Characterisation of the isometric composition operators on the Bloch space

Abstract

In this paper, we characterise the analytic functions ϕ mapping the open unit disk ▵ into itself whose induced composition operator C ϕ : f ↦ f ∘ ϕ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation ϕ = g B where g is a non-vanishing analytic function from Δ into the closure of ▵, and B is an infinite Blaschke product whose zeros form a sequence{ z n } containing 0 and a subsequence satisfying the conditions , and