Wigner kernel and Gabor matrix of operators

We exhibit the connection between the Wigner kernel and the Gabor matrix of a linear bounded operator T : $\mathcal{S}({\mathbb{R}}^d)\to {\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^d)$. The smoothing effect of the Gabor matrix is highlighted by basic examples. This connection allows a comparison between the classes of Fourier integral operators defined by means of the Gabor matrix and the Wigner kernel, showing the nice off-diagonal decay of the Gabor class with respect to the Wigner kernel one and suggesting further investigations. Modulation spaces containing the Sjöstrand class are the symbol classes of this study.