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. Under suitable assumptions on Ω and B(Ω), we establish a characterization of the composition operator and the weighted composition operator C_ϕ to be Fredholm operators on B(Ω). Our new approach utilizes the symbols of composition operators to construct a linearly independent function sequence, instead of the use of boundary behavior of reproducing kernels as those may not be applicable in general setting.