Isomorphism properties of Toeplitz operators and pseudo-differential operators between modulation spaces

We investigate the lifting property of modulation spaces and construct explicit isomorphisms between them. For each weight function ?? and suitable window function ??, the Toeplitz operator (or localization operator) Tp _?? (??) is an isomorphism from $M_{({\omega _0})}^{p,q}$ , onto $M_{({\omega _0}/\omega )}^{p,q}$ for every p, q ?? 1,??] and arbitrary weight function ?? ₀. The methods involve the pseudo-differential calculus of Bony and Chemin and the Wiener algebra property of certain symbol classes of pseudo-differential Operators.