### Abstract

We show that for an entire function φ belonging to the Fock space F^{2}(C^{n}) on the complex Euclidean space C^{n}, the integral operator S_{φ}F(z)=∫C^{n}F(w)e^{z⋅w¯}φ(z−w¯)dλ(w),z∈C^{n}, is bounded on F^{2}(C^{n}) if and only if there exists a function m∈L^{∞}(R^{n}) such that φ(z)=∫R^{n}m(x)e^{−2(x−[Formula presented]z)2 }dx,z∈C^{n}. Here dλ(w)=π^{−n}e^{−|w|2 }dw is the Gaussian measure on C^{n}. With this characterization we are able to obtain some fundamental results of the operator S_{φ}, including the normality, the C^{⁎} algebraic properties, the spectrum and its compactness. Moreover, we obtain the reducing subspaces of S_{φ}. In particular, in the case n=1, we give a complete solution to an open problem proposed by K. Zhu for the Fock space F^{2}(C) on the complex plane C (Zhu (2015) [30]).

Original language | English |
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Article number | 107001 |

Pages (from-to) | 1-33 |

Number of pages | 33 |

Journal | Advances in Mathematics |

Volume | 363 |

DOIs | |

Publication status | Published - 25 Mar 2020 |

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### Keywords

- Fock space
- Singular integral operator
- Bargmann transform
- Riesz transform
- Spectrum
- Reducing subspace

### Cite this

*Advances in Mathematics*,

*363*, 1-33. [107001]. https://doi.org/10.1016/j.aim.2020.107001