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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 C_{f} 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