Canonical integral operators on the Fock space
Xingtang Dong, Kehe Zhu
TL;DR
This work studies a two-parameter family of integral operators $T^{(s,t)}$ on the Fock space $F^2$, establishing exact boundedness and unitarity criteria in terms of $s$ and $t$ and identifying a Hilbert-Schmidt regime. Using the Bargmann transform, these operators realize a unitary projective representation of $SL(C\times C)$ (and thus $SL(2,\mathbb R)$) on $F^2$, unifying real-line linear canonical transforms within a complex-analytic framework. The authors derive explicit eigenvalues and eigenvectors in the unitary case, expressed via Hermite polynomials and Gaussian factors, and describe the spectral behavior depending on parameter choices. They further show that linear canonical transforms on $L^2(\mathbb R)$ correspond to $T^{(s,t)}$ on $F^2$ under Bargmann, yielding concrete realizations for fractional Fourier transforms, dilation, Fresnel, and chirp multiplication, thereby connecting operator theory on Fock space with classical signal-processing transforms.
Abstract
In this paper we introduce and study a two-parameter family of integral operators on the Fock space $F^2(C)$. We determine exactly when these operators are bounded and when they are unitary. We show that, under the Bargmann transform, these operators include the classical linear canonical transforms as special cases. As an application, we obtain a new unitary projective representation for the special linear group $SL(2,R)$ on the Fock space.