Let Un be the unit polydisc of ?n and φ=(φ1, ?, φ n ) a holomorphic self-map of Un. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Un) if and only if for ...Let Un be the unit polydisc of ?n and φ=(φ1, ?, φ n ) a holomorphic self-map of Un. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Un) if and only if for every ε > 0, there exists a δ > 0, such that $$\sum\limits_{k,1 = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial z_k }}(z)} \right|} \frac{{1 - |z_k |^2 }}{{1 - |\phi _l (z)|^2 }}< \varepsilon ,$$ whenever dist(φ(z), ?U n )<δ.展开更多
基金This work was supported in part by the National Natural Science Foundation of China ( Grant No. 19871081).
文摘Let Un be the unit polydisc of ?n and φ=(φ1, ?, φ n ) a holomorphic self-map of Un. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Un) if and only if for every ε > 0, there exists a δ > 0, such that $$\sum\limits_{k,1 = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial z_k }}(z)} \right|} \frac{{1 - |z_k |^2 }}{{1 - |\phi _l (z)|^2 }}< \varepsilon ,$$ whenever dist(φ(z), ?U n )<δ.
