We prove a new criterion that guarantees self-adjointness of Toeplitz
operator with unbounded operator-valued symbols. Our criterion applies, in
particular, to symbols with Lipschitz continuous derivatives, which is the
natural class of Hamiltonian functions for classical mechanics. For this we
extend the Berger-Coburn estimate to the case of vector-valued Segal-Bargmann
spaces. Finally, we apply our result to prove self-adjointness for a class of
(operator-valued) quadratic forms on the space of Schwartz functions in the
Schr\”odinger representation.
