GEOMETRIC PROPERTIES OF COMPOSITION OPERATORS BELONGING TO SCHATTEN CLASSES

We investigate the connection between the geometry of the image domain of an analytic function mapping the unit disk into itself and the membership of the composition operator induced by this function in the Schatten classes. The purpose is to provide solutions to Lotto’s conjectures and show a new compact composition operator which is not in any of the Schatten classes. 2000 Mathematics Subject Classification. 47B10, 47B33, 47B38, 30H05, 46E22.

where 0 < p < ∞.Let φ be an analytic function mapping D into itself.The composition operator C φ (induced by φ) on H p is defined by It is well known that C φ is a bounded linear operator on H p .The compactness of this operator is characterized in Shapiro [4] by the following criterion.Let dλ = (1 − |z| 2 ) −2 dA be the Mobius invariant measure on D. Let S p (H 2 ) be the Schatten ideal of operators on the Hilbert space H 2 for p > 0. D. H. Luecking and K. Zhu [3] proved the following theorem.(1.6) B. A. Lotto [2] began the investigation of the connection between the geometry of φ(D) and the membership of composition operators in the Schatten classes. Let , where g(z Then φ is a Riemann map from D onto the semi-disk with φ(1) = 1 (see Figure 1.1).Lotto proved that C φ is a compact operator but not a Hilbert-Schmidt (i.e., S 2 (H 2 )) operator.His investigation led to the following conjecture.

Lotto's conjecture 1. The composition operator C φ belongs to the Schatten ideal
Suppose ψ is a univalent map from D onto a crescent shaped region bounded by the semi-circle and a circular arc in the upper half of D that joins 0 to 1 (see Figure 1.2).Lotto proved that C ψ is a Hilbert-Schmidt operator and issued the following challenge.
Lotto's conjecture 2. Given p, 0 < p < ∞, there exists a simple example of a domain G p with G p ⊂ D, or there are easily verifiable geometric conditions on G p , such that the Riemann map from D onto G p induces a compact operator that is not in S p .
He described a way to produce a compact composition operator which is not in any Schatten ideal if his conjecture 2 is true.Here we want to point out that Tom Carrol and Carl C. Cowen gave such an example in [1].But the function φ that induces the desired compact non-Schatten class operator is described in terms of its "model for iteration."In general, it is hard to visualize the domain φ(D) and how the geometry of this domain prevents C φ to belong to S p for all p, 0 < p < ∞.
The goal of this paper is to prove both Lotto's conjectures.We establish Lotto's conjecture 1 in Section 3 and Lotto's conjecture 2 in Section 4. In Section 5, we follow Lotto's method to construct a Riemann map that induces a compact composition operator which is not in any Schatten ideals.

Background and terminology.
For infinite-dimensional Hilbert space H and compact operator T on H, we define singular numbers for T by ( We know that the compact operators are exactly those T for which s n (T ) → 0. By definition the finite rank operators are those for which s n is eventually zero.In between are the Schatten classes.Specifically, the Schatten p-class S p (H), 0 < p < ∞, consists of those T for which 2) The class S 1 (H) is the trace class and S 2 (H) is the famous Hilbert-Schmidt class.
We denote the set of bounded operators on H by B(H) and the set of compact operators on H by K(H).We have the following lemma.

Corollary 2.4. Suppose that Ω is the image of Ω 1 under an automorphism of the unit disk D and ρ is a univalent analytic function which maps D onto Ω. Then C ρ 1 ∈ S p (H) if and only if C ρ ∈ S p (H).
If φ is univalent, we have Thus, Shapiro's compactness criterion becomes the following corollary.
Corollary 2.5.Suppose that φ is a univalent selfmap of D. The composition operator C φ is compact on H 2 if and only if Luecking-Zhu theorem implies the following corollary.
Corollary 2.6.Suppose that φ is a univalent selfmap of D into itself.The composition operator C φ ∈ S p (H 2 ) if and only if We use Corollaries 2.5 and 2.6 to prove our theorems.

Proof of Lotto's conjecture 1
Theorem 3.1.Let φ be a Riemann map from D onto the semi-disk such that φ(1) = 1.Then the composition operator C φ induced by φ belongs to Schatten ideals S p for all p > 2.
Proof.Since ∂G contacts ∂D only at z = 1, we only need to consider what happens when z ∈ G closes to 1.For small ε > 0, let ∆(ε) = {z : |z − 1| < ε}.By Corollary 2.6, we need to prove (3.6) The last integral is finite if and only if the following integral: Each of the regions is bounded by the semi-circle and a circular arc that is inside of D joining 0 to 1 (see Figure 4.1).These two arcs form angles of απ at 0 and 1. Define φ α to be one of the Riemann maps from D onto G α such that φ α (1) = 1.Theorem 4.1 gives Lotto's conjecture 2 a positive answer: given any 0 < p < ∞, we can pick α = p/(p + 2) ∈ (0, 1), then G α ⊂ D is the domain for which the Riemann map from D onto G α induces a composition operator that is not in S p (H 2 ).
Case 1 (0 < α < 1/2).Define The last integral converges if and only if 2 − (1/α − 1)(p/2) < 1, that is, p > 2α/(1 − α).This simultaneously proves both parts (a) and (b) of Theorem 4.1 for The last integral is finite if and only if p > 2α/(1 − α).Using the same proof as in Case 1, we have T a,θ is an automorphism of D and maps G α onto Ω α,a,θ , where Ω α,a,θ can have different shapes and positions, depending on the parameters.By choosing appropriate parameters, we can apply Theorems 3.1 and 4.1 and Corollary 2.4 to different regions.We can also apply these theorems with Corollaries 2.2 and 2.3.In Section 5, we use Theorem 4.1 and Corollary 2.4 to construct a non-Schatten class compact composition operator.

A non-Schatten class composition operator.
Based on Lotto's suggestion, we successfully constructed a compact composition operator that is not in any Schatten ideals.Here is the process of the construction.
Let and a circular arc that is inside D joining 1 − 2r n to 1 (it is actually a line segment when n = 1).These two arcs form an angle of (n + 1)/(n + 3)π at 1. Let ) We have the following theorem.(2) Any Riemann map φ that maps D onto Ω induces a compact composition operator C φ .But C φ does not belong to any Schatten ideal S p (H 2 ), p > 0.
Proof.We estimate the distance between the centers c n−1 and c n of Ω n−1 and Ω n (n ≥ 2): (5.4) But the radius r n of Ω n is (1/2) sin(π /(n + 1)) ≥ 1/(1 + n).Thus Ω n−1 and Ω n overlap and Ω is simply connected. Since Ω n lies in the upper half of D. Therefore Ω ⊂ D is in the upper plane.By the construction of Ω, we know that Ω touches the boundary of D only at z n , n = 1, 2, 3,..., and 1.One can see that at z n , φ is not conformal (see [4,5]).Thus φ has no finite angular derivative at z n .Note that Ω ⊂ D is in the upper plane and z n → 1 as n → ∞, φ is not conformal at 1 either.By the angular derivative criterion for compactness (see [5]), we know C φ is compact.Let φ n be a Riemann map that maps D onto Ω n and C n be the induced composition operator.Let G α be the region defined in Section 4 and ψ α be a Riemann map from D onto G α .By Theorem 4.1, we know that the composition operator induced by ψ (n+1)/(n+3) does not belong to Schatten ideal S n+1 .Let Thus, according to Corollary 2.4, C n ∈ S n+1 (H 2 ).But Ω n ⊂ Ω for any positive integer n, C φ ∈ S n+1 (H 2 ) by Corollary 2.3 for any n.We know that S p (H 2 ) ⊂ S q (H 2 ) if p < q.Therefore C φ does not belong to any Schatten classes.This completes the proof of Theorem 5.1.

1 .
Introduction.Let D denote the unit disk in the complex plane C and let H p denote the Hardy space of functions f (z)

Shapiro's compactnessN
criterion.The operator C φ is compact if and only if lim |z|→1

0 1 1 Figure 5 . 1 Theorem 5 . 1 .
Figure 5.1Theorem 5.1.Suppose that Ω is defined by(5.3).Then we have (1) Ω is a simply connected domain contained in the upper half of D.(2) Any Riemann map φ that maps D onto Ω induces a compact composition operator C φ .But C φ does not belong to any Schatten ideal S p (H 2 ), p > 0.

Figure 5 . 2
Figure 5.2 2 , as z → 1.Where ≈ means comparable.A(t) and B(t) are comparable if there are positive constants C 1 and