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

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