Bounded Toeplitz operators and unbounded projected multiplication

Abstract

We construct a smooth, nonnegative radial symbol $g$ on $\mathbb{C}$ whose Toeplitz operator on Bargmann--Fock space is positive with singular values decaying faster than every exponential, while projected multiplication $f\mapsto P(gf)$ has no bounded extension from the ambient Gaussian $L^2$ space. All heat transforms of $g$ at positive times are Schwartz functions, and $g\in L^p(dA)$ for every $0<p<2$. We also construct smooth real symbols realizing every possible set of heat boundedness times. Any prescribed positive threshold may be included or excluded, with Schwartz transforms at every later time. All these symbols satisfy the coherent state domain condition, and all heat integrals converge absolutely.

arXiv comment16 pages