Berezin-Toeplitz Quantization on the Schwartz Space of Bounded Symmetric Domains

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 TfTg 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. In this paper, 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. Subject classification: Primary 22E30; Secondary 43A85, 47B35, 53D55.