Positivity, complex FIOs, and Toeplitz operators

TL;DR

This paper characterizes positive complex linear canonical transformations relative to certain weights and demonstrates that Toeplitz operator boundedness on the Bargmann space follows from their Weyl symbols' boundedness.

Contribution

It provides a new characterization of positivity for complex linear canonical transformations and links Toeplitz operator boundedness to Weyl symbol boundedness.

Findings

Characterization of positive complex linear canonical transformations.

Boundedness of Toeplitz operators implied by Weyl symbol boundedness.

Application to operators on the Bargmann space.

Abstract

We establish a characterization of complex linear canonical transformations that are positive with respect to a pair of strictly plurisubharmonic quadratic weights. As an application, we show that the boundedness of a class of Toeplitz operators on the Bargmann space is implied by the boundedness of their Weyl symbols.