A Boundedness Criterion for Singular Integral Operators of convolution type on the Fock Space

We show that for an entire function $\varphi$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, that the integral operator \begin{eqnarray*} S_{\varphi}F(z)= \frac{1}{\pi^n}\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} \varphi(z- \bar{w}) e^{-\left\vert w\right\vert^2}dw, \ \ \ \ \ z\in \mathbb{C}^n, \end{eqnarray*} is bounded on ${\mathscr F}^2(\mathbb{C}^n)$ if and only if there exists a function $m\in L^{\infty}(\mathbb{R}^n)$ such that $$ \varphi(z)=\int_{\mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)\cdot \left(x-\frac{i}{2} z \right)} dx, \ \ \ \ \ \ z\in \mathbb{C}^n. $$ With this characterisation we are able to obtain some fundamental results including the normaility, the algebraic property, spectrum and compactness of this operator $S_\varphi$. Moreover, we also obtain the reducing subspaces of $S_{\varphi}$. In particular, in the case $n=1$, we give a complete solution to an open problem proposed by K. Zhu for the Fock space ${\mathscr F}^2(\mathbb{C})$ (Integr. Equ. Oper. Theory {\bf 81} (2015), 451--454).