A Note on Weighted Composition Operators on the Fock Space

Abstract

Based on a new characterization of bounded and compact weighted composition operators on the Fock space obtained by Le T (Le T. Normal and isometric weighted composition operators on the Fock space. Bull. London. Math. Soc., 2014, 46: 847–856), this paper shows that a bounded weighted composition operator on the Fock space is a Fredholm operator if and only if it is an invertible operator, and if and only if it is a nonzero constant multiple of a unitary operator. The result is very different from the corresponding results on the Hardy space and the Bergman space.