A boundedness criterion for singular integral operators of convolution type on the Fock space

Guangfu Cao, Ji Li, Minxing Shen, Brett D. Wick, Lixin Yan*

*Corresponding author for this work

    Research output: Contribution to journalArticle

    Abstract

    We show that for an entire function φ belonging to the Fock space F2(Cn) on the complex Euclidean space Cn, the integral operator SφF(z)=∫CnF(w)ez⋅w¯φ(z−w¯)dλ(w),z∈Cn, is bounded on F2(Cn) if and only if there exists a function m∈L(Rn) such that φ(z)=∫Rnm(x)e−2(x−[Formula presented]z)2 dx,z∈Cn. Here dλ(w)=π−ne−|w|2 dw is the Gaussian measure on Cn. With this characterization we are able to obtain some fundamental results of the operator Sφ, including the normality, the C algebraic properties, the spectrum and its compactness. Moreover, we obtain the reducing subspaces of Sφ. In particular, in the case n=1, we give a complete solution to an open problem proposed by K. Zhu for the Fock space F2(C) on the complex plane C (Zhu (2015) [30]).

    Original languageEnglish
    Article number107001
    Pages (from-to)1-33
    Number of pages33
    JournalAdvances in Mathematics
    Volume363
    DOIs
    Publication statusPublished - 25 Mar 2020

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    Keywords

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

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