COMPOSITION OPERATORS BETWEEN
GENERALLY WEIGHTED BLOCH SPACES OF
POLYDISK
Composition Operators
Haiying Li, Peide Liu and Maofa Wang
vol. 8, iss. 3, art. 85, 2007
HAIYING LI, PEIDE LIU AND MAOFA WANG
School of Mathematics and Statistics
Wuhan University
Wuhan 430072, P.R. China
EMail: tslhy2001@yahoo.com.cn
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26 June, 2007
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Communicated by:
S.S. Dragomir
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2000 AMS Sub. Class.:
Primary 47B38.
Key words:
Holomorphic self-map; Composition operator; Bloch space; Generally weighted
Bloch space.
Received:
23 March, 2007
Accepted:
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Abstract:
Let φ be a holomorphic self-map of the open unit polydisk U n in Cn and
p
p, q > 0. In this paper, the generally weighted Bloch spaces Blog
(U n ) are
introduced, and the boundedness and compactness of composition operator Cφ
p
q
from Blog
(U n ) to Blog
(U n ) are investigated.
Acknowledgements:
Supported by the Natural Science Foundation of China (No.
10401027).
10671147,
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Contents
1
Introduction
3
2
Main Results and their Proofs
7
Composition Operators
Haiying Li, Peide Liu and Maofa Wang
vol. 8, iss. 3, art. 85, 2007
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1.
Introduction
Suppose that D is a domain in Cn and φ a holomorphic self-map of D. We denote
by H(D) the space of all holomorphic functions on D and define the composition
operator Cφ on H(D) by Cφ f = f ◦ φ.
The theory of composition operators on various classical spaces, such as Hardy
and Bergman spaces on the unit disk U in the finite complex plane C has been studied. However, the multivariable situation remains mysterious. It is well known in [3]
and [5] that the restriction of Cφ to Hardy or standard weighted Bergman spaces on
U is always bounded by the Littlewood subordination principle. At the same time,
Cima, Stanton and Wogen confirmed in [1] that the multivariable situation is much
different from the classical case (i.e., the composition operators on the Hardy space
of holomorphic functions on the open unit ball of C2 as well as on many other spaces
of holomorphic functions over a domain of Cn can be unbounded, even when n = 1
in [6]). Therefore, it would be of interest to pursue the function-theoretical or geometrical characterizations of those maps φ which induce bounded or compact composition operators. In this paper, we will pursue the function-theoretic conditions of
those holomorphic self-maps φ of U n which induce bounded or compact composition operators from a generally weighted p−Bloch space to a q−Bloch space with
p, q > 0.
For n ∈ N, we denote by U n the open unit polydisk in Cn :
Composition Operators
Haiying Li, Peide Liu and Maofa Wang
vol. 8, iss. 3, art. 85, 2007
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n
U = {z = (z1 , z2 , . . . , zn ) : |zj | < 1, j = 1, 2, . . . , n},
and
hz, wi =
n
X
zj wj ,
|z| =
p
hz, zi
j=1
for any z = (z1 , z2 , . . . , zn ), w = (w1 , w2 , . . . , wn ) in Cn . α = (α1 , α2 , . . . , αn ) is
Close
said to be an n multi-index if αi ∈ N, written by α ∈ Nn . For α ∈ Nn , we write
z α = z1α1 z2α2 · · · znαn and zi0 = 1, 1 ≤ i ≤ n for convenience. For z, w ∈ Cn , we
denote [z, w]j = z when j = 0, [z, w]j = w when j = n, and
[z, w]j = (z1 , z2 , . . . , zn−j , wn−j+1 , . . . , wn )
when j ∈ {1, 2, . . . , n − 1}. Then [z, w]n−j = w when j = 1, and [z, w]n−j+1 = w
when j = n + 1, for j = 2, 3, . . . , n,
Composition Operators
Haiying Li, Peide Liu and Maofa Wang
[z, w]n−j+1 = (z1 , z2 , . . . , zj−1 , wj , . . . , wn ).
vol. 8, iss. 3, art. 85, 2007
For any a ∈ C and zj0 = (z1 , z2 , . . . , zj−1 , zj+1 , . . . , zn ), we write
(a, zj0 ) = (z1 , z2 , . . . , zj−1 , a, zj+1 , . . . , zn ).
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0
Moreover, we adopt the notation (z [j ] )j 0 ∈N for an arbitrary subsequence of (z [j] )j∈N .
Recall that the Bloch space B(U n ) is the vector space of all f ∈ H(U n ) satisfying
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b1 (f ) = sup Qf (z) < ∞,
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z∈U n
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where
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Qf (z) =
|h∇f (z), ūi|
p
,
H(z, u)
u∈Cn \{0}
sup
∇f (z) =
∂f
∂f
(z), . . . ,
(z)
∂z1
∂zn
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and the Bergman metric H : U n × Cn → [0, ∞) on U n is
H(z, u) =
n
X
k=1
|uk |2
1 − |zk |2
(for example see [9], [15]). It is easy to verify that both |f (0)| + b1 (f ) and
n X
∂f
2
kf kB = |f (0)| + sup
(z)
∂zk (1 − |zk | )
z∈U n
k=1
are equivalent norms on B(U n ). In [10], [12] and [8], some characterizations of the
Bloch space B(U n ) have been given.
In a recent paper [2], a generalized Bloch space has been introduced, the p−Bloch
space: for p > 0, a function f ∈ H(U n ) belongs to the p−Bloch space B p (U n ) if
there is some M ∈ [0, ∞) such that
n X
∂f
2 p
∀z ∈ U n .
∂zk (z) (1 − |zk | ) ≤ M,
k=1
The references [13] to [7] studied these spaces and the operators in them.
Dana D. Clahane et al. in [2] proved the following two results:
Theorem A. Let φ be a holomorphic self-map of U n and p, q > 0. The following
statements are equivalent:
(a) Cφ is a bounded operator from B p (U n ) to B q (U n );
(b) There is M ≥ 0 such that
n X
∂φl (1 − |zk |2 )q
(1.1)
∂zk (z) (1 − |φl (z)|2 )p ≤ M,
Haiying Li, Peide Liu and Maofa Wang
vol. 8, iss. 3, art. 85, 2007
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∀z ∈ U n .
k, l=1
Theorem B. Let φ be a holomorphic self-map of U n and p, q > 0. If condition (1.1)
and
n X
∂φl (1 − |zk |2 )q
(1.2)
lim
∂zk (z) (1 − |φl (z)|2 )p = 0
φ(z)→∂U n
k, l=1
Composition Operators
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hold, then Cφ is a compact operator from B p (U n ) to B q (U n ).
p
Now we introduce the generally weighted Bloch space Blog
(U n ).
For p > 0, a function f ∈ H(U ) belongs to the generally weighted p−Bloch
p
space Blog
(U n ) if there is some M ∈ [0, ∞) such that
n X
∂f
2
2 p
(z)
∂zk (1 − |zk | ) log 1 − |zk |2 ≤ M,
∀z ∈ U n .
k=1
Its norm in
Haiying Li, Peide Liu and Maofa Wang
p
Blog
(U n )
p
kf kBlog
Composition Operators
is defined by
vol. 8, iss. 3, art. 85, 2007
n X
∂f
2
(1 − |zk |2 )p log
.
= |f (0)| + sup
(z)
∂zk
1 − |zk |2
z∈U n
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k=1
In this paper, we mainly characterize the boundedness and compactness of the
p
q
composition operators between Blog
(U n ) and Blog
(U n ), and extend some corresponding results in [2] and [11] in several ways.
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2.
Main Results and their Proofs
First, we have the following lemma:
p
Lemma 2.1. Let f ∈ Blog
(U n ) and z ∈ U n , then:
p , when 0 < p < 1;
(a) |f (z)| ≤ 1 + (1−p)n log 2 kf kBlog
P
n
4
1
1
p
(b) |f (z)| ≤ 2 log
+
k=1 log 1−|zk |2 kf kBlog , when p = 1;
2
2n log 2
P
n
1
2p−1
p
(c) |f (z)| ≤ n1 + (p−1)
k=1 (1−|zk |2 )p−1 kf kBlog , when p > 1.
log 2
p
p
Proof. Let p > 0, z ∈ U n , from the definition of k · kBlog
we have |f (0)| ≤ kf kBlog
and
p
p
kf kBlog
kf kBlog
∂f
≤
≤
(2.1)
(z)
∂zk (1 − |z |2 )p log 2
(1 − |zk |2 )p log 2
k
1−|zk |2
Composition Operators
Haiying Li, Peide Liu and Maofa Wang
vol. 8, iss. 3, art. 85, 2007
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n
for every z ∈ U and k ∈ {1, 2, . . . , n}. Notice that
f (z) − f (0) =
n
X
f ([0, z]n−k+1 ) − f ([0, z]n−k )
k=1
=
n
X
k=1
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Z
zk
0
1
∂f ([0, (tzk , zk0 )]n−k+1 )
dt
∂zk
and then from the inequality (2.1), it follows that
Z 1 kf k p
n
X
Blog
|zk |
|f (z)| ≤ |f (0)| +
dt
log 2 0 (1 − |tzk |2 )p
k=1
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p +
≤ kf kBlog
(2.2)
For p = 1, we have:
n Z |zk |
X
(2.3)
k=1
0
n Z
p
X
kf kBlog
log 2
k=1
|zk |
0
1
dt.
(1 − t2 )p
n
n
X1
1
1 + |zk | X 1
4
dt
=
.
log
≤
log
2
p
2
(1 − t )
2
1
−
|z
2
1
−
|z
k|
k|
k=1
k=1
If p > 0 and p 6= 1, then
n Z |zk |
n
n Z |zk |
X
X
X
1
1 − (1 − |zk |)1−p
1
(2.4)
dt
≤
dt
=
.
2 )p
p
(1
−
t
(1
−
t)
1
−
p
0
0
k=1
k=1
k=1
n Z
X
k=1
0
|zk |
1
1
dt ≤
.
2
p
(1 − t )
1−p
From (2.2) and (2.5), it follows that
|f (z)| ≤ 1 +
For (b), Since log
(2.6)
4
1−|zk |2
Haiying Li, Peide Liu and Maofa Wang
vol. 8, iss. 3, art. 85, 2007
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Now for (a), from (2.4),
(2.5)
Composition Operators
n
(1 − p) log 2
p .
kf kBlog
> log 4 = 2 log 2 for each k ∈ {1, 2, . . . , n}, then
1<
1
2n log 2
n
X
k=1
log
4
.
1 − |zk |2
Combining (2.2), (2.3) and (2.6) we get
X
n
1
1
4
p .
|f (z)| ≤
+
log
kf kBlog
2 log 2 2n log 2 k=1
1 − |zk |2
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For (c), from (2.4) we have
n Z |zk |
n
X
X
1
1 − (1 − |zk |)p−1
(2.7)
dt
≤
(1 − t2 )p
(p − 1)(1 − |zk |)p−1
k=1 0
k=1
≤
n
X
k=1
2p−1
.
(p − 1)(1 − |zk |2 )p−1
Composition Operators
By (2.2) and (2.7), we obtain
Haiying Li, Peide Liu and Maofa Wang
n
X
1
2p−1
p +
p
|f (z)| ≤ kf kBlog
kf kBlog
(p − 1) log 2 k=1 (1 − |zk |2 )p−1
X
n
2p−1
1
1
p .
≤
+
kf kBlog
n (p − 1) log 2 k=1 (1 − |zk |2 )p−1
Lemma 2.2. For p > 0, l ∈ {1, 2, . . . , n} and w ∈ U , the function fwl : U n → C,
Z zl
1
l
fw (z) =
2 dt
p
0 (1 − wt) log 1−wt
p
belongs to Blog
(U n ).
(2.8)
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n
∀z ∈ U , k 6= l
and
(2.9)
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Proof. Let k, l ∈ {1, 2, . . . , n} and w ∈ U , then
∂fwl
= 0,
∂zk
vol. 8, iss. 3, art. 85, 2007
1
∂fwl
(z) =
,
2
∂zl
(1 − wzl )p log 1−wz
l
∀z ∈ U n .
An easy estimate shows that there is 0 < M < +∞ such that
2
(1 − |wz|)p log 1−|wz|
2
|1 − wz|p log |1−wz|
≤ M,
∀z, w ∈ U.
Therefore, by (2.8) and (2.9), we have
n X
∂fwl
2
l
(1 − |zk |2 )p log
|fw (0)|+
(z)
∂zk
1 − |zk |2
k=1
=
Haiying Li, Peide Liu and Maofa Wang
vol. 8, iss. 3, art. 85, 2007
(1 − |zl |2 )p log 1−|z2 l |2
2
|1 − wzl |p | log 1−wz
|
l
2 p
≤
Composition Operators
(1 − |zl | ) log
2
1−|zl |2
2
1−|wzl |
(1 − |wzl |)p log
2p
≤
· M < +∞
pe log 2
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2
(1 − |wzl |) log 1−|wz
l|
2
p
|1 − wzl | log |1−wzl |
p
·
p
and thus {fwl : w ∈ U, l ∈ {1, 2, . . . , n}} ⊂ Blog
(U n ).
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Theorem 2.3. Let φ be a holomorphic self-map of the open unit polydisk U n and
p, q > 0, then the following statements are equivalent:
p
q
(a) Cφ is a bounded operator from Blog
(U n ) and Blog
(U n );
(b) There is an M > 0 such that
n X
log 1−|z2 k |2
∂φl (1 − |zk |2 )q
(2.10)
≤ M,
2
∂zk (z) · (1 − |φl (z)|2 )p · log
1−|φl (z)|2
k,l=1
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∀z ∈ U n .
Proof. Firstly, assume that (b) is true. By Lemma 2.1, there is a C > 0 such that for
p
all f ∈ Blog
(U n ),
p .
|f (φ(0))| ≤ Ckf kBlog
(2.11)
Then for all z ∈ U n ,
n X
∂(Cφ f ) 2
(1 − |zk |2 )q log
(z)
∂zk
1 − |zk |2
k=1
n X
∂f
2
2 p
≤
∂ξl (φ(z)) (1 − |φl (z)| ) log 1 − |φl (z)|2
l=1
n X
log 1−|z2 k |2
∂φl (1 − |zk |2 )q
×
2
∂zk (z) (1 − |φl (z)|2 )p · log
1−|φl (z)|2
k=1
p ,
≤ M kf kBlog
and (a) is obtained.
Conversely, let l ∈ {1, 2, . . . , n}, if (a) is true, i.e. there is a C ≥ 0 such that
p ,
kCφ f k ≤ Ckf kBlog
(2.12)
p
∀f ∈ Blog
(U n ),
then, by Lemma 2.2 and (2.12), there is a Q > 0 such that
n X
n
X
∂fwl
∂φl 2
(φ(z)) ·
(z) (1−|zk |2 )q log
≤ CQ,
2
∂ξ
∂z
1
−
|z
l
k
k|
k=1 l=1
1−|φl (z)|2
Haiying Li, Peide Liu and Maofa Wang
vol. 8, iss. 3, art. 85, 2007
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∀w ∈ U, z ∈ U n .
Letting w = φ(z), and using (2.8) and (2.9), we have
n X
log 1−|z2 k |2
∂φl (1 − |zk |2 )q
≤ CQ.
2
∂zk (z) · (1 − |φl (z)|2 )p · log
l,k=1
Composition Operators
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Lemma 2.4. Let φ : U n → U n be holomorphic and p, q > 0, then Cφ is comp
q
pact from Blog
(U n ) to Blog
(U n ) if and only if for any bounded sequence (fj )j∈N in
p
q
Blog (U n ), when fj → 0 uniformly on compacta in U n , then kCφ fj kBlog
→ 0 as
j → ∞.
p
Proof. Assume that Cφ is compact and (fj )j∈N is a bounded sequence in Blog
(U n )
with fj → 0 uniformly on compacta in U n . If the contrary is true, then there is a
q
subsequence (fjm )m∈N and a δ > 0 such that kCφ fjm kBlog
≥ δ for all m ∈ N. Due
to the compactness of Cφ , we choose a subsequence (fjml ◦ φ)l∈N of (Cφ fjm )m∈N =
p
(fjm ◦ φ)m∈N and some g ∈ Blog
(U n ), such that
(2.13)
q
lim kfjml ◦ φ − gkBlog
= 0.
Since Lemma 2.1 implies that for any compact subset K ⊂ U n , there is a Ck ≥ 0
such that
(2.14)
|fjml (φ(z)) − g(z)| ≤ Ck kfjml ◦ φ − gk
Haiying Li, Peide Liu and Maofa Wang
vol. 8, iss. 3, art. 85, 2007
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l→∞
q
Blog
Composition Operators
,
∀l ∈ N, z ∈ K.
By (2.13), fjml ◦φ−g → 0 uniformly on compact subset in U n . Since fjml φ(z) → 0
as l → ∞ for each z ∈ U n , and by (2.14), then g = 0; (2.13) shows
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lim kCφ (fjml )k
l→∞
q
Blog
= 0,
it gives a contradiction.
p
p
Conversely, assume that (gj )j∈N is a sequence in Blog
(U n ) such that kgj kBlog
≤
M for all j ∈ N. Lemma 2.1 implies that if (gj )j∈N is uniformly bounded on any
compact subset in U n and normal by Montel’s theorem, then there is a subsequence
(gjm )m∈N of (gj )j∈N which converges uniformly on compacta in U n to some g ∈
∂g
∂g
H(U n ). It follows that ∂zjm
→ ∂z
uniformly on compacta in U n for each l ∈
l
l
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p
p
p
{1, 2, . . . , n}. Thus g ∈ Blog
(U n ) with kgjm − gkBlog
≤ M + kgkBlog
< ∞ and
n
gjm − g converges to 0 on compacta in U , so by the hypotheses, gjm ◦ φ → g ◦ φ in
q
p
q
Blog
(U n ). Therefore Cφ is a compact operator from Blog
(U n ) to Blog
(U n ).
p
q
Lemma 2.5. If for every f ∈ Blog
(U n ), Cφ f belongs to Blog
(U n ), then φα ∈
q
n
Blog (U ) for each n-multi-index α.
Proof. As is well known, every polynomial pα : Cn → C defined by pα (z) = z α is
p
q
in Blog
(U n ). Thus, by the assumption Cφ (z α ) = φα ∈ Blog
(U n ).
Theorem 2.6. Suppose that p, q > 0, φ : U n → U n is a holomorphic self-map such
q
that φk ∈ Blog
(U n ) for each k ∈ {1, 2, . . . , n} and
(2.15)
n X
log 1−|z2 k |2
∂φl (1 − |zk |2 )q
= 0,
lim
2
∂zk (z) · (1 − |φl (z)|2 )p · log
φ(z)→∂U n
1−|φl (z)|2
k,l=1
then Cφ is a compact operator from
p
Blog
(U n )
Proof. Let (fj )j∈N be a sequence in
in U n and
p
Blog
(U n )
to
q
Blog
(U n ).
with fj → 0 uniformly on compacta
Composition Operators
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vol. 8, iss. 3, art. 85, 2007
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(2.16)
p
kfj kBlog
≤ C,
∀j ∈ N.
By Lemma 2.4, it suffices to show that
(2.17)
q
lim kCφ fj kBlog
= 0.
j→∞
q
Notice that if kφm kBlog
= 0 for all m ∈ {1, 2, . . . , n}, then φ = 0 and Cφ has
q
finite rank. Therefore, we can assume C > 0 and kφm kBlog
> 0 for some m ∈
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{1, 2, . . . , n}. Now let ε > 0, from (2.15), there is an r ∈ (0, 1) such that
n X
log 1−|z2 k |2
∂φl (1 − |zk |2 )q
ε
(2.18)
<
2
∂zk (z) · (1 − |φl (z)|2 )p · log
2C
1−|φl (z)|2
k,l=1
for all z ∈ U n satisfying d(φ(z), ∂U n ) < r. By using a subsequence and the chain
rule for derivatives, (2.16) and (2.18) guarantee that for all such z and j ∈ N,
n X
∂(Cφ f ) 2
2 q
∂zk (z)(1 − |zk | ) log 1 − |zk |2
k=1
n X
∂f
2
(1 − |φl (z)|2 )p log
≤
(φ(z))
∂ξl
1 − |φl (z)|2
l=1
n X
log 1−|z2 k |2
∂φl (1 − |zk |2 )q
×
2
∂zk (z) · (1 − |φl (z)|2 )p · log
1−|φl (z)|2
k=1
ε
ε
≤C·
= .
2C
2
n
∂fj
∂zl
j∈N
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vol. 8, iss. 3, art. 85, 2007
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n
To obtain the same estimate in the case d(φ(z), ∂U ) ≥ r, let Er = {w : d(w, ∂U ) ≥
r}. Since E
r iscompact, by the hypothesis, (fj )j∈N and the sequences of partial
derivatives
Composition Operators
converge to 0 uniformly on Er , respectively. Then
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n X
∂(Cφ fj ) 2
2 q
∂zk (z)(1 − |zk | ) log 1 − |zk |2
k=1
n n X
∂fj
X ∂φl 2
2 q
≤
∂ξl (φ(z)) ·
∂zk (z) · (1 − |zk | ) log 1 − |zk |2
k=1
k=1
n X
∂fj
q
≤
∂ξl (φ(z)) · kφl kBlog
l=1
n
X
∂fj
ε
q
≤
sup (w) · kφl kBlog
≤
2
w∈Er ∂ξl
l=1
(as j → +∞).
q
Since {φ(0)} is compact, we have fj (φ(0)) → 0 as j → ∞, and kCφ fj kBlog
→ 0 as
p
q
n
n
j → ∞, thus Cφ is a compact operator from Blog (U ) to Blog (U ).
n
Composition Operators
Haiying Li, Peide Liu and Maofa Wang
Theorem 2.7. Let φ be a holomorphic self-map of U and p, q > 0. If conditions
p
q
(2.10) and (2.15) hold, then Cφ is a compact operator from Blog
(U n ) to Blog
(U n ).
vol. 8, iss. 3, art. 85, 2007
p
q
Proof. If (2.10) is true, then Cφ is bounded from Blog
(U n ) to Blog
(U n ) by Theorem
q
2.3, and φk ∈ Blog
(U n ) for each k ∈ {1, 2, . . . , n} by Lemma 2.5. The proof follows
on applying (2.15) and Theorem 2.6.
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References
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Composition Operators
Haiying Li, Peide Liu and Maofa Wang
vol. 8, iss. 3, art. 85, 2007
Title Page
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[11] R. YONEDA, the composition operator on weighted Bloch space, Arch. Math.,
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Composition Operators
Haiying Li, Peide Liu and Maofa Wang
vol. 8, iss. 3, art. 85, 2007
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