On the isometric composition operators on the Bloch space in

Let φ be a holomorphic self-map of a bounded homogeneous domain D in C n . In this work, we show that the composition operator C φ : f ↦ f ○ φ is bounded on the Bloch space B of the domain and provide estimates on its operator norm. We also give a sufficient condition for φ to induce an isometry on 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 φ to be an isometry on 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.