Analysis and Mathematical Physics

Berger and Coburn conjectured that a Toeplitz operator on Bargmann--Fock space is bounded if and only if the heat transform of its symbol at time $t=1/4$ is bounded, the borderline time singled out by the Weyl calculus under the Bargmann transform. We disprove this conjecture in every dimension by constructing a symbol whose Toeplitz operator is bounded yet whose heat transform at $t=1/4$ is unbounded on $\mathbb C^n$. The counterexample is assembled from translated blocks with summable Toeplitz norms but fixed-size heat profiles, built from oscillatory Weyl symbols whose operator norms are controlled by a Hilbert--Schmidt estimate. Time permitting, we discuss a counterexample showing the failure of the converse direction as well.