The Range of Localization Operators and Lifting Theorems for Modulation and Bargmann-Fock Spaces

TL;DR

This paper investigates the range of localization operators on modulation and Bargmann-Fock spaces, introduces a lifting theorem, and characterizes the range of Gabor multipliers and Toeplitz operators using advanced spectral invariance and inequalities.

Contribution

It provides a new lifting theorem for localization operators and characterizes the range of Gabor multipliers and Toeplitz operators in complex analysis and time-frequency analysis.

Findings

Established a lifting theorem for localization operators.

Characterized the range of Gabor multipliers.

Characterized the range of Toeplitz operators on Bargmann-Fock spaces.

Abstract

We study the range of time-frequency localization operators acting on modulation spaces and prove a lifting theorem. As an application we also characterize the range of Gabor multipliers, and, in the realm of complex analysis, we characterize the range of certain Toeplitz operators on weighted Bargmann-Fock spaces. The main tools are the construction of canonical isomorphisms between modulation spaces of Hilbert-type and a refined version of the spectral invariance of pseudodifferential operators. On the technical level we prove a new class of inequalities for weighted gamma functions.