Compactness of Composition Operators from the p-Bloch Space to the q-Bloch Space on the Classical Bounded Symmetric Domains

In this paper, we introduce the weighted Bloch spaces ( ) ( ) , β p I m n R on the first type of classical bounded symmetric domains ( ) , I m n R , and prove the equivalence of the norms 1, p f and ( ) 2, + 2 p+ m n f . Furthermore, we study the compactness of composition operator φ C from ( ) ( ) , β p I m n R to ( ) ( ) , β q I m n R , and obtain a sufficient and necessary condition for ( ) ( ) ( ) ( ) : , , φ β β p q I I C m n m n R → R to be compact.


Introduction
Let Ω be a bounded homogeneous domain in n  .The class of all holomorphic functions on Ω will be denoted by ( ) H Ω .For φ a holomorphic self-map of Ω and ( ) , and C φ is called the composition operator with symbol φ .The composition operators as well as related operators known as the weighted composition operators between the weighted Bloch spaces were investigated in [1] [2] in the case of the unit disk, and in [3]- [7] for the case of the unit ball.The study of the weighted composition operators from the Bloch space to the Hardy space H ∞ was carried out in [8] [9] for the unit ball.Characterizations of the boundedness and the compactness of the composition operators and the weighted ones between the Bloch spaces were given in [10]- [12] for the polydisc case, and in [13]- [18] for the case of the bounded symmetric domains.Furthermore, we will give some results about the composition operators for the case of the weighted Bloch space on the bounded symmetric domains.
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 [19]: ( ) { } T , : is an complex matrix, 0 where m n ≤ and m I is the m m × identity matrix, T Z is the transpose of Z ; ( ) { } T : is a symmetric matrix, , 0 : is a antisymmetric matrix, , 0 .
Now we define a holomorphic function f to be in the p-Bloch space ( ) ( ) where , , , , , , .
We can prove that ( ) ( ) . We are concerned here with the question of when . In this work,we shall denote by C a positive constant, not necessarily the same on each occurrence.
In Section 2, we prove the equivalence of the norms defined in this paper and in [20].
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 φ from the p-Bloch space ( ) ( ) ) to be compact, where 0 p ≥ and 0 q ≥ .Specifically,we prove the following result: Theorem 1.1.Let φ be a holomorphic self-map of ( ) . The compactness of the composition operators for the weighted Bloch space on the bounded symmetric domains of ( ) ( ) ℜ is similar with the case of ( ) ; we omit the details.

The Equivalence of the Norms
Denote [20] ( ) 2, 2, , sup det and 0 where B is any m n × matrix and satisfies Proof.The metric matrix of ( ) ) ( ) 1 .
Therefore, the proof is completed.□

Some Lemmas
Here we state several auxiliary results most of which will be used in the proof of the main result.Lemma 3.1.[18] Let N ⊂   be a bounded homogeneous domain.Then there exists a constant C , depending only on  , such that for each z ∈  whenever φ holomorphically maps  into itself.Here ( ) denotes the Jacobian matrix of φ .
Lemma 3.2.Let φ be a holomorphic self-map of ( ) and K a compact subset of ( ) For any compact ( ) , there exists a constant ( ) . Then there exists ( ) and equality holds if and only if A is a diagonal matrix.Lemma 3.4.Let ( ) ( ) Proof.For any ( ) Proof.We can get the conclusion by the process of the proof on Theorem 2.1.□ Lemma 3.6.[18] Let ( ) ( ) where m m U × and n n V × are unitary matrices and (2) ( ) ( ) ( ) 0 0; ( ) ( ) : , , that converges to 0 uniformly on compact subsets of ( ) The proof is trial by using the normal methods.□

Proof of Theorem 1.1
( ) By the chain rule, we have ( It follows from (4.1) and (4.2) that ( ) ( ) We assume that { } k f converges to 0 uniformly on compact subsets of ( ) converges to 0 uniformly on compact subsets of ( ) . Thus, for given 0 ε > , there exists k large enough such that for any . Then by inequalities (4.3) and (4.5) and Lemma 3.2, it follows that, for k large enough, for all 1, 2, j =  .Now we will construct a sequence of functions { } j f satisfying the following three conditions : tends to 0 uniformly on any compact subset of ( ) (III) 0 as .

→ ∞ 
The existence of this sequence will contradict the compactness of C φ .
We will construct the sequence of functions { } j f according to the following four parts: A -D.
Part A: Suppose that ( ) 11 , 1, 2,  where kl E is the m n × matrix whose element at the kth row and lth column is 1 and the other elements are 0. Since φ maps ( ) We construct the sequence of functions { } j f according to the following three different cases.Case 1.If for some j , ( ) max , where a is any positive number.
To begin with, we will prove the sequence of functions ( ) { } j f Z defined by (4.10) satisfies the three conditions.We can get that ( )  Let E be any compact subset of ( ) . Then there exists a ( ) ) a r p j p j a r j r a r This proves that 0  as j → ∞ , which means that the sequence of functions ( ) { } j f Z defined by (4.10) satisfies condition (III).
We can prove that the sequence of functions ( ) { } j f Z defined by (4.12) or (4.14) satisfies the conditions (I) -(III) by using the analogous method as above.
Part B: Now we assume that ( ) ( ) ( ) Using formula (1.1), we have We construct the sequence of functions { } j f according to the following six different cases.Case 1.If for some j , ( ) max , , , , Case 2. If for some j , ( ) max , , , , , Case 3. If for some j , ( ) max , , , , Case 4. If for some j , ( ) max , , , , Case 5.If for some j , ( ) max , , , , , Case 6.If for some j , ( ) max , , , , For any ( ) ii ii     Let E be any compact subset of ( ) . Since there exists a ( ) 1 e 1 e , 1,2. 1 1 1 converges to 0 uniformly on E. Therefore,the sequence of { } j f converges to 0 uniformly on E as j → ∞ .Thus, the sequence of functions ( ) 1 e 1 e q j j j j j j j j j p p q j j j j j j j j j j j j j 1 e j j p p j j j a r a r j j r r a r r This proves that 0  as j → ∞ , which means that the sequence of functions ( ) { } j f Z defined by (4.19) satisfies condition (III).
, where the sequence of functions { } j f is the sequence obtained in Part A. We have , , , where ( ) , , We prove that the sequence of functions { } j g is a bounded sequence in ( ) ( ) Next we prove that { } j g converges to 0 uniformly on any compact subset E of ( ) then by the definition of ( ) j Ψ and Lemma 3.6, we can get a calculation directly that , 1, , . 1 , .
product A B × of A and B is defined as the ms nt × matrix ( ) ikjl C c = such that the element at the ik -th row and jl -th column ikjl ij kl c a b = [19].Then the Bergman metric of ( ) be a classical bounded symmetric domain, and ( ) , T z z denote its metric matrix.Then a holomorphic function f on ( ) the condition (1.3) fails.Then there exist an 0 0 ε > , a sequence { } j Z in ( ) on E as j → ∞ .Thus, the sequence of functions we can use the same methods as in Part A to construct a sequence of functions .23) By using the same methods as in Part A, we can prove the sequences of functions 18)-(4.23)satisfying conditions (I) -(III).Now, as an example,we will prove that the sequence of functions by(4.19)satisfying the conditions (I) (III).


, and combining the discussion in Part A,we can get that 0 as j → ∞ ; that means the sequence of functions { } j g satisfies condition (III).
construct the sequence of functions { } j f according to the following three different cases.Case 1.If for some j ,

Case 3 .
If for some j , (

.
methods as in Part A and Part B, we can prove the sequences of functions 29)-(4.31)satisfying conditions (I) -(III).Part D: In the general situation.For ( ) ( ) Of course, P is an m m × unitary matrix, Q is an n n × unitary matrix, and where the sequence of { } j f are the functions obtained in Part C.From the same discussion as that in Part B, we know that tends to 0 uniformly on E .Thus, { } j g satisfies condition (II).□