On the isometric composition operators on the Bloch space in \(\mathbb{C}^n\)

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 a...

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Bibliographic Details
Published in:arXiv.org 2022-07
Main Authors: Allen, Robert F, Colonna, Flavia
Format: Article
Language:English
Subjects:
Composition
Operators
Online Access:Get full text
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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 \(\cal{B}\). 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 \(\cal{B}\) 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.
ISSN:2331-8422
DOI:10.48550/arxiv.0809.3438