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

TL;DR

This paper explores the isomorphism properties of Toeplitz and pseudo-differential operators on modulation spaces, providing explicit constructions and conditions under which these operators are isomorphisms, thus advancing the understanding of their structure.

Contribution

The paper constructs explicit isomorphisms between modulation spaces using Toeplitz operators, extending the pseudo-differential calculus and Wiener algebra techniques.

Findings

Toeplitz operators act as isomorphisms between weighted modulation spaces.

Explicit conditions for isomorphism involving weight functions and window functions.

Application of pseudo-differential calculus to establish these isomorphisms.

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) $\tp_\fy (\omega)$ is an isomorphism from $M^{p,q}_{(\omega_0)}$ onto $M^{p,q}_{(\omega_0/\omega)}$ 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.