On the compactness of the Weyl operator in ${\mathcal{S}}_{\omega }$

We characterize, using time-frequency analysis, the continuity and compactness of the Weyl operator in global classes of ultradifferentiable functions ${\mathcal{S}}_{\omega }$, for weight functions $\omega$ in the sense of Braun, Meise and Taylor. As a consequence, we give results about the compactness of the localization operator in ${\mathcal{S}}_{\omega }$, in relation with the spaces of $\omega$-multipliers and $\omega$-convolutors of ${\mathcal{S}}_{\omega }$. Moreover, we provide several examples that complement our investigation.