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

Abstract

We show that for an entire function φ belonging to the Fock space F2(Cn) on the complex Euclidean space Cn, the integral operatorSφF(z)=∫CnF(w)ez⋅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 thatφ(z)=∫Rnm(x)e−2(x−i2z)2dx,z∈Cn. Here dλ(w)=π−ne−|w|2dw is the Gaussian measure on Cn. With this characterization we are able to obtain some fundamental results of the operator Sφ, including the normality, the C⁎ algebraic properties, the spectrum and its compactness. Moreover, we obtain the reducing subspaces of Sφ.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]).