For a unitary matrix X of order n over the field of complex numbers and an entire function φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document} belonging to the Fock space F2:=F2(Cn)\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {F}^2:=\mathfrak {F}^2(\mathbb {C}^n)$$\end{document}, we define an integral operator on F2(Cn)\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {F}^2(\mathbb {C}^n)$$\end{document} of the form (HφXf)(z)=∫Cnf(w)φ(z+X∗Xw¯)ezw¯dλ(w).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (H_\varphi ^X f)(z) = \int _{\mathbb {C}^{n}} f(w)\varphi (z+X^*\overline{Xw})e^{z\overline{w}} d\lambda (w). \end{aligned}$$\end{document}Here dλ(z)=π-ne-|z|2dz\documentclass[12pt]{minimal}
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\begin{document}$$d\lambda (z) = \pi ^{-n} e^{-\vert z\vert ^2}dz$$\end{document} is a Gaussian measure on Cn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {C}^n$$\end{document}. We characterize all the symbols φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document} for which the operator HφX\documentclass[12pt]{minimal}
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\begin{document}$$H_\varphi ^X$$\end{document} is bounded. Next, we consider integral operator on F2\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {F}^2$$\end{document} defined by (Rφf)(z)=∫Cnf(w)φ(z⋆w¯)dλ(w)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (R_\varphi f)(z) = \int _{{\mathbb C^n}} f(w) \varphi (z\star \bar{w})d\lambda (w) \end{aligned}$$\end{document}for φ∈F2\documentclass[12pt]{minimal}
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\begin{document}$$\varphi \in \mathfrak {F}^2$$\end{document}, where ⋆\documentclass[12pt]{minimal}
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\begin{document}$$\star $$\end{document} is a coordinatewise multiplication. We give a complete characterization for the symbols φ∈F2(Cn)\documentclass[12pt]{minimal}
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\begin{document}$$\varphi \in \mathfrak {F}^2(\mathbb {C}^n)$$\end{document} so that the operator Rφ\documentclass[12pt]{minimal}
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\begin{document}$$R_\varphi $$\end{document} is bounded on F2\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak {F}^2$$\end{document}. In addition to boundedness, we also obtain some fundamental results for the operators HφX\documentclass[12pt]{minimal}
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\begin{document}$$H_\varphi ^X$$\end{document} and Rφ\documentclass[12pt]{minimal}
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\begin{document}$$R_\varphi $$\end{document} such as normality, the C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-algebra properties, the spectrum and the compactness. Moreover, we characterize the common reducing subspaces for each of the collections BX={HφX∈B(F2):φ∈F2},R={Rφ∈B(F2):φ∈F2},\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathfrak {B}^X&= \Big \{H_\varphi ^X \in \mathcal {B}(\mathfrak {F}^2) : \varphi \in \mathfrak {F}^2 \Big \} ,~~ \mathfrak {R} = \Big \{R_\varphi \in \mathcal {B}(\mathfrak {F}^2) : \varphi \in \mathfrak {F}^2\Big \}, \end{aligned}$$\end{document}respectively.
