Berezin-Toeplitz quantization on the Schwartz space of bounded symmetric domains

摘要

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 v. 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 axe also given.