Journal of Operator Theory

Volume 94, Issue 2, Autumn 2025 pp. 427-443.

Fredholm composition operators on Banach spaces of holomorphic functions

Authors: Guangfu Cao (1), Li He (2), Ji Li (3)
Author institution: (1) School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China
(2) School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China
(3) School of Mathematical and Physical Sciences, Macquarie University, NSW, 2109, Australia

Summary: Let $B(\Omega)$ be a Banach space of holomorphic functions on a\break bounded connected domain $\Omega$ in $\mathbb C^n$, which contains the ring of polynomials on $\Omega $. Under suitable assumptions on $\Omega$ and $B(\Omega)$, we establish a characterization of the composition operator and the weighted composition operator $C_\varphi$ to be Fredholm operators on $B(\Omega)$. 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.

DOI: http://dx.doi.org/10.7900/jot.2024jan09.2477
Keywords: Banach space of holomorphic functions, evaluation function, composition operator, Fredholm operator, automorphism

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