Closed Range Composition Operators on the Bloch Space of Bounded Symmetric Domains

Abstract

Let \({\mathbb {B}}_X\) and \({\mathbb {B}}_Y\) be bounded symmetric domains realized as the unit balls of \(\hbox {JB}^*\)-triples X and Y, respectively. In this paper, we generalize the Landau theorem to holomorphic mappings on \({\mathbb {B}}_X\) using the Schwarz–Pick lemma for holomorphic mappings on \({\mathbb {B}}_X\). Next, we give a necessary condition for the composition operators \(C_{\varphi }\) between the Bloch spaces on \({\mathbb {B}}_X\) and \({\mathbb {B}}_Y\) to be bounded below by using a sampling set for the Bloch space, where \(\varphi \) is a holomorphic mapping from \({\mathbb {B}}_X\) to \({\mathbb {B}}_Y\). We also obtain other necessary conditions for the composition operators \(C_{\varphi }\) between the Bloch spaces in the case \({\mathbb {B}}_Y\) is a complex Hilbert ball \({\mathbb {B}}_H\). We give a sufficient condition for the composition operators \(C_{\varphi }\) between the Bloch spaces on \({\mathbb {B}}_X\) and \({\mathbb {B}}_Y\) to be bounded below using a sampling set for the Bloch space. In the case \(\dim X=\dim Y<\infty \), we also give another sufficient condition for the composition operators \(C_{\varphi }\) between the Bloch spaces on \({\mathbb {B}}_X\) and \({\mathbb {B}}_Y\) to be bounded below.

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Acknowledgements

The author would like to give thanks to the referee for useful suggestions which improved the paper.

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Correspondence to Hidetaka Hamada.

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Hamada, H. Closed Range Composition Operators on the Bloch Space of Bounded Symmetric Domains. Mediterr. J. Math. 17, 104 (2020). https://doi.org/10.1007/s00009-020-01543-1

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Keywords

  • Bloch space
  • bounded below
  • bounded symmetric domain
  • closed range
  • composition operator
  • Landau theorem

Mathematics Subject Classification

  • Primary 32A18
  • Secondary 32M15
  • 47B38
  • 30H30