Quantum harmonic analysis (QHA) is well-suited to study Toeplitz operators on the Fock space, as Toeplitz operators can be written as QHA convolutions.
We discuss an operator analog of the heat semigroup and then provide a very short proof of the well-known Berger-Coburn theorem for boundedness of Toeplitz operators (and also of the related theorems that discuss Schatten-class membership and compactness of Toeplitz operators).
We also share some insights into the Berger-Coburn conjecture. We note that the analog of the Berger-Coburn conjecture that classifies Hilbert-Schmidt Toeplitz operators is indeed true.
This talk is based on joint work with Mishko Mitkovski.