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.KeywordsCompact OperatorToeplitz OperatorPseudodifferential OperatorTrigonometric PolynomialBergman SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.