I will present recent advances and perspectives on the study of non-self-adjoint Berezin--Toeplitz quantization, and in particular, the role played by the geometry of Mabuchi space. The technical core of these results is semiclassical analysis in real-analytic regularity, as originally developed by Sjöstrand and refined in the framework of Kähler geometry.

Together, these tools allow to reach new results about the spectrum and dynamics of non-self-adjoint operators in a high frequency or small parameter regime.