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Compactness of composition operators on the Bloch space in classical bounded symmetric domains. (English) Zbl 1044.47021
In [J. Shi and L. Luo, Acta Math. Sin., Engl. Ser. 16, 85–98 (2000; Zbl 0967.32007) and Z. Zhou and J. Shi, Complex Variables Theory Appl. 46, 73–88 (2001; Zbl 1026.47018)], the authors proved that for a holomorphic self-map $$\phi=(\phi_1, \dots, \phi_n)$$ of a bounded homogeneous domain $$\Omega$$, the composition operator $$C_\phi$$ is compact on the Bloch space $$\beta(\Omega)$$ if for every $$\epsilon >0$$, there exists a $$\delta > 0$$ such that $\frac{H_\phi(z) ( J \phi(z)u, J \phi(z)u)}{H_z(u,u)} < \epsilon \tag{$$*$$}$ for all $$u \in \mathbb{C}^n-\{0\}$$ whenever $$\text{dist}(\phi(z), \partial \Omega) <\delta$$. Here $$H_z(u,u)$$ is the Bergman metric of $$\Omega$$ and $$J\phi(z)$$ denotes the Jacobian matrix of $$\phi$$. Specifically, if  $$\Omega$$ is the unit ball $$B_n$$ or the unit polydisc $$U^n$$, the condition $$(*)$$ is necessary.
In the present paper, the authors prove that the condition $$(*)$$ is a sufficient and necessary condition for the composition operator to be compact on the Bloch space of a classical bounded symmetric domain.
Reviewer: Jinkee Lee (Pusan)

MSC:
 47B33 Linear composition operators 32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
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