My research interests include operator theory, C*-algebras, and quantum mechanics from the viewpoint of deformations of C*-algebras. The operators on which I focus are usually of Toeplitz-type and act on the square- integrable holomorphic functions on phase-space. I am interested in C*-algebras of these operators with various interesting "symbols" and the relation of such algebras to algebras of pseudo-differential operators which have been studied classically.
Recently, I have become interested in the structure of the Berezin symbol calculus of general operators on Bergman reproducing kernel Hilbert spaces. This calculus serves as a model for "quantization" and has been the object of considerable attention since it was introduced by Berezin in the 1970's. My most recently published paper, written jointly with Bo Li, "Directional derivative estimates for Berezin's operator calculus", appears in the Proceedings of the AMS 136 (2008) pp. 641-649. This paper is a sequel to "Sharp Berezin Lipschitz estimates" ( Proceedings of the AMS 135 (2007) pp. 1163-1168) and "A Lipschitz estimate for Berezin's operator calculus" (Proceedings of the AMS 133 (2005) pp. 127-131). These papers show that the Berezin symbols of general bounded operators must satisfy certain severe Bloch-type growth limitations which were not previously known. In his Ph.D thesis, Bo Li has extended these results further, to obtain Bloch-type estimates on the higher derivatives of general Berezin symbols.
Mr. Bo Li should receive his Ph.D. in 2008 under my supervision.