The Lifting Property for Frame Multipliers and Toeplitz Operators

TL;DR

This paper investigates the invertibility of frame multipliers, showing that positive symbols induce isomorphisms between coorbit spaces, with applications to Gabor and Fock spaces, using localized frames and matrix algebra techniques.

Contribution

It establishes that positive-symbol frame multipliers are Banach space isomorphisms, extending lifting theorems to modulation and Fock spaces.

Findings

Positive-symbol frame multipliers are Banach space isomorphisms.

New lifting theorem between modulation spaces via Gabor frames.

Isomorphisms between weighted Fock spaces.

Abstract

Frame multipliers are an abstract version of Toeplitz operators in frame theory and consist of a composition of a multiplication operator with the analysis and synthesis operators. Whereas the boundedness properties of frame multipliers on Banach spaces associated to a frame, so-called coorbit spaces, are well understood, their invertibility is much more difficult. We show that frame multipliers with a positive symbol are Banach space isomorphisms between the corresponding coorbit spaces. The results resemble the lifting theorems in the theory of Besov spaces and modulation spaces. Indeed, the application of the abstract lifting theorem to Gabor frames yields a new lifting theorem between modulation spaces. A second application to Fock spaces yields isomorphisms between weighted Fock spaces. The main techniques are the theory of localized frames and existence of inverse-closed matrix algebras.