Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey type pseudo-differential operators

We deduce one-parameter group properties for pseudo-differential operators $\mathrm{Op}(a)$, where $a$ belongs to the class $\Gamma ^{(\omega 0)}$ of certain Gevrey symbols. We use this to show that there are pseudo-differential operators $\mathrm{Op}(a)$ and $\operatorname{Op} (b)$whichareinversestoeachothers,where$a\in \Gamma ^{(\omega 0)}$and$b\in \Gamma ^{(1/\omega 0)}*$. We apply these results to deduce lifting property for modulation spaces and construct explicit isomorpisms between them. For each weight functions $\omega ,\omega _0$ moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) $\mathrm{Tp}({\omega }_0)$ is an isomorphism from $M_{(\omega )}^{p,q}$ onto $M_{(\omega \mathrm{/}{\omega }_0)}^{p,q}$ for every $p,q\in (0,\mathrm{\infty }]$.