Compactness for pseudo-differential and Toeplitz operators on modulation spaces

We deduce various norm equivalences, and convolution estimates for the modulation space $M^{\sharp ,q}_{(\omega )}$ consisting of all $f\in M^{\infty ,q}_{(\omega )}$ such that $|V_\phi f \cdot \omega |$ satisfies a mild vanishing condition at infinity. We prove that $M^{\sharp ,q}_{(\omega )}$ is the completion of the Gelfand-Shilov space $\Sigma _1$ under the $M^{\infty ,q}_{(\omega )}$ norm. We use these results to deduce compactness for $\Psi$DO $\op (\mathfrak a )$, with $\mathfrak a \in M^{\sharp ,q}_{(\omega )}$, $0<q\le 1$, when acting on a broad family of modulation spaces.