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.

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