power bounded composition operators

Computational Methods and Function TheoryVolume 12 (2012), No. 1, 105–117

Power Bounded Composition Operators

Elke Wolf

(Communicated by Edward B. Saff)

Abstract. Let φ be an analytic self-map of the open unit disk D. Such amap induces a composition operator Cφ acting on weighted Bergman spacesof infinite order. We study when such operators are power bounded and whenthey are uniformly mean ergodic.

Keywords. Weighted composition operator, weighted Bergman space of in-finite order, power bounded, uniformly mean ergodic.

2000 MSC. 47B33, 47B38.

1. Introduction

Let D denote the open unit disk in the complex plane, H(D) the set of all analyticfunctions on D and φ an analytic self-map of D. Such a map induces throughcomposition the so-called composition operator

Cφ : H(D) → H(D), f �→ f ◦ φ.

The study of such operators acting between various spaces of analytic functionshas quite a long and rich history since these operators occur naturally in a varietyof problems (for instance: the study of multiplication operators and the theory ofdynamical systems). Thus, many properties of composition operators have beeninvestigated, see e.g. [21, 13, 17, 9, 11, 10, 16, 20]. This can only be a sample ofarticles.

In this paper we are interested in composition operators acting in the followingsetting. We say that a function v : D → (0,∞) is a weight on D, if it is boundedand continuous. If v is in addition radial (i.e. v(z) = v(|z|) for every z ∈ D),decreasing and satisfies lim|z|→1 v(z) = 0, then v is called typical.

We consider the weighted Bergman spaces of infinite order

H∞v := {f ∈ H(D) : ‖f‖v := sup

z∈D

v(z)|f(z)| < ∞},

Received December 3, 2010, in revised form September 9, 2011.Published online November 19, 2011.

ISSN 1617-9447/$ 2.50 c© 2012 Heldermann Verlag

106 E. Wolf CMFT

endowed with norm ‖ · ‖v. Such spaces arise naturally in functional analysis(spectral theory, functional calculus), complex analysis, partial differential equa-tions and convolution equations as well as in distribution theory. They have beenstudied intensively in several articles. For further information, see e.g. [2, 5, 3, 4].

For a Banach space X, we denote the space of all continuous linear operatorsfrom X into itself by L(X) and assume that L(X) is equipped with the operatornorm topology. Given T ∈ L(X), its Cesaro means are defined by

T[n] :=1

n

n∑m=1

Tm, n ∈ N.

The following equality is well-known and can be checked easily

1

nT n = T[n] − n − 1

nT[n−1], n ∈ N,

where T[0] := I is the identity operator on X. An operator T ∈ L(X) is uniformlymean ergodic if (T[n])n is a convergent sequence in L(X). Moreover, it is powerbounded if and only if there is C > 0 such that

supn∈N

‖T n‖ ≤ C.

We say that an operator T on X is similar to a contraction if we can find aninvertible operator S on X and a contraction C on X such that

T = S−1 ◦ C ◦ S.

We call a contraction C on X strict if ‖C‖ < 1.

A good reference for information on ergodic theory is the monograph [15]. In [8]Bonet and Ricker studied when multiplication operators acting on weightedBergman spaces of infinite order are power bounded and when they are uniformlymean ergodic. Additionally, interesting articles related to this topic are [1, 7].

In what follows we investigate under which conditions on the weights and onthe inducing maps composition operators are power bounded uniformly meanergodic, respectively.

2. Basic facts

For an introduction to as well as for an in-depth study of the concept of compo-sition operators we refer the reader to the monographs [12, 22]. In the setting ofweighted spaces of holomorphic functions the so-called associated weights playan important role. For a weight v we can define the associated weight as follows:

v(z) :=1

sup{|f(z)| : f ∈ H∞v , ‖f‖v ≤ 1} =

1

‖δz‖H∞v

′,

where δz denotes the point evaluation of z. By [4] the associated weight v iscontinuous, v ≥ v > 0 and for every z ∈ D we can find fz ∈ H∞

v with ‖fz‖v ≤ 1

12 (2012), No. 1 Power Bounded Composition Operators 107

such that |fz(z)| = 1/v(z). Furthermore, it is well-known that if a weight v isradial and satisfies the Lusky condition

(2.1) infn

v(1 − 2−n−1)

v(1 − 2−n)> 0,

then v and v are equivalent, which means that we can find a constant k > 0 with

kv(z) ≥ v(z) ≥ v(z) for every z ∈ D.

Weights with this property are called essential. Since often it is quite difficult tocompute the associated weight, it is very useful to know under which conditionsweights are essential.

Moreover, by [9] we know that the norm of a composition operator Cφ acting onH∞

v is given by

‖Cφ‖ = supz∈D

v(z)

v(φ(z)).

Obviously, Cnφf = Cφnf for every f ∈ H(D), where

φn := φ ◦ · · · ◦ φ︸︷︷︸n−times

.

Thus,

‖Cφn‖ = supz∈D

v(z)

v(φn(z)).

Finally, some geometric data of the unit disk are required. Recall, that thepseudohyperbolic distance of two points z, w ∈ D is given by

ρ(z, w) = |ϕw(z)|,where

ϕw(z) =w − z

1 − wzdenotes the Mobius transformation that interchanges w and 0. Throughout thispaper we will need the following theorem, given in [10].

Theorem 1 (Bonet-Lindstrom-Wolf [10]). Let v be a typical weight satisfyingcondition (2.1) and φ, ψ : D → D analytic. Then the norm of the differenceCφ − Cψ : H∞

v → H∞v is equivalent with the expression

supz∈D

max

{v(z)

v(φ(z)),

v(z)

v(ψ(z))

}ρ(φ(z), ψ(z)).

We close this section by stating the famous Denjoy-Wolff Theorem which willplay an important role in this article.

Theorem 2 (Denjoy-Wolff Theorem). Let φ be an analytic self-map of D. If φis not the identity and not an automorphism with exactly one fixed point, thenthere is a unique point p ∈ D such that (φn)n converges to p uniformly on thecompact subsets of D.

108 E. Wolf CMFT

3. Power bounded and uniformly mean ergodiccomposition operators

Lemma 3. Let v be a typical weight with (2.1) and φ be an analytic self-mapof D. If φ has a fixed point in D, then Cφ : H∞

v → H∞v is similar to a contraction.

Proof. Let a ∈ D be a fixed point of φ. Now, put

ψ := ϕa ◦ φ ◦ ϕa,

where

ϕa(z) :=a − z

1 − azfor every z ∈ D. It is well-known that for the holomorphic automorphism ϕa

we have that ϕ−1a = ϕa. Moreover, ψ(0) = 0. The Schwarz Lemma yields that

|ψ(z)| ≤ |z| for every z ∈ D. Then we obviously have that

supz∈D

v(z)

v(ψ(z))≤ 1.

Since Cφ = Cϕa ◦ Cψ ◦ Cϕ−1a

, we obtain the claim.

Theorem 4. Let v be a typical weight satisfying (2.1) and φ be an analytic self-map but not a conformal automorphism of D. The operator Cφ : H∞

v → H∞v is

power bounded if and only if it is similar to a contraction.

Proof. By Lemma 3 it is enough to show that φ has a fixed point in D. We provethis indirectly and assume that φ has no fixed point in D. By the Denjoy-WolffTheorem we know that in this case the sequence (|φn|)n tends to 1 uniformly onthe compact subsets of D. Hence

‖Cφn‖ = ‖Cφn‖ = sup

z∈D

v(z)

v(φn(z))≥ v(0)

v(φn(0))→ ∞,

as n → ∞. This is a contradiction.

Conversely, by hypothesis, there exists an invertible operator S on H∞v such

that Cφ = S−1 ◦ C ◦ S, where C is a contraction on H∞v , i.e. ‖C‖ ≤ 1. Hence

Cnφ = S−1 ◦ Cn ◦ S and thus ‖Cn

φ‖ ≤ ‖S−1‖‖S‖ for every n ∈ N. This meansthat Cφ is power bounded.

Remark 5. Let v be a typical weight with (2.1) and φ an analytic self-mapbut not a conformal automorphism of D. Note that, Cφ : H∞

v → H∞v is power

bounded if and only if φ has a fixed point in D.

Example 6. We consider the composition operator Cφ : H∞v → H∞

v induced by

φ(z) =z + 1

2and v(z) = 1 − |z|


for every z ∈ D. Thus, we obtain

φn(z) =z + 1 + · · · + 2n−1

2n

for every n ∈ N and every z ∈ D. We show this by induction. Obviously we havethat

φ2(z) = φ(φ(z)) =z+12

+ 1

2=

z + 3

4=

z + 1 + 21

22.

Next, we assume that the claim is true for some n ∈ N. Then, we get that

φn+1(z) = φ(φn(z)) =z+1+···+2n−1

2n + 1

2=

z + 1 + · · · + 2n−1 + 2n

2n+1.

Hence the claim follows. Easy calculations show that

‖Cnφ‖ = sup

z∈D

v(z)

v(φn(z))= sup

z∈D

1 − |z|1 − |φn(z)| = 2n.

Thus, Cφ obviously is not power bounded.

Example 7. Let v be an arbitrary typical weight. Then obviously each analyticself-map φ of D that has a fixed point induces a power bounded compositionoperator Cφ : H∞

v → H∞v , e.g. consider

v(z) = e−1/(1−|z|) and φ(z) =z

2+

1

3

for every z ∈ D.

Next, we turn our attention to the investigation when a composition operator isuniformly mean ergodic.

Proposition 8. Let v be a weight and φ be an analytic self-map of D. IfCφ : H∞

v → H∞v is similar to a strict contraction, then Cφ is uniformly mean

ergodic.

Proof. We will show that ‖(Cφ)[n]‖ → 0 as n → ∞. By hypothesis, we can findan invertible operator S on H∞

v and an operator C on H∞v with ‖C‖ < 1 such

that Cφ = S−1 ◦ C ◦ S. Thus, we arrive at the following estimate

‖(Cφ)[n]‖ ≤ 1

n

n∑m=1

‖Cnφ‖ ≤ 1

n

n∑m=1

‖S−1‖‖C‖m‖S‖ ≤ 1

n‖S−1‖‖S‖M → 0

as n → ∞, where

M :=∞∑

m=1

‖C‖m =1

1 − ‖C‖ < ∞.

The converse is not true, as the following trivial example shows.

110 E. Wolf CMFT

Example 9. If we take

v(z) = 1 − |z| and φ(z) = id(z) = z

for every z ∈ D we obtain φn(z) = z for every n ∈ N. Obviously we have that‖Cφ‖ = 1 and

(Cφ)[n] =1

n

n∑m=1

Cmφ = Cφ

for every n ∈ N. Hence Cφ is uniformly mean ergodic.

Theorem 10. Let v be a typical weight with (2.1) and φ be an analytic self-mapbut not a conformal automorphism of D. Let us assume that φ has an attractingfixed point a in D, i.e. φ′(a) = 0, then Cφ : H∞

v → H∞v is uniformly mean ergodic.

Proof. Model maps for analytic self-maps as described above are given by

φ(z) = λz

for every z ∈ D with |λ| < 1. (One can change variables analytically in aneighbourhood of a and conjugate φ to the map λz for λ = φ′(a), for details seee.g. [19]. Originally this was shown by Koenigs in [14].) Obviously we have thatφn(z) = λnz for every z ∈ D and every n ∈ N as well as ‖Cφn‖ = ‖Cφ‖ = 1for every n ∈ N. If C0 is the composition operator defined by C0 : H∞

v → H∞v ,

(C0f)(z) = f(0) for every z ∈ D we obtain by using Theorem 1

‖(Cφ)[n] − C0‖ =

∥∥∥∥∥1

n

n∑m=1

Cφm − C0

∥∥∥∥∥ ≤ 1

n

n∑m=1

‖Cφm − C0‖

≤ C1

n

n∑m=1

supz∈D

max

{v(z)

v(φm(z)),v(z)

v(0)

}ρ(φm(z), 0)

≤ C1

n

n∑m=1

supz∈D

|φm(z)| ≤ C1

n

n∑m=1

|λ|m

→ 0

since |λ| < 1. Hence, in this case, ((Cφ)[n])n∈N tends to C0 in L(H∞v ). Thus, Cφ

is uniformly mean ergodic, and the claim follows.

Theorem 11. Let v(z) = 1 − |z| for every z ∈ D. Moreover, let φ be ananalytic self-map but not a conformal automorphism of D such that φ has asuper-attracting fixed point a ∈ D, i.e. φ′(a) = 0. Then Cφ : H∞

v → H∞v is

uniformly mean ergodic.

Proof. Model maps for this type of analytic self-maps of D are φ(z) = zn for

every z ∈ D, n ≥ 2. Hence, the iterates are given by φk(z) = znkfor every

z ∈ D, k ∈ N. (One can change variables analytically in a neighbourhood ofa and conjugate φ to the map zn for some n ≥ 2 in a neighbourhood of a, for


details see again e.g. [19]. The proof of this fact goes back to Bottcher [6]). Wewill show that the sequence ((Cφ)[k])k tends to C0 with respect to the operatornorm ‖ · ‖, where C0 is given by (C0f)(z) = f(0) for every z ∈ D.

The function

f : [0, 1) → R, f(r) =1 − r

1 − rnk

is monotone decreasing since

f ′(r) =−1 + (1 − nk)rnk

+ nkrnk−1

(1 − rnk)2≤ 0 for every r ∈ [0, 1).

Moreover, we have that

limr→1

1 − r

1 − rnk = limr→1

1

nkrnk−1=

1

nk

and ∞∑k=1

1

nk=

1

n − 1.

Hence there has to be 0 < r0 < 1 such that∞∑

k=1

1 − r0

1 − rnk

0

= M < ∞.

Now, we choose such an 0 < r0 < 1 and obtain with an application of Theorem 1

‖(Cφ)[k] − C0‖ ≤ 1

k

k∑m=1

‖Cφm − C0‖

≤ 1

k

k∑m=1

sup|z|≤r0

max

{1 − |z|

1 − |z|nm , 1 − |z|}

ρ(φm(z), 0)

+1

k

k∑m=1

sup|z|>r0

max

{1 − |z|

1 − |z|nm , 1 − |z|}

ρ(φm(z), 0)

≤ 1

k

k∑m=1

|r0|nm

+1

k

k∑m=1

1 − r0

1 − r0nm

≤ 1

k

1

1 − r0n

+1

kM → 0

as k → ∞. Thus, the claim follows.

Remark 12. If v is a typical weight with (2.1) such that v(r)/v(rn) is monotonedecreasing with respect to r and such that there is C < 1 with

limr→1

v(r)

v(rn)≤ Cn

112 E. Wolf CMFT

for every n ∈ N, then — with the same proof as above — we can show thatan analytic self-map of D with a super-attracting fixed point a ∈ D induces auniformly mean ergodic composition operator Cφ : H∞

v → H∞v . An example of

this is the weight

v(z) =1

1 − ln(1 − |z|) .

Proposition 13. Let v be a weight and φ be an analytic self-map of D. Moreover,let us assume that there is α > 0 such that

supz∈D

v(z)

v(φn(z))≥ α

(supz∈D

v(z)

v(φ(z))

)n

for every n ∈ N. If Cφ : H∞v → H∞

v is uniformly mean ergodic, then

supz∈D

v(z)

v(φ(z))≤ 1,

i.e. Cφ is a contraction and hence power bounded.

Proof. If Cφ is uniformly mean ergodic, then obviously we have that

1

n‖Cφn‖ → 0 as n → ∞.

Thus, for a fixed ε > 0 we can find n0 ∈ N such that for every n ≥ n0 we havethat ‖Cφn‖ ≤ nε. Hence we get

supz∈D

v(z)

v(φ(z))≤

(nε

α

)1/n

for every n ≥ n0. Thus,

supz∈D

v(z)

v(φ(z))≤ 1.

Next we construct examples of analytic self-maps φ of D and weights v such that

supz∈D

v(z)

v(φn(z))=

(supz∈D

v(z)

v(φ(z))

)n

for every n ∈ N.

Proposition 14. Let a, b ∈ N and a > b. We put

φ(z) :=az + b

a + bz

for every z ∈ D. Then φ is an analytic self-map of D and, if n ∈ N is an evennumber,

φn(z) =

(an +

(n2

)an−2b2 + · · · + bn

)z +

(n1

)an−1b + · · · + (

nn−1

)abn−1((

n1

)an−1b + · · · + (

nn−1

)abn−1

)z + an +

(n2

)an−2b2 + · · · + bn


for every z ∈ D. Moreover, if n is an odd number,

φn(z) =

(an +

(n2

)an−2b2 + · · · + (

nn−1

)abn−1

)z +

(n1

)an−1b + · · · + bn((

n1

)an−1b + · · · + bn

)z + an +

(n2

)an−2b2 + · · · + (

nn−1

)abn−1

for every z ∈ D.

Proof. We will prove this by induction. Easy calculations show that

φ2(z) =(a2 + b2)z + 2ab

(a2 + b2) + 2abz,

φ3(z) =(a3 + 3ab2)z + (3a2b + b3)

(3a2b + b3)z + (a3 + 3ab2)

for every z ∈ D.

We assume that the claim is true for n ∈ N. First we suppose that n is an evennumber. We have to determine φn+1(z) for every z ∈ D.

φn+1(z) = φ(φn(z))

=a(an+(n

2)an−2b2+···+bn)z+(n1)an−1b+···+( n

n−1)abn−1

((n1)an−1b+···+( n

n−1)abn−1)z+an+(n2)an−2b2+···+bn

+ b

b(an+(n

2)an−2b2+···+bn)z+(n1)an−1b+···+( n

n−1)abn−1

((n1)an−1b+···+( n

n−1)abn−1)z+an+(n2)an−2b2+···+bn

+ a

=

(an+1 + · · · + ((

nn

)+

(n

n−1

))abn

)z +

((n1

)+

(n0

))anb + · · · + bn+1(((

n1

)+

(n0

))anb + · · · + bn+1

)z + an+1 + · · · + ((

nn

)+

(n

n−1

))abn

=

(an+1 +

(n+1

2

)an−2b2 + · · · + (

n+1n

)abn

)z +

(n+1

1

)anb + · · · + bn+1((

n+11

)anb + · · · + bn+1

)z + an+1 + · · · + (

n+1n

)abn

.

If n is an odd number, we can show the claim analogously.

Proposition 15. Let a, b ∈ N with a > b. We consider the analytic self-map

φ(z) =az + b

a + bz

for every z ∈ D and the weight v(z) = 1 − |z|. Then

supz∈D

v(z)

v(φn(z))=

(supz∈D

v(z)

v(φ(z))

)n

.

Proof. Calculations show that

supz∈D

v(z)

v(φ(z))=

−b − a

b − a.

114 E. Wolf CMFT

Hence, from the above formula we get

supz∈D

v(z)

v(φn(z))=

−an − (n2

)an−2b2 − · · · − bn − (

n1

)an−1b − · · · − (

nn−1

)abn−1(

n1

)an−1b + · · · + (

nn−1

)abn−1 − an − (

n2

)an−2b2 − · · · − bn

=

(−b − a

b − a

)n

=

(supz∈D

v(z)

v(φ(z))

)n

,

if n is an even number. For an odd number n we can do analogous calculationsand the claim follows.

We thank Alejandro Miralles for communicating the following to us.

Proposition 16. Let φ : D → D be an analytic map with fixed point 0 andv(z) = 1− |z| for every z ∈ D. Then Cφ is a non-strict contraction and satisfies

‖Cφn‖ = supz∈D

v(z)

v(φn(z))=

(supz∈D

v(z)

v(φ(z))

)n

= 1

for every n ∈ N.

Proof. The fact that Cφ is a contraction follows directly from Schwarz’s Lemma.The first equality is clear since v = v in this case. Next, since, φn(0) = 0 forevery n ∈ N, by Schwarz’s Lemma we know that either |φ′

n(0)| < 1 for everyn ∈ N or there exists n0 such that |φ′

n0(0)| = 1, so for every n ≥ n0 we have that

φn(z) = anz, where an ∈ C and |an| = 1.

Now, if φn(z) = az, |a| = 1, then

v(z)

v(φn(z))= 1

for every z ∈ D, so the supremum is equal to 1.

For φn with |φ′n(0)| < 1, we have that the supremum is attained at z = 0 and is

equal to 1 since |φ(z)| < |z| for every z = 0 by Schwarz’s Lemma.

Since the methods involving the Denjoy-Wolff Theorem exclude the automor-phisms of the unit disk we now turn our attention to this case and start withconsidering the Mobius transformations which interchange a point a ∈ D with 0,i.e. maps given by

ϕa(z) =a − z

1 − azfor every z ∈ D.

In [9] Bonet, Domanski, Lindstrom and Taskinen showed that if H∞v is generated

by a typical weight satisfying condition (2.1), then for every a ∈ D the composi-tion operator Cϕa : H∞

v → H∞v is bounded. We will use this fact to obtain the

following proposition.

Proposition 17. Let v be a typical weight satisfying (2.1).


(i) For every a ∈ D, the operator Cϕa : H∞v → H∞

v is power bounded.(ii) For every a ∈ D, the operator Cϕa : H∞

v → H∞v is uniformly mean ergodic.

Proof. Let us start with (i). It is easy to show that

Cnϕa

=

{Cϕa if n ∈ N is even

Cid if n ∈ N is odd,

where Cid is defined as (Cidf)(z) = f(z) for every z ∈ D. Hence

supn∈N

‖Cnϕa‖ ≤ max{‖Cid‖, ‖Cϕa‖} = max{1, ‖Cϕa‖},

and Cϕa is power bounded.

It remains to show (ii). We will prove that∥∥∥∥(Cϕa)[n] − 1

2(Cϕa + Cid)

∥∥∥∥ → 0 as n → ∞.

Calculations prove

(Cϕa)[n] − 1

2(Cϕa + Cid) =

⎧⎨⎩

1

2n(Cϕa − Cid) as n ∈ N is odd

0 as n ∈ N is even.

Hence∥∥∥∥(Cϕa)[n] − 1

2(Cϕa + Cid)

∥∥∥∥ ≤ 1

2n‖Cϕa − Cid‖ ≤ 1

nmax{‖Cϕa‖, ‖Cid} → ∞,

as n → ∞. Thus, the claim follows.

Since all automorphisms of the unit disk are given by

fa(z) = eiΘ a − z

1 − az

for a ∈ D and θ ∈ R we will study rotations next. Obviously, each rotation isa contraction and hence power bounded. Thus, it remains to investigate whenrotations are uniformly mean ergodic.

Proposition 18. Let k ∈ N0 and φk(z) = eiπ/kz for every z ∈ D. ThenCφk

: H∞v → H∞

v is uniformly mean ergodic.

Proof. We show that∥∥∥∥∥(Cφk)[n] − 1

2k

2k∑r=1

Crφ

∥∥∥∥∥ → 0 as n → ∞.

116 E. Wolf CMFT

Calculations show that

(Cφ)[n] − 1

2k

2k∑r=1

Crφ =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2k − 1

2knCφ − 1

2kn

2k∑r=2

Crφ, if n = 2kl + 1

2k − 2

2kn(Cφ + C2

φ) − 2

2kn

2k∑r=3

Crφ, if n = 2kl + 2

...

1

2kn

2k−1∑r=1

Crφ − 2k − 1

2knC2k

φ , if n = 2kl + (k − 1)

0 if n = 2kl.

Since all the involved operators are bounded, this sequence tends to zero asn → ∞.

Proposition 19. Let Θ = p/q ∈ [0, 2) be rational and φΘ(z) = eiΘπz for everyz ∈ D. Then CφΘ

: H∞v → H∞

v is uniformly mean ergodic.

Proof. Obviously, we need 2q steps to get back to the operator Cφ. Thus,proceeding as in the previous proposition we obtain the claim.

Next, it remains open, whether each rotation is uniformly mean ergodic. Thus,we pose the following suggestion which — unfortunately — we are not able toprove.

Suggestion 20. Let Θ ∈ [0, 2) and φΘ(z) = eiΘπz for every z ∈ D. ThenCφΘ

: H∞v → H∞

v is uniformly mean ergodic, and the sequence ((Cφ)[n])n tendsto the operator T : H∞

v → H∞v , given by

(Tf)(z) =1

2

∫ 2

0

(CφΘf)(z) dΘ =

1

2

∫ 2

0

f(eiΘπz) dΘ.

Acknowledgement. I would like to thank Jose Bonet for the interesting andfruitful discussion I had with him on this topic. Moreover, I would like to thankthe members of the Instituto Universitario de Matematica Pura y Aplicada ofthe Universidad Politecnica de Valencia for their hospitality and kindness.

References

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Elke Wolf E-mail: [email protected]: Universitat Paderborn, Mathematisches Institut, 33095 Paderborn, Germany.

power bounded composition operators

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