Compactness for pseudo-differential and Toeplitz operators on modulation spaces

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