Characterizing boundedness of metaplectic Toeplitz operators

TL;DR

This paper investigates the boundedness and compactness of certain Toeplitz operators on the Bargmann space, establishing necessary conditions and confirming the Berger-Coburn conjecture for these operators.

Contribution

It proves that boundedness of Weyl symbols is necessary for bounded Toeplitz operators and confirms the Berger-Coburn conjecture in this setting.

Findings

Boundedness of Weyl symbols is necessary for operator boundedness.

Compactness corresponds to Weyl symbols vanishing at infinity.

Completes proof of Berger-Coburn conjecture for this class of operators.

Abstract

We study Toeplitz operators on the Bargmann space, with Toeplitz symbols given by exponentials of complex quadratic forms. We show that the boundedness of the corresponding Weyl symbols is necessary for the boundedness of the operators, thereby completing the proof of the Berger-Coburn conjecture in this case. We also show that the compactness of such Toeplitz operators is equivalent to the vanishing of their Weyl symbols at infinity.