\(\mathcal{L}\)-invariant and radial singular integral operators on the Fock space. (English) Zbl 1514.42013
Summary: For a unitary matrix \(X\) of order \(n\) over the field of complex numbers and an entire function \(\varphi\) belonging to the Fock space \(\mathfrak{F}^2:=\mathfrak{F}^2(\mathbb{C}^n)\), we define an integral operator on \(\mathfrak{F}^2(\mathbb{C}^n)\) of the form
\[
(H_\varphi^X f)(z) = \int_{\mathbb{C}^n} f(w)\varphi (z+X^\ast\overline{Xw})e^{z\overline{w}} d\lambda (w).
\]
Here \(d\lambda (z) = \pi^{-n} e^{-\vert z\vert^2}dz\) is a Gaussian measure on \(\mathbb{C}^n\). We characterize all the symbols \(\varphi\) for which the operator \(H_\varphi^X\) is bounded. Next, we consider integral operator on \(\mathfrak{F}^2\) defined by
\[
(R_\varphi f)(z) = \int_{\mathbb{C}^n} f(w) \varphi (z\star \bar{w})d\lambda (w)
\]
for \(\varphi \in \mathfrak{F}^2\), where \(\star\) is a coordinatewise multiplication. We give a complete characterization for the symbols \(\varphi \in \mathfrak{F}^2(\mathbb{C}^n)\) so that the operator \(R_\varphi\) is bounded on \(\mathfrak{F}^2\). In addition to boundedness, we also obtain some fundamental results for the operators \(H_\varphi^X\) and \(R_\varphi\) such as normality, the \(C^\ast\)-algebra properties, the spectrum and the compactness. Moreover, we characterize the common reducing subspaces for each of the collections
\[
\mathfrak{B}^X = \Big \{H_\varphi^X \in \mathcal{B}(\mathfrak{F}^2) : \varphi \in \mathfrak{F}^2 \Big \}, \mathfrak{R} = \Big \{R_\varphi \in \mathcal{B}(\mathfrak{F}^2) : \varphi \in \mathfrak{F}^2\Big \},
\]
respectively.
MSC:
42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
42B15 | Multipliers for harmonic analysis in several variables |
42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
30H20 | Bergman spaces and Fock spaces |
47G10 | Integral operators |
47A15 | Invariant subspaces of linear operators |
Keywords:
Fock space; Fourier transform; Bargmann transform; \(\mathcal{L}\)-invariant operator; radial operator; singular integral operator; multiplication operator; reducing subspaceReferences:
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