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

TL;DR

This paper investigates when composition operators induced by holomorphic self-maps are isometries on the Bloch space in complex domains, providing conditions, examples, and spectral analysis.

Contribution

It establishes boundedness and norm estimates for composition operators on the Bloch space, and characterizes isometric operators in various complex domains.

Findings

Boundedness of composition operators on the Bloch space

Sufficient conditions for isometric composition operators

Complete spectral description in the unit disk case

Abstract

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.