Information Geometry & Physics Seminar

Toeplitz operators on Bargmann-Fock space provide a meeting point between complex analysis, operator theory, and the Weyl calculus. An early-1990s conjecture of Berger and Coburn predicted that boundedness of such an operator should be detected exactly by a borderline heat transform of its symbol—the time naturally selected by the Bargmann correspondence between Weyl quantisation operators and Toeplitz operators. 

I will explain why this natural endpoint principle fails. The counterexample is built from oscillatory Weyl symbols whose quantizations are small in Hilbert–Schmidt norm while their endpoint heat profiles remain large. By translating and summing these blocks, one obtains a bounded, indeed compact, Toeplitz operator whose endpoint Weyl symbol is unbounded. I will also discuss the broader pseudodifferential lesson: boundedness is not governed by a single scalar heat profile but is, however, governed by a matrix condition.

Time permitting, we mention various situations in which the original endpoint principle does hold.

Also available on Zoom: https://caltech.zoom.us/j/8430173681