Singular Integral Operators on the Fock Space

Abstract

It can be shown that the Hilbert transform is unitarily equivalent to the following integral operator on the Fock space F ²:

$$S_{\varphi} f(z)\,=\,\int_{\mathbb{c}}\, f(w)e^{z\overline{w}} \varphi\, \left(z\, -\, \overline{w}\right)\,d\lambda(w),$$

where ${d\lambda(z)=\pi^{-1}e^{-|z|^2} \,dA(z)}$ is the Gaussian measure and ${\varphi}$ is a certain function in F ². We pose the general problem of characterizing entire functions ${\varphi}$ such that the operator above is bounded on F ².