Analysis Seminar

Berger and Coburn proposed an endpoint boundedness criterion for Toeplitz operators on the Bargmann--Fock space in which the decisive quantity is the heat transform of the symbol at the borderline time $t=1/4$, the time naturally singled out by the Weyl calculus under the Bargmann transform. We show that this criterion fails for general measurable symbols in every complex dimension $n\ge 1$. Concretely, we construct a symbol satisfying a coherent state admissiibility condition, and the associated Toeplitz operator extends to a bounded operator on Bargmann space, but the heat transform is unbounded on $\mathbb C^n$. The example is obtained by summing translated bounded "blocks'' whose Toeplitz norms are summable while their critical-time heat profiles have fixed size. The blocks are produced by combining a Hilbert--Schmidt estimate for Weyl quantization with the Bargmann correspondence between Weyl and Toeplitz operators.