Berezin-Toeplitz Quantization

Abstract

A version of Toeplitz operators which "interpolates" between classical Toeplitz operators on the circle and pseudo-differential operators on ℝn was introduced in a series of papers by F. A. Berezin [B1, B2, B3]. Subsequently, C. A. Berger and I undertook a detailed analytic study of Berezin's operators in order to find an analog of the classical symbol calculus of pseudo-differential operators. In a series of papers [BC1, BC2, BC3], we dissected the largest class of bounded "symbols" for which Berezin—Toeplitz operators on ℂn have a "good" symbol calculus and commute modulo the compact operators. Then joined by K. H. Zhu and D. Békollé, we settled the corresponding problem for Berezin—Toeplitz operators on bounded symmetric domains Ω in ℂn [BCZ1, BCZ2, BBCZ]. Roughly speaking, to admit a good Berezin—Toeplitz symbol calculus, the symbols must satisfy a "vanishing mean oscillation" condition. On ℂn, exp $$\left( {i\sqrt {{|z|}} } \right)$$ is good while exp $$\left( {i|z|} \right)$$ is not good.