Analysis Seminar

This talk is about a recent proof that Berger and Coburn's 1991 boundedness criterion for Toeplitz operators on Fock space fails in every dimension. The criterion asks whether the heat transform of the symbol at time t=1/4, the borderline time singled out by the Weyl calculus, characterizes boundedness. We construct a counterexample: a symbol whose Toeplitz operator is bounded, yet whose heat transform is unbounded on Cⁿ. The construction sums translated "blocks" with summable Toeplitz norms but fixed-size heat profiles, built via Hilbert–Schmidt estimates for Weyl quantization and the Bargmann correspondence.