H2 (Hd ) slightly different from that of Section 3. For
α ∈ Nd−1 , we denote by Eα the set of functions f of H2 (Hd ) which may be written f (z, w) =
F (z)wα , so that we can decompose H2 (Hd ) as ⊕α∈Nd−1 Eα . Each Eα is stable under Cψ0 , so
that
span ker rCψ0 |Eα − eiθ , θ ∈ R is dense in Eα .
Now, for f (z, w) = F (z)wα , one has
λd/2 eiθ
F (z).
r
This means that we have obtained the following corollary:
rCψ0 f = eiθ f ⇐⇒ F (λz + b) =
Corollary 4.8. For each α ∈ Nd−1 , for each r > λ−d/2 , for each b ∈ C with =m(b) > 0,
!
d/2 eiθ
λ
Eα,r,b := span F (z)wα ∈ H2 (Hd ); ∃θ ∈ R, F (λz + b) =
F (z)
r
is dense in Eα .
If we now start with ψ0 (z) = (λz,
√
λw), then we get
18
Corollary 4.9. For each α ∈ Nd−1 , for each r ∈ (λ−d/2 , λd/2 ),
Eα,r,0
λd/2 eiθ
:= span F (z)w ∈ H (Hd ); ∃θ ∈ R, F (λz) =
F (z)
r
α
!
2
is dense in Eα .
4.3. Proof of non-hypercyclicity. We suppose that ψ ∈ LF M (Hd ) is given by
ψ(z, u, v) = (λz + b, Du, Av + c)
with dim v > 0 and we have to prove that Cψ is not hypercyclic. This is the easy part of the
proof which depends on the following lemma. The notations are that of Section 3.
Lemma 4.10. There exists N ∈ N such that, if we denote by PN the orthogonal projection
on KN , then kPN ◦ Cψ|KN k < 1.
Proof. The proof goes along the same lines as that of Lemma 3.5. Recall that Cψ is uppertriangular in the decomposition H2 (Hd ) = ⊕|α|<N Hα ⊕ KN . Let A = U ΣV be the singular
value decomposition of A. We factorize Cψ as Cψ = Cτ1 ◦ CψΣ ◦ Cτ2 = Cτ2 ◦ψΣ ◦τ1 where

 τ1 : (z, w) 7→ (z, u, V v)
ψΣ : (z, w) 7→ λz + b, Du, Σv + U −1 c .

τ2 : (z, w) 7→ (z, u, U v)
⊥,
Since Cτ1 and Cτ2 are isometries of H2 (Hd ) and since they both preserve KN and KN
it suffices to establish kPN ◦ CψΣ |KN k < 1 for N large enough. Let us finally introduce
ψ0 (z, w) = (λz + b, Du, Σv). Clearly, PN ◦ CψΣ |KN = Cψ0 |KN . We can now conclude because,
as in Lemma 3.5, the sequence kCψ0 |KN k) goes to 0 as N goes to infinity.
Let us now deduce that Cψ is not hypercyclic. Suppose that the contrary holds and let
⊥ and f ∈ K , with N ≥ 0. The set
f = f1 + f2 be a hypercyclic vector with f1 ∈ KN
2
N
n
{PN Cψ f ; n ≥ 0} has to be dense in KN . Since Cψ is upper-triangular in the decomposition
⊥ ⊕ K , this means that {(P ◦ C )n f ; n ≥ 0} has to be dense. When N is large enough,
KN
2
N
N
ψ
this is impossible, since the sequence ((PN ◦ Cψ )n f2 )n≥0 goes to zero.
4.4. Proof of supercyclicity. We suppose now that ψ is given by the formula
ψ(z, u, v) = (λz + b, Du, Av + c)
with =m(b) > 0 and we intend to show that Cψ is supercyclic. We will apply the outer
supercyclicity criterion with Y = ⊕α∈Nd−1 Eα,r,b for some fixed r > λ−d/2 . Y is dense in
H2 (Hd ) by Corollary 4.8.
By linearity it suffices, for each y = F (z)wα ∈ Eα,r,b , with F (λz + b) = r−1 λd/2 eiθ F (z) and
α ∈ Nd−1 , to find a subset Xy ⊂ H2 (Hd ) and a sequence of maps (Sn ) satisfying the assumptions of the outer supercyclicity criterion. The idea is that, although Cψ is not invertible, it
will be invertible on a finite-dimensional space containing y, allowing the construction of the
sequence (Sn y).
From now on, we will rather write
ψ (z, w) = (λz + b, M w + c)
D 0
is an upper triangular matrix. Let also N = |α| and let us consider
0 A
the following finite-dimensional subspace of H2 (Hd ):


X

Z=
aβ F (z)wβ ; aβ ∈ C .


where M =
|β|≤N
19
Z is contained in H2 (Hd ), since F (z)wβ belongs to H2 (Hd ) as soon as F (z)wα belongs to
H2 (Hd ), for any |β| ≤ |α|. We endow Nd−1 with the order defined in Section 3. Hence,
we get immediately that Z is mapped into itself by Cψ and that the matrix of Cψ|Z with
respect to the canonical basis (F (z)wβ )|β|≤N is upper-triangular with non-zero coefficients
on the diagonal. This implies that the map Cψ|Z : Z → Z is invertible. In other words,
there exists a constant C (which depends on y) such that, for any f ∈ Z, we may find g ∈ Z
with Cψ (g) = f and kgk ≤ Ckf k. It is now easy to construct by induction a sequence (Sn y)
satisfying kSn yk ≤ C n kyk and Cψn Sn y = y.
Let us now proceed with the construction of Xy . Let us consider γ ∈ Nd−1 . By linearity, it
is sufficient to construct a dense subspace Xy,γ of Eγ such that, for any x ∈ Xy,γ , kCψn xk·kSn yk
goes to zero. We shall prove that Eγ,ρ,b , for a sufficiently large ρ, is convenient. By linearity
again, it is sufficient to deal with any x = G(z)wγ with G(λz + b) = ρ−1 λd/2 eiδ G(z). We
point out that the condition we will impose on ρ must be independent of x (nevertheless, it
can depend on
γ).
nP
o
β ; a ∈ C , endowed with the `2 -norm:
Let Z0 :=
a
w
β
|β|≤|γ| β
2
X
X
β
aβ w =
|aβ |2 .
|β|≤|γ|
|β|≤|γ|
Let also T be the self-map of Z0 defined by T (wβ ) = (M w + c)β . T defines a bounded
operator on the finite-dimensional space Z0 , and kT k just depends on |γ|. Furthermore,
Z
λdn
|G(z)|2 |T n (wγ )|2
2
n
kCψ (x)k =
dσ
ρ2n ∂Hd
|z + i|2d
P
Z
|G(z)|2 β |aβ |2 |wβ |2
λdn
=
dσ
ρ2n ∂Hd
|z + i|2d
P
P
|aβ |2 ≤ kT k2n , so that
where (aβ )|β|≤|γ| is defined by T n (wγ ) = β aβ wβ . In particular,
there exists a constant D (which depends on x) such that
kCψn (x)k ≤ D
kT kn λdn/2
.
ρn
This yields
kSn yk · kCψn (x)k ≤ D
C n kT kn λdn/2
.
ρn
It suffices to choose ρ > CkT kλd/2 (and this choice is independent of x) to end up the proof
of the supercyclic part of Theorem 4.2.
4.5. Proof of cyclicity and non-supercyclicity. In this last subsection, we work with
ψ ∈ LF M (Hd ) univalent which may be written
ψ(z, u, v) = (λz, Du, Av)
with dim(v) > 0 and we have to prove that Cψ is cyclic and not supercyclic.
We prove the non-supercyclicity of Cψ by spectral considerations. Indeed, let H0 = H0 ,
H1 = ⊕1≤|α|<N Hα and H2 = KN for some large N . Cψ is diagonal with respect to the
decomposition H2 (Hd ) = H0 ⊕ H1 ⊕ H2 . Moreover, the work of Section 3 shows, at least if
N is large enough, that there exists R0 > R2 such that, for any f0 ∈ H0 and any f2 ∈ H2 ,
kCψn f0 k ≥ R0n kf0 k and kCψn f2 k ≤ R2n kf2 k.
20
That Cψ is non-supercyclic follows now from a standard argument in linear dynamics, similar
to that which proves that each component of the spectrum of a supercyclic operator has to
intersect the unit circle (see [3, Theorem 1.24]).
To prove that Cψ is cyclic, it is convenient to write it as
ψ(z, w) = (λz, M w).
We fix µ ∈ Cd−1 , α ∈ Nd−1 and we observe that M defines, by composition, a linear map L on
the finite-dimensional space of homogeneous polynomials of degree |α|. Moreover, since M is
invertible, it is easy to prove that L is also invertible (for instance, since M is upper-triangular,
P
L is also upper-triangular in the basis (wβ )|β|=|α| ordered with ≺). Let Q(X) = j bj X j be
the minimal polynomial of L. Q does not vanish at 0 since L is invertible and it satisfies
P
j
α
−d/2 , λd/2 ) such that rµ/λd/2 eiθ is never a root
j bj (M w) = 0. We consider now r ∈ (λ
of Q. Let finally F (z)wα ∈ H2 (Hd ) be such that F (λz) = r−1 λd/2 eiθ F (z) and let us set
P (X) = Q rλ−d/2 e−iθ X . Then
j j
X P (Cψ ) (F (z)wα ) =
bj rλ−d/2 e−iθ · r−1 λd/2 eiθ F (z)(M j w)α = 0
j
S
and P (µ) 6= 0. This means that Eα,r,0 ⊂ P ∈Pµ ker P (Cψ ) for any µ ∈ C. Theorem 4.5 and
Corollary 4.9 allow us to conclude that Cψ is cyclic.
5. Applications to general composition operators
5.1. Introduction. As mentioned in the introduction of the paper, one of the reasons to
introduce linear fractional composition operators was the hope to use them to deduce properties of composition operators associated to general holomorphic self-maps of the ball. This
is called the transference principle, which has been proved to be very efficient in the onedimensional setting. Here is one of the results that we may obtain on D (see [6]). Let
φ : D → D be holomorphic and satisfying the following properties:
(i) φ extends to a continuous and univalent self-map of D;
(ii) φ(D\{1}) ⊂ D;
(iii) φ has Denjoy-Wolff point at +1 with φ0 (1) ∈ (0, 1) and φ has C 1+ε -smoothness at +1.
Then Cφ is hypercyclic.
The strategy applied by Bourdon and Shapiro to prove this result can be summarized as
follows. They first build a linear-fractional model for φ; namely, they identify two self-maps
of D, denoted by σ and φλ , such that φλ is a hyperbolic linear-fractional map of D and the
intertwining equation σ ◦ φ = φλ ◦ σ holds true.
Next, they transfer the hypercyclicity of Cφλ to Cφ via Cσ ; precisely, if f is any hypercyclic
vector of Cφλ , then Cσ (f ) is a hypercyclic vector of Cφ .
A general theory of linear fractional models in Bd has been developped in [1] (see also [8]).
However, two difficulties arise in the several-variables context when we try to imitate the last
part of the proof. First, not every self-map of Bd induces a bounded composition operator
on H 2 (Bd ). Second, because of the lack of Mergelyan theorem in Cd , it is rather difficult to
prove that a composition operator Cσ has dense range in H 2 (Bd ).
At this stage, we do not expect to be able to prove a theorem which is as general as Bourdon
and Shapiro theorem. However, we will show that a small perturbation of a hyperbolic linearfractional map preserves the dynamical properties of the associated composition operator.
Here is the main statement of this section.
Theorem 5.1. Let ψ : Hd → Hd be a holomorphic self-map of Hd which may be written
ψ(z, w) = (λz + R(z), M w + c) = (ψ1 (z), ψ2 (w))
with
•
•
•
•
•
21
ψ1 extends continuously to a continuous and univalent self-map of P+ ∪ {∞};
R(z) = o(|z|1−ε ), ε > 0, when |z| → +∞, z ∈ P+ ;
λ > 1, inf z∈P+ =m(R(z)) ≥ d > 0;
Q = λI − M M ∗ is a hermitian positive semi-definite matrix;
d − kck2 > hQ+ M ∗ c, M ∗ ci.
Suppose moreover that Cψ is a bounded operator on H2 (Hd ). Then Cψ is supercyclic on
H2 (Hd ).
Let us comment this theorem. The two first conditions are regularity properties which
are required on ψ1 . They are already present in the one-variable setting. The three last
conditions are here to ensure that ψ is a hyperbolic self-map of Hd . Finally, we have to add
that Cψ defines a bounded composition operator on H2 (Hd ), since this is (in general) not
automatic. If we had imposed more regularity on ψ near ∞, we would obtain the continuity
of Cψ by applying Wogen’s theorem (see [24]).
Our proof that Cψ is supercyclic does not consist in the transference of a supercyclic vector.
We will rather transfer the assumptions of the supercyclicity criterion. More precisely, we
assume for a while that the two following claims are true:
Claim 1. There exists σ = (E, D) : Hd → Hd and b ∈ C with =m(b) > 0 so that
σ ◦ ψ = ψλ,b,M,c ◦ σ
with ψλ,b,M,c (z, w) = (λz + b, M w + c) is a hyperbolic linear fractional map of Hd .
Claim 2. For any α ∈ Nd−1 , Cσ maps Eα continuously into Eα , and Cσ (Eα ) is dense in Eα .
Let us show how we can deduce from this that Cψ is hypercyclic by applying the outer
supercyclicity criterion. Let r > λ−d/2 and Fα,r,b = Cσ (Eα,r,b ). By Claim 2 and since Eα,r,b is
dense in Eα , the subset Y = ⊕α∈Nd−1 Fα,r,b is dense in H2 (Hd ). Let us fix y = F ◦ E(z, w)wα ∈
Fα,r,b and let us set


X

Z=
aβ F ◦ E(z, w)wβ ; aβ ∈ C


|β|≤N
nP
o
β
with |α| = N . Let us also consider Z =
|β|≤N aβ F (z)w ; aβ ∈ C like in Section 4.4.
The intertwining map σ induces a commutative diagram
Cψλ,b,M,c
/Z
Z
Cσ
Z
Cψ
Cσ
/Z
Since Z and Z have finite dimension, and since Cσ is one-to-one, it is an isomorphism from
Z onto Z. Therefore, the work done in Section 4.4 carries on into this new context, and we
can find a sequence (Sn y) in Z and a positive constant C such that kSn yk ≤ C n kyk and
Cψn Sn y = y.
The construction of Xy follows the same idea. We shall prove that, for any γ ∈ Nd−1 , for
ρ := ρ(γ, y) large enough, any vector x of Fγ,ρ,b , which is dense in Eγ , will satisfy kSn yk ·
kCψn xk → 0. The proof done above can be reproduced mutatis mutandis. We may then apply
the outer supercyclicity criterion since ⊕γ∈Nd−1 Fγ,ρ,b is dense in H2 (Hd ).
Therefore, our last task is to prove Claim 1 and Claim 2.
22
5.2. The intertwining map. We suppose that ψ1 (z) = λz + R(z) with inf =m(R(z)) ≥
d > 0 and R(z) = o(|z|1−ε ) for some ε > 0. We also assume that ψ1 is continuous and
[n]
univalent on P+ ∪ {∞}. Let us set z(n) = ψ1 (z) = ψ1 ◦ . . . ψ1 (z). It is straightforward to
observe that
n−1
X R(z(j))
z(n)
=
z
+
.
λn
λj+1
j=0
Pn−1 R(z(j))
converges
The work done in the proof of Theorem 4.7 in [6] shows that the series j=0
λj+1
uniformly on compact subsets of P+ to a holomorphic function H. The function z 7→ z +H(z)
is continuous and univalent on P+ ∪ {∞}. Moreover, the imaginary part of H is bounded
below. Indeed,
+∞
X
d
d
=m(H(z)) ≥
=
> 0.
j+1
λ
λ−1
j=0
Let us set
E(z) = z + H(z) −
iθd
λ−1
with θ ∈ (0, 1). Then
z(n + 1)
iθd
lim
−
n
n→+∞
λ
λ−1
iθd
iθd
−
= λ E+
λ−1
λ−1
= λE + iθd.
E ◦ ψ1 (z) =
Thus, if we set D(z, w) = w, σ = (E, D) and b = iθd, then σ is a self-map of Hd which
satisfies σ ◦ ψ = ψλ,M,b,c ◦ σ. Moreover, if we choose θ sufficiently close to 0, then =m(b) −
kck2 > hQ+ M ∗ c, M ∗ ci, so that ψλ,M,b,c becomes a linear fractional map of Hd . Thus Claim
1 is proved. However, to prove Claim 2, we shall need several properties of E, which are
summarized below:
(Q1): E maps P+ into P+ and is univalent on P+ ∪ {∞};
(Q2): There exist two constants C1 , C2 > 0 such that
∀z ∈ P+ , C1 |z + i| ≤ |E(z) + i| ≤ C2 |z + i|.
Property (Q1) has already been proved above. Moreover, it implies that (Q2) is true on
compact subsets of P+ . So we just need to prove this last property when |z| is large enough.
Now, since |R(z)| = o(|z|1−ε ), for any µ > λ, we can find M > 0 such that, for any z ∈ P+
with |z| ≥ M ,
|z| ≤ |ψ1 (z)| ≤ µ|z|.
We start with such a z and we iterate ψ1 . We find |z(j)| ≤ µj |z|, so that
|H(z)| ≤
+∞ j(1−ε)
X
µ
|z|1−ε
j=0
λj+1
≤ C|z|1−ε
if µ1−ε is smaller than λ. This gives immediately (Q2).
5.3. The intertwining composition operator. We conclude this section by the proof of
Claim 2. In this context, it is easier to work directly in Bd . So we set
Fα = f (z)wα ∈ H 2 (Bd ) = {g ◦ σc ; g ∈ Eα } .
23
We have to prove that Cσc−1 ◦σ◦σc maps Fα into itself continuously and with dense range. For
notational simplicity, we denote by σ the map σc−1 ◦ σ ◦ σc . However, its coordinate functions
are now denoted by σ1 and σ2 , so that
σ1 (z, w) =
E ◦ σc (z, w) − i
2iw
w
:=
and σ2 (z, w) =
.
E ◦ σc (z, w) + i
G(z)
(1 − z) E ◦ σc (z, w) + i
Observe that E ◦ σc (z, w) depends only on z. Thus, for any F (z, w) = f (z)wα in Fα ,
F ◦ σ(z, w) =
f ◦ σ1 (z) α
w
G(z)|α|
can be written g(z)wα . To compute the norm of F ◦ σc in H 2 (Bd ), we use the following
formula (see [25, Lemma 1.10]):
2
Z
2 d−2
kF ◦ σk = C
Z
(1 − |z| )
|f ◦ σ1 (z)|2
|w|2 =1−|z|2
|z|≤1
|wα |2
dm(w)dA(z).
|G(z)|2|α|
Now, it is easy to show that G = (E ◦ σc + i)/(σc + i), so that, by (Q2), there exists some
C > 0 such that
1
≤ C for any z ∈ D.
|G(z)|2|α|
We then find
Z
Z
2
2 d−2
kF ◦ σk .
(1 − |z| )
|f ◦ σ1 (z)|2 |wα |2 dm(w)dA(z)
|z|≤1
|w|2 =1−|z|2
Z
.
(1 − |z|2 )d−2+|α|+(d−1)/2 |f ◦ σ1 (z)|2 dA(z).
|z|≤1
Now, σ1 is a self-map of D, so it defines a continuous operator on the Bergman space
Ad−2+|α|+(d−1)/2 (D). Hence,
Z
kF ◦ σk2 .
(1 − |z|2 )d−2+|α|+(d−1)/2 |f (z)|2 dA(z)
|z|≤1
Z
Z
2 d−2
2
.
(1 − |z| ) |f (z)|
|wα |2 dm(w)dA(z)
|z|≤1
2
|w|2 =1−|z|2
. kF k .
Let us now prove that Cσ (Fα ) is dense in Fα . More precisely, we shall prove that {σ1p σ2α ; p ≥
m α
0} is dense in Fα . Let
P z w p∈ Fαα and let us approximate it by a finite linear fractional
combination F (z) = p≥0 ap σ1 (z)σ2 (z, w). Using the same formula, we infer that
kF − z m wα k2
2
X
|wα |2
p
2 d−2
|α|
m
ap σ1 (z) − G(z) z .
(1 − |z| )
dm(w)dA(z)
2|α|
|z|≤1
|w|2 =1−|z|2 p≥0
|G(z)|
2
X
Z
p
2 d−2+|α|+(d−1)/2 |α| m .
(1 − |z| )
a
σ
(z)
−
G(z)
z
p
1
dA(z).
|z|≤1
p≥0
Z
Z
We conclude because σ1 is an univalent and continuous self-map of D. By Walsh’s theorem,
the polynomials are dense in the algebra A(σ1 (D)). This in turn implies that the polynomials
in σ1 are dense in the Bergman space Ad−2+|α|+(d−1)/2 (D). Observe that, this G is bounded,
the function z 7→ G(z)|α| z m belongs to this space.
24
5.4. Conclusion. We hope that Theorem 5.1 could be extended to a more general context.
As mentioned before, at least on B2 , a linear fractional model does exist for a wide class
of self-maps of B2 . Assuming some regularity on ψ near ∞, it is reasonable to expect that
the transfer map σ will have some regularity near ∞. We then can hope to apply Wogen’s
theorem to ensure that Cσ defines a continuous operator on H2 (Hd ). The main problem
seems to be how to prove that Cσ has dense range. A famous example of Wermer [23] points
out that a domain can be biholomorphic to the bidisc without being a Runge domain (the
polynomials are not dense in H(G)). That was the main reason to restrict ourselves to a map
ψ = (ψ1 , ψ2 ) where ψ1 just depends on z and ψ2 just depends on w.
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HYPERBOLIC COMPOSITION OPERATORS ON THE BALL 1

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