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:S(Rd)→S′(Rd). 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 in Cordero et al. (J Math Pures Appl 99(2):219-233, 2013) and the Wigner kernel in Cordero et al. (Nonlinear Differ Equ Appl 31:69, 2024. https://doi.org/10.3761007/s00030-024-00961-4), showing the nice off-diagonal decay of the Gabor class with respect to the Wigner kernel and suggesting further investigations. Modulation spaces containing the Sjöstrand class are the symbol classes of this study.