Isometric composition operators on the Bloch space in the polydisk. (English) Zbl 1163.32002
Hibschweiler, Rita A. (ed.) et al., Banach spaces of analytic functions. AMS special session, University of New Hampshire, Durham, NH, USA, April 22–23, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4268-3/pbk). Contemporary Mathematics 454, 9-21 (2008).
Let \(\Delta^{n}=\left\{z=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}:|z_{k}|<1,k=1,\ldots,n\right\}\) be the unit polydisk. Denote by \(\mathcal{B}\) the space of all Bloch functions on \(\Delta^{n}\).
Let \(f\) be a complex-valued holomorphic function on \(\Delta^{n}\). A holomorphic function \(\phi\) mapping \(\Delta^{n}\) into itself induces on \(\mathcal{B}\) the composition operator \(C_{\phi}(f)=f\circ\phi\).
In the paper, the authors obtain some necessary conditions on \(\phi\) for \(C_{\phi}\) to be an isometry on \(\mathcal{B}\). Non-trivial examples of isometric composition operators on \(\mathcal{B}\) are also given.
For the entire collection see [Zbl 1135.30003].
Let \(f\) be a complex-valued holomorphic function on \(\Delta^{n}\). A holomorphic function \(\phi\) mapping \(\Delta^{n}\) into itself induces on \(\mathcal{B}\) the composition operator \(C_{\phi}(f)=f\circ\phi\).
In the paper, the authors obtain some necessary conditions on \(\phi\) for \(C_{\phi}\) to be an isometry on \(\mathcal{B}\). Non-trivial examples of isometric composition operators on \(\mathcal{B}\) are also given.
For the entire collection see [Zbl 1135.30003].
Reviewer: Dorina Raducanu (Brasov)
MSC:
| 32A18 | Bloch functions, normal functions of several complex variables |
| 30D45 | Normal functions of one complex variable, normal families |
| 32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |
| 47B33 | Linear composition operators |
