Volume 2011, Article ID 164843,26pages doi:10.1155/2011/164843
Research Article
Toeplitz Operators on the Bergman Space of Planar Domains with Essentially Radial Symbols
Roberto C. Raimondo
1, 21Division of Mathematics, Faculty of Statistics, University of Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milan, Italy
2Department of Economics, University of Melbourne, Parkville, VIC 3010, Australia
Correspondence should be addressed to Roberto C. Raimondo,[email protected] Received 25 February 2011; Revised 5 June 2011; Accepted 6 June 2011
Academic Editor: B. N. Mandal
Copyrightq2011 Roberto C. Raimondo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the problem of the boundedness and compactness ofTφ when φ ∈ L2ΩandΩis a planar domain. We find a necessary and sufficient condition while imposing a condition that generalizes the notion of radial symbol on the disk. We also analyze the relationship between the boundary behavior of the Berezin transform and the compactness ofTφ.
1. Introduction
LetΩbe a bounded multiply-connected domain in the complex planeC, whose boundary
∂Ωconsists of finitely many simple closed smooth analytic curvesγj j1,2, . . . , nwhereγj are positively oriented with respect toΩandγj∩γi ∅ifi /j. We also assume thatγ1is the boundary of the unbounded component ofC\Ω. LetΩ1be the bounded component ofC\γ1, andΩj j 2, . . . , nthe unbounded component ofC\γj, respectively, so thatΩ ∩nj1Ωj.
Fordν 1/πdxdy, we consider the usualL2-spaceL2Ω L2Ω, dν. The Bergman spaceL2aΩ, dν, consisting of all holomorphic functions which areL2-integrable, is a closed subspace ofL2Ω, dνwith the inner product given by
f, g
Ωfzgzdνz 1.1 forf, g∈L2Ω, dν. The Bergman projection is the orthogonal projection
P :L2Ω, dν−→L2aΩ, dν. 1.2
It is well-known that for anyf∈L2Ω, dν, we have
P fw
ΩfzKΩz, wdνz, 1.3
whereKΩis the Bergman reproducing kernel ofΩ. Forϕ∈L∞Ω, dν, the Toeplitz operator Tϕ :L2aΩ, dν → L2aΩ, dνis defined byTϕP Mϕ, whereMϕis the standard multiplication operator. A simple calculation shows that
Tϕfz
ΩϕwfwKΩw, zdνw. 1.4
For square-integrable symbols, the Toeplitz operator is densely defined but is not necessarily bounded; therefore, the problem of finding necessary and sufficient conditions on the function ϕ ∈ L2Ω, dν for the Toeplitz operators Tϕ to be bounded or compact is a natural one, and it has been studied by many authors. Several important results have been established when the symbol has special geometric properties. In fact, in the context of radial symbols on the disk, many papers have been written with quite surprising resultssee1 of Grudsky and Vasilevski,2of Zorboska, and3of Korenblum and Zhushowing that operators with unbounded radial symbols can have a very rich structure. In fact, in the case of a continuous symbol, the compactness of the Toeplitz operators depends only on the behavior of the symbol on the boundary of the disk and this is similar to what happens in the Hardy space case, even though in the case of Bergman space, the Toeplitz operator with continuous radial symbol is a compact perturbation of a scalar operator and in the Hardy space case a Toeplitz operator with radial symbol is just a scalar operator. In the case of unbounded radial symbols, a pivotal role is played by the fact that in the Bergman space setting, contrary to the Hardy space setting, there is an additional direction that Grudsky and Vasileski term as inside the domain direction: symbols that are nice with respect to the circular direction may have very complicated behavior in the radial direction. Of course, in the context of arbitrary planar domains, it is not possible to use the notion of radial symbol. We go around this difficulty by making two simple observations. To start, it is necessary to notice that the structure of the Bergman kernel suggests that there is in any planar domain an internal region that we can neglect when we are interested in boundedness and compactness of Toeplitz operators with square integrable symbols, therefore the inside the domain direction counts up to a certain point. The second observation consists in exploiting the geometry of the domain and conformal equivalence in order to partially recover the notion of radial symbol. For these reasons, we study the problem for planar domains when the Toeplitz operator symbols have an almost-radial behavior and, for this class, we give a necessary and sufficient condition for boundedness and compactness. We also address the problem of the characterization of compactness by using the Berezin transform. In fact, under a growth condition for the almost- radial symbol, we show that the Berezin transform vanishes to the boundary if and only if the operator is compact.
The paper is organized as follows. In Section2, we describe the setting where we work, give the relevant definitions, and state our main result. In Section3, we collect results about the Bergman kernel for a planar domain and the structure ofL2aΩ, dν. In Section4, we prove the main result and study several important consequences.
2. Preliminaries
LetΩbe the bounded multiply-connected domain given at the beginning of Section1, that is, Ω ∩nj1Ωj, whereΩ1 is the bounded component ofC\γ1, andΩj j 2, . . . , nis the unbounded component ofC\γj. We use the symbolΔto indicate the punctured disk{z∈C| 0<|z|<1}. LetΓbe any one of the domainsΩ,Δ,Ωj j2, . . . , n.
We call KΓz, w the reproducing kernel of Γ and we use the symbol kΓz, w to indicate the normalized reproducing kernel, that is,kΓz, w KΓz, w/KΓw, w1/2.
For anyA∈ BL2aΓ, dν, we defineA, the Berezin transform of A, by Aw AkΓw, kΓw
ΓAkwΓzkwΓzdνz, 2.1 wherekwΓ· KΓ·, wKΓw, w−1/2.
If ϕ ∈ L∞Γ, then we indicate with the symbol ϕ the Berezin transform of the associated Toeplitz operatorTϕ, and we have
ϕw
ΓϕzkΓwz2dνz. 2.2 We remind the reader that it is well known thatA∈ C∞b Γ, and we have A ∞≤ A BL2Ω. It is possible, in the case of bounded symbols, to give a characterization of compactness using the Berezin transformsee4,5.
We remind the reader that anyΩbounded multiply-connected domain in the complex planeC, whose boundary∂Ωconsists of finitely many simple closed smooth analytic curves γj j 1,2, . . . , n, is conformally equivalent to a canonical bounded multiply-connected domain whose boundary consists of finitely many circles see 6. This means that it is possible to find a conformally equivalent domainD∩ni1DiwhereD1{z∈C:|z|<1}and Dj {z∈C:|z−aj|> rj}forj 2, . . . , n. Hereaj ∈D1and 0< rj <1 with|aj−ak|> rjrk
ifj /kand 1− |aj|> rj. Before we state the main results of this paper we need to give a few definitions.
Definition 2.1. LetΩ ∩ni1Ωi be a canonical bounded multiply-connected domain. We say that the set ofn1 functionsP{p0, p1, . . . , pn}is a∂-partition forΩif
1for everyj0,1, . . . , n, pj:Ω → 0,1is a Lipschitz,C∞-function,
2for everyj 2, . . . , n, there exists an open set Wj ⊂ Ω and anj > 0 such that Uj {ζ∈Ω:rj <|ζ−aj|< rjj}, and the support ofpjis contained inWjand
pjζ 1, ∀ζ∈Uj, 2.3
3forj 1, there exists an open setW1 ⊂ Ωand an1 > 0 such thatU1 {ζ ∈Ω : 1−1<|ζ|<1}and the support ofp1is contained inW1and
p1ζ 1, ∀ζ∈U1, 2.4
4for everyj, k 1, . . . , n, Wj∩Wk ∅, the setΩ\n
j1Wjis not empty and the function
p0ζ 1, ∀ζ∈
⎛
⎝n
j1
Wj
⎞
⎠
c
∩Ω, p0ζ 0, ∀ζ∈Uk, k1, . . . , n,
2.5
5for anyζ∈Ω, the following equation:
n k0
pkζ 1 2.6
holds.
We need to point out two facts about the definition above:ithat near each connected component of the boundary there is only one function which is different from zeronote that this implies that the function must be equal to 1, andiifar away from the boundary only the functionp0is different from zero.
Definition 2.2. A functionϕ:Ω ∩ni1Ωi → Cis said to be essentially radial if there exists a conformally equivalent canonical bounded domainD∩ni1Di, such that if the mapΘ:Ω → Dis the conformal mapping fromΩontoD, then
1for everyk2, . . . , nand for somek>0, we have
ϕ◦Θ−1z ϕ◦Θ−1|z−ak|, 2.7
whenz∈Uk {ζ∈Ω:rk<|ζ−ak|< rkk}, 2fork1 and for some1>0, we have
ϕ◦Θ−1z ϕ◦Θ−1|z|, 2.8
whenz∈U1{ζ∈Ω: 1−1<|ζ|<1}.
The reader should note that in the case where it is necessary to stress the use of a specific conformal equivalence, we will say that the map ϕ is essentially radial via Θ :
∩n1Ω → ∩n1D.
Before we proceed, the reader should notice that the definition, in the case of the disk, just says that, when we are near to the boundary, the values depend only on the distance from the center of the disk, so the function is essentially radial. In the general case, to formalize the fact that the values depend essentially on the distance from the boundary, we can simplify our analysis if we use the fact that this type of domain is conformally equivalent to a canonical
bounded multiply-connected domain whose boundary consists of finitely many circles. For this type of domain the idea of essentially radial symbol is quite natural. For this reason, we use this simple geometric intuition to give the general definition.
Before we state the main result, we stress that in what follows, when we are working with a general multiply-connected domain and we have a conformal equivalence Θ :
∩n1Ω → ∩n1D, we always assume that the∂-partition is given on∩n1Dand transferred to∩n1ΩthroughΘin the natural way.
At this point, we can state the main result.
Theorem 2.3. Letϕ ∈L2Ωbe an essentially radial function viaΘ : ∩n1Ω → ∩n1D, if one definesϕjϕ·pj, wherej 1, . . . , nandPis a∂-partition forΩ, then the following are equivalent:
1the operator
Tϕ :L2aΩ, dν−→L2aΩ, dν 2.9
is bounded (compact).
2for anyj 1, . . . , nthe sequencesγϕj {γϕjm}m∈Nare in∞Zc0Zwhere, by definition, ifj 2, . . . , n,
γϕjm rj
∞
rj
ϕj◦Θ−1
rj2m1/2m1s1/2m1aj
1
s2ds, ∀m∈Z, 2.10
and ifj 1
γϕ1m 1
0
ϕ1◦Θ−1
s1/2m1
ds, ∀m∈Z. 2.11
3. The Structure of L
2aΩ and Some Estimates about the Bergman Kernel
From now on, we will assume thatΩ ∩nj1ΩjwhereΩ1{z∈C:|z|<1}andΩj{z∈C:
|z−aj|> rj}forj 2, . . . , n. Here,aj ∈Ω1and 0< rj <1 with|aj−ak|> rjrkifj /kand 1− |aj|> rj. We will indicate with the symbolΔ0,1the punctured diskΩ1\ {0}.
With the symbolsKΩjz, w, KΩz, w, KΔz, w, we denote the Bergman kernel on Ωj, Ω, andΔ, respectively.
In order to gain more information about the kernel of a planar domain, it is important to remind the reader that for the the punctured diskΔ0,1and the diskΩ1, we haveLpaΔ0,1
LpaΩ1, ifp≥2, and, for anyz, w∈Δ2, KΔz, w KΩ1z, w see7,8. This fact has an important and simple consequence. In fact, if we considerΔa,r {z∈C: 0<|z−a|< r}and Oa,r {z∈C:|z−a|> r}, we can conclude that
KOa,rz, w r2
r2−z−a·w−a2, ∀z, w∈Oa,r×Oa,r. 3.1
To see this, we use the well-known fact that the reproducing kernel of the unit disk is given by1−zw−2, therefore we have
KΔ0,1z, w 1
1−z·w2, ∀z, w∈Δ0,1×Δ0,1. 3.2 This implies, by conformal mapping, that the reproducing kernel ofΔa,ris
KΔa,rz, w r2
r2−z−a·w−a2, ∀z, w∈Δa,r×Δa,r. 3.3
Now, we defineϕ:Δa,r → Oa,rby
ϕz z−a−1r2a, 3.4
and we use the well-known fact that the Bergman kernels of Δa,r and ψΔa,r Oa,r are related via
KOa,r
ϕz, ϕw
ϕzϕw KΔa,rz, w 3.5
to obtain that
KOa,rz, w r2
r2−z−a·w−a2, ∀z, w∈Oa,r×Oa,r. 3.6
SinceΩ1O0,1and, forj2, . . . , n, Oaj,rj Ωj, then the last equation implies that
KΩ1z, w 1 1−z·w2, KΩjz, w rj2
rj2−
z−aj
·
w−aj2
3.7
ifj2, . . . , n.
We also note that if we define
EΩz, w KΩz, w−n
j1
KΩjz, w, 3.8
we can prove the following.
Lemma 3.1. (1)EΩis conjugate symmetric about z and w. For eachw ∈Ω, EΩ·, wis conjugate analytic onΩandEΩ∈C∞Ω×Ω.
(2) There are neighborhoodsUjof∂Ωj j1, . . . , nand a constantC >0 such thatUj∩Uk
is empty ifj /kand
KΩz, w−KΩjz, w< C, 3.9
forz∈Ωandw∈Uj. (3)EΩ∈L∞Ω×Ω.
Proof. a Since the Bergman kernels KΩ and KΩj have these properties see 9, by the definition ofEΩ, we get1.
bThe proof is given in7,8.
cUsing the fact that
KΩ1z, w 1 1−z·w2, KΩjz, w rj2
rj2−
z−aj
·
w−aj2,
3.10
forj2, ..., nand1and2, we get3.
We observe that we can chooseRj > rj forj 2, . . . , nandR1 <1 such thatGj {z: rj <|z−aj|< Rj}j 2, . . . , nandG1{z:R1 <|z|<1}, then we haveGj ⊂Uj, whereUj
is the same as in Lemma3.1. We also have the following.
Lemma 3.2. There are constantsD>0 andM>0 such that 1for anyz, w∈Gi×Ω∪Ω×Gi, one has
KΩz, w< DKΩjz, w,
KΩjz, w<KΩz, wM, 3.11
2for anyz∈Ω, one hasKΩjz, z< KΩz, z.
Proof. By the explicit formula of the Bergman kernelsKΩi, there are constantsCiandMisuch that
KΩiz, w≥Ci, 3.12 forz, w∈Gi×Ω∪Ω×Giand
KΩiz, w≤Mi 3.13
ifz, w∈/Gi×Gifori1,2, . . . , n. From the last Lemma, it follows that KΩz, w≤KΩiz,wC≤
1 C
Ci
KΩiz, w,
KΩiz, w≤KΩz, wEΩz, w
j /i
KΩjz, w
<KΩz, w EΩ ∞
i /j
Mj,
3.14
wheneverz, w∈Gi×Ω∪Ω×Gi. If we callDthe biggest number among{1C/Cj}and we letM EΩ ∞n
j1Mj, then we get the first claimed estimate. The proof of2can be found in8,10.
It is clear from what we wrote so far that we put a strong emphasis on the fact that the domain under analysisΩis actually the intersection of other domains, that is,Ω ∩nj1Ωj. This also suggests that we should look for a representation of the elements of L2aΩ that reflects this fact. For this reason, we give the following.
Definition 3.3. GivenΩ ∩nj1ΩjwithΩ1 {z∈C:|z|<1}andΩj {z∈C:|z−aj|> rj}, for anyf ∈L2aΩ, we definen1 functionsP0f, P1f, P2f, . . . , Pnf as follows: ifz ∈Ω, then we set, forj1,
P1fz 1 2πi·
γ1
fζ
ζ−zdζ, 3.15
forj2,3, . . . , n,
Pjf 1 2πi·
γj
fζ
ζ−zdζ− 1 2πi·
γj
fζdζ, 3.16
and forj0,
P0fn
j2
1 2πi·
γj
fζdζ 1
z−aj, 3.17
whereγj j1, . . . , nare the circles which center ataj a10and lie inGjsee Lemma3.2, respectively, so thatzis exterior toγj j2, . . . , nand interior toγ1.
It is important that the reader notices that the Cauchy theorem implies that our definition is independent from how we chooseγ1, . . . ,γn. Moreover, it is important to notice that the domains of the functions P2f, . . . , Pnf are actually the setsΩ2, . . . ,Ωn. In the next Lemma, we give more information about this representation.
Lemma 3.4. Forf∈L2aΩ, one can write it uniquely as
fz n
j1
Pjf z
P0f
z, 3.18
with Pjf ∈ L2aΩj, P0f ∈ L2aΩ∩C∞Ω, PkPjf 0 if j /k, and moreover, there exists a constantM1such that, forj 0,1, . . . , n, one has
Pjf
Ω≤Pjf
Ωj ≤M1f
Ω. 3.19
In particular, iff ∈L2aΩi, thenPif fand f
Ωi ≤M1f
Ω, 3.20
fori1, . . . , n.
Proof. Letf be any function analytic onΩ. For anyz ∈Ω, letγi i 1, . . . , nbe the circles which center ataia1 0and lie inGi, respectively, so thatzis exterior toγi i2, . . . , n and interior toγ1. Using Cauchy’s Formula, we can write
fz n
j1
1 2πi·
γj
fζ
ζ−zdζ. 3.21
Let
fjz 1 2πi·
γj
fζ
ζ−zdζ. 3.22
By Cauchy’s Formula, the valuefjzdoes not depend on the choice ofγjif 1 ≤ j ≤ nand
fz n
jfjz. Of course, each fj is well defined for all z ∈ Ωj and analytic in Ωj. In addition, ifj /1, we have thatfjz → 0 as|z| → ∞. Writing the Laurent expansion atajof fj, we have
f1z ∞
k0
α1,kzk, 3.23
and, forj /1,
fjz −∞
k−1
αj,k
z−ajk
, 3.24
and these series converge to fj uniformly and absolutely on any compact subset of Ωj, respectively. We remark that the coefficients are given by the following formula:
αj,k 1 2πi
γj
fζ ζ−aj
k1dζ, 3.25
wherek≥0 ifj1 andk≤ −1 ifj /1 andγj ⊂Gj, 1≤j≤n. Moreover, iffis holomorphic in someΩjandfz → 0 as|z| → ∞wheni /1, thenαjk 0 for allj /iby Cauchy’s theorem and, therefore,fj 0.
Now, we defineP1ff1and
Pjfz −∞
k−2
αjk
z−ajk
, 3.26
forj2,3, . . . , nand
P0fz n
j2
αj,−1 z−aj
−1
, 3.27
thenfz n
i0Pifzfor allz∈ΩandPkPjf 0 if 0/k /j /0 as we have proved above.
We claim thatf ∈ L2aΩimplies that Pif ∈ L2aΩj forj 1,2, . . . , n, respectively.
Indeed, since each annulusGj is contained inΩ, f ∈L2aΩimplies thatfis an element of L2aGifor alli1,2, . . . , n.
For any fixedi, note thatPjf 0/j /iandP0f−αj,−1·z−aj−1 are analytic onGi∪ C/Ωiand lim|z| → ∞Pjfz 0 forj /1. Expanding them as Laurent series, it follows that:
1ifi1, thenPjf ∞
k1βjk/zkforj /1, 2ifi /1, then
Pjfz ∞
k0
βjkz−aik, 3.28
for 0/j /iand
P0fz ∞
k0
β0kz−aik αi,−1
z−ai. 3.29
It is obvious that, in any case, these series converge uniformly and absolutely on Gi. Observing that eachGiis an annulus atai, we have, by direct computation, that
f, f
Gi ≥
Pif, Pif
Gi|αi,−1|2lnRi−lnri 3.30
ifi /1 and
f, f
G1 ≥
P1f, P1f
G1. 3.31
Therefore, for anyi1, . . . , n, there exists a constantMsuch that Pif
Gi ≤f
Gi ≤fΩ, ∗
|αi,−1| ≤M·f
Ω. ∗∗
From the definition ofPjf, we derive
P1f2
G1 ∞
0
|α1k|2
1−R2k21
k1 ,
Pif2
Gi −∞
k−2
|α|2ik
ri2k2−R2k2i
k1 ,
3.32
fori2, . . . , n. The convergence of these series is guaranteed by the conditions∗and∗∗.
SinceR1<1 andri< Ri, it follows thatPif∈L2aΩiand P1f2
Ω1∞
0
|α1k|2 k1, Pif2Ω
i −∞
k−2
|α1k|2ri2k2 k1 ,
3.33
fori2, . . . , n. Comparing the expression of Pif Ωi with the expression of Pif Gi, it follows that Pif Ωi < M· Pif Gifor some constantMfori1, . . . , n. Hence, Pif Ωi < M· Pif Ω. Moreover, if we defineMMax{ z−ai−1 Ω}, from the inequalities Pif Gi ≤ f Gi ≤ f Ω
and|αi,−1| ≤M· f Ωand the definition ofP0, it follows that P0f Ω ≤n·M·M· f Ω. Iff ∈ L2aΩi for somei ∈ {1,2, . . . , n}, note that limfz 0 as|z| → ∞ fori /1, thenfz Pifz αi,−1z−ai−1 ifi /1 andP1f f ifi 1. Fori /1, sincef ∈L2aΩi ⊂ L2aΩimplies thatPif ∈L2aΩi, thenαi,−1·z−ai−1 ∈L2aΩi. We must haveαi,−1 0 and, consequently,P0f 0. Hence, in any case,f∈L2aΩiimpliesfPifandPjf 0 ifi /j, and this remark completes our proof.
Lemma 3.5. If{fn}is a bounded sequence inL2aΩandfn → 0 weakly inL2aΩ, thenPjfn → 0 weakly onL2aΩjforj 1, . . . , nandP0fn → 0 uniformly onΩ.
Proof. By the previous Lemma, we know that the linear transformations{Pj}are bounded operators, thenfn → 0 weakly inL2aΩimplies thatPjfn → 0 weakly on L2aΩjforj
1, . . . , n. For the same reason,P0fn → 0 weakly inL2aΩand thenP0fnζ → 0 for anyζ∈Ω.
Since
P0fmn
i2
αi,−1m
ζ−ai, 3.34 by the estimates given in the last lemma, we have that|αi,−1m|< M fm Ω. The boundedness of{ fm Ω}implies that the family of continuous functions{P0fm}is uniformly bounded and equicontinuous onΩ, then, by Arzela-Ascoli’s Theorem, we have thatP0fm → 0 uniformly onΩ.
4. Canonical Multiply-Connected Domains and Essentially Radial Symbols
In this section, we investigate, with the help of the results established in the previous section, necessary and sufficient conditions on the essentially radial function ϕ ∈ L2Ω, dνfor the Toeplitz operatorTϕto be bounded or compact.
Before we state the next Theorem, we remind the reader that
KΩζ, z EΩζ, z n
1
KΩζ, z, 4.1
whereEΩ∈L∞Ω×Ωand, for all1, . . . , n, we have
KΩζ, z KΩζ, z, ∀ζ, z∈Ω×Ω, 4.2
whereKΩ is the reproducing kernel ofΩ. If we use the symbolKΩ0 to indicateEΩ, we can write
KΩζ, z n
0
KΩζ, z. 4.3
We also remind the reader that ifI:L2aΩ → L2aΩis the identity operator, then
In
0
P, 4.4
whereP :L2aΩ → L2aΩis a bounded operator for all0,1, . . . , nwithPf ∈L2aΩif 1, . . . , nandP0f ∈ C∞ΩandPkP 0 ifk /see Lemma3.4.
In order to make our notation a little simpler, when we use a kernel operator we will denote it by the name of its kernel function. For example, the Bergman projection will be denoted by the symbolKΩ.
We are now in a position to prove the following result.
Lemma 4.1. Letϕ∈L2Dbe an essentially radial function whereD∩nj1DjwithD1{z∈C:
|z|<1}andDj{z∈C:|z−aj|> rj}forj2, . . . , n. If one definesϕjϕ·pjwherej1, . . . , n andP{p0, p1, . . . , pn}is a∂-partition forD,then the following are equivalent:
1the operator
Tϕ:L2aD, dν−→L2aD, dν 4.5
is bounded (compact);
2for anyj1, . . . , n, the operators Tϕj :L2a
Dj, dν
−→L2a Dj, dν
4.6
are bounded (compact).
Proof. Let{p0, p1, . . . , pn}be a partition of the unit onD∩nj1Dj, which is a canonical domain.
Now, we notice that for allf ∈L2Dand for allw∈D, we have the following:
Tϕfw
D
ϕzfzKDz, wdvz
n
j0
D
ϕzfzKDjz, wdvz
n
j0
n k0
D
ϕzpkzfzKDjz, wdvz
n
j0
n k0
Tjkfw,
4.7
where, by definition, we have
Tjkfw
D
ϕzpkzKDjz, wfzdvwdvz. 4.8
Claim 1. The operatorTj0is Hilbert-Schmidt for anyj0,1, . . . , n.
Proof. We observe that, by definition, we have
Tj0fw
D
ϕzp0zKDjz, wfzdvz, 4.9
therefore, if we define I1
D
ϕzp0zKDjz, w2dvzdvw, 4.10
we have
I1
D
ϕzp0z2
D
KDjz, w2dvw
dvz
≤
D
ϕzp0z2KDjz, zdvz
≤
z∈supppMax0p0z2KDjz, z
D
ϕjz2dvz
≤
z∈supppMax0p0z2KDjz, z
·ϕ2
D,2
<∞.
4.11
This implies that for anyt0,1, . . . , n,Tt0is Hilbert-Schmidt. Therefore, the operator n
t0
Tt0 4.12
is Hilbert-Schmidt, and this completes the proof of the claim.
Claim 2. The operatorT0kis Hilbert-Schmidt for anyk0,1, . . . , n.
Proof. We observe that, by definition, we have
T0kfw
D
ϕzpkzKD0z, wfzdvz, 4.13
therefore, if we define I2
D
ϕzpkzKD0z, w2dvzdvw, 4.14
we have
I2
D
ϕzp0z2KD0z, w2dvwdvz
≤
z,w∈D×DMax
KD0z, w2
·vD·
D
ϕzp0z2dvz
≤
z,w∈D×DMax
KD0z, w2
·vD·ϕ2
D,2
<∞.
4.15
This implies that for anyt0,1, . . . , n,T0tis Hilbert-Schmidt. Therefore, the following n
t0
T0t 4.16
is Hilbert-Schmidt, and this completes the proof of the claim.
Claim 3. The operatorTijis Hilbert-Schmidt ifi /j /0 andj, i1, . . . , n.
Proof. We observe that
Tjkfw
D
ϕzpkzKDjz, wfzdvwdvz. 4.17
To start, we give the following:
Njiz, wdef ϕjz·KDiz, w. 4.18
We will show that Fubini theorem and the properties of the∂-partition imply that
D
Njiz, w2dvwdvz<∞. 4.19
In fact, we have
D
Njiz, w2
D
D
Njiz, w2dvw
dvz
D
ϕjz2KDiz, w2dvwdvz
D
ϕjz2
D
KDiz, w2dvw
dvz
D
ϕjz2KDiz, zdvz
D
ϕz2pjz2KDiz, zdvz
≤
z∈supppMaxjpjz2KDiz, z
· ϕ 2D,2
<∞.
4.20
Therefore, we can write that
Tϕ Kn
1
Tϕ, 4.21
whereKis a compact operator.
We also observe that Lemma3.4implies thatTϕ n
j0TϕPj, and we prove that the operatorTϕPjis compact ifj /andj, 1, . . . , n.
Proof. In order to simplify the notation, we define the operatorRj,TϕPj KDMϕpPj. To prove our statement, it is enough to prove that if we take a bounded sequence{fn}inL2D such thatfn → 0 weakly, then we can prove that Rj,fn 2 → 0. We know that the continuity ofP implies thatPjfk → 0 weakly onH2Dl, and{ Pjfk D}is bounded by Lemma3.5.
Since it is a sequence of holomorphic functions, we know that{Pjfk}is uniformly bounded on any compact subset ofD. Therefore, the sequence{Pjfk}is a normal family of functions.
SincePjfkζ → 0 for anyζ∈Dj, thenPjfkconverges uniformly on any compact subset of Dj and consequently onF suppp. To complete the proof, we remind the reader that if we define the operatorsQ :L2D → L2D, for 1,2, . . . , n, in this way
Qfz
D
fζKDζ, zdvζ. 4.22
It is possible to prove, with the help of Schur’s testsee11 , thatQis a bounded operator see5. Now, we observe that
Rj,fkζ≤SupPjfkζ:ζ∈F
·QjXFϕpsζ, 4.23
then, by using the fact thatQis bounded, we have Rj,fk
D≤SupPjfkζ:ζ∈F
·M·ϕ1ps
D,2−→0, 4.24
and this completes the proof of our claim. Notice also that using the same strategy, we can prove that eachTϕP0is compact.
Therefore, we have
TϕKn
1
Tϕ
KK1n
1
TϕP,
4.25
whereK, K1 are compact operators. SincePt2 Pt, PtPs 0 and ifj /, thenTϕis bounded compactif and only if the operatorsTϕPare boundedcompactoperators.
SincePL2aD L2aD, then it follows that the operatorTϕPis boundedcompact if and only ifTϕis boundedcompact.
We are finally, with the help of1’s main result, in a position to prove the main result of this paper.
Theorem 4.2. Letϕ∈L2Dbe an essentially radial function whereD ∩nj1DjwithD1 {z ∈ C : |z| < 1}andDj {z ∈ C : |z−aj| > rj}forj 2, . . . , n. If one definesϕj ϕ·pj where j1, . . . , nandP{p0, p1, . . . , pn}is a∂-partition forDthen the following are equivalent:
1the operator
Tϕ:L2aD, dν−→L2aD, dν 4.26
is bounded (compact).
2for anyj 1, . . . , n, the sequencesγϕj {γϕjm}m∈Nare in∞Zc0Zwhere, by definition, ifj 2, . . . , n
γϕjm rj
∞
rj
ϕj
rj2m1/2m1s1/2m1aj
1
s2ds ∀m∈Z, 4.27
and forj1,
γϕ1m 1
0
ϕ1
s1/2m1
ds, ∀m∈Z. 4.28
Proof. In the previous theorem, we proved that the operator under examination is bounded compactif and only if for anyj1, . . . , nthe operators
Tϕj :L2 Dj, dν
−→L2a Dj, dν
4.29 are boundedcompact. Ifj 2, . . . , n, we observe that if we consider the following sets Δ0,1 {z∈C: 0<|z−a|<1}andΔaj,rj {z∈C: 0<|z−aj|< rj}and the following maps
Δ0,1−−−→α Δaj,rj
−−−→β Dj, 4.30
whereαz ajrjzandβw w−aj−1rj2ajand we use Proposition 1.1 in8, we can claim that
Tϕj Vβ◦α−1Tϕj◦β◦αVβ◦α, 4.31
where Vβ◦α : L2Δ0,1 → L2Dj is an isomorphism of Hilbert spaces. Therefore, Tϕj is boundedcompactif and only ifTϕj◦β◦α is boundedcompact. We also know that this, in turn, is equivalent to the fact that the sequence
γϕj
γϕjm
m∈N 4.32
is in∞Zc0Z, where
γϕjm 1
0
ϕj◦β◦α
r1/2m1
dr, ∀m∈Z. 4.33
To complete the proof, we observe that sinceϕjis radial andβ◦αr r−1rjajthen, after a change of variable, we can rewrite the last integral, and therefore the formula
γϕjm rj ∞
rj
ϕj
rj2m1/2m1s1/2m1aj1
s2ds, ∀m∈Z 4.34
must hold for anyj 2, . . . , n. The casej1 is immediate.
Now, we can prove the following.
Theorem 4.3. Letϕ ∈ L2Ωbe an essentially radial function via the conformal equivalenceΘ : Ω → D, define ϕj ϕ·pj wherej 1, . . . , n and Pis a ∂-partition forΩ, then the following conditions are equivalent:
1the operator
Tϕ :L2aΩ, dν−→L2aΩ, dν 4.35 is bounded (compact);
2for anyj 1, . . . , n, the sequencesγϕj {γϕjm}m∈Nare in∞Zc0Zwhere, by definition, ifj 2, . . . , n
γϕjm rj ∞
rj
ϕj◦Θ−1
rj2m1/2m1s1/2m1aj1
s2ds, ∀m∈Z, 4.36
and forj1
γϕ1m 1
0
ϕ1◦Θ−1
s1/2m1
ds, ∀m∈Z. 4.37
Proof. We know that Ω is a regular domain, and therefore if Θ is a conformal mapping from Ω onto D then the Bergman kernels of Ω and ΘΩ D, are related via KDΘz,ΘwΘzΘw KΩz, w, and the operatorVΘf Θ·f ◦Θis an isometry fromL2DontoL2Ω see Proposition 1.1 in8. In particular, we haveVΘPDPΩVΘand this implies thatVΘTϕ Tϕ◦Θ−1VΘ. Therefore, the operatorTϕ is bounded compactif and only if the operatorTϕ◦Θ−1 : L2D, dν → L2aD, dνis boundedcompact. In the previous theorem we proved that the operator in exam is bounded compactif and only if for any j1, . . . , nthe operators
Tϕj◦Θ−1 :L2a Dj, dν
−→L2a Dj, dν
4.38 are boundedcompact. Hence, we can conclude that the operator is boundedcompactif and only if for anyj 1, . . . , nthe sequencesγϕj {γϕjm}m∈Nare in∞Zc0Zwhere, by definition, ifj 2, . . . , n, we have
γϕjm rj ∞
rj
ϕj◦Θ−1
rj2m1/2m1s1/2m1aj1
s2ds, ∀m∈Z, 4.39
