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# A note on the Königs domain of compact composition operators on the Bloch space

- Matthew M Jones
^{1}Email author

**2011**:31

https://doi.org/10.1186/1029-242X-2011-31

© Jones; licensee Springer. 2011

**Received:**31 January 2011**Accepted:**10 August 2011**Published:**10 August 2011

## Abstract

Let
be the unit disk in the complex plane. We define
to be the little Bloch space of functions *f* analytic in
which satisfy lim_{|z|→1}(1 *-* |*z*|^{2})|*f*'(*z*)| = 0. If
is analytic then the composition operator *C*_{
φ
} : *f* ↦ *f* ∘ *φ* is a continuous operator that maps
into itself. In this paper, we show that the compactness of *C*_{
φ
} , as an operator on
, can be modelled geometrically by its principal eigenfunction. In particular, under certain necessary conditions, we relate the compactness of *C*_{
φ
} to the geometry of
, where *σ* satisfies Schöder's functional equation *σ* ∘ *φ = φ*'(0)*σ*.

**2000 Mathematics Subject Classification**: Primary 30D05; 47B33 Secondary 30D45.

## Keywords

- Unit Disk
- Hardy Space
- Composition Operator
- Connected Domain
- Hyperbolic Geometry

## 1 Introduction

*f*, analytic in with

This space has many important applications in complex function theory, see [1] for an overview of many of them. We denote by
the little Bloch space of functions in
that satisfy lim_{|z|→1}(1 *-* |*z*|^{2})|*f* '(*z*)| = 0. This space coincides with the closure of the polynomials in
.

*C*

_{ φ }, acting on as

*f*↦

*f*∘

*φ*. It was shown in [2] that every such operator maps continuously into itself. Moreover, it was proved that

*C*

_{ φ }is compact on if and only if

*φ*satisfies

where the infimum is taken over all sufficiently smooth arcs that have endpoints *z* and *w*.

Here,
is the Poincaré density of
. The hyperbolic derivative of *φ* is given by *φ*'(*z*)*/*(1 *-* |*φ*(*z*)|^{2}) and functions that satisfy (1) are called little hyperbolic Bloch functions or written
.

Note that this is just the eigenfunction equation for *C*_{
φ
} . Kœnigs' theorem states that if *φ* has fixed point at the origin then (2) has a unique solution for *γ* = *φ*'(0) which we call the *Kœnigs function* and denote by *σ* from here on. In the study of the geometric properties of *φ* in relation to the operator theoretic properties of *C*_{
φ
} , it has become evident that the Kœnigs function is much more fruitful to study than *φ* itself. In particular, see [3] for a discussion of the Kœnigs function in relation to compact composition operators on the Hardy spaces.

If we let
be the *Kœnigs domain* of *φ*, then (2) may be interpreted as implying that the action of *φ* on
is equivalent to multiplication by *γ* on Ω. It is due to this that the pair (Ω, *γ* ) is often called the geometric model for *φ*.

which is independent of the choice of *f*. Since this equation, in terms of differentials, is
(for *w* = *f* (*z*)), we see that the hyperbolic distance on
defined above carries over to a hyperbolic distance on Ω. For a more thorough treatment of the hyperbolic metric, see [4].

In [5], the Königs domain of a compact composition operator on the Hardy space was studied and the following result was proved.

**Theorem A**. *Let φ be a univalent self-map of*
*with a fixed point in*
*. Suppose that for some positive integer n*_{0}*there are at most finitely many points of*
*at which*
*has an angular derivative. Then the following are equivalent*.

- 1.
*Some power of C*_{ φ }*is compact on the Hardy space H*^{2}; - 2.
*σ lies in H*^{ p }*for every p <*∞; - 3.

for some *ε* > 0 and all *w* ∈ Γ, where *δ*_{Ω} is the distance from *w* to the boundary of Ω as defined below. The purpose of this paper is to provide a similar result to this in the context of the Bloch space.

## 2 Simply connected domains

*σ*represents the Riemann mapping of onto Ω with

*σ*(0) = 0 and

*σ*'(0) > 0. We will also define

*φ*via the Schröder functional equation. Throughout we let

so that *δ*_{Ω}(*w*) is the Euclidean distance from *w* to the boundary of Ω.

**Theorem 1**. *Let φ be a univalent function mapping*
*into*
, *φ*(0) = 0. *Suppose that the closure of*
*intersects*
*only at finitely many fixed points and is contained in a Stolz angle of opening no greater than απ there*.

*If* |*φ*'(0)| > 16 tan(*απ/* 2) *then the following are equivalent*

Remark: It has recently been shown by Smith [6] that compactness of *C*_{
φ
} on
is equivalent to compactness of *C*_{
φ
} on
, *BMOA* and *VMOA* when *φ* is univalent and so in the above theorem, the first condition could read: *C*_{
φ
} is compact on
,
, *BMOA* and *VMOA* Before proceeding, we prove the following lemma.

**Lemma 1**. *Under the hypotheses of the theorem, w and γw tend to the same prime end at* ∞, *and ∂γ* Ω ⊂ Ω.

*Proof*. The first assertion follows from the fact that the closure of touches only at fixed points. Suppose now that the second assertion is false and there are distinct prime ends

*ρ*

_{1}and

*ρ*

_{2}with

*ρ*

_{1}=

*γρ*

_{2}. Then under the boundary correspondence given by

*σ*there are distinct points

*η*, with

It follows that
and therefore *ζ* is a fixed point of *φ*. Hence, we have the contradiction *ρ*_{1} = *ρ*_{2}. □

*Proof*. We first prove that 1 is equivalent to 2.

*C*

_{ φ }is compact on if and only if

By the Lemma, we see that *γw* → ∂Ω means *w* → ∞ and *w* ∈ *γ* Ω, and we have shown that 1 and 2 are equivalent.

Suppose that 2 holds and let *ε* > 0 be given. Then we can find a *R* > 0 so that *δ*_{Ω} (*w*) *<εδ*_{Ω} (*γw*) for all |*w*| > *R*, since there are only a finite number of prime ends at *∞*. Choose *w* ∈ Ω arbitrarily with modulus greater than *R* and let *n* satisfy |*γ*|^{
-n
}*R* < |*w*| ≤ |*γ* |^{-n -1}*R*.

*w*→ ∞ in

*γ*Ω,

*γ*

^{ n }

*w*lies in a closed set properly contained in Ω and therefore

*δ*

_{Ω}(

*γ*

^{ n }

*w*) is bounded below by a constant independent of

*w*. We thus have that

and since *ε* was arbitrary, the left-hand side of the above inequality must tend to ∞. Hence, we have shown that lim_{w→∞}|*w*| ^{
β
} *δ*_{Ω} (*w*) = 0 for every *β* > 0.

Now
may be interpreted geometrically as lim_{w→∂Ω}*n*|*w*| ^{n-1}*δ*_{Ω} (*w*) = 0 and this follows from the above argument. Therefore, 2 implies 3.

then 2 holds.

To complete the proof, we require the following lemma whose proof we merely sketch.

and the assertion follows. □

as *w* → ∞ and so 2 holds. □

It is of interest to consider the growth of *σ* since condition 3 would imply that it has very slow growth. The following corollary follows from 3 and the fact that functions in
grow at most of order log 1/(1 *-* |*z*|).

**Corollary 1**.

*Suppose that φ satisfies the hypotheses of the Theorem and that any of the equivalent conditions holds, then for r*= |

*z*|.

We also provide the following restatement of the hypotheses of Theorem 1 to illustrate the main properties of the Königs domain.

**Corollary 2**.

*Let*Ω

*be an unbounded domain in ℂ with γ*Ω ⊂ Ω

*and*0 ∈ Ω

*. Suppose that has*Ω

*only finitely many prime ends at*∞

*and*

*In addition, suppose that* ∂*γ* Ω ⊂ Ω. *If*
, *σ*(0) = 0, *σ*'(0) > 0, *and φ is defined by Schröder's equation, then the following are equivalent*.

*γ*Ω ⊂ Ω, then we deduce from the proof of the Theorem that is equivalent to

In this situation, the finite part of the boundary of Ω plays a complicated role in the behaviour of *φ*. We conclude this section by constructing a domain that displays very bad boundary properties. This answers a question of Madigan and Matheson in [2].

In [2] it was shown that if ∂*φ*(*D*) touches
in a cusp, then
. However, it is not sufficient that ∂*φ*(*D*) touches
at an angle greater that 0. The question was raised of whether or not it is possible that
can be infinite.

With the hypothesis that ∂*γ* Ω ⊂ Ω the prime ends at ∞ correspond to points of
that touch
. Therefore,
is at most countable. A natural question to ask is whether or not
can ever be positive, where Λ represents linear measure.

This example is well known in the setting of the unit disk, see [7, Corollary 5.3]. We describe here the construction in terms of the Königs domain.

**Theorem 2**. *There is a univalent function*
*such that*
.

*Proof*. We construct the domain Ω so that it satisfies (4). Let 0 <

*γ*< 1 be given. We will define a nested sequence ,

*n*= 1, 2, ... so that

where Θ _{
n
} ⊂ Θ_{n+1}for all *n* = 1, 2, ....

First let *N* > 2 be chosen arbitrarily and let Θ_{1} = {2*πk*/*N* : *k* = 0, ..., *N -* 1}.

_{ n }has been defined, then let Θ

_{n+1}be such that Θ

_{ n }⊂ Θ

_{n+1}and whenever

*θ*∈ Θ

_{ n }is isolated, we define a sequence

*θ*

_{ k }∈ Θ

_{n+1},

*k*= 1, 2, ..., so that

*θ*

_{ k }→

*θ*as

*k*→ ∞ and for each

*k*there is a

*j*so that

*θ - θ*

_{ k }=

*θ*

_{ j -θ }. Moreover, assume that

In this way, we define the sequence of sets Θ_{
n
}, *n* = 1, 2, .... We will, furthermore, assume that for each
, there is a sequence *θ*_{
n
} ∈ Θ_{
n
}, *n* = 1, 2, ...., such that *θ*_{
n
} → *θ*.

We claim that this gives the desired domain Ω with boundary defined by (5).

To see this, let *γw* ∈ Ω be arbitrary, then by construction, we may find a *ζ* ∈ ∂Ω so that *δ*_{Ω} (*γw*) = |*ζ - γw*|. It is readily seen that for such *ζ*, there is an *n* so that *ζ* ∈ {*re*^{iθ}: *r* ≥ *γ*^{-n}} for some *θ* ∈ Θ _{
n
} and moreover, *θ* is isolated in Θ_{
n
}.

If we now consider *w*, we may find a sequence *θ*_{
k
} → *θ* as *k* → ∞ so that
for all *k* hence we may fix a *k* so that *δ*_{Ω} (*w*) = |*w* - *n*| for
.

*w*,

*η*] by the arc of joining

*w*to

*η*, we see that

*δ*

_{Ω}(

*w*) ≍ |

*w*||

*α - θ*

_{ k }| where

*w*=

*re*

^{iα}.Therefore, we have the estimate

*δ*

_{Ω}(

*w*) ≤ |

*w*||

*θ*

_{k+1}-

*θ*

_{ k }|. By a similar argument, we deduce the estimate

*δ*

_{Ω}(

*γw*) ≍ |

*γw*||

*θ - θ*

_{ k }| and so

by (6) and so the construction is complete.

We claim that if
is defined as usual and *φ* is given by Schröder's equation, then
.

In fact, if *θ* ∈ Θ _{
n
} is isolated, then the ray *R* = {*re*^{iθ}: *r* ≥ *γ*^{-n-1}} is contained in a single prime end of Ω. Therefore, to each such ray, there exists a point
that corresponds to *R* under *σ*. Since *γR* ⊂ ∂Ω, we thus have that *ζ* corresponds to a prime end *p* under *φ* with
.

On the other hand, if *θ* ∈ Θ _{
n
} is isolated, then *R*' = {*re*^{iθ}: *γ*^{-n}≤ *r* < *γ*^{-n-1}} satisfies *γR*'∩ ∂Ω = ∅, and so there is an arc
such that *σ* (*ρ*_{
θ
} ) = *R*' and *ρ*_{
θ
} has an end-point in
.

The result follows. □

## 3 Multiply connected domains

*γ*Ω ⊂ Ω for some . Let

*σ*be a universal covering map of onto Ω with

*σ*(0) = 0. Then

*σ*'(0) ≠ 0 and we may define

*φ*via (2). Now we have

However, if Ω is not simply connected, then *σ* is an infinitely sheeted covering of Ω and therefore the equation *σ* (*z*) = 0 has infinitely many distinct solutions, *z*_{
n
} , *n* = 0, 1, ....

for all *n* ≥ 0, we see that
. Thus, we have proved the following result.

**Proposition 1**. *Suppose that* Ω ⊂ ℂ *is a domain satisfying* 0 ∈ Ω *and γ* Ω ⊂ Ω, *and let*
*be a universal covering map with σ*(0) = 0.

## Declarations

## Authors’ Affiliations

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