Toeplitz operators and analytic Bergman kernels
In the first part of the talk, we shall discuss some joint work with L. Coburn and J. Sjöstrand, dealing with continuity conditions for Toeplitz operators acting on quadratic Bargmann spaces of entire functions, in connection with a conjecture by C. Berger and L. Coburn, relating Toeplitz and Weyl quantizations. The second part of the talk will be concerned with the semiclassical asymptotics for Bergman projections in exponentially weighted spaces of holomorphic functions, with real analytic strictly plurisubharmonic weights. Here a result due to O. Rouby, J. Sjöstrand, S. Vu Ngoc, and to A. Deleporte, establishes that one can describe the Bergman projection up to an exponentially small error. We shall discuss a direct approach to the construction of asymptotic Bergman projections in the analytic case, developed jointly with A. Deleporte and J. Sjöstrand, which allows us to give a simplified proof of this result.