Gaussian Integral Operator between Fock Spaces

We study Gaussian integral operators $$(Tf)(z)=\frac{1}{\pi }{\int }_{\mathbb{C}}{e}^{\alpha z^2+\beta {\bar{w}}^2+\gamma z\bar{w}+\eta z+\delta \bar{w}}f(w)e^{-\mathrm{\mid }w{\mathrm{\mid }}^2}dA(w)$$ acting between Fock spaces ${\mathcal{F}}^p$ and ${\mathcal{F}}^q$, where $0<p,q\le \mathrm{\infty }$. We give complete characterizations of their boundedness and compactness in terms of explicit conditions on the parameters of the Gaussian kernel. The proofs are based on a unified approach combining reproducing-kernel estimates, atomic decompositions, and sampling sequences, together with a reduction to diagonal operators on sequence spaces. We also study Schatten-class membership on ${\mathcal{F}}^2$ and compute the Hilbert–Schmidt norm explicitly.