asymptotic behaviour of the powers of composition operators on banach spaces … · 2017-08-08 ·...

ASYMPTOTIC BEHAVIOUR OF THE POWERS OFCOMPOSITION OPERATORS ON BANACH SPACES

OF HOLOMORPHIC FUNCTIONS

W. ARENDT, I. CHALENDAR, M. KUMAR AND S. SRIVASTAVA

Abstract. We study the asymptotic behaviour of the powers Tn

of a composition operator T on an arbitrary Banach space X ofholomorphic functions on the open unit disc D of C. We showthat for composition operators, one has the following dichotomy:either the powers converge uniformly or they do not converge evenstrongly. We also show that uniform convergence of the powers ofan operator T ∈ L(X) is very much related to the behaviour ofthe poles of the resolvent of T on the unit circle T of C and thatall poles of the resolvent of the composition operator T on X arealgebraically simple. Our results are applied to study the asymp-totic behaviour of semigroups of composition operators associatedwith holomorphic semiflows.

1. Introduction

Composition operators on spaces of holomorphic functions have beena subject which attracted attention for many years and several mono-graphs are devoted to it (see [12, 23]). One reason is the interestinginterplay between the theory of holomorphic functions and operatortheory, another is the universal character of composition operators.

1991 Mathematics Subject Classification. 47B33, 30A05, 47A10, 47D06.Key words and phrases. Composition operators, Banach spaces of analytic func-

tions, asymptotic behaviour, poles of the resolvent, mean ergodicity, holomorphicsemiflows, strongly continuous semigroups.

Corresponding author : Isabelle Chalendar, Universite Paris Est Marne-la-Vallee 5, Bd Descartes, Champs sur Marne 77454 Marne-la-Vallee, Cedex 2, [email protected].

Wolfgang Arendt, Institute of Applied Analysis, University of Ulm, 89069, Ulm,Germany. [email protected].

Mahesh Kumar, Lady Shri Ram College For Women, Department of Mathemat-ics, University of Delhi, Delhi, India. [email protected].

Sachi Srivastava, Department of Mathematics, University of Delhi, South Cam-pus, Delhi, India. sachi [email protected].

1

2 W. ARENDT, I. CHALENDAR, M. KUMAR AND S. SRIVASTAVA

One operator theoretical topic which seems to be neglected so far isthe asymptotic behaviour of the powers T n of a composition operatorT . This is the subject we treat in the present paper.

Given an operator T ∈ L(X), where X is a complex Banach space,we may distinguish the following modes of convergence which are or-dered in decreasing strength.

(U) Uniform convergence: limn→∞ Tn exists in L(X);

(S) Strong convergence: limn→∞ Tnx exists in X for all x ∈ X;

(W) Weak convergence: T nx converges weakly in X for all x ∈ X;(E) Ergodicity (Cesaro convergence): limn→∞

1n

∑n−1k=0 T

kx exists inX for all x ∈ X;

(D) Decomposition property: X = fix(T )⊕Im(I − T ), where fix(T ) =ker(I − T ) = x ∈ X : Tx = x.

In other words,

(U)⇒ (S)⇒ (W )⇒ (E)⇒ (D),

and in any of these cases, the limit of T n in the uniform, strong, weakor Cesaro sense is the projection onto fix(T ) along the decomposition(D).

If T is power bounded (i.e. supn∈N ‖T n‖ < ∞), then (D) ⇐⇒ (E)(see [21, Theorem 1.3, p. 26] and [26, Ch.VIII, §3]). Moreover, if X isreflexive and T is power bounded, then (E) and (D) hold automatically.

Uniform convergence is closely related to properties of the essentialspectrum (see Proposition 3.1). Also, for strong convergence, spectralcriteria exist, but are more difficult to prove (see [2, Theorem 5.1]).

It turns out, that for composition operators, the situation is veryspecial: If the powers converge strongly, then they converge alreadyuniformly. In other words, one has the following dichotomy: either thepowers converge uniformly or they do not converge even strongly. Thisis our main result in Section 4.

Uniform convergence of the powers of an operator T ∈ L(X) is verymuch related to the behaviour of the poles of the resolvent of T on theunit circle T of C.

One of our main results is true for composition operators on anarbitrary Banach space of holomorphic functions on the open unit discD of C (see Section 2 for a precise definition).

Theorem 1.1. Let ϕ : D→ D be holomorphic such that f ϕ ∈ X forall f ∈ X. Then all poles of the resolvent of the composition operatorCϕ : f 7→ f ϕ on X are algebraically simple.

Recall that if λ0 is a pole of the resolvent of an operator T (for short,λ0 is a pole of T ), then the algebraic multiplicity of λ0 is the dimension

ASYMPTOTIC BEHAVIOUR OF POWERS OF COMPOSITION OPERATORS 3

of the spectral projection associated with λ0. In the finite dimensionalcase, this is the multiplicity of λ0 as a root of the characteristic poly-nomial associated with T . Of course, algebraically simple means thatthe algebraic multiplicity is one. Our main example of spaces X is theHardy space Hp(D), 1 ≤ p < ∞. We will see that for all holomorphicfunctions ϕ : D→ D, the operator Cϕ on Hp(D) given by Cϕf = f ϕsatisfies (U) if and only ϕ has a fixed point in D and ϕ is not inner.Actually, if ϕ is inner, then (S) is violated. However, (W ) is alwaystrue. The situation in the disc algebra A(D), the Wiener algebra W (D)and H∞(D) is also dichotomical. For these spaces, ergodicity has beeninvestigated recently by Beltran-Meneu, Gomez-Collado, Jorda, andJornet (see [7]). Concerning uniform convergence, we obtain the fol-lowing result.

Theorem 1.2. Let X ∈ A(D),W (D), H∞(D). Let ϕ : D → D beholomorphic such that ϕ ∈ A(D) if X = A(D) and sup

n∈N‖ϕn‖W (D) < ∞

if X = W (D). Then CϕX ⊂ X and Cϕ satisfies (U) on X if and onlyif Cϕ satisfies (S).

Our results can be applied to holomorphic semiflows. Let G ∈∩0<p<1H

P (D).Then the radial limits G(eiθ) := limr→1G(reiθ) exist al-most everywhere. Assume that

(1) Re(zG(z)) ≤ 0 a.e. on T.Consider the operator A on H2(D) given by D(A) = f ∈ H2(D) :Gf ′ ∈ H2(D), Af = Gf ′. Then by the results of [5] which extend ageneration theorem for holomorphic flows by Berkson and Porta, theoperator A generates a C0-semigroup of composition operators (Ct)t≥0on H2(D). We will show the following.

Theorem 1.3. Assume in addition to (1) that

a) Rez∈ω(zG(z)) < 0 for some measurable set ω ⊂ T of positiveLebesgue measure, and that

b) G(α) = 0 for some α ∈ D.Then limt→∞ ‖Ct − P‖L(H2(D)) = 0, where P is the projection given byPf = f(α)1D.

This is proved in Section 6, where also the necessity of Conditionsa) and b) is shown.

All our results depend, of course, on a good knowledge of the be-haviour of the iterates ϕn := ϕ · · · ϕ (n times) expressed by theDenjoy–Wolff theorem (see Theorem 4.1 below), as well as a goodknowledge of the essential spectrum provided by Shapiro, Matache andmany others (see [24], [19]).


The paper is organized as follows. In Section 2, we describe thoseBanach spaces which can be injected continuously into what we denoteby Hol(D), the space of all holomorphic functions defined on D. Polesare investigated in Section 3, where Theorem 1.1 is proved. In Section4, we show that strong convergence implies uniform convergence inmany cases. Section 5 is concerned with specific algebras such as theWiener algebra, the disc algebra and H∞(D). Finally, Section 6 isdevoted to semigroups of composition operators on the Hardy spaceH2(D).

2. Banach spaces of holomorphic functions

Let Ω ⊂ C be a connected open set, and let Hol(Ω) denote the spaceof all holomorphic functions on Ω. Let X be a subspace of Hol(Ω),which is a Banach space endowed with its proper norm ‖ · ‖X . Thefollowing proposition describes the compatibility of the norm ‖ · ‖Xwith the topology of Hol(Ω).

If z ∈ Ω, then we denote by δz : Hol(Ω) → C, the evaluation at z;i.e.

〈δz, f〉 := f(z), f ∈ Hol(Ω).

We write δz ∈ X ′, if there exists c ≥ 0 such that

|〈δz, f〉| ≤ c‖f‖X for all f ∈ X.

Proposition 2.1. The following assertions are equivalent:

(i) δz ∈ X ′ for all z ∈ Ω;

(ii) fn → f in X implies that f(m)n (z)→ f (m)(z) uniformly on each

compact subset of Ω and for all m ∈ N0, where N0 = N ∪ 0;(iii) there exists a compact set K ⊂ Ω such that z ∈ K : δz ∈ X ′

is infinite.

If these three equivalent conditions are satisfied, we write X →Hol(Ω) and call X, a Banach space of holomorphic functions.

Proof of Prop. 2.1. (iii) ⇒ (ii) Let K1 ⊂ Ω be compact such thatK ⊂ K1. Then, consider the restriction map:

J : f ∈ X → f|K1 ∈ C(K1),

where C(K1) denotes the Banach space of continuous functions on K1

endowed with the supremum norm. Then J is linear and has a closedgraph. Indeed, let fn → f in X such that fn|K1

→ g in C(K1). Byassumption (iii), we get f(z) = g(z) for infinitely many points havinga limit point in K. It follows that f = g by the isolated zeros theorem


of holomorphic functions. The closed graph theorem now implies thatJ is continuous. Consequently, there exists c1 ≥ 0 such that

‖f|K1‖C(K1) ≤ c1‖f‖X , f ∈ X.This implies (ii) for m = 0, and it is true for all m ∈ N via Cauchy’sintegral formula, as is well-known.

The implications (ii)⇒ (i)⇒ (iii) are obvious.

One may ask, whether the equivalent conditions in Proposition 2.1,are automatically satisfied. This is false as following result shows.

Proposition 2.2. Assume that X is an infinite dimensional complexBanach space such that X → Hol(Ω). Then there exists a norm ‖ · ‖1on X such that (X, ‖ · ‖1) is complete and such that for each compactK ⊂ Ω, the set z ∈ K : δz ∈ (X, ‖ · ‖1)′ is finite.

Proof. There exists a bijective linear map S : X → X, equal to itsinverse and which is not continuous (take φ : X → C linear and notcontinuous, u ∈ X such that φ(u) = 1 and set S(x) := x − 2φ(x)u,see [4]). Thus ‖f‖1 := ‖Sf‖X defines a norm on X which is notequivalent to ‖ · ‖X . Let K be a compact subset of Ω. Assume thatz ∈ Ω : δz ∈ (X, ‖·‖1)′ is infinite. As, in the proof of Proposition 2.1,we deduce that the identity operator from (X, ‖ · ‖X) to (X, ‖ · ‖1) isbounded. This implies the equivalence of the two norms by the theoremof the continuous inverse, a contradiction.

The paper [1] by Alpay and Mills contains a similar result for theHardy space.

If X is a Hilbert space such that X → Hol(Ω), then for each z ∈ Ω,we find kz ∈ X such that

f(z) = 〈f, kz〉 for all f ∈ X.One frequently calls, the family kz, z ∈ Ω, a reproducing kernel. Thefunction z ∈ Ω→ kz ∈ X is holomorphic, as a vector-valued mapping.More generally, we have the following result.

Proposition 2.3. Let X be a Banach space such that X → Hol(D).Then the mapping δ : z ∈ Ω→ δz ∈ X ′ is holomorphic.

Proof. Let K be compact subset of Ω so that

‖f‖C(K) := supz∈K|f(z)| ≤ cK‖f‖X , f ∈ X,

as in the proof of Proposition 2.1. Thus, for z ∈ K,

‖δz‖X′ = sup|f(z)| : f ∈ X, ‖f‖X ≤ 1 ≤ cK .


Thus, the mapping δ is locally bounded and then the claim followsfrom [3, Proposition A.3(ii)].

Let D be the open unit disc of the complex plane C. For 1 ≤ p ≤ ∞,the Hardy spaces Hp(D) are the classical Banach spaces of holomorphicfunctions, whereas A(D) denotes the disc algebra, i.e. the algebra ofbounded holomorphic functions on D continuous on the closure of D.

Another class form the so called weighted Hardy spaces H2(β) [12,Section 2.1] which we want to consider also for p 6= 2. Let β = (βn)n≥0be a sequence of positive reals and consider the Banach spaces

`p(β) :=

(an)n≥0 :

∑n≥0

|an|pβpn <∞

,

endowed with the norm ‖(an)n≥0‖p`p(β) =∑

n≥0 |an|pβpn for 1 ≤ p <∞.

Let f ∈ Hol(D). Then there exist unique an ∈ C such that∑∞

n=0 anzn =

f(z) for all z ∈ D. We set f(n) := an. For 1 ≤ p < ∞, we define thespace

Hp(β) :=f ∈ Hol(D) : (f(n))n≥0 ∈ `p(β)

.

Obviously, Hp(β) is a normed space for ‖f‖Hp(β) := ‖(f(n))n≥0‖`p(β)containing the polynomials. An extra condition on β is required inorder to make Hp(β) a Banach space.

Proposition 2.4. Let 1 ≤ p < ∞ and β = (βn)n≥0 a sequence ofpositive reals. The following assertions are equivalent:

(i) Hp(β) is complete;(ii) the power series

∑∞n=0 anz

n has radius of convergence greateror equal to 1 for all (an)n≥0 ∈ `p(β);

(iii) lim infn→∞ β1/nn ≥ 1;

(iv) δz ∈ Hp(β)′ for all z ∈ D.

We omit the proof and refer to the arguments given in [12, Section2.1].

If ρ := lim infn→∞ β1/nn > 1, then the radius of convergence of each

power series in Hp(β) is greater or equal to ρ. Thus, also points zoutside D, define a continuous functional δz ∈ Hp(β)′. We now givea condition which implies that only for z ∈ D one has δz ∈ Hp(β)′.In fact, δz defines a multiplicative functional on the polynomials. Wewant to make sure by a condition, that they extend continuously toHp(β), only if z ∈ D. The following result extends [12, Thm 2.15] tothe various spaces we have introduced.


Proposition 2.5. Let 1 ≤ p < ∞ and (βn)n≥0 a sequence of positive

reals such that lim infn→∞ β1/nn ≥ 1. Assume that

∑n≥0 β

−qn = ∞ if

p > 1, with 1/p+ 1/q = 1, and infn≥0 βn = 0 if p = 1. Let L ∈ Hp(β)′

be such that

〈L, en〉 = 〈L, e1〉n for all n ∈ N,where en(z) = zn. Then there exists z0 ∈ D such that L = δz0.

Proof. There exists a sequence of complex numbers (bn)n≥0 such that

〈L, f〉 =∑n≥0

anbnβpn

for all f ∈ Hp(β), f(z) =∑

n≥0 anzn, where

∑n≥0 |bn|qβpn < ∞ if

p > 1, and supn≥0 |bn| < ∞ if p = 1. Let z0 = 〈L, e1〉 ∈ C. Then, byassumption,

〈L, f〉 =N∑n=0

anzn0 ,

for all polynomials f(z) =∑N

n=0 anzn. Therefore zn0 = bnβ

pn for all

n ∈ N.If p > 1, then we get∑

n≥0

|z0|nqβ−pqn βpn =∑n≥0

|bn|qβpn <∞.

Since −pq + p = −q and∑

n≥0 β−qn =∞, it follows that z0 ∈ D.

If p = 1, then supn∈N |bn| <∞, and thus supn∈N|z0|nβn

<∞. It follows

from the assumptions that z0 ∈ D.Since 〈L, f〉 = f(z0) for all polynomials f , the conclusion follows

from the density of the polynomials.

Remark 2.6. Assume that infn≥0 βn > 0 if p = 1 and∑

n≥01βqn< ∞

if 1 < p < ∞, with 1/p + 1/q = 1. Then lim infn→∞ β1/nn ≥ 1 and

δz ∈ Hp(β)′ for all z ∈ D.

For p = 2, the spaces H2(β) comprises several classical Hilbertspaces. For example, if βn = 1 for all n, then H2(β) = H2(D).

If β0 = 1 and βn =√n for n ≥ 1, then H2(β) is the Dirichlet space

D, whereas if βn = 1√n+1

, H2(β) is the Bergman space.

In the case where β = 1, i.e. βn = 1 for all n ≥ 0, the space Hp(1)is isometrically isomorphic to `p if one identifies (an)n≥0 in `p withthe power series

∑n≥0 anz

n. In particular, H2(1) = H2(D). However,Hp(1) 6= Hp(D) for p 6= 2!


The case p = 1 is of special interest. Actually, H1(1) = W (D), theWiener algebra of all power series with absolute convergent coefficients.This is a commutative Banach algebra with unit.

3. Simple poles and uniform convergence

Our aim is to study poles of the resolvent of a composition operator.In fact, the widely developed spectral theory of composition operatorsshows that in many cases the essential spectral radius is strictly lessthan 1. This has important consequences for the asymptotic behaviourof the powers, if the poles are algebraically simple. Our main resultshows that the poles are always algebraically simple for compositionoperators under the most general assumption on the underlying Banachspace.

We start recalling the characterization of uniform convergence of thepowers by spectral properties.

Let X be a Banach space and T ∈ L(X). Assume that λ0 is anisolated point of the spectrum σ(T ) of T . Then the residuum of theLaurent development of the resolvent R(λ, T ) = (λI−T )−1 at the pointλ0 is the same as the spectral projection P with respect to the set λ0.We refer to [20, A-III, §3.6] for more details. If λ0 is a pole, then λ0is an eigenvalue of T . The algebraic multiplicity of λ0 is defined as thedimension of PX. In finite dimension, it coincides with the multiplicityof λ0 as a root of the characteristic polynomial of T . We say that λ0is simple if dimPX = 1. Then also dim ker(T − λ0I) = 1, i.e. also thegeometric multiplicity of λ0 is one.

The essential spectral radius of T is denoted by re(T ). Denotingby T the unit circle in the complex plane and by σp(T ) the set of alleigenvalues of T , we have the following general result on the asymptoticbehaviour of the powers T n as n→∞.

Proposition 3.1. Let T ∈ L(X). The following assertions are equiv-alent:

(i) P := limn→∞ Tn exists in L(X) and P is of finite rank;

(ii) (a) re(T ) < 1, (b) σp(T ) ∩ T ⊂ 1 and (c) if 1 is in thespectrum it is a pole of order 1.

In that case, P is the residue at 1.

For the proof see the arguments given in [20, A-III, §3.7].

Now we turn our attention to composition operators. The essen-tial spectrum has been studied extensively, and many cases are known


where re(Cϕ) < 1 (see Section 4). In view of Proposition 3.1, it remainsto investigate the pole order at 1.

Let X → Hol(D) and let ϕ : D → D be holomorphic such thatCϕX ⊂ X. Consequently, Cϕ ∈ L(X) by the closed graph theorem.

We consider the iterates ϕn = ϕϕ · · · ϕ (n times) and we assumethroughout the remainder of this section that there exists α ∈ D suchthat

(2) limn→∞

ϕn(z) = α for all z ∈ D.

In fact, we will see later, that in many spaces, even the weaker modeof convergence (E) fails, if (2) is not satisfied. Note that ϕ(α) = α.At first, we identify the projection given by the limits of T n, if theyexist. One should keep in mind, the modes of convergence (U), (S),(W ), (E) and (D) described in the introduction.

Proposition 3.2. Assume that

(D) X = fix(Cϕ)⊕ (I − Cϕ)X.

(a) Assume that 1D ∈ X. Then the projection P along (D) is given

by Pf = f(α)1D. Moreover, fix(Cϕ) = C 1D and (I − Cϕ)X =Xα := f ∈ X : f(α) = 0.

(b) If 1D /∈ X, then X = Xα.

Proof. If f ∈ fix(Cϕ), then f(z) = f(ϕ(z)) = f(ϕn(z)) for all n ∈ N.

Sending n → ∞ yields f(z) = f(α) for all z ∈ D. If f ∈ (I − Cϕ)X,then we find gn ∈ X such that f = limn→∞(gn − Cϕgn). Since α =ϕ(α), it follows that f(α) = 0. The projection P onto C 1D along thedecomposition Hol(D) = C 1D ⊕ Holα(D) is given by Pf = f(α)1D,where Holα(D) = g ∈ Hol(D) : g(α) = 0. Since fix(Cϕ) ⊂ C 1D and

(I − Cϕ)X ⊂ Holα(D), the claim follows.

Lemma 3.3. If 1 is a pole of the resolvent of Cϕ, then the residue isgiven by Pf = f(α)1D.

Proof. Assume that 1 is a pole of the resolvent of Cϕ. Thus, 1 ∈ σp(Cϕ).Recall that Ker(I−Cϕ) = fix(Cϕ) = C 1D, by Proposition 3.2. Assumethat the pole order of 1 is larger than 1. Looking at the Jordan normalform of T0 := Cϕ|X0

, where X0 = PX, P the residue, we see that there

exists f ∈ X such that (I−Cϕ)f = 1D. Evaluating at z = α, we obtaina contradiction. Thus, the pole order is 1. It follows that P is theprojection onto ker(Cϕ − I) = C 1D along Xα := f ∈ X : f(α) = 0.Thus, Pf = f(α)1D, by Proposition 3.2.

From Lemma 3.3 and Proposition 3.1, we deduce the following


Theorem 3.4. The following assertions are equivalent:

(i) Cnϕ converges in L(X) as n→∞;

(ii) re(Cϕ) < 1.

Proof. (i) ⇒ (ii) Let P = limn→∞Cnϕ in L(X). Then Pf = f(α)1D,

by Proposition 3.2. It follows from Proposition 3.1 that re(Cϕ) < 1.(ii) ⇒ (i) Assume that re(Cϕ) < 1. Then 1 is a pole of order 1

or 1 /∈ σ(Cϕ), by Lemma 3.3. Thus, (a) and (c) of Proposition 3.2are fulfilled. In order to show (b), let Cϕf = eiθf , eiθ 6= 1. Thenf(ϕn(z)) = einθf(z) for all z ∈ D. Since f(ϕn(z)) → f(α), it followsthat f(z) = 0 for all z ∈ D. Thus, σp(Cϕ) ∩ T ⊂ 1. Now the claimfollows from Proposition 3.1.

Next we investigate the other possible poles of the resolvent of Cϕ.This will allow us to give stronger estimates of the behaviour of Cn

ϕ

as n → ∞ (see Corollary 3.8). We keep assumption (2) and assumein addition that ϕ is not an automorphism or constant. Then oneknows precisely the eigenvalues and eigenfunctions of the compositionoperator considered on the entire space Hol(D). In fact, one frequentlycalls the equation

(3) f ϕ = λf

Schroder’s Equation. The following theorem describes its solutions inHol(D).

Proposition 3.5. [23, Konig’s Theorem 1884]

(a) If ϕ′(α) = 0, then (3) has a non zero solution only if λ = 1. Inthat case, each solution is constant.

(b) Assume that ϕ′(α) 6= 0. Then (3) has a non zero solution f ifand only if λ = ϕ′(α)n =: λn for some n ∈ N0.

(c) There exists exactly one function σ ∈ Hol(D) such that

σ ϕ = λ1σ, σ′(α) = 1.

One has σ(α) = 0.(d) Thus, σn ϕ = λnσ

n.(e) If n ∈ N0 and f ∈ Hol(D) such that f ϕ = λnϕ, then f = cσn

for some c ∈ C.

Our next aim is to describe all possible poles of (λ − Cϕ)−1, whereCϕ ∈ L(X), as before. It follows from Konig’s Theorem that σp(Cϕ) ⊂1, if ϕ′(α) = 0 and σp(Cϕ) ⊂ λn : n ∈ N0, if ϕ′(α) 6= 0, wherewe define λn := ϕ′(α)n. Recall that |ϕ′(α)| < 1, since ϕ is not anautomorphism. Here, we denote by σp(T ), the set of all eigenvalues of

ASYMPTOTIC BEHAVIOUR OF POWERS OF COMPOSITION OPERATORS11

an operator T. Since each pole is an eigenvalue, only the λn are possiblepoles.

Theorem 3.6. Let n ∈ N0. Assume that λn is a simple pole of theresolvent of Cϕ. Then the residue Pn at λn has dimension 1. In fact,PnX = Cσn.

Proof. It follows from Konig’s theorem that ker(λnI − Cϕ) = Cσn is1-dimensional. Thus, it remains to show that S := (Cϕ − λnI) is zeroon PnX. If not, considering the Jordan form, we find a basis f1, . . . , fmof PnX such that Sfk = fk−1 for k = 2, . . . ,m and Sf1 = 0. Thus,f1 ∈ ker(λn − Cϕ) and we can assume f1 = σn.

Assume that m ≥ 2. We want to show that this leads to a contra-diction. Let g = f2. Then (Cϕ − λn)g = σn, i.e.

(4) g(ϕ(z))− λng(z) = σ(z)n for all z ∈ D.We have already treated the case n = 0 in Lemma 3.3. In order to makethe proof more transparent, we add the case n = 2, before proving thegeneral case. For n = 2, we have λ2 = ϕ′(α)2 = λ21. Deriving (4) yields

(5) g′(ϕ(z))ϕ′(z)− λ2g′(z) = 2σ(z)σ′(z).

For z = α, we obtain

g′(α)[ϕ′(α)− λ2] = 2σ(α)σ′(α) = 0.

Since λ2 = ϕ′(α)2 6= ϕ′(α), we conclude that g′(α) = 0. Deriving (5),we obtain

(6) g′′(ϕ(z))ϕ′(z)2 + g′(ϕ(z))ϕ′′(z)− λ2g′′(z) = 2σ′(z)2 + 2σ(z)σ′′(z).

For z = α, this gives

0 = g′′(α)ϕ′(α)2 − λ2g′′(α) = 2,

a contradiction. For the general case, we use di Bruno’s formula,

dm

dzm(g ϕ)(z) =

∑ m!

k1! · · · km!g(k)(ϕ(z))

(ϕ′(z)

1!

)k1· · ·(ϕ(m)(z)

m!

)km,

where the sum is taken over all (k1, . . . , km) ∈ Nm0 such that k1 + 2k2 +

· · · + mkm = m, and where k = k1 + k2 + · · · + km. Letting z = αin (4) yields (1 − λn)g(α) = σ(α)n = 0. Thus, g(α) = 0. We claimthat g(m)(α) = 0 for m = 0, 1, . . . , n − 1. For m = 0, this is proved.Let m ≤ n − 1 be maximal such that g(k)(α) = 0 for k = 0, 1, . . . ,m.Assume that m < n − 1. Deriving (4) m + 1 times, we obtain by diBruno’s formula(d

dz

)m+1

|z=αg(ϕ(z)) =

∑ (m+ 1)!

k1! · · · km+1!g(k)(α)

(ϕ′(α)

1!

)k1· · ·(ϕ(m+1)(α)

(m+ 1)!

)km+1


summed over all (k1, . . . , km+1) such that k1+2k2+ · · ·+(m+1)km+1 =m + 1, where k = k1 + k2 + · · · + km+1. Recall that g(k)(α) = 0 fork ≤ m. If k ≥ m+ 1, then it follows that

k2 + 2k3 + · · ·+mkm+1 ≤ 0.

Thus, k2 = k3 = · · · = km+1 = 0. Hence, k1 = m+ 1. It follows that(d

dz

)m+1

|z=αg(ϕ(z)) = g(m+1)(α)ϕ′(α)m+1.

Thus, by (4), in view of our assumption that m + 1 < n, and sinceσ(α) = 0, we obtain

g(m+1)(α)ϕ′(α)m+1 − λng(m+1)(α) =

(d

dz

)m+1

|z=ασ(z)n = 0.

Since λn = ϕ′(α)n 6= ϕ′(α)m+1, it follows that g(m+1)(α) = 0 in contrastto the choice of m. We have proved that g(m)(α) = 0 for m = 0, 1, . . .,n− 1. Again, by di Bruno’s formula, we deduce that(

d

dz

)n|z=α

g(ϕ(z)) = g(n)(α)ϕ′(α)n = λng(n)(α).

It follows from (4) that

0 = λng(n)(α)− λng(n)(α) =

(d

dz

)n|z=α

σ(z)n = n!,

since σ(α) = 0, σ′(α) = 1. This is a contradiction.

It is desirable to compute the residue Pn when λn is a pole. We dothis for n = 0, 1, 2, 3. From Theorem 3.6, we know that

Pnf = C(f)σn (f ∈ X),

where C : X → C is a continuous linear form which we want to deter-mine.

Theorem 3.7. Assume that λn is a simple pole of the resolvent of Cϕwith residue Pn. Then

P0f = f(α)1D for n = 0;

P1f = f ′(α)σ for n = 1;

P2f =1

2!

[f ′′(α) +

ϕ′′(α)

λ2 − λ1f ′(α)

]σ2 for n = 2;

P3f =1

3!

[f ′′′(α) +

3λ1ϕ′′(α)

λ3 − λ2f ′′(α) +

(3λ1ϕ

′′(α)2

λ3 − λ1+

ϕ′′′(α)

λ3 − λ1

)f ′(α)

]σ3


for n = 3.

Proof. Let f ∈ X, gλ = (λ− Cϕ)−1f for 0 < |λ− λn| < ε. Then

(7) Cn(f)σn = Pnf = limλ→λn

(λ− λn)gλ

in X. Consequently, convergence in (7) is also pointwise and even forall derivatives, i.e.

(8) Cn(f)

(d

dz

)mσ(z)n = lim

λ→λn(λ− λn)g

(m)λ (z).

Since

(9) λgλ(z)− gλ(ϕ(z)) = f(z),

it follows that

gλ(α) =1

λ− 1f(α).

From (8) for m = 0, n = 0, we obtain

C0(f) = C0(f)σ(α)0 = limλ→λ0

(λ− λ0)gλ(α) = f(α),

since λ0 = 1. This shows the formula for n = 0. Deriving (9), weobtain

(10) λg′λ(z)− g′λ(ϕ(z))ϕ′(z) = f ′(z).

For z = α, this gives g′λ(α)(λ− λ1) = f ′(α), hence g′λ(α) = 1λ−λ1f

′(α).

Thus, from (8) for m = 1, n = 1, we obtain

C1(f) = C1(f)σ′(α) = limλ→λ1

(λ− λ1)g′λ(α) = f ′(α).

This proves the case n = 1. Deriving (10) again gives

(11) λg′′λ(z)− g′′λ(ϕ(z))ϕ′(z)2 − g′λ(ϕ(z))ϕ′′(z) = f ′′(z).

Hence,

(λ− λ2)g′′λ(α) = f ′′(α) +1

λ− λ1f ′(α)ϕ′′(α).

Since σ(α) = 0, σ′(α) = 1, we have (σ2)′′(α) = 2. Hence by (8),

2C2(f) = C2(f)

(d

dz

)2

|z=ασ(z)2

= limλ→λ2

(λ− λ2)g′′λ(α)

= f ′′(α) +1

λ2 − λ1f ′(α)ϕ′′(α).

This implies the formula for P2.


Finally, we consider n = 3. Deriving (11) gives

f ′′′(z) = λg′′′λ (z)− g′′′λ (ϕ(z))ϕ′(z)3 − g′′λ(ϕ(z))2ϕ′(z)ϕ′′(z)

− g′′λ(ϕ(z))ϕ′(z)ϕ′′(z)− g′λ(ϕ(z))ϕ′′′(z).

Hence,

(λ− λ3)g′′′λ (α) = g′′λ(α)2ϕ′(α)ϕ′′(α) + g′′λ(α)ϕ′(α)ϕ′′(α)

+ g′λ(α)ϕ′′′(α) + f ′′′(α)

=1

λ− λ2

(f ′′(α) +

1

λ− λ1f ′(α)ϕ′′(α)

)3λ1ϕ

′′(α)

+1

λ− λ1f ′(α)ϕ′′′(α) + f ′′′(α).

Since(ddz

)3|z=α

σ(z)3 = 6, it follows from (8) for m = n = 3 that

6C3(f) =1

λ3 − λ2

(f ′′(α) +

1

λ3 − λ1f ′(α)ϕ′′(α)

)3λ1ϕ

′′(α)

+1

λ3 − λ1f ′(α)ϕ′′′(α) + f ′′′(α).

This yields the formula for P3.

The fact that all possible poles are algebraically simple can be usedto give a more precise description of the asymptotic behaviour of Cn

ϕ

as n → ∞. We continue to suppose that ϕ is not an automorphismand limn→∞ ϕn(z) = α for all z ∈ D. We assume that ϕ′(α) 6= 0.Then |ϕ′(α)| < 1 and the possible eigenvalues of Cϕ are λn := ϕ′(α)n,n = 0, 1, . . .. Thus, |λn| → 0 as n→∞.

Corollary 3.8. Consider Cϕ on X and assume that re(Cϕ) < 1. Letre(Cϕ) < ρ < 1 such that |λn| 6= ρ for all n ∈ N. Let m ∈ N0 be thenumber such that |λn| > ρ for n = 0, . . . ,m and |λn| < ρ for n > m.Denote by Pn the residue at λn, n = 0, . . . ,m. Then there exists M ≥ 0such that

‖Cnϕ − P0 − λn1P1 − λn2P2 − · · · − λnmPm‖ ≤Mρn

for all n ∈ N.

Proof. Consider the spectral projectionQ associated with λ0, . . . , λm,let X1 = QX, X2 = (I − Q)X, Tj = Cϕ|Xj

, j = 1, 2. Then r(T2) < ρ

and

T n2 = P0 + λn1P1 + · · ·+ λnmPm

for all n ∈ N0.


Many results are known on the essential spectrum of Cϕ on severalspaces. We refer to [12] and [23]. In the context of Corollary 3.8, itis of interest that a famous formula due to Shapiro expresses re(Cϕ)in terms of the Nevanlinna counting function. We will discuss severalconcrete cases in the next section.

Here we just add that in the case where ϕ is an automorphism oneknows in many cases that the spectrum of Cϕ is a ring (see [12, Ch.7,§7.1]) and so there are no poles. Of course, if ϕ is an elliptic automor-phism, then ϕ is conjugate to a rotation. If the angle is rational, thenCnϕ = I for some n ∈ N, where I is the identity operator. In this case,

the spectrum of Cϕ consists of the nth roots of the unity and they areall poles of order 1.

4. Strong Convergence implies Uniform Convergence

Convergence of sequences of composition operators has been stud-ied by Schwartz [25] and by Matache [18] on the space H2(D). Theyobtain in particular results for powers of a composition operator (seeTheorem 3 in [18] which we use in Subsection D below). Our pointis to compare the diverse modes of convergence (U)-(D) and to allowgeneral spaces of analytic functions. In [18] also, convergence in theHilbert-Schmidt norm is considered.

Let ϕ : D → D be holomorphic. In this section, we will describethe asymptotic behaviour of Cn

ϕ on diverse spaces X, according to ourhierarchy (U)⇒ (S)⇒ (W )⇒ (E)⇒ (D), see Section 1. We will seethat in many case (S) implies (U), which is a very particular property ofcomposition operators. Our aim is to study the asymptotic property ofCϕ in terms of properties of ϕ. At first, we want to exclude the somehowtrivial case of an elliptic automorphism, i.e. an automorphism fixing apoint of D. The rotations ρθ(z) = eiθz, are the automorphisms fixing 0.Given b ∈ D, we let ψb(z) = b−z

1−bz . Then ψb ∈ Aut(D) is involutive, i.e.

ψb ψb = id, where id is the identity map on D. Moreover, ψb(0) = b,ψb(b) = 0. The ψb are exactly the involutive automorphisms. Thus, ϕis an elliptic automorphism if and only if there exist θ ∈ R, b ∈ D suchthat ϕ = ψb ρθ ψb. Thus, Cϕ = Cψb Cρθ Cψb . Since Cn

ρθe1 = einθe1,

Cnρθ

does not converge weakly in X, if e1 ∈ X and eiθ 6= 1. In theremainder of this section, we assume that ϕ is not constant and not anelliptic automorphism. The asymptotic behaviour of ϕ is then describedby the Denjoy-Wolff Theorem.


Theorem 4.1. [12, Denjoy-Wolff] There exists a unique point α ∈ Dsuch that the iterates ϕn of ϕ converge to α uniformly on compactsubsets of D.

The point α is called the Denjoy-Wolff point of ϕ. If b ∈ D is a fixedpoint of ϕ, then b is the Denjoy-Wolff point of ϕ. We now considerdiverse cases involving the Denjoy-Wolff point.

A. Denjoy-Wolff point in ∂D. If the Denjoy-Wolff point is on theboundary then we will see that Cϕ is not power bounded on manyspaces.

Let X be a Banach space such that X → Hol(D) and en ∈ X forall n ≥ 1 (where en(z) = zn). Assume furthermore that the pointevaluations are the only “multiplicative” linear forms on X, i.e. weassume that for L ∈ X ′,

(12) 〈L, en〉 = 〈L, e1〉n implies that L = δz for some z ∈ D.

Example 4.2. Examples of Banach spaces satisfying the conditionsabove are

(a) Hp(D), 1 ≤ p <∞;(b) zHp(D), 1 ≤ p <∞;(c) Hp(β), 1 ≤ p < ∞, where β satisfies the condition of Proposi-

tion 2.5.

Theorem 4.3. Assume that the Denjoy-Wolff point α of ϕ is in ∂D.Let X be a Banach space satisfying the conditions above. If Cϕ(X) ⊂X, then supn∈N ‖Cn

ϕ‖L(X) =∞.

Proof. Assume on the contrary that ‖Cnϕ‖L(X) ≤ M (n ∈ N). Let

f ∈ spanen : n ∈ N. Then

|f(α)| = limn→∞

|f(ϕn(0))| ≤ ‖δ0‖ supn‖f ϕn‖X ≤ ‖δ0‖M‖f‖X .

Thus, by the Hahn-Banach Theorem, there exists L ∈ X ′ such that〈L, f〉 = f(α) for all f ∈ spanen : n ∈ N. It follows that 〈L, en〉 =αn = 〈L, e1〉n. By our hypothesis on X, L = δz for some z ∈ D. Thusz = 〈L, e1〉 = α, a contradiction.

Since in the situation of Theorem 4.3, Cϕ is not power bounded, thepowers do not even converge weakly. Thus, in the following, we willassume that α ∈ D.


B. α ∈ D and weak convergence. We start by the weakest conver-gence modes.

Theorem 4.4. Let X → Hol(D) be reflexive such that Cϕ(X) ⊂ Xand supn∈N ‖Cn

ϕ‖L(X) <∞. Assume that α = ϕ(α) ∈ D. Then

(13) Cnϕf f(α)1D (n→∞) for all f ∈ X.

Here denotes weak convergence. We need the following

Lemma 4.5. Let X → Hol(D) be reflexive and let (fn)n∈N be a boundedsequence in X. If limn→∞ fn(z) exists for all z ∈ D, then (fn)n∈Nconverges weakly in X as n→∞.

Proof. Let fnk f in X. Then

f(z) = limk→∞

fnk(z) = limn→∞

fn(z) for all z ∈ D.

Thus, each subsequence has a subsequence, converging weakly to f ,where f ∈ X. This implies the claim.

Remark 4.6. It suffices that limn→∞ fn(z) exists for all z ∈ K, whereK is subset of uniqueness of D.

Proof of Theorem 4.4. Let f ∈ X. Then

limn→∞

(Cnϕf)(z) = lim

n→∞f(ϕn(z)) = f(α) for all z ∈ D.

It follows from Lemma 4.5 that Cnϕf f(α)1D.

Remark 4.7. If in the context of Theorem 4.4, 1D /∈ X, then the proofshows that f(α) = 0 for all f ∈ X. Thus, (13) means that Cn

ϕf 0as n→∞ in this case.

If ϕ : D → D is holomorphic, then Cϕ (Hp(D)) ⊂ Hp(D), 1 ≤ p <∞. If ϕ(0) = 0, then Cϕ is a contraction. If the Denjoy-Wolff pointα is in D, then consider ψ = ψα ϕ ψα. Then ψ(0) = 0. Thus,‖Cψ‖ ≤ 1. Moreover, Cϕ is similar to Cψ and so, Cϕ is automaticallypower bounded on Hp(D). From Theorem 4.4, we deduce that (Cn

ϕ)n∈Nconverges weakly, if 1 < p < ∞. This remains true even for p = 1 aswe will prove below. So, we have the following result.

Theorem 4.8. Assume that the Denjoy-Wolff point α of ϕ is in D.Let 1 ≤ p <∞, X = Hp(D). Then Cn

ϕf f(α)1D (n→∞) weakly inHp(D) for all f ∈ Hp(D).

Proof. We only have to consider the case p = 1. We may assumethat α = 0, considering a similarity transform otherwise. We identifyH1(D) with H1(T) := f ∈ L1(T) : f(n) = 0 for all n < 0, where


f(n) =∫T f(z)zn dz as usual. For r > 0, the set Kr := f ∈ L1(T) :

|f(z)| ≤ r a.e. is weakly compact in L1(T) (see [22, II.5.10]). The

set Kr := H1(T) ∩ Kr is weakly closed in Kr and so weakly compactin the closed subspace H1(T) of L1(T).

(a) Let f ∈ H∞(D). We show that Cnϕf → f(0)1D weakly. Let r > 0

such that |f(z)| ≤ r for all z ∈ D. Then f ∈ Kr and Cnϕf ∈ Kr for

all n ∈ N . By Eberlein’s Theorem, each sequence in Kr has a weaklyconvergent subsequence. Let nk < nk+1 such that Cnk

ϕ f h in H1(D).Then f(ϕnk(z)) = (Cnk

ϕ f)(z) → h(z). Since f(0) = limk→∞ f(ϕnk(z)),it follows that h = f(0)1D. We have shown that each subsequence of(Cn

ϕf)n∈N has a subsequence, converging weakly to f(0)1D. This showsthat Cn

ϕf f(0)1D.

(b) Since H∞(D) is dense in H1(D), the result follows from (a).

Corollary 4.9. Under the hypotheses of Theorem 4.4 or 4.8, one hasσp(Cϕ) ∩ T ⊂ 1, σp(C

′ϕ) ∩ T ⊂ 1, where σp denotes the point

spectrum and C ′ϕ the adjoint of Cϕ.

This is an extension of [12, Proposition 7.32].

C. ϕ is inner, α = ϕ(α) ∈ D. Assume that the Denjoy-Wolff point αof ϕ is in D.

Theorem 4.10. Let X = Hp(D), 1 ≤ p < ∞. If ϕ is an innerfunction, then (Cn

ϕ)n∈N does not converge strongly.

Proof. Let f ∈ X, f 6= 0 such that f(α) = 0. If α = 0, then by [12,Theorem 3.8], Cϕ is an isometry. Thus, ‖Cn

ϕf‖Hp(D) = ‖f‖Hp(D) doesnot converge to 0 = f(0)1D. If α 6= 0, then Cϕ is similar to an isometry,i.e. T = SCϕS

−1 is isometric, where S ∈ L(Hp(D)). Thus,

‖Cnϕf‖Hp(D) = ‖S−1T nSf‖Hp(D) ≥ ‖S‖−1L(Hp(D))‖T

nSf‖Hp(D)

= ‖S‖−1L(Hp(D))‖Sf‖Hp(D) =: δ > 0

for all n ∈ N. Thus, Cnϕ does not converge to 0 = f(α)1D.

Corollary 4.11. Assume that ϕ is inner. Let X → Hol(D) such thatX ⊂ H1(D). Assume that there exists 0 6= f ∈ X such that f(α) = 0.Assume that Cϕ(X) ⊂ X. Then (Cn

ϕ)n∈N does not converge strongly.

Proof. By the closed graph theorem, there exist c > 0 such that ‖g‖H1(D) ≤c‖g‖X for all g ∈ X. In particular,

c‖Cnϕf‖X ≥ ‖Cn

ϕf‖H1(D) ≥ δ,

where δ > 0 is from the proof of Theorem 4.10. Thus, (Cnϕ)n∈N does

not converge strongly.


D. ϕ is not inner, α = ϕ(α) ∈ D. If ϕ is not inner, with Denjoy-Wolff point in D, then the powers of Cϕ, converge uniformly on severalinteresting spaces. Recall that ϕ is not an automorphism, if ϕ is notinner.

Theorem 4.12. Assume that ϕ is not inner with Denjoy-Wolff pointα ∈ D. Let 1 ≤ p <∞, X = Hp(D). Then

‖Cnϕ − P‖L(X) → 0 (n→∞),

where Pf = f(α)1D.

Proof. (a) Let p = 2. We may assume that α = 0 (considering thesimilarity transform Cψα otherwise). The result follows from Theorem 3in [18].

(b) Let 1 ≤ p < ∞ be arbitrary. Then it follows from [9, Theorem3.8] that

(re,Hp(Cϕ))p ≤ (re,H2(Cϕ))2.

So, the claim follows from (a).

As summary of our results, we obtain the remarkable property ofcomposition operators that strong convergence implies uniform con-vergence. Here, ϕ is just a holomorphic function from D to D.

Corollary 4.13. Let ϕ : D→ D be holomorphic. Consider the operatorCϕ on Hp(D), where 1 ≤ p < ∞. If (Cn

ϕ)n∈N converges strongly, thenthe Denjoy-Wolff point α of ϕ is in D and ‖Cn

ϕ − P‖L(Hp(D)) → 0 asn→∞, where Pf = f(α)1D.

Proof. Assume that (Cnϕ)n∈N converges strongly. If ϕ is an elliptic au-

tomorphism, then ϕ = id (see the beginning of this section). Thus, wemay assume that ϕ is not an elliptic automorphism. It follows fromTheorem 4.3 that the Denjoy-Wolff point α of ϕ is in D. It followsfrom Theorem 4.10 that ϕ is not inner. Now Theorem 4.12 implies theresult.

We may extend Theorem 4.12 to other spaces containing H2(D) withthe help of the following result by Chalendar-Partington [11].

We consider H2(β), where β satisfies the conditions of Proposition2.5 and in addition, βn ≥ βn+1 > 0. Note that H2(D) ⊂ H2(β) in thiscase.

Proposition 4.14. [11, Corollary 2.2] Let S ∈ L(H2(D)) be lower-triangular, i.e. Sen ∈ spanen, en+1, . . . for all n ∈ N. Then there

exists a unique S ∈ L(H2(β)) such that S|H2(D)= S. Moreover, ‖S‖ ≤

‖S‖.


Theorem 4.15. Let ϕ : D→ D be holomorphic, not inner, with ϕ(0) =0. Then Cϕ(H2(β)) ⊂ H2(β) and on H2(β)

‖Cnϕ − P‖L(H2(β)) → 0 (n→∞),

where Pf = f(0)1D.

Proof. Since ϕ(0) = 0, ϕ(z) = zϕ1(z), where ϕ1 : D → D is holomor-phic, and therefore Cϕ and its powers are lower-triangular with respectto the orthonormal basis (zn)n≥0 of H2(D). Since P : f 7→ f(0)1D isalso a lower-triangular bounded operator on H2(D), the result followsfrom Proposition 4.14 and Corollary 4.13.

In order to consider symbols ϕ with a non-zero Denjoy-Wolff pointα ∈ D, an extra hypothesis on the space H2(β) is required, namelyto be automorphism invariant, which fails for certain H2(β) which areeither too large or too small [15, 17]. See also [14] for upper and lowerbounds of the norm of composition operators by automorphisms onH2(β) when βn = 1

(n+1)ν, where 0 ≤ ν ≤ 2.

Corollary 4.16. Let ϕ : D → D be holomorphic, not inner, withDenjoy-Wolff point α ∈ D, α 6= 0. Suppose also that H2(β) is auto-morphism invariant. Then Cϕ(H2(β)) ⊂ H2(β) and on H2(β)

‖Cnϕ − P‖L(H2(β)) → 0 (n→∞),

where Pf = f(α)1D.

Proof. Set ϕα(z) = α−z1−αz and observe that ψ := ϕαϕϕα is a holomor-

phic map from D to D such that ψ(0) = 0. Therefore, by Theorem 4.15,we have

‖Cnψ − P‖L(H2(β)) → 0 (n→∞),

where Pf = f(0)1D. Moreover, since ϕα ϕα(z) = z for all z ∈ D, weget

Ckϕ = CϕαC

kψCϕα ,

for all k ∈ N. The conclusion follows from the continuity of Cϕα = C−1ϕαin H2(β) (by hypothesis) and the fact that CϕαPCϕα(f) = f(α)1D.

5. Wiener algebra, A(D) and H∞(D)

In this section, we will show that on the disc algebra A(D), theWiener algebra W (D) and on H∞(D), ergodicity already implies uni-form convergence, provided the Denjoy-Wolff point lies in the interior.We use a recent result of [7].


Theorem 5.1. Let ϕ ∈ A(D) with Denjoy-Wolff point α in D. Thefollowing assertions are equivalent:

(i) Cϕ is ergodic;

(ii) there exists n ∈ N such that ϕn(D) ⊂ D;(iii) limn→∞ ‖Cn

ϕ − P‖L(A(D)) = 0, where Pf = f(α)1D.

Proof. We can assume that α = 0.(i)⇒ (ii) It is shown in [7, Theorem 3.4] that limn→∞ ϕn(z) = 0 for

all z ∈ D. Since |ϕn(z)| is decreasing, by Dini’s theorem ‖ϕn‖C(D) → 0

as n→∞. This implies (ii).(ii) ⇒ (iii) Let n ∈ N such that ϕn(D) ⊂ D. Then by [12, Propo-

sition 3.11], Cϕn is compact. This implies that re(Cϕ) = 0. Now (iii)follows from Theorem 3.4 and Propostion 3.2.

Next, we consider the Wiener algebra:

W (D) :=

f : D→ C : ∃ (an)n≥0 ∈ `1, f(z) =

∞∑n=0

anzn

,

which is isomorphic to `1. Let aj ∈ `1, j = 0, 1, . . .. Then Tx :=∑∞j=0 xja

j defines a bounded operator on `1 if and only if supj∈N0‖aj‖`1

<∞. The operator T is compact if and only if ‖aj‖`1 → 0 as j →∞.Now let ϕ ∈ W (D). Since Cϕej = ϕj (the j−th power of ϕ and not

the iterate), we have Cϕ(W (D)) ⊂ W (D) if and only if supj≥0 ‖ϕj‖W (D)< ∞. Moreover, Cϕ is compact on W (D) if and only if ‖ϕj‖W (D) → 0as j →∞.

Theorem 5.2. Let ϕ ∈ W (D) with Denjoy-Wolff point α in D. As-sume that supn ‖ϕn‖W (D) <∞. The following assertions are equivalent:

(i) Cϕ is ergodic;

(ii) there exists n ∈ N such that ϕn(D) ⊂ D;(iii) limn→∞ ‖Cn

ϕ − P‖L(W (D)) = 0, where Pf = f(α)1D.

Proof. We can assume that α = 0.(i)⇒ (ii) can be proved as [7, Theorem 3.4 (i)⇒ (iii)].(ii)⇒ (iii) Let ψ = ϕn, where ϕn(D) ⊂ D. Then there exists r < 1

such that ‖ψ‖C(D) < r < 1. By Wiener’s Theorem, the spectral radius

r(ψ) in W (D) equals ‖ψ‖C(D). Thus, limm→∞ ‖ψm‖1/mW (D) < r. Hence,

limm→∞ ‖ψm‖W (D) = 0. It follows from the remark above that Cψ iscompact. Thus, Cn

ϕ = Cϕn is compact. It follows that re(Cϕ) = 0, and(iii) follows from Theorem 3.4 and Propostion 3.2.

Finally, we consider H∞(D).


Theorem 5.3. Let ϕ : D → D be holomorphic. Assume that theDenjoy-Wolff point α of ϕ is in D. Consider the composition operatorCϕ on H∞(D). The following assertions are equivalent:

(i) Cϕ is ergodic;(ii) limn→∞ ‖ϕn − α‖L∞(D) = 0;(iii) limn→∞ ‖Cn

ϕ − P‖L(H∞(D)) = 0, where Pf = f(α)1D.

Proof. Again, we can assume that α = 0. Then (i)⇒ (ii) follows from[7, Theorem 3.3].

(ii) ⇒ (iii) There exists n0 such that ϕn0(D) ⊂ rD with 0 < r < 1.It follows from [12, Proposition 3.11] that Cn0

ϕ = Cϕn0 is compact.Thus, re(Cϕ) = 0, and (iii) follows from Theorem 3.4 and Propostion3.2.

Remark 5.4. If the Denjoy-Wolff point is in ∂D, then the compositionoperator Cϕ on H∞(D) is not mean ergodic. However, if ϕ ∈ A(D),then Cϕ is not uniformly mean ergodic on A(D), but could be mean er-godic on A(D). Similarly, on W (D), Cϕ is not uniformly mean ergodicon W (D), but could be mean ergodic on W (D)(see [7, Theorem 3.6 andTheorem 3.7] for more details).

6. Semigroups of composition operators: an outlook

The aim of this section is to study the asymptotic behaviour of asemigroup of composition operators on H2(D). Such semigroups areassociated with holomorphic semiflows (ϕt)t≥0 defined below.

Definition 6.1. A family (ϕt)t≥0 is called a holomorphic semiflow onD if

(1) ϕt : D→ D is holomorphic for all t ≥ 0;(2) ϕ0(z) = z for all z ∈ D;(3) ϕt+s(z) = ϕt ϕs(z) for all t, s ≥ 0, z ∈ D;(4) limt→0 ϕt(z) = z for all z ∈ D.

Well-known properties of holomorphic semiflows are the following:

(1) For all t ≥ 0, ϕt is injective;(2) limt→0 ϕt = id uniformly on every compact subset of D;(3) If there exists t0 > 0 such that ϕt0 is an automorphism (resp.

elliptic automorphism), then ϕt is an automorphism (resp. el-liptic automorphism) for all t > 0;

(4) For all semiflows which are not elliptic automorphisms, thereexists a unique α ∈ D such that limt→∞ ϕt(z) = α uniformly onevery compact subset of D. This α is called the Denjoy-Wolffpoint of the semiflow.


In 1978, Berkson and Porta (see [8]) proved that a semiflow of holo-morphic functions has an infinitesimal generator, namely the holomor-phic function G : D→ C defined by

G(z) = limt→0

ϕt(z)− zt

.

They proved that G is of the form

(14) G(z) = (α− z)(1− αz)F (z),

where α is the Denjoy-Wolff point of (ϕt)t≥0 and where F : D→ C+ :=z ∈ C : Re(z) ≥ 0 is holomorphic. Such F ∈ ∩p<1H

p and thereforeit follows that G has radial limits almost everywhere on the unit circle(see [13, Theorem 3.2 and 2.2]).

In 2015, C. Avicou, I. Chalendar and J. Partington (see [5]) gaveanother characterization of G avoiding the Denjoy-Wolff point. Theyproved that G ∈ ∩p<1H

p is an infinitesimal generator of a holomorphicsemiflow if and only if Re(zG(z)) ≤ 0 a.e. on T.

Let (ϕt)t≥0 be a holomorphic semiflow on D. Then Ctf = f ϕt,defines a quasi-contractive C0-semigroup on H2(D). Its generator A isgiven by

Af = Gf ′,

D(A) = f ∈ H2(D) : Gf ′ ∈ H2(D).

Theorem 6.1. The following assertions are equivalent:

(i) Ct converges strongly as t→∞;(ii) the semiflow does not consist of automorphisms and the Denjoy-

Wolff point α is in D;(iii) G has a zero in D and there exists a set of positive measure

Ω ⊂ T such that Re(zG(z)) < 0 on Ω.

In that case, there exist M > 0, ε > 0 such that

(15) ‖Ct − P‖L(H2(D)) ≤Me−εt for all t > 0,

where Pf = f(α)1D.

Proof. a) By [6, Proposition 2.10], (Ct)t≥0 is a group if and only ifRe(zG(z)) = 0 a.e. on T. Moreover, (Ct)t≥0 is a group if and only if Ctis invertible for one (equivalently all) t > 0. By [12], this is equivalentto ϕt being an automorphism for all t > 0. Also, note that G has azero in D if and only if α the Denjoy-Wolff point of (ϕt)t≥0 is in D using(14).This proves the equivalence of (ii) and (iii).

b) Now assume that (ii) holds. Since ϕt is injective for all t > 0, noϕt is inner for t > 0. By Theorem 4.12, Cnt = Cn

t converges to P asn→∞ in L(H2(D)), where Pf = f(α)1D.


The space X := (I − P )H2(D) is invariant under Ct. Let Tt =Ct|X . Denote by B, the generator of (Tt)t≥0 and by ω(B) := infω :∃M such that ‖Tt‖L(X) ≤Meωt for all t ≥ 0 its growth bound. Since

the spectral radius of Tt is given by r(Tt) = eω(B)t, it follows thatω(B) < 0. This proves (15), since CtP = P for all t ≥ 0. We haveshown that (ii) implies (15), and hence also (i).

Conversely, if (i) holds, then ϕt is not inner for any t > 0, by Theorem4.10. Moreover, by Theorem 4.3, the Denjoy-Wolff point α is in D. Thisproves (ii).

In [10], a complete description of semiflows of linear fractional mapsfrom D to D is given, in terms of a so-called Konig function h, fromwhich, one can easily recover the expression of the semiflow. It followsthat a semiflow (ϕt)t≥0 of linear fractional maps, whose Denjoy-Wolffpoint α is in D, has the following form:

(16) ϕt(z) = ϕα ψt ϕα,

where ϕα(z) = α−z1−αz and ψt(z) = e−ctz

1+z(e−ct−1)/β , with Re(c) ≥ 0 and

β ∈ (C∪∞) \D. From this explicit expression, we deduce that ϕt isan automorphism if and only if β =∞ and Re(c) = 0. Again, considerthe semigroup (Ct)t≥0 on H2(D), where Ctf = f ϕt. By Theorem 6.1,we get:

Corollary 6.2. Let (ϕt)t≥0 be a semiflow of linear fractional mapsfixing α ∈ D. The following assertions are equivalent:

(i) (Ct)t≥0 strongly converges to P , where Pf = f(α)1D;(ii) (Ct)t≥0 uniformly converges;

(iii) β 6=∞ or Re(c) > 0, where β and c are defined in (16).

Acknowledgments. The authors are grateful to the referee for somehelpful comments.

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