In this paper, we introduce the weighted Bloch spaces on the first type of classical bounded symmetric domains , and prove the equivalence of the norms and . Furthermore, we study the compactness of composition operator from to , and obtain a sufficient and necessary condition for to be compact.
Let
The composition operators as well as related operators known as the weighted composition operators between the weighted Bloch spaces were investigated in [
In 1930s all irreducible bounded symmetric domains were divided into six types by E. Cartan. The first four types of irreducible domains are called the classical bounded symmetric domains, the other two types, called exceptional domains, consist of one domain each (a 16 and 27 dimensional domain).
The first three types of classical bounded symmetric domains can be expressed as follows [
where
Let
matrix
where
Following Timoney’s approach (see [
Now we define a holomorphic function
where
We can prove that
with the case on
Let
Let
In Section 2, we prove the equivalence of the norms defined in this paper and in [
In Section 3, we state several auxiliary results most of which will be used in the proofs of the main results.
Finally, in Section 4, we establish the main result of the paper. We give a sufficient and necessary condition for the composition operator Cf from the p-Bloch space
Theorem 1.1. Let
for all
The compactness of the composition operators for the weighted Bloch space on the bounded symmetric domains of
Denote [
Lemma 2.1. (Bloomfield-Watson) [
where
Theorem 2.1.
Proof. The metric matrix of
For any
Denote
Thus
Hence
Furthermore,
Since
Thus
For
then we have
Combining (2.2) and (2.3),
Next,
and
Therefore, the proof is completed. □
Here we state several auxiliary results most of which will be used in the proof of the main result.
Lemma 3.1. [
for each
on
Lemma 3.2. Let
for all
Proof. For
For any compact
Thus
Combining Lemma 3.1 with (3.3) shows that (3.2) holds. □
Lemma 3.3. (Hadamard) [
and equality holds if and only if
Lemma 3.4. Let
Proof. For any
Thus we have
It follows from Lemma 3.3 that
Lemma 3.5. Let
If (3.6) holds, then
Proof. We can get the conclusion by the process of the proof on Theorem 2.1. □
Lemma 3.6. [
where
Denote
(3)
(4)
(5)
(6)
Lemma 3.7.
any bounded sequence
Proof. The proof is trial by using the normal methods. □
Proof. Let
Suppose (1.3) holds. Then for any
for all
By the chain rule, we have
If
It follows from (4.1) and (4.2) that
whenever
On the other hand, there exists a constant
So if
We assume that
for any
whenever
Combining (4.4) and (4.6) shows that
For the converse, arguing by contradiction, suppose
the condition (1.3) fails. Then there exist an
for all
Now we will construct a sequence of functions
(I)
(II)
The existence of this sequence will contradict the compactness of
We will construct the sequence of functions
Part A: Suppose that
where
Denote
Denote
then
We construct the sequence of functions
Case 1. If for some
then set
where
Case 2. If for some
then set
where
Case 3. If for some
then set
Next, we will prove that the sequences of functions
To begin with, we will prove the sequence of functions
It follows from Lemma 3.5 that
This proves that the sequence of functions
Let
for any
Since
But
Now (4.8) and (4.9) mean that
Combining (4.7) and (4.16), we have
Since
This proves that
We can prove that the sequence of functions
Part B: Now we assume that
It is clear that
If
Using formula (1.1), we have
Denote
Then,
We construct the sequence of functions
Case 1. If for some
then set
Case 2. If for some
then set
Case 3. If for some
then set
Case 4. If for some
then set
Case 5. If for some
then set
Case 6. If for some
then set
By using the same methods as in Part A, we can prove the sequences of functions
Now, as an example,we will prove that the sequence of functions
For any
Thus
By Lemma
It follows from Lemma 3.5 and (4.25) that
Let
Since
So
formly on E. Therefore,the sequence of
For case 2,
Combining (4.7) and (4.26), we have
Since
This proves that
If
If we denote
Denote
where
and
It is clear that
We prove that the sequence of functions
Since
So
Next we prove that
It is clear that
Since
morphic on
by the definition of
Hence
Part C: Assume that
where
Just as in Part B, we can use the same methods to prove the conclusion. And for
Using formula (1.1), we have
Denote
then,
We construct the sequence of functions
Case 1. If for some
then set
Case 2. If for some
then set
Case 3. If for some
then set
Using the same methods as in Part A and Part B, we can prove the sequences of functions
Part D: In the general situation. For
We may assume that
that
verges uniformly to
Let
From the same discussion as that in Part B, we know that
set
We thank the Editor and the referee for their comments. Research is funded by the National Natural Science Foundation of China (Grant No. 11171285) and the Postgraduate Innovation Project of Jiangsu Province of China (CXLX12-0980).
JianbingSu,HuijuanLi,XingxingMiao,RuiWang, (2014) Compactness of Composition Operators from the p-Bloch Space to the q-Bloch Space on the Classical Bounded Symmetric Domains. Advances in Pure Mathematics,04,649-664. doi: 10.4236/apm.2014.412074