Fredholm composition operators on Banach spaces of holomorphic functions

Let B(Ω) be a Banach space of holomorphic functions on a bounded connected domain Ω in Cn, which contains the ring of polynomials on Ω. When n > 1, we may assume that Ω is simply connected since any holomorphic function on Ω can be analytically extended to the holes in Ω by Hartogs theorem. Given a holomorphic selfmap φ : Ω → Ω, the composition operator Cφ : B(Ω) → B(Ω) is defined by Cφf=f◦φ,∀f∈B(Ω). We characterize the composition operator Cφ to be a Fredholm operator on B(Ω).