Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 127431, 19 pages
doi:10.1155/2009/127431
Research Article
Differences of Weighted Composition Operators on
Hα∞ BN ∗
Jineng Dai1, 2, 3 and Caiheng Ouyang1
1
Wuhan Institute of Physics and Mathematics, The Chinese Academy of Science, Wuhan 430071, China
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
2
Correspondence should be addressed to Jineng Dai, daijineng@163.com
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 of CN .
Copyright q 2009 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
Let CN denote the Euclidean space of complex dimension NN ≥ 1. For z z1 , . . . , zN and
w w1 , . . . , wN in CN , we denote the inner product of z and w by
z, w z1 w1 · · · zN wN ,
1.1
and we write |z| z, z |z1 |2 · · · |zN |2 . Let BN {z ∈ CN : |z| < 1} be the open unit
ball of CN and let HBN be the space of all holomorphic functions on BN . For a holomorphic
self-map of the unit ball ϕ : BN → BN and u ∈ HBN , we define a weighted composition
operator Wϕ,u by
Wϕ,u f u · f ◦ ϕ
1.2
for f ∈ HBN . As for u ≡ 1, the weighted composition operator Wϕ,1 is the usual composition
operator, denoted by Cϕ . When ϕ is the identity mapping I, the operator WI,u is also called
the multiplication operator. During the past few decades much effort has been devoted to the
2
Journal of Inequalities and Applications
research of such operators on different Banach spaces of holomorphic functions see 1–9.
The general ideal is to explain the operator-theoretic behavior of Wϕ,u such as boundendness
and compactness, in terms of the function-theoretic properties of the symbols ϕ and u. For a
comprehensive overview of the field, we refer to the books by Cowen and MacCluer 10 and
Shapiro 11.
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.
12 Hosokawa et al. 13 on bounded spaces H ∞ ; by Moorhouse 14 on weighted Bergman
spaces, and by Hosokawa and Ohno 15 and Nieminen 16 on Bloch spaces. In 1, the
authors investigated the boundedness and compactness of Cϕ − Cψ on weighted Banach
spaces. In 16, Nieminen characterized the compactness of Wϕ,u − Wψ,v when two weighted
composition operators Wϕ,u and Wψ,v are bounded operators on weighted Banach spaces.
Lindström and Wolf 17 generalized 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 of 16, 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 inequality see Lemma 3.2 which
is useful to characterize the boundedness of differences of weighted composition operators on
weighted Banach spaces in Section 3. In Section 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 Hα∞ be the weighted Banach space of holomorphic functions f on BN
satisfying
α f ∞ sup 1 − |z|2 fz < ∞.
Hα
z∈BN
2.1
Denote by Bα the Bloch-type space of holomorphic functions f on BN such that
α sup 1 − |z|2 ∇fz < ∞,
z∈BN
2.2
Journal of Inequalities and Applications
3
where ∇fz ∂f/∂z1 z, . . . , ∂f/∂zN z. When α 1, the space B1 is the usual Bloch
space. We call the function
1
Kz, z 1 − |z|2
N1
2.3
the Bergman kernel of BN and denote the Bergman matrix by
1
Bz N1
∂2
log Kz, z
∂zi ∂zj
.
2.4
z ∈ BN .
2.5
N×N
For f ∈ HBN , we define
∇fz, w N
Qf z sup ,
: 0
/w ∈ C
Bzw, w
It is well known that see 21 or 22
1 − |z|2 |w|2 |z, w|2
.
Bzw, w 2
1 − |z|2
2.6
Moreover, for α > 1/2, if a holomorphic function is f ∈ Bα , then we have
α α−1
Qf z.
sup 1 − |z|2 ∇fz ≈ sup 1 − |z|2
z∈BN
z∈BN
2.7
Here and below we use the abbreviated notation A ≈ B to mean that there exists a positive
constant C such that C−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 space Hα∞ can be identified with the Bloch-type space Bα1 . Thus, we
can easily see that if f ∈ Hα∞ for α > 0, then
α
f ∞ ≈ f0 sup 1 − |z|2 Qf z.
Hα
z∈BN
2.8
For any point a ∈ BN − {0}, we define
ϕa z a − Pa z − sa Qa z
,
1 − z, a
z ∈ BN ,
2.9
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Journal of Inequalities and Applications
where sa 1 − |a|2 , Pa is the orthogonal projection from CN onto the one-dimensional
subspace a generated by a, and Qa I−Pa is the projection onto the orthogonal complement
of a, that is
Pa z z, a
|a|
2
Qa z z −
a,
z, a
|a|2
a,
z ∈ BN .
2.10
When a 0, we simply define ϕa z −z. It is well known that each ϕa is a homeomorphism
of the closed unit ball BN onto BN . Let
ρa, z ϕa z.
2.11
Then ρ is a metric on BN and is invariant under automorphisms. The metric ρ is called the
pseudohyperbolic metric.
For any two points z and w in BN , let γt γ1 t, . . . , γN t : 0, 1 → BN be a
smooth curve to connect z and w. Define
l γ 1
B γt γ t, γ t dt.
2.12
0
The infimum of the set consisting of all lγ is denoted by βz, w, where γ is a smooth curve
in BN from z and w. We call β the Bergman metric on BN . It is known that
βz, w 1 ρz, w
1
log
.
2
1 − ρz, w
2.13
3. The Boundedness of Wϕ,u − Wψ,v
In this section, we will characterize the boundedness of the operator Wϕ,u −Wψ,v : Hα∞ → Hβ∞ .
For this purpose, we state some useful lemmas.
Lemma 3.1. For z and w in BN , then
1 − ρz, w
1 ρz, w
1 − |z|2
≤
.
≤
2
1 ρz, w 1 − |w|
1 − ρz, w
3.1
Proof. Set ϕw z a. Since ϕw is an involution, it follows that ϕw a z. Thus, from the
identity
1 − |w|2 1 − |a|2
2
,
1 − ϕw a 1 − |z|2 |1 − w, a|2
3.2
Journal of Inequalities and Applications
5
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|2
1 |a|
≤
.
≤
2
1 |a| |1 − w, a|
1 − |a|
3.4
1 − ϕw z
1 ϕw z
1 − |z|2
≤
,
≤
1 ϕw z 1 − |w|2 1 − ϕw z
3.5
Therefore, it follows that
which, together with 2.11, yields the desired estimate.
The following lemma can be found in 16, 17 for the one variable case.
Lemma 3.2. Let f ∈ Hα∞ . Then
α
α
1 − |z|2 fz − 1 − |w|2 fw ≤ Cf Hα∞ ρz, w
3.6
for all z, w in BN .
Proof. Fix any two points z and w in BN . Let γ γt0 ≤ t ≤ 1 be a smooth curve in BN from
w to z. Then
1 α
α
2 α 2
2
1 − |z| fz − 1 − |w| fw d 1 − γt f γt
0
1
0
2 α
f γt d 1 − γt
1
2 α 1 − γt df γt .
0
3.7
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Journal of Inequalities and Applications
Since f ∈ Hα∞ , we get
N 1 2 α−1 2 α 1
γk tγk t γk tγk t dt
− αf γt 1 − γt
f γt d 1 − γt
0
0
k1
1
2 α−1 γt, γ t dt
≤ 2α f γt 1 − γt
0
1 γt, γ t 2 dt
Hα∞
0 1 − γt
≤ C f ≤ Cf Hα∞
1
B γt γ t, γ t dt,
0
3.8
where the last inequality comes from 2.6.
On the other hand,
N
1
2 α 2 α 1 ∂f 1 − γt
1 − γt
df γt γk t
γt dt.
0
0
∂zk
k1
3.9
From the definition of Qf we see that
N
∂f
γt ≤ Qf γt
B γt γ t, γ t .
γk t
k1
∂zk
3.10
Thus, by 2.8 it follows that
1
1
2 α 2 α 1 − γt df γt ≤
1 − γt Qf γt
B γt γ t, γ t dt
0
0
1
B γt γ t, γ t dt.
Hα∞
≤ C f 3.11
0
Therefore, we have proved that
1
α
α
2
2
f
≤
C
1
−
fz
−
1
−
fw
B γt γ t, γ t dt.
|z|
|w|
Hα∞
3.12
0
Since γ γt 0 ≤ t ≤ 1 is an arbitrary smooth curve in BN from w to z, by the definition of
βz, w, we have
α
α
1 − |z|2 fz − 1 − |w|2 fw ≤ Cf Hα∞ βz, w.
3.13
Journal of Inequalities and Applications
7
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
α
α
1 − |z|2 fz − 1 − |w|2 fw ≤ 2f Hα∞ .
3.14
α
α
1 − |z|2 fz − 1 − |w|2 fw ≤ Cf Hα∞ ρz, w.
3.15
Thus, we have
Remark 3.3. From the proof of Lemma 3.2, it is not hard to see that for any z, w ∈ rBN {z ∈
BN : |z| < r < 1}, then
α
α
1 − |z|2 fz − 1 − |w|2 fw ≤ Cfr Hα∞ ρz, w
3.16
for any f ∈ Hα∞ , where fr Hα∞ sup 1 − |z|2 α |fz|.
z∈rBN
Now we provide a characterization of the boundedness of Wϕ,u −Wψ,v from Hα∞ to Hβ∞ .
For that purpose, consider the following three conditions:
β
1 − |z|2 |uz| sup 2 α ρ ϕz, ψz < ∞,
z∈BN
1 − ϕz
3.17
β
1 − |z|2 |vz| sup 2 α ρ ϕz, ψz < ∞,
z∈BN
1 − ψz
3.18
β
β
1 − |z|2 uz
1 − |z|2 vz sup 2 α − 2 α < ∞.
z∈BN 1 − ϕz
1 − ψz
3.19
Theorem 3.4. The following statements are equivalent.
i Wϕ,u − Wψ,v : Hα∞ → Hβ∞ is bounded.
ii The conditions 3.17 and 3.19 hold.
iii The conditions 3.18 and 3.19 hold.
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Journal of Inequalities and Applications
Proof. First, we prove the implication ii → iii. Assume that the conditions 3.17 and 3.19
hold. Note that the pseudohyperbolic metric ρ is less then 1. Then we have
β
β
2
1
−
1 − |z|2 |vz| |z|
|uz| 2 α ρ ϕz, ψz ≤ 2 α ρ ϕz, ψz
1 − ψz
1 − ϕz
β
β
1 − |z|2 uz
1 − |z|2 vz 2 α − 2 α ρ ϕz, ψz ,
1 − ψz
1 − ϕz
3.20
which implies that 3.18 holds.
Next, we show the implication iii → i. Let f ∈ Hα∞ . Assume that the conditions
3.18 and 3.19 hold; by Lemma 3.2, we have
β 1 − |z|2 Wϕ,u − Wψ,v fz
β 1 − |z|2 f ϕz uz − f ψz vz
⎡
⎤
β
β
1 − |z|2 uz
1 − |z|2 vz
α 2
⎢
⎥
1 − ϕz f ϕz ⎣ 2 α − 2 α ⎦
1 − ϕz
1 − ψz
β
1 − |z|2 vz 2 α 2 α f ϕz − 1 − ψz
f ψz 2 α 1 − ϕz
1 − ψz
β
β
β
1 − |z|2 uz
1 − |z|2 vz 1 − |z|2 |vz|
2 α − 2 α C f Hα∞ ρ ϕz, ψz 2 α
Hα∞ 1 − ψz
1 − ψz
1 − ϕz
≤ f ≤ Cf Hα∞ ,
from which it follows that Wϕ,u − Wψ,v : Hα∞ → Hβ∞ is bounded.
3.21
Journal of Inequalities and Applications
9
Finally, we prove the implication i → ii. Assume that Wϕ,u − Wψ,v is bounded. Fix
w ∈ BN ; consider the function fw defined by
ϕψw z, ϕψw ϕw
fw z α ·
ϕψw ϕw 1 − z, ϕw
1
3.22
for z ∈ BN . It is easy to check that fw ∈ Hα∞ with fw Hα∞ ≤ 2α . Note that
fw
ρ ϕw, ψw
ϕw 2 α ,
1 − ϕw
fw ψw 0.
3.23
By the boundedness of Wϕ,u − Wψ,v , we have
∞ > Cfw Hα∞ ≥ Wϕ,u − Wψ,v fw H ∞
β
β sup 1 − |z|2 fw ϕz uz − fw ψz vz
z∈BN
β ≥ 1 − |w|2 fw ϕw uw − fw ψw vw
3.24
β 1 − |w|2 ρ ϕw, ψw |uw|
2 α
1 − ϕw
for any w ∈ BN . This proves 3.17. Now we prove that 3.19 is also true. For given w ∈ BN
instead of the function fw , we consider the function gw given by
gw z 1
α .
1 − z, ψw
3.25
Clearly, gw ∈ Hα∞ with gw Hα∞ ≤ 2α . Thus, we have
β ∞ > Wϕ,u − Wψ,v gw H ∞ ≥ 1 − |w|2 gw ϕw uw − gw ψw vw
β
|Iw Jw| ,
3.26
10
Journal of Inequalities and Applications
where
⎡
⎤
β
β
2
2
1
−
1
−
uw
vw
|w|
|w|
2 α ⎢
⎥
Iw 1 − ψw gw ψw ⎣ 2 α − 2 α ⎦
1 − ϕw
1 − ψw
β
β
1 − |w|2 uw
1 − |w|2 vw
2 α − 2 α ,
1 − ϕw
1 − ψw
3.27
β
1 − |w|2 uw α α ϕw2 gw ϕw − 1 − ψw2 gw ψw .
1
−
Jw 2 α
1 − ϕw
By 3.17 that has been proved as before and Lemma 3.2, we conclude that |Jw| < ∞ for all
w ∈ BN , which implies that |Iw| < ∞ for all w ∈ BN . Thus, the condition 3.19 is proved.
The whole proof is complete.
Corollary 3.5. The weighted composition operator Wϕ,u : Hα∞ → Hβ∞ is bounded if and only if
β
1 − |z|2 |uz|
sup 2 α < ∞.
z∈BN
1 − ϕz
3.28
Proof. The result follows from the simple case in which v ≡ 0 of Theorem 3.4.
Corollary 3.6. The operator uCϕ − uCψ : Hα∞ → Hβ∞ is bounded if and only if the following two
conditions hold:
β
1 − |z|2 |uz| sup 2 α ρ ϕz, ψz < ∞,
z∈BN
1 − ϕz
2
β
3.29
1 − |z| |uz| sup 2 α ρ ϕz, ψz < ∞.
z∈BN
1 − ψz
Proof. The necessity is clear by Theorem 3.4. To prove the sufficiency, we only need to show
that if the conditions 3.29 hold, then
β
β
1 − |z|2 uz
1 − |z|2 uz 2 α − 2 α < ∞
1 − ψz
1 − ϕz
3.30
Journal of Inequalities and Applications
11
for all z ∈ BN . In fact, we easily see that 3.30 holds for z ∈ BN satisfying ρϕz, ψz > 1/2
by 3.29. If z ∈ BN such that ρϕz, ψz ≤ 1/2, by Lemma 3.1 we have
α
α
α
2 α 1 − |z|2 uz
1 − |z|2 |uz| 1 − |z|2 uz ϕz
1
−
α − α α 1 −
2
2
2
2
1 − ψz
1 − ψz
1 − ϕz
1 − ϕz
α
α 1 − |z|2 |uz| 1 ρ ϕz, ψz
≤ 2 α 1 − 1 − ρϕz, ψz 1 − ϕz
3.31
α
1 − |z|2 |uz| ≤C 2 α ρ ϕz, ψz ,
1 − ϕz
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 operators Wϕ,u
and Wψ,v are unbounded on Hα∞ while the operator Wϕ,u − Wψ,v is bounded on Hα∞ .
Choose the analytic functions ϕz 1 z/2 and ψz ϕz tz − 13 in the unit
disk, where t is real and positive and t is so small that ψ maps the unit disk D into D. Let
uz vz 1 − z−1 , α β > 0. It is easy to see that when 0 < r < 1,
α
1 − r 2 ur
α −→ ∞,
1 − ϕ2 r
α
1 − r 2 vr
α −→ ∞
1 − ψ 2 r
3.32
as r → 1. This shows that Wϕ,u and Wψ,v are unbounded on Hα∞ by Corollary 3.5. On the
other hand, we know that ρϕz, ψz ≤ Ct|1 − z| when t is small enough see 12, Example
1. By the Schwarz-Pick lemma, we get
α
1 − |z|2 |uz| Ct|1 − z|
sup 2 α ρ ϕz, ψz ≤ sup |1 − z| < ∞,
z∈D
z∈D
1 − ϕz
2
α
1 − |z| |vz| Ct|1 − z|
sup 2 α ρ ϕz, ψz ≤ sup |1 − z| < ∞.
z∈D
z∈D
1 − ψz
So, it follows that Wϕ,u − Wψ,v is bounded on Hα∞ from Corollary 3.6.
3.33
12
Journal of Inequalities and Applications
4. The Compactness of Wϕ,u − Wψ,v
In this section, we give a characterization of the compactness of the operator Wϕ,u − Wψ,v :
Hα∞ → Hβ∞ . Before doing this, we need the following lemma whose proof is an easy
modification of that of 10, 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 Hα∞ with fn → 0 uniformly on compact subsets of BN , and then
Wϕ,u − Wψ,v fn H ∞ → 0.
β
Here we consider the following conditions:
β
1 − |z|2 uz 2 α ρ ϕz, ψz −→ 0 as ϕz −→ 1,
1 − ϕz
4.1
β
1 − |z|2 vz 2 α ρ ϕz, ψz −→ 0 as ψz −→ 1,
1 − ψz
β
β
1 − |z|2 uz
1 − |z|2 vz
2 α − 2 α −→ 0
1 − ϕz
1 − ψz
as ϕz −→ 1, ψz −→ 1.
4.2
4.3
In one complex variable case, Nieminen 16 characterized the compactness of Wϕ,u −
Wψ,v : Hα∞ → Hβ∞ under the assumption that Wϕ,u and Wψ,v are bounded from Hα∞ to Hβ∞ .
Here, a necessary and sufficient condition for the compactness of Wϕ,u − Wψ,v : Hα∞ → Hβ∞ is
completely obtained in the case of several variables without any assumption.
Theorem 4.2. The operator Wϕ,u − Wψ,v : Hα∞ → Hβ∞ is compact if and only if Wϕ,u − Wψ,v :
Hα∞ → Hβ∞ is bounded and the conditions 4.1–4.3 hold.
Proof. First, we prove the sufficiency. Assume that Wϕ,u − Wψ,v : Hα∞ → Hβ∞ is bounded and
the conditions 4.1–4.3 hold. Then the conditions 3.17–3.19 hold by Theorem 3.4. For
ε > 0, there exists 0 < r < 1 such that
β
1 − |z|2 |vz| 2 α ρ ϕz, ψz ≤ ε
1 − ψz
when ψz > r,
4.4
β
1 − |z|2 |uz| 2 α ρ ϕz, ψz ≤ ε when ϕz > r,
1 − ϕz
β
β
1 − |z|2 uz
1 − |z|2 vz 2 α − 2 α ≤ ε when ϕz > r, ψz > r.
1 − ψz
1 − ϕz
4.5
4.6
Journal of Inequalities and Applications
13
To prove that Wϕ,u − Wψ,v : Hα∞ → Hβ∞ is compact, we will apply Lemma 4.1. Suppose that
{fn } is a sequence in Hα∞ such that fn Hα∞ ≤ 1 and fn → 0 uniformly on compact subsets of
BN . We need only to show that Wϕ,u − Wψ,v fn H ∞ → 0. We write
β
1 − |z|2
β fn ϕz uz − fn ψz vz |In z Jn z|,
4.7
where
⎡
⎤
β
β
2
1 − |z|2 vz
2 α ⎢ 1 − |z| uz
⎥
In z 1 − ϕz fn ϕz ⎣ 2 α − 2 α ⎦,
1 − ϕz
1 − ψz
2
β
4.8
1 − |z| vz α α ϕz2 fn ϕz − 1 − ψz2 fn ψz .
Jn z 1
−
2 α
1 − ψz
We divide the argument into a few cases.
Case 1 |ψz| ≤ r and |ϕz| ≤ r. Clearly, by 3.19, In z converges to 0 uniformly for all z
with |ϕz| ≤ r and |ψz| ≤ r. On the other hand, from Remark 3.3 and 3.18, we have
β
1 − |z|2 |vz| α 2
fn z ≤ ε
1
−
ρ
ϕz,
ψz
sup
|z|
|Jn z| ≤ C 2 α
z∈rBN
1 − ψz
4.9
for sufficiently large n.
Case 2 |ψz| > r and |ϕz| ≤ r. As in the proof of Case 1, In z → 0 uniformly. As regards
Jn z, by Lemma 3.2 and inequality 4.4, we have |Jn z| ≤ ε for sufficiently large n.
Case 3 |ψz| > r and |ϕz| > r. The inequality 4.6 implies that |In z| ≤ ε for n sufficiently
large. Meanwhile, Jn z → 0 uniformly by Lemma 3.2 and inequality 4.4.
Case 4 |ψz| ≤ r and |ϕz| > r. Then we rewrite
1 − |z|2
β fn ϕz uz − fn ψz vz |Pn z Qn z|,
4.10
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Journal of Inequalities and Applications
where
⎡
⎤
β
β
2
1 − |z|2 vz
2 α ⎢ 1 − |z| uz
⎥
Pn z 1 − ψz fn ψz ⎣ 2 α − 2 α ⎦,
1 − ϕz
1 − ψz
2
4.11
β
1 − |z| uz α 2 α fn ϕz − 1 − ψz2 fn ψz .
Qn z 2 α 1 − ϕz
1 − ϕz
The desired result follows by an argument analogous to that in the proof of Case 2. Thus,
together with the above cases, we conclude
Wϕ,u − Wψ,v fn Hβ∞
β sup 1 − |z|2 fn ϕz uz − fn ψz vz ≤ ε
4.12
z∈BN
for sufficiently large n.
Now we will prove the necessity. Suppose that Wϕ,u − Wψ,v : Hα∞ → Hβ∞ is compact.
Since the compactness implies the boundedness, we only need to show that 4.1–4.3 hold.
Let {zn } be a sequence of points in BN such that |ϕzn | → 1 as n → ∞. Consider the
functions fn defined by
2
ϕψzn z, ϕψzn ϕzn 1 − ϕzn .
fn z α1 ·
ϕψz ϕzn 1 − z, ϕzn n
4.13
Clearly, fn converges to 0 uniformly on compact subsets of BN as n → ∞ and fn ∈ Hα∞ with
fn Hα∞ ≤ C for all n. Moreover,
ρ ϕzn , ψzn fn ϕzn 2 α ,
1 − ϕzn fn ψzn 0.
4.14
By the compactness of Wϕ,u − Wψ,v and Lemma 4.1, it follows Wϕ,u − Wψ,v fn H ∞ → 0. On
β
the other hand, we have
Wϕ,u − Wψ,v fn Hβ∞
β sup 1 − |z|2 fn ϕz uz − fn ψz vz
z∈BN
≥ 1 − |zn |2
β fn ϕzn uzn − fn ψzn vzn β 1 − |zn |2 ρ ϕzn , ψzn |uzn |
,
2 α
1 − ϕzn 4.15
Journal of Inequalities and Applications
15
which shows that 1 − |zn |2 β ρϕzn , ψzn |uzn |/1 − |ϕzn |2 α → 0 as |ϕzn | → 1. This
proves 4.1. The condition 4.2 also holds by similar arguments. Now we show that the
condition 4.3 holds. Assume that {zn } is a sequence of points in BN such that |ϕzn | → 1
and |ψzn | → 1 as n → ∞. Define the function
2
1 − ψzn gn z α1 .
1 − z, ψzn 4.16
It is easy to check that gn converges to 0 uniformly on compact subsets of BN as n → ∞ and
gn ∈ Hα∞ with gn Hα∞ ≤ C for all n. Furthermore, gn ψzn 1 − |ψzn |2 −α . By Lemma 4.1
we have Wϕ,u − Wψ,v gn H ∞ → 0. Meanwhile, we have
β
Wϕ,u − Wψ,v gn Hβ∞
β ≥ 1 − |zn |2 gn ϕzn uzn − gn ψzn vzn |Izn Jzn |,
4.17
where
⎡
⎤
β
β
2
2
1
−
1
−
uz
vz
|z
|z
|
|
n
n
n
n
2 α ⎢
⎥
Izn 1 − ψzn gn ψzn ⎣ 2 α − 2 α ⎦
1 − ϕzn 1 − ψzn β
β
1 − |zn |2 uzn 1 − |zn |2 vzn 2 α − 2 α ,
1 − ϕzn 1 − ψzn 4.18
β
1 − |zn |2 uzn 2 α 2 α Jzn 1 − ϕzn gn ϕzn − 1 − ψzn gn ψzn .
2 α
1 − ϕzn By Lemma 3.2 and the condition 4.1 that has been proved, we conclude Jzn → 0, which
implies that Izn → 0 as n → ∞. This shows that 4.3 is true. The whole proof is complete.
Corollary 4.3. The weighted composition operator Wϕ,u : Hα∞ → Hβ∞ is compact if and only if Wϕ,u
is bounded and
β
1 − |z|2 uz
2 α −→ 0
1 − ϕz
as ϕz −→ 1.
4.19
16
Journal of Inequalities and Applications
Corollary 4.4. If β ≥ α, then the composition operator Cϕ : Hα∞ → Hβ∞ is bounded. If β > α, then
Cϕ : Hα∞ → Hβ∞ is compact.
Proof. By the Schwarz-Pick lemma in the unit ball see 23
2
1 − ϕ0
1 ϕ0
1 − |z|2
,
2 ≤ ≤
1 − ϕz, ϕ0 2 1 − ϕ0
1 − ϕz
4.20
we obtain
β
1 − |z|2
2 β−α
.
2 α ≤ C 1 − ϕz
1 − ϕz
4.21
Therefore, the desired results follow from Corollaries 3.5 and 4.3.
The following result appears in 1 when u ≡ 1 in one-dimensional case.
Corollary 4.5. The operator uCϕ −uCψ : Hα∞ → Hβ∞ is compact if and only if uCϕ −uCψ is bounded
and the following conditions hold:
β
1 − |z|2 uz 2 α ρ ϕz, ψz −→ 0
1 − ϕz
β
1 − |z|2 uz 2 α ρ ϕz, ψz −→ 0
1 − ψz
as ϕz −→ 1,
4.22
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
conditions 4.22 and 4.23 imply that
β
β
1 − |z|2 uz
1 − |z|2 uz
a
2 α − 2 α −→ 0 as ϕz −→ 1, ψz −→ b 1.
1 − ϕz
1 − ψz
4.24
Suppose that 4.24 is not true. Then there exist ε0 > 0 and a sequence of points {zn } ⊂ BN
such that |ϕzn | → 1 and |ψzn | → 1 as n → ∞ and
β
β
1 − |zn |2 uzn 1 − |zn |2 uzn 2 α − 2 α ≥ ε0 .
1 − ψzn 1 − ϕzn 4.25
Journal of Inequalities and Applications
17
We deduce that ρϕzn , ψzn → 0 as n → ∞. If not, there exists a subsequence {znk } in
{zn } such that ρϕznk , ψznk → a > 0. By 4.22 and 4.23, we obtain
β
1 − |znk |2 uznk 2 α −→ 0,
1 − ϕznk β
1 − |znk |2 uznk 2 α −→ 0,
1 − ψznk 4.26
which contradicts 4.25. Thus, we may assume ρϕzn , ψzn ≤ 1/2 for all n. Therefore, by
Lemma 3.1 and 4.22 we obtain
β
β
β
1 − |zn |2 uzn 1 − |zn |2 |uzn | 1 − ϕzn 2 α
1 − |zn |2 uzn −
−
1
α
α α 2
2
2
1 − ψzn 2
1 − ψzn 1 − ϕzn 1 − ϕzn ≤C
1 − |zn |2
β
4.27
|uzn |ρ ϕzn , ψzn −→ 0,
2 α
1 − ϕzn which contradicts 4.25. The proof is complete.
Corollary 4.6. Let λ be a complex number and λ / 0, 1. Suppose the operator Cϕ −Cψ is bounded from
Hα∞ to Hβ∞ . Then Cϕ − λCψ : Hα∞ → Hβ∞ is compact if and only if Cϕ and Cψ are compact from Hα∞
to Hβ∞ .
Proof. Assume that Cϕ and Cψ are compact from Hα∞ to Hβ∞ . It is clear that Cϕ − λCψ : Hα∞ →
Hβ∞ is compact. Conversely, by Theorem 4.2, we can see that 4.22 and 4.23 hold for u ≡ 1 if
Cϕ −λCψ : Hα∞ → Hβ∞ is compact. Thus, it follows that Cϕ −Cψ is compact from Corollary 4.5.
So, we conclude that Cψ 1/1 − λCϕ − λCψ − Cϕ − Cψ is also compact, which implies
the compactness of Cϕ 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 in Example 3.7. Let β ≥ α > 0, uz vz 1 − zα−β . Clearly, when 0 < r < 1,
β
1 − r 2 ur
α −→ 2β ,
1 − ϕ2 r
β
1 − r 2 vr
α −→ 2β
1 − ψ 2 r
4.28
18
Journal of Inequalities and Applications
as r → 1. Therefore, from Corollary 4.3 it follows that Wϕ,u and Wψ,v are not compact from
Hα∞ to Hβ∞ . By the Schwarz-Pick lemma, we see that Wϕ,u − Wψ,v : Hα∞ → Hβ∞ is bounded
from Corollary 3.6. Note that ρϕz, ψz → 0 as z → 1, and so we have
β
1 − |z|2 |uz| 2 α ρ ϕz, ψz ≤ Cρ ϕz, ψz −→ 0
1 − ϕz
2
as ϕz −→ 1,
β
4.29
1 − |z| |vz| 2 α ρ ϕz, ψz ≤ Cρ ϕz, ψz −→ 0 as ϕz −→ 1.
1 − ψz
Thus, by Corollary 4.5 we conclude that Wϕ,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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