Evaluation functions and composition operators on Banach spaces of holomorphic functions

Let $B(\mathrm{\Omega })$ be the Banach space of holomorphic functions on a bounded connected domain $\mathrm{\Omega }$ in ${\mathbb{C}}^n$, which contains the ring of polynomials on $\mathrm{\Omega }$. In this paper, we first establish a criterion for $B(\mathrm{\Omega })$ to be reflexive via evaluation functions on $B(\mathrm{\Omega })$, that is, $B(\mathrm{\Omega })$ is reflexive if and only if the evaluation functions span the dual spaces $(B(\mathrm{\Omega }){)}^{\ast }$. Moreover, under suitable assumptions on $\mathrm{\Omega }$ and $B(\mathrm{\Omega })$, we establish a characterization of the composition operator $C_{\phi }$ to be a Fredholm operator on $B(\mathrm{\Omega })$ via the property of the holomorphic self-map $\phi :\mathrm{\Omega }\to \mathrm{\Omega }$. Our new approach utilizes the symbols of composition operators to construct a linearly independent function sequence, which bypasses the use of boundary behavior of reproducing kernels as those may not be applicable in our general setting.