Compactness of localization operators on modulation spaces of $\omega$-tempered distributions

We give sufficient conditions for compactness of localization operators on modulation spaces $\textbf{M}^{p,q}_{m_{\lambda}}( \mathbb{R}^{d})$ of $\omega$-tempered distributions whose short-time Fourier transform is in the weighted mixed space $L^{p,q}_{m_\lambda}$ for $m_\lambda(x)=e^{\lambda\omega(x)}$.