Evaluation functions and composition operators on Banach spaces of holomorphic functions

Abstract

Let $B(\Omega )$ be the Banach space of holomorphic functions on a bounded connected domain $\Omega$ in ${\mathbb{C}}^n$, which contains the ring of polynomials on $\Omega$. In this paper, we first establish a criterion for $B(\Omega )$ to be reflexive via evaluation functions on $B(\Omega )$, that is, $B(\Omega )$ is reflexive if and only if the evaluation functions span the dual spaces $(B(\Omega ){)}^{\ast }$. Moreover, under suitable assumptions on $\Omega$ and $B(\Omega )$, we establish a characterization of the composition operator $C_{\phi }$ to be a Fredholm operator on $B(\Omega )$ via the property of the holomorphic self-map $\phi :\Omega \to \Omega$. 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.

Improved the result in the previous version