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

TL;DR

This paper uses time-frequency analysis to characterize the conditions for the Weyl operator's continuity and compactness within ultradifferentiable function classes, impacting localization operators and multiplier spaces.

Contribution

It provides a novel characterization of Weyl operator compactness in ultradifferentiable spaces using time-frequency analysis, with implications for localization operators.

Findings

Characterization of Weyl operator compactness in $\

Relations between localization operators and $\

Examples illustrating the theoretical results.

Abstract

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.