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A note on the Königs domain of compact composition operators on the Bloch space
Journal of Inequalities and Applicationsvolume 2011, Article number: 31 (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.
1 Introduction
Let be the unit disk in the complex plane and its boundary. We define the Bloch space to be the Banach space of functions, 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 .
Suppose now that is analytic, then we may define the operator, 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
Recall that the hyperbolic geometry on is defined by the distance
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 .
The Schröder functional equation is the equation
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.
In this paper, we study the geometry of Ω when . In order to do this, we will use the hyperbolic geometry of Ω. If is a universal covering map and Ω is a hyperbolic domain in ℂ, then the Poincaré density on Ω is derived from the equation
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 selfmap ofwith a fixed point in. Suppose that for some positive integer n_{0}there are at most finitely many points ofat whichhas 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.
does not contain a twisted sector.
Here, Ω is said to contain a twisted sector if there is an unbounded curve Γ ∈ Ω with
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
Throughout this section, we assume that Ω is an unbounded simply connected domain in ℂ with 0 ∈ Ω. As in the previous section, 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 mappinginto, φ(0) = 0. Suppose that the closure ofintersectsonly 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
1. C_{ φ } is compact on;
2. ;
3. For every n > 0, .
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.
By the results of Madigan and Matheson [2], and Smith [6] cited above C_{ φ } is compact on if and only if
However, by Schröder's equation
Since Ω is simply connected, λ_{Ω} (w) ≍ 1/δ_{Ω} (w) and so C_{ φ } is compact on if and only if
Since γ Ω ⊂ Ω, γw → ∂Ω implies that w → ∂Ω. Therefore, (3) holds 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.
Then we have that δ_{Ω} (w) < ε^{n}δ_{Ω} (γ^{n} w) and hence
Now as 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 lefthand 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}nw ^{n1}δ_{Ω} (w) = 0 and this follows from the above argument. Therefore, 2 implies 3.
To show that 3 implies 2, we need to show that if
then 2 holds.
To complete the proof, we require the following lemma whose proof we merely sketch.
Lemma 2. Under the hypotheses of the theorem,
Sketch of Proof. First note that
Now if lies in a nontangential angle of opening απ at ζ, then a short calculation shows that
and the assertion follows. □
Now with f defined above, we have
for large enough w. Hence,
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.
1. C_{ φ } is compact on;
2. ;
3. For every n > 0, .
The hypothesis on the boundary of Ω is vital. If we do not assume that ∂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 functionsuch 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} = {2πk/N : k = 0, ..., N  1}.
Suppose now that Θ _{ 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 .
By estimating the line segment [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) ≤ wθ_{k+1} θ_{ k } . By a similar argument, we deduce the estimate δ_{Ω} (γw) ≍ γwθ  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^{n1}} 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^{n1}} satisfies γR'∩ ∂Ω = ∅, and so there is an arc such that σ (ρ_{ θ } ) = R' and ρ_{ θ } has an endpoint in .
Hence, each is contained in a prime end of and
The result follows. □
3 Multiply connected domains
The geometric arguments of the previous section potentially lend themselves to multiply connected domains in the following way. Suppose that Ω is a domain in ℂ with 0 ∈ Ω and 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, ....
Now, since
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 letbe a universal covering map with σ(0) = 0.
If φ, as defined by (2) is inthen Ω is simply connected.
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Keywords
 Unit Disk
 Hardy Space
 Composition Operator
 Connected Domain
 Hyperbolic Geometry