Bounded Berezin-Toeplitz operators on the Segal-Bargmann space

TL;DR

This paper investigates the boundedness of Berezin-Toeplitz operators on a generalized Segal-Bargmann space, providing deformation estimates for their compositions using pseudodifferential calculus and heat flow methods.

Contribution

It introduces new boundedness criteria and deformation estimates for Berezin-Toeplitz operators on a generalized Fock space, extending previous results to broader settings.

Findings

Established boundedness conditions for Berezin-Toeplitz operators.

Derived deformation estimates for operator compositions.

Applied pseudodifferential calculus and heat flow techniques.

Abstract

We discuss the boundedness of Berezin-Toeplitz operators on a generalized Segal-Bargmann space (Fock space) over the complex $n$-space. This space is characterized by the image of a global Bargmann-type transform introduced by Sj\"ostrand. We also obtain the deformation estimates of the composition of Berezin-Toeplitz operators whose symbols and their derivatives up to order three are in the Wiener algebra of Sj\"ostrand. Our method of proofs is based on the pseudodifferential calculus and the heat flow determined by the phase function of the Bargmann transform.