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GEOMETRIC PROPERTIES OF COMPOSITION OPERATORS
BELONGING TO SCHATTEN CLASSES
YONGSHENG ZHU
(Received 9 August 1999)
Abstract.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 compo-sition 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.
1. Introduction. Let D denote the unit disk in the complex plane Cand let Hp denote the Hardy space of functions
f (z)= ∞
n=0
anzn (1.1)
analyticinDsuch that
fpp= sup 0≤r <1
1 2π
2π
0
f
r eiθpdθ <∞, (1.2)
where 0< p <∞. Letφbe an analyticfunction mappingDinto itself. The composition operatorCφ(induced byφ) onHpis defined by
Cφ(f )(z)=fφ(z), z∈D. (1.3)
It is well known thatCφis a bounded linear operator onHp. The compactness of this operator is characterized in Shapiro [4] by the following criterion.
Shapiro’s compactness criterion. The operatorCφis compact if and only if
lim |z|→1
Nφ(z)
log1/|z|=0, (1.4)
where
Nφ(w)= z∈φ−1(w)
log 1
|z|, w∈D−
φ(0). (1.5)
Figure1.1
0 1
Figure1.2
Luecking-Zhu theorem. The composition operatorCφ∈Sp(H2)if and only if
Nφ(z)
log1/|z|∈Lp/2(dλ). (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
φ(z)=1−ig(z)1 , whereg(z)=
i1−1+zz. (1.7)
Thenφis a Riemann map fromDonto the semi-disk
z: Im(z) >0 andz−12<12
(1.8)
withφ(1)=1 (seeFigure 1.1).
Lotto proved thatCφis a compact operator but not a Hilbert-Schmidt (i.e.,S2(H2)) operator. His investigation led to the following conjecture.
Lotto’s conjecture1. The composition operatorCφbelongs to the Schatten ideal
Sp,ifp >2.
Supposeψis a univalent map fromDonto a crescent shaped region bounded by the semi-circle
z: Im(z)≥0 andz−12=12
, (1.9)
and a circular arc in the upper half ofDthat joins 0 to 1 (seeFigure 1.2).
Lotto’s conjecture2. Givenp, 0< p <∞,there exists a simple example of a domainGpwithGp⊂D, or there are easily verifiable geometric conditions onGp, such that the Riemann map fromDontoGpinduces a compact operator that is not inSp.
He described a way to produce a compact composition operator which is not in any Schatten ideal if his conjecture2is true. Here we want to point out that Tom Carrol and Carl C. Cowen gave such an example in [1]. But the functionφthat 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 domainφ(D)and how the geometry of this domain preventsCφto belong toSpfor allp, 0< p <∞.
The goal of this paper is to prove both Lotto’s conjectures. We establish
Lotto’s conjecture 1in Section 3and Lotto’s conjecture 2in Section 4. In Section 5, we follow Lotto’s method to construct a Riemann map that induces a compact com-position operator which is not in any Schatten ideals.
2. Background and terminology. For infinite-dimensional Hilbert space H and compact operatorT onH, we define singular numbers forT by
sn=sn(T)=infT−K:Khas rank≤n. (2.1)
We know that the compact operators are exactly thoseT for whichsn(T)→0. By def-inition the finite rank operators are those for which snis eventually zero. In between are the Schatten classes. Specifically, the Schattenp-classSp(H), 0< p <∞, consists of thoseT for which
∞
n=0
snp<∞. (2.2)
The class S1(H)is the trace class and S2(H) is the famous Hilbert-Schmidt class. Clearly,Sp(H)⊂Sq(H), if 0< p < q <∞.
We denote the set of bounded operators on H by B(H) and the set of compact operators onHbyK(H). We have the following lemma.
Lemma2.1(see [3]). The Schatten classSp(H),0<p <∞, is a two-sided ideal inB(H). LetΩn(n=0,1,2)be simply connected domains such thatΩ0⊂Ω1⊂Ω2⊂Dand ρnbe univalent maps fromDontoΩn, respectively. The following useful corollaries are easy consequences of this lemma (see [2]).
Corollary2.2. IfCρ1∈Sp(H)for somep,0< p <∞, thenCρ0∈Sp(H). Corollary2.3. IfCρ1∈Sp(H)for somep,0< p <∞, thenCρ2∈Sp(H).
Corollary2.4. Suppose thatΩis the image ofΩ1under an automorphism of the
unit diskDandρis a univalent analytic function which mapsDontoΩ. ThenCρ1∈ Sp(H)if and only ifCρ∈Sp(H).
Ifφis univalent, we have
that is equivalent to 1− |φ−1(w)|, as |φ−1(w)| →1. Thus, Shapiro’s compactness criterion becomes the following corollary.
Corollary2.5. Suppose thatφis a univalent selfmap ofD. The composition oper-atorCφis compact onH2if and only if
lim |z|→1
1−φ(z)
1−|z| = ∞. (2.4)
Luecking-Zhu theorem implies the following corollary.
Corollary2.6. Suppose thatφis a univalent selfmap ofDinto itself. The compo-sition operatorCφ∈Sp(H2)if and only if
χφ(D)·1−1−|φ−1z(z)| ∈Lp/2(dλ). (2.5)
We use Corollaries2.5and2.6to prove our theorems.
3. Proof of Lotto’s conjecture1
Theorem3.1. Letφbe a Riemann map fromDonto the semi-disk
G= z: Im(z) >0andz−1 2 <1
2
, (3.1)
such thatφ(1)=1. Then the composition operatorCφinduced byφbelongs to Schatten idealsSpfor allp >2.
Proof. Since∂Gcontacts∂Donly atz=1,we only need to consider what happens whenz∈Gcloses to 1. For smallε >0,let∆(ε)= {z:|z−1|< ε}. ByCorollary 2.6, we need to prove
G∩∆(ε)
1−φ−1(z) 1−|z|
p/2
dA(z)
1−|z|22<∞, ifp >2. (3.2)
It is not difficult to find suchφso thatφ−1(z)=(z2−(z−1)2i)/(z2+(z−1)2i)and
1−φ−1(z)2= −4Imz(z¯ −1)2
z2+(z−1)2i. (3.3)
Letz=a+bi∈G, then Im(z(z¯ −1))2≈2ab(a−1)≈Im(z−1)2, asz→1. Where≈ means comparable.A(t)andB(t)are comparable if there are positive constantsC1 andC2such thatC1A(t)≤B(t)≤C2A(t). So we have
1−φ−1(z)2≈ −Im(z−1)2, asz →1, z∈G. (3.4)
Letz=1−r e−iθ∈G. Then 1−z=r e−iθ∈Gwhich implies thatr2≤rcosθ. Thus
0 1 0 1 0 1
Figure4.1
Now we use (3.4) and (3.5) to show that (3.2) is true. In fact,
G∩∆(ε)
1−φ−1(z) 1−|z|
p/2
dA(z)
1−|z|22
≈
G∩∆(ε)
1−φ−1(z)2 1−|z|
p/2
dA(z)
1−|z|2
≈
G∩∆(ε)
−Im(z−1)2 1−|z|
p/2 dA(z)
1−|z|2
≈
G∩∆(ε)
r2sin2θ rcosθ
p/2 r dr dθ
(rcosθ)2, z=1−r e−iθ.
(3.6)
The last integral is finite if and only if the following integral:
G
r2sin2θ
rcosθ
p/2 r dr dθ
(rcosθ)2 (3.7)
is finite. But
G
r2sin2θ rcosθ
p/2 r dr dθ (rcosθ)2=2p/2
π/2
0
cosθ
0 (rsinθ)
p/2 dr dθ rcos2θ
≈ π/2
0
dθ
(π/2−θ)2−p/2<∞, if and only ifp >2. (3.8)
This completes the proof ofTheorem 3.1.
4. Geometric characterization of Schatten ideals. For 0< α <1, letGαrepresents one of the shaded regions inFigure 4.1.
Each of the regions is bounded by the semi-circle
z: Im(z)≥0 andz−12=12
, (4.1)
Aα Aα Aα
Figure4.2
Theorem4.1. Suppose thatCφα is the composition operator induced byφα. Then
(a) Cφαdoes not belong to Schatten idealS2α/(1−α)(H2).
(b) Cφα∈Sp(H2)for anyp >2α/(1−α).
Theorem 4.1givesLotto’s conjecture 2a positive answer: given any 0< p <∞, we can pickα=p/(p+2)∈(0,1), thenGα⊂Dis the domain for which the Riemann map fromDontoGαinduces a composition operator that is not inSp(H2).
Lemma4.2. Letτ(z)=i(1/z−1). ThenτmapsGαonto the sectorAα(seeFigure 4.2), where the two sides of the sector form an angle ofαπ. Moreover,
1−|z|2≈ρ2+2ρsinθ, ifτ(z)=ρeiθ, ρ →0. (4.2)
Proof. It is easy to verify thatτ(Gα)=Aαand
1−|z|2=1− 1 1−iτ
2
≈ρ2+2ρsinθ, ifτ(z)=ρeiθ, ρ →0. (4.3)
Lemma4.3. Suppose thatw(σ )=(1+iσ )/(1−iσ ), thenw maps the upper-half planeH+ontoDand
1−|w| ≈Imσ , asσ →0(orw →1), σ∈H+. (4.4)
Proof. Letw=r eiθ. Then
σ=i1−r eiθ 1+r eiθ ≈
1 2θ+
1
2(1−r )i, asw →1. (4.5) Therefore,
Imσ ≈1−r . (4.6)
On the other hand, we have
1−|w|2≈ −iθ+(1−r )i+iθ−(1−r )i=2(1−r )≈Imσ . (4.7)
Now we can begin to proveTheorem 4.1. Proof ofTheorem4.1. Letψα=φ−1
α . We can decomposeψα
ByCorollary 2.6, we need to show that
Gα
1−ψα(z) 1−|z|
p/2
dA(z)
1−|z|22 (4.9)
is finite forp >2α/(1−α). Actually we only need to consider what happens to (4.9) whenzcloses to 1. So we consider
Gα∩∆(ε)
1−ψα(z) 1−|z|
p/2
dA(z)
1−|z|22
≈
Gα∩∆(ε)
1−w(z) 1−|z|
p/2 dA(z)
1−|z|2
≈
τ(Gα∩∆(ε))
Imσ
ρ2+2ρsinθ
p/2 ρ dρ dθ
ρ2+2ρsinθ2,
(4.10)
whereσ is in the upper-half plane,(ρ,θ)∈τ(Gα∩∆(ε))⊂Aαandσ =τ1/α. Without loss of generality, we may assume thatτ(Gα∩∆(ε))= {(ρ,θ)∈Aα:ρ <1}. Now we consider three cases: 0< α <1/2;α=1/2; and 1/2< α <1.
Case1(0< α <1/2). Define
Ω1=(ρ,θ)∈τGα∩∆(ε): 0< ρ≤2sinθ,
Ω2=(ρ,θ)∈τGα∩∆(ε):ρ >2sinθ,
Gα
1−ψα(z) 1−|z|
p/2
dA(z)
1−|z|22
≈
Ω1
Imσ
ρ2+2ρsinθ
p/2 ρ dρ dθ
ρ2+2ρsinθ2
+
Ω2
Imσ ρ2+2ρsinθ
p/2
ρ dρ dθ
ρ2+2ρsinθ2
≈
Ω1
ρ1/αsin(θ/α)
ρsinθ
p/2 ρ dρ dθ (ρsinθ)2
+
Ω2
ρ1/αsin(θ/α)
ρ2
p/2ρ dρ dθ
ρ22
≈ απ
0 dθ
2sinθ
0
ρ1/αsin(θ/α) ρsinθ
p/2 ρ dρ (ρsinθ)2
+ απ
0 dθ
1
2sinθ
ρ1/αsin(θ/α) ρ2
p/2ρ dρ dθ
ρ22
≈ απ
0
1
θ2−(1/α−1)(p/2)dθ.
(4.11)
Case2(α=1/2). That is what exactlyTheorem 3.1is about. Case3(1/2< α <1). Define
Ω1= (ρ,θ)∈τGα∩∆(ε): 0< ρ≤2sinθandθ <π4
,
Ω2=(ρ,θ)∈τGα∩∆(ε):ρ >2sinθ,
Ω3= (ρ,θ)∈τGα∩∆(ε): 0< ρ≤2sinθandθ≥π4
,
Ω3
1−ψα(z) 1−|z|
p/2
dA(z)
1−|z|22≈
Ω3
Imσ
ρ2+2ρsinθ
p/2 ρ dρ dθ
ρ2+2ρsinθ2
≈ απ
π/4 1 θ2dθ
1
0ρ
(1/α−1)(p/2−2)dρ.
(4.12)
The last integral is finite if and only ifp >2α/(1−α). Using the same proof as in
Case 1, we have
Ω1
1−ψα(z) 1−|z|
p/2
dA(z)
1−|z|22<∞ if and only ifp > 2α 1−α,
Ω2
1−ψα(z) 1−|z|
p/2
dA(z)
1−|z|22<∞ if and only ifp > 2α 1−α.
(4.13)
This completes the proof ofTheorem 4.1. Let
Ta,θ(z)=1+z+aza eiθ, −1< a <1,0≤θ <2π. (4.14) Ta,θ is an automorphism ofD and maps Gα ontoΩα,a,θ, where Ωα,a,θ can have different shapes and positions, depending on the parameters. By choosing appropriate parameters, we can apply Theorems3.1and4.1andCorollary 2.4to different regions. We can also apply these theorems with Corollaries2.2and 2.3. InSection 5, we use
Theorem 4.1andCorollary 2.4to construct a non-Schatten class compact composition operator.
5. 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.
Letθn=π/(n+1), zn=eiθn, rn =(1/2)sinθn, and cn =(1−rn)zn, wheren=
1,2,3,....DefineΩnto be the region bounded by the semi-circle
z: Im(z)≥0 andz−cn=rn, (5.1) and a circular arc that is insideDjoining 1−2rnto 1 (it is actually a line segment whenn=1). These two arcs form an angle of(n+1)/(n+3)πat 1. Let
Ω
n=zeiθn:z∈Ωn, (5.2)
Ω= ∪∞
n=1Ωn. (5.3)
0 1 1
Figure5.1
Theorem5.1. Suppose thatΩis defined by (5.3). Then we have (1)Ωis a simply connected domain contained in the upper half ofD.
(2)Any Riemann mapφthat mapsDontoΩinduces a compact composition operator
Cφ. ButCφdoes not belong to any Schatten idealSp(H2),p >0.
Proof. We estimate the distance between the centerscn−1 and cn ofΩn−1 and Ω
n(n≥2):
cn−1−cn=1−rn
zn−1−rn−1zn−1
=
1−12sinnπ+1
ei(π/(n+1))−1−1 2sin
π n
ei(π/n)
=O
1
n2
.
(5.4)
But the radius rn of Ω
n is (1/2)sin(π/(n+1))≥1/(1+n). Thus Ωn−1 and Ωn overlap andΩis simply connected.
Since
Imcn=1−rnImzn=1−rnsinnπ+1>12sinnπ+1=rn. (5.5) Ω
nlies in the upper half ofD. ThereforeΩ⊂Dis in the upper plane. By the con-struction ofΩ, we know thatΩtouches the boundary ofDonly atzn,n=1,2,3,..., and 1. One can see that atzn, φis not conformal (see [4,5]). Thusφhas no finite angular derivative atzn. Note thatΩ⊂Dis in the upper plane andzn→1 asn→ ∞,φ is not conformal at 1 either. By the angular derivative criterion for compactness (see [5]), we knowCφis compact. Letφnbe a Riemann map that mapsDontoΩ
nandCnbe the induced composition operator. LetGαbe the region defined inSection 4andψα be a Riemann map fromDontoGα. ByTheorem 4.1, we know that the composition operator induced byψ(n+1)/(n+3)does not belong to Schatten idealSn+1. Let
ηn(z)= z+
1−2rn
1+1−2rnzeiθ, z∈D. (5.6)
Thenηnis an automorphism ofD. Moreover,ηn(G(n+1)/(n+3))=Ωn(seeFigure 5.2). Thus, according toCorollary 2.4,Cn∈Sn+1(H2). ButΩn⊂Ωfor any positive integer n,Cφ∈Sn+1(H2)byCorollary 2.3for anyn. We know thatSp(H2)⊂Sq(H2)ifp < q.
ThereforeCφ does not belong to any Schatten classes. This completes the proof of
D
φn
ηn
ψn
0 1
Figure5.2
Acknowledgements. It is a great pleasure for me to express my sincere grati-tude to my advisor, Dr David Stegenga, for his understanding, patient guidance, and valuable suggestions. Thanks also go to Dr Wayne Smith and Dr George Csordas.
References
[1] T. Carroll and C. C. Cowen,Compact composition operators not in the Schatten classes, J. Operator Theory26(1991), no. 1, 109–120.MR 94c:47045. Zbl 792.47031.
[2] B. A. Lotto,A compact composition operator that is not Hilbert-Schmidt, Studies on Com-position Operators, Contemporary Mathematics, vol. 213, Amer. Math. Soc., Rhode Island, 1998, pp. 93–97.MR 98j:47071. Zbl 898.47025.
[3] D. H. Luecking and K. H. Zhu,Composition operators belonging to the Schatten ideals, Amer. J. Math.114(1992), no. 5, 1127–1145.MR 93i:47032. Zbl 792.47032.
[4] J. H. Shapiro,The essential norm of a composition operator, Ann. of Math. (2)125(1987), no. 2, 375–404.MR 88c:47058. Zbl 642.47027.
[5] ,Composition Operators and Classical Function Theory, Universitext: Tracts in Math-ematics, Springer-Verlag, New York, 1993.MR 94k:47049. Zbl 791.30033.
Yongsheng Zhu: Fair Isaac2150Shattuck Avenue, Berkeley, CA94704, USA E-mail address:[email protected]
