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 ${\mathscr F}^2(\mathbb{C}^n)$ if and only if there exists a function $m\in L^{\infty}(\mathbb{R}^n)$ such that $$ \varphi(z)=\int_{\mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)\cdot \left(x-\frac{i}{2} z \right)} dx, \ \ \ \ \ \ z\in \mathbb{C}^n. $$ Here $d\lambda(w)= \pi^{-n}e^{-\left\vert w\right\vert^2}dw$ is the Gaussian measure on $\mathbb C^n$. With this characterization we are able to obtain some fundamental results including the normaility, the algebraic property, spectrum and compactness of this operator $S_\varphi$. Moreover, we 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 ${\mathscr F}^2(\mathbb{C})$ on the complex plane ${\mathbb C}$ (Integr. Equ. Oper. Theory {\bf 81} (2015), 451--454).

中文翻译：

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Fock空间上卷积型奇异积分算子的有界准则

我们证明，对于属于复欧几里得空间 $\mathbb{C}^n$ 上的 Fock 空间 ${\mathscr F}^2(\mathbb{C}^n)$ 的整个函数 $\varphi$，积分运算符 \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*} 有界于 ${\mathscr F}^2(\ mathbb{C}^n)$ 当且仅当存在一个函数 $m\in L^{\infty}(\mathbb{R}^n)$ 使得 $$ \varphi(z)=\int_{\ mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)\cdot \left(x-\frac{i}{2} z \右)} dx, \ \ \ \ \ \ z\in \mathbb{C}^n。$$ 这里 $d\lambda(w)= \pi^{-n}e^{-\left\vert w\right\vert^2}dw$ 是 $\mathbb C^n$ 上的高斯测度。通过这种表征，我们能够获得一些基本结果，包括规范性、代数性质、这个算子 $S_\varphi$ 的频谱和紧凑性。此外，我们获得了 $S_{\varphi}$ 的约简子空间。特别是在 $n=1$ 的情况下，我们给出了 K. Zhu 提出的一个开放问题的完整解，用于复平面上的 Fock 空间 ${\mathscr F}^2(\mathbb{C})$ ${\mathbb C}$ (Integr. Equ. Oper. Theory {\bf 81} (2015), 451--454)。