A boundedness criterion for singular integral operators of convolution type on the Fock space

We show that for an entire function φ belonging to the Fock space F²(Cⁿ) on the complex Euclidean space Cⁿ, the integral operator S_φF(z)=∫CⁿF(w)e^z⋅w¯φ(z−w¯)dλ(w),z∈Cⁿ, is bounded on F²(Cⁿ) if and only if there exists a function m∈L^∞(Rⁿ) such that φ(z)=∫Rⁿm(x)e^{−2(x−[Formula presented]z)2} dx,z∈Cⁿ. Here dλ(w)=π⁻ⁿe^−|w|2 dw is the Gaussian measure on Cⁿ. With this characterization we are able to obtain some fundamental results of the operator S_φ, including the normality, the C^⁎ algebraic properties, the spectrum and its compactness. Moreover, we obtain the reducing subspaces of S_φ. In particular, in the case n=1, we give a complete solution to an open problem proposed by K. Zhu for the Fock space F²(C) on the complex plane C (Zhu (2015) [30]).