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.
Categories: math.FA