Characterizations of Bounded Singular Integral Operators on the Fock Space and Their Berezin Transforms

There is a singular integral operators S phi on the Fock space F2(C), which originated from the unitarily equivalent version of the Hilbert transform on L2(R). In this paper, we give an analytic characterization of functions phi with finite zeros such that the integral operator S phi is bounded on F2(C) using Hadamard's factorization theorem. As an application, we obtain a complete characterization for such symbol functions phi such that the Berezin transform of S phi is bounded while the operator S phi is not. Also, the corresponding problem in higher dimensions is considered.