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

Abstract

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

27 pages