Singular Integral Operators on the Fock Space

Abstract: In this talk we will discuss the recent solution of a question raised by K. Zhu about characterizing a class of singular integral operators on the Fock space. 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

\[ S_{\varphi}F(z)=\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} \varphi(z- \bar{w})\,d\lambda(w), \quad z\in\mathbb{C}^n, \]

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

\[ \varphi(z)=\int_{\mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)^2} dx, \quad \in\mathbb{C}^n. \] 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 of the operator $S_\varphi$, including the normality, the $C^*$ algebraic properties, the spectrum and its compactness. Moreover, we obtain the reducing subspaces of $S_{\varphi}$.

In particular, in the case $n=1$, this gives a complete solution to the question 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).

This talk is based on joint work with Guangfu Cao, Ji Li, Minxing Shen, and Lixin Yan.

mathematical physicsanalysis of PDEsclassical analysis and ODEscombinatoricscomplex variablesfunctional analysisinformation theorymetric geometryoptimization and controlprobability

Audience: researchers in the topic

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