Differences of Weighted Composition Operators on

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Volume 2009, Article ID 127431,19pages doi:10.1155/2009/127431

Research Article

Differences of Weighted Composition Operators on

H

_α

∞

BN

∗

Jineng Dai

1, 2, 3

_{and Caiheng Ouyang}

1

1_{Wuhan Institute of Physics and Mathematics, The Chinese Academy of Science, Wuhan 430071, China} 2_{Hubei Province Key Laboratory of Intelligent Robot, School of Science, Wuhan Institute of Technology,}

Wuhan 430073, China

3_{Graduate School of the Chinese Academy of Science, Beijing 100039, China}

Correspondence should be addressed to Jineng Dai,[email protected]

Received 21 August 2009; Revised 29 October 2009; Accepted 16 November 2009

Recommended by Donal O’Regan

We study the boundedness and compactness of diﬀerences of weighted composition operators on weighted Banach spaces in the unit ball ofCN_.

Copyrightq2009 J. Dai and C. Ouyang. 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.

1. Introduction

LetCN_{denote the Euclidean space of complex dimension}_N_N_≥_{1. For}_z _z

1, . . . , zNand

w w1, . . . , wNinCN, we denote the inner product ofzandwby

z, wz1w1· · ·zNwN, 1.1

and we write|z|z, z|z1|2· · ·|zN|2. LetBN{z∈CN:|z|<1}be the open unit ball ofCN_{and let}_H_B

Nbe the space of all holomorphic functions onBN. For a holomorphic self-map of the unit ballϕ :BN → BNandu ∈HBN, we define a weighted composition operatorWϕ,uby

Wϕ,u

fu·f◦ϕ 1.2

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research of such operators on diﬀerent Banach spaces of holomorphic functionssee1–9.

The general ideal is to explain the operator-theoretic behavior ofWϕ,usuch as boundendness and compactness, in terms of the function-theoretic properties of the symbolsϕandu. For a comprehensive overview of the field, we refer to the books by Cowen and MacCluer10and Shapiro11.

The study of diﬀerences of two composition operators was first started on Hardy spaces. The primary motivation for this is to understand the topological structure of the set of composition operators on Hardy spaces. After that, such related problems have been studied on several spaces of holomorphic functions by many authors: by MacCluer et al. 12Hosokawa et al.13on bounded spacesH∞; by Moorhouse14on weighted Bergman spaces, and by Hosokawa and Ohno 15and Nieminen 16on Bloch spaces. In 1, the

authors investigated the boundedness and compactness of Cϕ −Cψ on weighted Banach spaces. In16, Nieminen characterized the compactness ofWϕ,u−Wψ,vwhen two weighted composition operators Wϕ,u and Wψ,v are bounded operators on weighted Banach spaces. Lindstr öm and Wolf17generalized Nieminen’s results on more general weighted Banach spaces. Furthermore, they estimated the essential norm of differences of two weighted composition operators. These works concerned with differences of weighted composition operators mainly focused on the setting of one variable. Recently, Toews 18, Gorkin et

al.19, and Aron et al. 20 extended the results of 12 to the case of several variables, respectively. In this paper, we study the boundedness and compactness of diﬀerences of weighted composition operators on weighted Banach spaces in the setting of several variables and extend some results of16,17. Due to the diﬀerence between one variable

and several variables, some special constructive techniques are applied. After collecting some preliminary results in the next section, we give an elegant inequalityseeLemma 3.2which is useful to characterize the boundedness of diﬀerences of weighted composition operators on weighted Banach spaces inSection 3. InSection 4, we continue to describe the compactness of diﬀerences of weighted composition operators on these spaces and obtain some interesting corollaries.

2. Preliminaries

For 0 < α < ∞, let H_α∞ be the weighted Banach space of holomorphic functionsf onBN satisfying

_f

Hα∞ sup

z∈BN

1− |z|2αfz<∞. _2.1

Denote byBα_{the Bloch-type space of holomorphic functions}_f_on_B

Nsuch that

sup z∈BN

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where∇fz ∂f/∂z1z, . . . ,∂f/∂zNz. Whenα1, the spaceB1is the usual Bloch space. We call the function

Kz, z 1

1− |z|2N1 2.3

the Bergman kernel ofBNand denote the Bergman matrix by

Bz 1 N1

∂2

∂zi∂zjlogKz, z

N×N

. 2.4

Forf∈HBN, we define

Qfz sup

∇fz, w

Bzw, w : 0/w∈C

N

, z∈BN. 2.5

It is well known thatsee21or22

Bzw, w

1− |z|2|w|2|z, w|2

1− |z|22

. 2.6

Moreover, forα >1/2, if a holomorphic function isf ∈Bα_{, then we have}

sup z∈BN

1− |z|2α∇fz≈ sup z∈BN

1− |z|2α−1Qfz. 2.7

Here and below we use the abbreviated notationA ≈Bto mean that there exists a positive constantCsuch thatC−1_B_≤_A_≤_CB_{. Throughout this paper, constants are denoted by}_C_and they are positive finite quantities and not necessarily the same in each occurrence. Note that the weighted Banach spaceH_α∞ can be identified with the Bloch-type spaceBα1_{. Thus, we} can easily see that iff ∈H_α∞forα >0, then

_f Hα∞ ≈

_f₀_sup z∈BN

1− |z|2αQfz. 2.8

For any pointa∈BN− {0}, we define

ϕaz a−Paz−saQaz

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where sa

1− |a|2_,_P_a _{is the orthogonal projection from}_CN _{onto the one-dimensional} subspaceagenerated bya, andQaI−Pais the projection onto the orthogonal complement ofa, that is

Paz z, a

|a|2 a, Qaz z−

z, a

|a|2 a, z∈BN. 2.10

Whena0, we simply defineϕaz −z. It is well known that eachϕais a homeomorphism of the closed unit ballBNontoBN. Let

ρa, z ϕaz. 2.11

Thenρis a metric onBN and is invariant under automorphisms. The metricρis called the pseudohyperbolic metric.

For any two points zand w in BN, let γt γ1t, . . . , γNt : 0,1 → BN be a smooth curve to connectzandw. Define

lγ

₁

0

Bγtγt, γtdt. 2.12

The infimum of the set consisting of alllγis denoted byβz, w, whereγis a smooth curve inBNfromzandw. We callβthe Bergman metric onBN. It is known that

βz, w 1

2log

1ρz, w

1−ρz, w. 2.13

3. The Boundedness of

W

ϕ,u

−

W

ψ,v

In this section, we will characterize the boundedness of the operatorWϕ,u−Wψ,v:Hα∞ → Hβ∞. For this purpose, we state some useful lemmas.

Lemma 3.1. ForzandwinBN, then

1−ρz, w

1ρz, w ≤

1− |z|2

1− |w|2 ≤

1ρz, w

1−ρz, w. 3.1

Proof. Setϕwz a. Since ϕw is an involution, it follows that ϕwa z. Thus, from the identity

1−ϕwa21− |z|2

1− |w|21− |a|2

|1− w, a|2 ,

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we have

1− |z|2

1− |w|2

1− |a|2

|1− w, a|2. 3.3

On the other hand,

1− |a|

1|a| ≤

1− |a|2

|1− w, a|2 ≤

1|a|

1− |a|. 3.4

Therefore, it follows that

1−ϕwz 1ϕwz

≤ 1− |z|2 1− |w|2 ≤

1ϕwz 1−ϕwz

, 3.5

which, together with2.11, yields the desired estimate.

The following lemma can be found in16,17for the one variable case.

Lemma 3.2. Letf ∈H_α∞. Then

1− |z|2αfz−1− |w|2αfw≤Cf_H∞

αρz, w 3.6

for allz, winBN.

Proof. Fix any two pointszandwinBN. Letγγt0≤t≤1be a smooth curve inBNfrom

wtoz. Then

1− |z|2αfz−1− |w|2αfw

1

0

d1−γt2αfγt

₁

0

fγtd1−γt2α

₁

0

1−γt2αdfγt.

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Sincef ∈H_α∞, we get

₁

0

fγtd1−γt2α

₁

0

−αfγt1−γt2α−1

N

k1

γktγ_kt γktγkt

dt

≤2α

1

0 _f

γt1−γt2α−1γt, γtdt

≤Cf_H∞

α

₁

0

γt, γt

1−γt2 dt

≤Cf_H∞

α

₁

0

Bγtγt, γtdt,

3.8

where the last inequality comes from2.6.

On the other hand,

₁

0

1−γt2αdfγt

₁

0

1−γt2α N

k1

γ_kt∂f ∂zk

γtdt. 3.9

From the definition ofQf we see that

N

k1

γ_kt∂f ∂zk

γt≤Qf

γtBγtγt, γt. 3.10

Thus, by2.8it follows that

₁

0

1−γt2αdfγt≤

₁

0

1−γt2αQf

γtBγtγt, γtdt

≤Cf_H∞

α

₁

0

Bγtγt, γtdt.

3.11

Therefore, we have proved that

1− |z|2αfz−1− |w|2αfw≤Cf_H∞

α

1

0

Bγtγt, γtdt. 3.12

Sinceγγt 0≤t≤1is an arbitrary smooth curve inBNfromwtoz, by the definition of

βz, w, we have

1− |z|2αfz−1− |w|2αfw≤Cf_H∞

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If ρz, w < 1/2, routine estimates show that βz, w ≤ ρz, w. If ρz, w ≥ 1/2, then 4ρz, w≥2. Meanwhile, note that

1− |z|2αfz−1− |w|2αfw≤2f_H∞

α. 3.14

Thus, we have

1− |z|2αfz−1− |w|2αfw≤Cf_H∞

αρz, w. 3.15

Remark 3.3. From the proof ofLemma 3.2, it is not hard to see that for anyz, w∈rBN {z∈

BN:|z|< r <1}, then

1− |z|2αfz−1− |w|2αfw≤Cfr_H∞

αρz, w 3.16

for anyf∈H_α∞, wherefrHα∞ sup

z∈rBN

1− |z|2α|fz|.

Now we provide a characterization of the boundedness ofWϕ,u−Wψ,vfromHα∞ toHβ∞. For that purpose, consider the following three conditions:

sup z∈BN

1− |z|2β|uz|

1−ϕz2α

ρϕz, ψz <∞, 3.17

sup z∈BN

1− |z|2β|vz|

1−ψz2α

ρϕz, ψz<∞, 3.18

sup z∈BN

1− |z|2βuz

1−ϕz2α

−

1− |z|2βvz

1−ψz2α

<∞. 3.19

Theorem 3.4. The following statements are equivalent.

iWϕ,u−Wψ,v:Hα∞ → Hβ∞ is bounded. iiThe conditions3.17and3.19hold.

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Proof. First, we prove the implicationii→iii. Assume that the conditions3.17and3.19

hold. Note that the pseudohyperbolic metricρis less then 1. Then we have

1− |z|2β|vz|

1−ψz2α

ρϕz, ψz≤

1− |z|2β|uz|

1−ϕz2α

ρϕz, ψz

1− |z|2βuz

1−ϕz2α

−

1− |z|2βvz

1−ψz2α

ρ

ϕz, ψz,

3.20

which implies that3.18holds.

Next, we show the implicationiii → i. Letf ∈ H_α∞. Assume that the conditions 3.18and3.19hold; byLemma 3.2, we have

1− |z|2βWϕ,u−Wψ,v

fz

1− |z|2βfϕzuz−fψzvz

1−ϕz2αfϕz

⎡ ⎢ ⎣

1− |z|2βuz

1−ϕz2α

−

1− |z|2βvz

1−ψz2α

⎤ ⎥ ⎦

1− |z|2βvz

1−ψz2α

1−ϕz2αfϕz−1−ψz2αfψz

≤f_H∞

α

1− |z|2βuz

1−ϕz2α

−

1− |z|2βvz

1−ψz2α

C

f_H∞

αρ

ϕz, ψz

1− |z|2β|vz|

1−ψz2α

≤Cf_H∞

α,

3.21

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Finally, we prove the implicationi→ ii. Assume thatWϕ,u−Wψ,v is bounded. Fix

w∈BN; consider the functionfwdefined by

fwz 1

1−z, ϕwα ·

ϕψwz, ϕψw

ϕw

ϕψwϕw 3.22

forz∈BN. It is easy to check thatfw∈Hα∞withfw_H∞

α ≤2

α_{. Note that}

fw

ϕw ρ

ϕw, ψw

1−ϕw2α , fw

ψw0. _3.23

By the boundedness ofWϕ,u−Wψ,v, we have

∞> Cfw_H∞

α ≥

_W_ϕ,u₋_W_ψ,v

fw_H∞

β

sup z∈BN

1− |z|2βfw

ϕzuz−fw

ψzvz

≥1− |w|2βfw

ϕwuw−fw

ψwvw

1− |w|2βρϕw, ψw|uw|

1−ϕw2α

3.24

for anyw ∈BN. This proves3.17. Now we prove that3.19is also true. For givenw ∈BN instead of the functionfw, we consider the functiongwgiven by

gwz

1

1−z, ψwα. 3.25

Clearly,gw∈Hα∞withgw_H∞

α ≤2

α_{. Thus, we have}

∞>Wϕ,u−Wψ,v

gw_H∞

β ≥

1− |w|2βgw

ϕwuw−gw

ψwvw

|Iw Jw|,

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where

Iw 1−ψw2αgw

ψw

⎡ ⎢ ⎣

1− |w|2βuw

1−ϕw2α

−

1− |w|2βvw

1−ψw2α

⎤ ⎥ ⎦

1− |w|2βuw

1−ϕw2α

−

1− |w|2βvw

1−ψw2α ,

Jw

1− |w|2βuw

1−ϕw2α

1−ϕw2αgw

ϕw−1−ψw2αgw

ψw.

3.27

By3.17that has been proved as before andLemma 3.2, we conclude that|Jw|<∞for all

w ∈BN, which implies that|Iw|<∞for allw ∈BN. Thus, the condition3.19is proved. The whole proof is complete.

Corollary 3.5. The weighted composition operatorWϕ,u:Hα∞ → Hβ∞is bounded if and only if

sup z∈BN

1− |z|2β|uz|

1−ϕz2α

<∞. 3.28

Proof. The result follows from the simple case in whichv≡0 ofTheorem 3.4.

Corollary 3.6. The operatoruCϕ−uCψ :Hα∞ → Hβ∞is bounded if and only if the following two conditions hold:

sup z∈BN

1− |z|2β|uz|

1−ϕz2α

ρϕz, ψz<∞,

sup z∈BN

1− |z|2β|uz|

1−ψz2α

ρϕz, ψz<∞.

3.29

Proof. The necessity is clear byTheorem 3.4. To prove the suﬃciency, we only need to show

that if the conditions3.29hold, then

1− |z|2βuz

1−ϕz2α

−

1− |z|2βuz

1−ψz2α

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for allz∈BN. In fact, we easily see that3.30holds forz∈BNsatisfyingρϕz, ψz>1/2 by3.29. Ifz∈BNsuch thatρϕz, ψz≤1/2, byLemma 3.1we have

1− |z|2αuz

1−ϕz2α

−

1− |z|2αuz

1−ψz2α

1− |z|2α|uz|

1−ϕz2α

1− 1−ϕz 2

1−ψz2

α

≤

1− |z|2α|uz|

1−ϕz2α

1−

1ρϕz, ψz

1−ρϕz, ψz

α

≤C

1− |z|2α|uz|

1−ϕz2α

ρϕz, ψz,

3.31

which implies that 3.30 holds for z ∈ BN satisfying ρϕz, ψz ≤ 1/2. The proof is complete.

Example 3.7. We give a nontrivial example such that the weighted composition operatorsWϕ,u andWψ,vare unbounded onHα∞while the operatorWϕ,u−Wψ,vis bounded onHα∞.

Choose the analytic functionsϕz 1z/2 andψz ϕz tz−13in the unit disk, wheretis real and positive andtis so small thatψ maps the unit disk DintoD. Let

uz vz 1−z−1,αβ >0. It is easy to see that when 0< r <1,

1−r2α_u_r

1−ϕ2_rα −→ ∞,

1−r2α_v_r

1−ψ2_rα −→ ∞ 3.32

asr → 1. This shows thatWϕ,u andWψ,v are unbounded on Hα∞ byCorollary 3.5. On the other hand, we know thatρϕz, ψz≤Ct|1−z|whentis small enoughsee12, Example 1. By the Schwarz-Pick lemma, we get

sup z∈D

1− |z|2α|uz|

1−ϕz2α

ρϕz, ψz≤sup z∈D

Ct|1−z|

|1−z| <∞,

sup z∈D

1− |z|2α|vz|

1−ψz2α

ρϕz, ψz≤sup z∈D

Ct|1−z|

|1−z| <∞.

3.33

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4. The Compactness of

W

ϕ,u

−

W

ψ,v

In this section, we give a characterization of the compactness of the operatorWϕ,u −Wψ,v :

H_α∞ → H_β∞. Before doing this, we need the following lemma whose proof is an easy modification of that of10, Proposition 3.11.

Lemma 4.1. The operator Wϕ,u −Wψ,v : Hα∞ → Hβ∞ is compact if and only if whenever {fn} is a bounded sequence in H_α∞ with fn → 0 uniformly on compact subsets of BN, and then Wϕ,u−Wψ,vfn_H∞

β → 0.

Here we consider the following conditions:

1− |z|2βuz

1−ϕz2α

ρϕz, ψz−→0 as ϕz−→1, 4.1

1− |z|2βvz

1−ψz2α

ρϕz, ψz−→0 asψz−→1, 4.2

1− |z|2βuz

1−ϕz2α

−

1− |z|2βvz

1−ψz2α

−→0 asϕz−→1, ψz−→1. 4.3

In one complex variable case, Nieminen16characterized the compactness ofWϕ,u−

Wψ,v :Hα∞ → Hβ∞under the assumption thatWϕ,uandWψ,v are bounded fromHα∞toHβ∞. Here, a necessary and suﬃcient condition for the compactness ofWϕ,u−Wψ,v:Hα∞ → Hβ∞is completely obtained in the case of several variables without any assumption.

Theorem 4.2. The operatorWϕ,u −Wψ,v : Hα∞ → Hβ∞is compact if and only ifWϕ,u −Wψ,v :

H_α∞ → H_β∞is bounded and the conditions4.1–4.3hold.

Proof. First, we prove the suﬃciency. Assume thatWϕ,u−Wψ,v :Hα∞ → Hβ∞is bounded and the conditions4.1–4.3hold. Then the conditions3.17–3.19hold byTheorem 3.4. For

ε >0, there exists 0< r <1 such that

1− |z|2β|vz|

1−ψz2α

ρϕz, ψz≤ε whenψz> r, 4.4

1− |z|2β|uz|

1−ϕz2α

ρϕz, ψz≤ε whenϕz> r, 4.5

1− |z|2βuz

1−ϕz2α

−

1− |z|2βvz

1−ψz2α

≤ε when

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To prove thatWϕ,u−Wψ,v :Hα∞ → Hβ∞is compact, we will applyLemma 4.1. Suppose that {fn}is a sequence inHα∞such thatfn_H∞

α ≤1 andfn → 0 uniformly on compact subsets of BN. We need only to show thatWϕ,u−Wψ,vfn_H∞

β → 0. We write

1− |z|2βfn

ϕzuz−fn

ψzvz|Inz Jnz|, 4.7

where

Inz

1−ϕz2αfn

ϕz

⎡ ⎢ ⎣

1− |z|2βuz

1−ϕz2α

−

1− |z|2βvz

1−ψz2α

⎤ ⎥ ⎦,

Jnz

1− |z|2βvz

1−ψz2α

1−ϕz2αfn

ϕz−1−ψz2αfn

ψz.

4.8

We divide the argument into a few cases.

Case 1|ψz| ≤r and|ϕz| ≤r. Clearly, by3.19,Inzconverges to 0 uniformly for allz with|ϕz| ≤rand|ψz| ≤r. On the other hand, fromRemark 3.3and3.18, we have

|Jnz| ≤C

1− |z|2β|vz|

1−ψz2α

ρϕz, ψzsup z∈rBN

1− |z|2αfnz≤ε 4.9

for suﬃciently largen.

Case 2|ψz|> rand|ϕz| ≤r. As in the proof of Case1,Inz → 0 uniformly. As regards

Jnz, byLemma 3.2and inequality4.4, we have|Jnz| ≤εfor suﬃciently largen.

Case 3|ψz|> rand|ϕz|> r. The inequality4.6implies that|Inz| ≤εfornsuﬃciently large. Meanwhile,Jnz → 0 uniformly byLemma 3.2and inequality4.4.

Case 4|ψz| ≤rand|ϕz|> r. Then we rewrite

1− |z|2βfn

ϕzuz−fn

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where

Pnz

1−ψz2αfn

ψz

⎡ ⎢ ⎣

1− |z|2βuz

1−ϕz2α −

1− |z|2βvz

1−ψz2α

⎤ ⎥ ⎦,

Qnz

1− |z|2βuz

1−ϕz2α

1−ϕz2αfn

ϕz−1−ψz2αfn

ψz.

4.11

The desired result follows by an argument analogous to that in the proof of Case2. Thus, together with the above cases, we conclude

Wϕ,u−Wψ,v

fn_H∞

β _zsup_∈_B N

1− |z|2βfn

ϕzuz−fn

ψzvz≤ε _4.12

for suﬃciently largen.

Now we will prove the necessity. Suppose thatWϕ,u−Wψ,v :Hα∞ → Hβ∞is compact. Since the compactness implies the boundedness, we only need to show that4.1–4.3hold. Let {zn} be a sequence of points in BN such that |ϕzn| → 1 as n → ∞. Consider the functionsfndefined by

fnz

1−ϕzn2

1−z, ϕzn α1 ·

ϕψznz, ϕψzn

ϕzn _ϕ_ψ_z

n

ϕzn

. 4.13

Clearly,fnconverges to 0 uniformly on compact subsets ofBNasn → ∞andfn∈Hα∞with fn_H∞

α ≤Cfor alln. Moreover,

fn

ϕzn

ρ

ϕzn, ψzn

1−ϕzn2

α, fn

ψzn

0. _4.14

By the compactness ofWϕ,u−Wψ,vandLemma 4.1, it followsWϕ,u−Wψ,vfn_H∞

β → 0. On

the other hand, we have

_W_ϕ,u₋_W_ψ,v

fn_H∞

β _zsup_∈_B N

1− |z|2βfn

ϕzuz−fn

ψzvz

≥1− |zn|2 β

fn

ϕzn

uzn−fn

ψzn

vzn

1− |zn|2 β

ρϕzn, ψzn

|uzn|

1−ϕzn2

α ,

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which shows that1− |zn|2βρϕzn, ψzn|uzn|/1− |ϕzn|2α → 0 as|ϕzn| → 1. This proves4.1. The condition4.2also holds by similar arguments. Now we show that the condition4.3holds. Assume that{zn}is a sequence of points inBNsuch that|ϕzn| → 1 and|ψzn| → 1 asn → ∞. Define the function

gnz

1−ψzn2

1−z, ψzn

α1. 4.16

It is easy to check thatgnconverges to 0 uniformly on compact subsets ofBNasn → ∞and

gn∈Hα∞withgnH∞α ≤Cfor alln. Furthermore,gnψzn 1− |ψzn|

2−α_{. By}_{Lemma 4.1} we haveWϕ,u−Wψ,vgn_H∞

β → 0. Meanwhile, we have

_W_ϕ,u₋_W_ψ,v

gn_H∞

β ≥

1− |zn|2 β

gn

ϕzn

uzn−gn

ψzn

vzn|Izn Jzn|, 4.17

where

Izn

1−ψzn2 α

gn

ψzn

⎡ ⎢ ⎣

1− |zn|2 β

uzn

1−ϕzn2 α −

1− |zn|2 β

vzn

1−ψzn2 α ⎤ ⎥ ⎦

1− |zn|2 β

uzn

1−ϕzn2 α −

1− |zn|2 β

vzn

1−ψzn2 α ,

Jzn

1− |zn|2 β

uzn

1−ϕzn2 α

gn

ϕzn

−1−ψzn2 α

gn

ψzn

.

4.18

ByLemma 3.2and the condition4.1that has been proved, we concludeJzn → 0, which implies thatIzn → 0 asn → ∞. This shows that4.3is true. The whole proof is complete.

Corollary 4.3. The weighted composition operatorWϕ,u:Hα∞ → Hβ∞is compact if and only ifWϕ,u is bounded and

1− |z|2βuz

1−ϕz2α −→

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Corollary 4.4. Ifβ≥α, then the composition operatorCϕ :Hα∞ → Hβ∞is bounded. Ifβ > α, then

Cϕ:Hα∞ → Hβ∞is compact.

Proof. By the Schwarz-Pick lemma in the unit ballsee23

1− |z|2

1−ϕz2 ≤

1−ϕ02

₁₋

ϕz, ϕ02 ≤

1ϕ0

1−ϕ0, 4.20

we obtain

1− |z|2β

1−ϕz2α

≤C1−ϕz2β−α. 4.21

Therefore, the desired results follow from Corollaries3.5and4.3.

The following result appears in1whenu≡1 in one-dimensional case.

Corollary 4.5. The operatoruCϕ−uCψ :Hα∞ → Hβ∞is compact if and only ifuCϕ−uCψis bounded and the following conditions hold:

1− |z|2βuz

1−ϕz2α

ρϕz, ψz−→0 asϕz−→1, 4.22

1− |z|2βuz

1−ψz2α

ρϕz, ψz−→0 as ψz−→1. 4.23

Proof. This is the case in which u ≡ v of Theorem 4.2. We need only show that the two conditions4.22and4.23imply that

1− |z|2βuz

1−ϕz2α

−

1− |z|2βuz

1−ψz2α

−→0 asϕz−→1, ψz−→ a

b1. 4.24

Suppose that4.24is not true. Then there existε0 > 0 and a sequence of points{zn} ⊂ BN such that|ϕzn| → 1 and|ψzn| → 1 asn → ∞and

1− |zn|2 β

uzn

1−ϕzn2 α −

1− |zn|2 β

uzn

1−ψzn2 α

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We deduce thatρϕzn, ψzn → 0 asn → ∞. If not, there exists a subsequence{znk}in

{zn}such thatρϕznk, ψznk → a >0. By4.22and4.23, we obtain

1− |znk|

2β_u_z nk

1−ϕznk

2α −→0,

1− |znk|

2β_u_z nk

1−ψznk

2α −→0, 4.26

which contradicts4.25. Thus, we may assumeρϕzn, ψzn≤1/2 for alln. Therefore, by Lemma 3.1and4.22we obtain

1− |zn|2 β

uzn

1−ϕzn2 α −

1− |zn|2 β

uzn

1−ψzn2 α

1− |zn|2 β

|uzn|

1−ϕzn2 α

1−ϕzn 2

1−ψzn2 α

−1

≤C

1− |zn|2 β

|uzn|ρ

ϕzn, ψzn

1−ϕzn2

α −→0, 4.27

which contradicts4.25. The proof is complete.

Corollary 4.6. Letλbe a complex number andλ /0,1. Suppose the operatorCϕ−Cψis bounded from

H_α∞toH_β∞. ThenCϕ−λCψ :Hα∞ → Hβ∞is compact if and only ifCϕandCψare compact fromHα∞ toH_β∞.

Proof. Assume thatCϕandCψ are compact fromHα∞toHβ∞. It is clear thatCϕ−λCψ :Hα∞ →

H_β∞is compact. Conversely, byTheorem 4.2, we can see that4.22and4.23hold foru≡1 if

Cϕ−λCψ :Hα∞ → Hβ∞is compact. Thus, it follows thatCϕ−Cψis compact fromCorollary 4.5. So, we conclude thatCψ 1/1−λCϕ−λCψ−Cϕ−Cψis also compact, which implies the compactness ofCϕ Cϕ−Cψ Cψ. This completes of the proof.

Example 4.7. We give an example of noncompact composition operators such that their diﬀerence is compact. Choose two analytic functions ϕz and ψz in the unit disk, as previously inExample 3.7. Letβ≥α >0,uz vz 1−zα−β. Clearly, when 0< r <1,

1−r2β_u_r

1−ϕ2_rα −→2 β_,

1−r2β_v_r

1−ψ2_rα −→2

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asr → 1. Therefore, fromCorollary 4.3it follows thatWϕ,uandWψ,v are not compact from

H_α∞toH_β∞. By the Schwarz-Pick lemma, we see thatWϕ,u−Wψ,v : Hα∞ → Hβ∞is bounded fromCorollary 3.6. Note thatρϕz, ψz → 0 asz → 1, and so we have

1− |z|2β|uz|

1−ϕz2α

ρϕz, ψz≤Cρϕz, ψz−→0 asϕz−→1,

1− |z|2β|vz|

1−ψz2α

ρϕz, ψz≤Cρϕz, ψz−→0 asϕz−→1.

4.29

Thus, byCorollary 4.5we conclude thatWϕ,u−Wψ,v:Hα∞ → Hβ∞is compact.

Acknowledgments

The authors would like to thank the referee for valuable comments and suggestions. This work was supported in part by the National Natural Science Foundation of China No. 10971219

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