Journal of Lie Theory

Borthwick, Lesniewski and Upmeier [Nonperturbative deformation quantization of Cartan domains, J. Funct. Anal. 113 (1993) 153--176] proved that on any bounded symmetric domain (Hermitian symmetric space of non-compact type), for any compactly supported smooth functions f and g, the product of the Toeplitz operators T_fT_g on the standard weighted Bergman spaces can be asymptotically expanded into a series of another Toeplitz operators multiplied by decreasing powers of the Wallach parameter ν. This is the Berezin-Toeplitz quantization.
We remove the hypothesis of compact support and show that their result can be extended to functions f, g in a certain algebra which contains both the space of all smooth functions whose derivatives of all orders are bounded and the Schwartz space. Applications to deformation quantization are also given.