On the isometric composition operators on the Bloch space in \(\mathbb C^n\). (English) Zbl 1166.32300
Summary: Let \(\varphi \) be a holomorphic self-map of a bounded homogeneous domain \(D\) in \(\mathbb C^n\). In this work, we show that the composition operator \(C_\varphi :f\mapsto f\circ \varphi \) is bounded on the Bloch space \(\mathcal B\) of the domain and provide estimates on its operator norm. We also give a sufficient condition for \(\varphi \) to induce an isometry on \(\mathcal B\). This condition allows us to construct non-trivial examples of isometric composition operators in the case when \(D\) has the unit disk as a factor. We then obtain some necessary conditions for \(C_\varphi \) to be an isometry on \(\mathcal B\) when \(D\) is a Cartan classical domain. Finally, we give the complete description of the spectrum of the isometric composition operators in the case of the unit disk and for a wide class of symbols on the polydisk.
MSC:
| 32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |
| 47B33 | Linear composition operators |
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