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

Abstract

Let $\phi$ be a holomorphic self-map of a bounded homogeneous domain $D$ in ${\mathbb{C}}^n$. In this work, we show that the composition operator $C_{\phi }:f\mapsto f\circ \phi$ 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 $\phi$ 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 $C_{\phi }$ 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.

References in corpus (1)

• Isometries and spectra of multiplication operators on the Bloch space

Cited by in corpus (3)

• Weighted composition operators from $H^{\infty }$ to the Bloch space of a bounded homogeneous domain
• Multiplication operators on the Bloch space of bounded homogeneous domains
• Boundedness and compactness of composition operators on Segal-Bargmann spaces