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)


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